Multi-Rate Models for Dimensioning and Performance ... - CiteSeerX

COST 242

Multi-Rate Models for Dimensioning and Performance Evaluation of ATM Networks Edited by Michael Ritter Phuoc Tran-Gia

University of Wurzburg Institute of Computer Science Am Hubland D-97074 Wurzburg

Interim Report June 1994

Preface COST stands for European COoperation in the Field of Scienti c and Technical Research. The current project COST 242 is the follow-on project of COST 224 which ended in October 1991. It is entitled \Methods for the Performance Evaluation and Design of Multiservice Broadband Networks". The project began ocially on 30 April 1992 with the rst signatures of the Memorandum of Understanding (MOU) and is planned to last four years until 29 April 1996. Sixteen European countries have signed the MOU, viz Czechoslovakia, Denmark, Finland, France, Germany, Greece, Hungary, Ireland, Italy, the Netherlands, Norway, Poland, Spain, Sweden, Switzerland and the United Kingdom. The scope of action of the COST 242 project encompasses queueing models for end-to-end performance, trac control strategies, network design and routing strategies, evaluation of high speed LAN and MAN proposals, interconnection of heterogeneous multi-service networks as well as trac characterization. Until now, the participants at the Management Committee meetings agreed to prepare two interim reports. This decision has been made due to the fact that the COST 224 Final Report has met with a very favorable reaction internationally. With these interim reports it is intended to draft intermediate results on speci c subjects derived within the COST 242 project and to outline already published literature on these subjects. The rst one of the two interim reports is the present one entitled \Multi-Rate Models used for Dimensioning and Performance Evaluation of ATM Networks", the second one is concerned with \Cell Delay Variation". These interim reports are prepared by groups of editors and are to be presented at the mid-term seminar in September 1994 which takes place in the middle of the project lifetime. This interim report is written as a monograph; however, several Management Committee members and experts have provided valuable contributions or comments to speci c parts of this report. These are in detail: M. Azmoodeh who provided diverse material on trunk reservation and multi-service networks, R. Gibbens, Nigel Bean and Stan Zachary who delivered a section on trunk reservation, P. Key who contributed to control issues of multiservice networks, C. Lavrijsen who provided the part concerning the comparison of single link models, K. Lindberger who provided thoughts on grade of service formulations as well as U. Mocci, C. Scoglio and A. Tonietti who delivered interesting material on service integration and dimensioning of multi-service networks. The editors would also like to thank F. Hubner who with K. Lindberger provided the initial input for drafting this interim report. Special thanks also to S.-E. Tidblom and J. Virtamo for constructive comments which helped us to improve the quality of the i

present report. Finally, we would like to thank Prof. Richard Harris, Prof. Frank Kelly and Dr. James Roberts for carefully reviewing the manuscript. Wurzburg, June 1994 M. Ritter P. Tran-Gia

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Contents 1. Introduction

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2. Grade of service formulations

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2.1 General comments and some key questions : : : : : : : : : : : : : : : : : : 5 2.2 Grade of service : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6

3. Equivalent bandwidth estimations for VBR calls

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3.1 The complete equivalent bandwidth formulae : : : : : : : : : : : : : : : : : 10 3.2 Concluding remarks : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15

4. Call blocking in multi-service systems 4.1 Exact algorithms : : : : : : : : : : : : : : : : 4.1.1 Product form solution : : : : : : : : : 4.1.2 Recursive solution : : : : : : : : : : : : 4.2 Simple approximations : : : : : : : : : : : : : 4.3 Approximations for peaked trac sources : : : 4.3.1 System description : : : : : : : : : : : 4.3.2 Justi cation of the system assumptions 4.3.3 Approximation methods : : : : : : : : 4.3.4 Numerical results : : : : : : : : : : : : 4.3.5 Comparison results : : : : : : : : : : :

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5. Trunk reservation and forming of call blocking

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Rule for equalizing call blocking : : : : : : : : : : : : Accurate approximations : : : : : : : : : : : : : : : : Simple approximations : : : : : : : : : : : : : : : : : Heavy trac approximation : : : : : : : : : : : : : : 5.4.1 Analysis of the model : : : : : : : : : : : : : : 5.4.2 Discussion : : : : : : : : : : : : : : : : : : : : 5.4.3 Examples : : : : : : : : : : : : : : : : : : : : 5.5 In uence of holding time : : : : : : : : : : : : : : : : 5.5.1 In uence of the mean holding time : : : : : : 5.5.2 In uence of the holding time distribution type 5.1 5.2 5.3 5.4

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6.1 Approximations for ON/OFF-sources : : : : : : 6.1.1 Blocking probabilities : : : : : : : : : : : 6.1.2 State probabilities : : : : : : : : : : : : 6.2 Approximations for CBR and ON/OFF-sources 6.2.1 Blocking probabilities : : : : : : : : : : : 6.2.2 State probabilities : : : : : : : : : : : :

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7. Extensions to multi-service networks 7.1 7.2 7.3 7.4

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Fixed point model without trunk reservation Fixed point model with trunk reservation : : Numerical results : : : : : : : : : : : : : : : Conclusions and further studies : : : : : : :

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8.1.2 Investigation of integration policies : : : : : : : : : : : 8.1.3 Trac clustering rules : : : : : : : : : : : : : : : : : : 8.2 Design of virtual paths : : : : : : : : : : : : : : : : : : : : : : 8.2.1 VP networks designed as a network infrastructure : : : 8.2.2 Design of ATM networks integrating VPs and VCs : : 8.3 Control in multi-service networks : : : : : : : : : : : : : : : : 8.3.1 Framework : : : : : : : : : : : : : : : : : : : : : : : : 8.3.2 Reversible controls : : : : : : : : : : : : : : : : : : : : 8.3.3 Asymptotic results and adaptive control : : : : : : : : 8.3.4 Markov Decision Theory and state-dependent controls 8.3.5 Single link models and trunk reservation : : : : : : : : 8.3.6 Discussion : : : : : : : : : : : : : : : : : : : : : : : : :

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9. Concluding remarks

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COST documents

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List of Figures 3.1 Flow chart for the assignment of the equivalent bandwidth k. : : : : : : : : 15 4.1 Basic link model with complete sharing. : : : : : : : : : : : : : : : : : : : 4.2 State space example for product form solution (N = 2, C1 = C=5; C2 = C=2:5). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.3 State space reduction for the recursive solution (M = 5, m1 = 1, m2 = 2). : 4.4 Call blocking probabilities in a multi-service system. : : : : : : : : : : : : : 4.5 Link utilization sharing in a multi-service system. : : : : : : : : : : : : : : 4.6 Oscillation of call blocking in multi-service systems. : : : : : : : : : : : : : 4.7 Reference con guration. : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4.8 Blocking probabilities as a function of the rate-ratio. : : : : : : : : : : : : 4.9 Blocking probability for dierent peakedness factors. : : : : : : : : : : : : 4.10 Blocking probability for dierent trac mixes. : : : : : : : : : : : : : : : : 5.1 Basic link model for trunk reservation mode. : : : : : : : : : : : : : : : : : 5.2 Trunk reservation mode: example for the approximate state space. : : : : : 5.3 Blocking probability equalization by trunk reservation and approximation accuracy. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.4 Link utilization sharing with trunk reservation. : : : : : : : : : : : : : : : 5.5 Blocking probability equalization in a multi-service system. : : : : : : : : : 5.6 Link utilization sharing in a multi-service system. : : : : : : : : : : : : : : 5.7 Analytical and simulation results with varying trunk reservation. : : : : : : 5.8 Analytical and simulation results with varying trunk reservation. : : : : : : 5.9 Analytical and simulation results with varying trunk reservation. : : : : : : vi

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5.10 Sensitivity to changes in mean holding time ratio (heavy load). : : : : : : : 62 5.11 Sensitivity to changes in mean holding time ratio (moderate load). : : : : : 63 5.12 Sensitivity of GOS of call type 1 and its trunk reservation parameter to changes in mean holding time while maintaining GOS of call type 2. : : : : 64 6.1 6.2 6.3 6.4 6.5

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List of Tables 3.1 Results of the function y = y(q; Ploss). : : : : : : : : : : : : : : : : : : : : : 12 3.2 Coecients a and b as functions of the cell loss probability Ploss . : : : : : : 13 3.3 Comparison of N1, N2 and N3 for x = 0:01. : : : : : : : : : : : : : : : : : : 14 4.1 Trac mixtures to study the approximation accuracy. : : : : : : : : : : : : 29 4.2 Accuracy of the approximate blocking probabilities. : : : : : : : : : : : : : 30 4.3 Basic multi-slot formulae. : : : : : : : : : : : : : : : : : : : : : : : : : : : 34 5.1 5.2 5.3 5.4

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Dimensioning for equal GOS requirements in normal conditions. : : Dimensioning for unequal GOS requirements in normal conditions. : Service class characteristics. : : : : : : : : : : : : : : : : : : : : : : Total costs (B1 = B2 = B3 = 0:01). : : : : : : : : : : : : : : : : : : Total costs (B1 = 0:01; B2 = B3 = 0:05). : : : : : : : : : : : : : : : viii

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1. Introduction Future broadband integrated services digital networks (B-ISDNs) are rapidly emerging. The basic underlying technology is the asynchronous transfer mode (ATM). While transmission techniques have experienced enormous advances, most of the tools and methods needed to dimension ATM networks, in particular functionalities such as connection admission control (CAC), usage parameter control (UPC) etc., have still to be developed and their functions to be evaluated. Telecommunication services have evolved from telephony, where a single circuit or channel is held for the life of a call for exclusive use of a customer having a telephone connection. In the current digital world, the term channel often represents a slot or a number of slots in a time-division multiplex (TDM) frame. For example, a standard telephone call occupies one slot and transmits information at 64 kbps. In conventional telephone systems it is not surprising that mathematical models have concentrated on a resource model with integer capacity where demands occupy a single slot. The analysis of such systems is relatively mature, and has evolved from Erlang's celebrated loss formula to tools and techniques which allow networks to be modeled and designed and which can use alternative or dynamic routing strategies with state dependent controls. This can be considered already as classical. However, the analysis when demands require more than one unit of resource or have noninteger requirements, as in ATM systems with emerging services, is less developed. This leads to the class of models dealing with multi-rate trac (or multi-bit-rate trac), which is the main subject of this monograph. This monograph is an interim report of the COST 242 Management Committee. The target of the COST 242 project is the investigation and development of Methods for the Performance Evaluation and Design of Multi-Service Broadband Networks. The material in this monograph re ects the status of discussions, i.e. technical documents of COST 242 members and experts on the subject of multi-rate modeling after the rst two years of the project (1992-1993). This report has been drafted under the responsibility of two editors. However, various sections and parts of chapters have been contributed by other authors working in the COST 242 project, as indicated in the preface to this report. Since ATM principles and submodels of system and network components have appeared in a number of reports (e.g. the nal report of the previous COST project 224 [15]), we will brie y summarize the main subject of this report in the following.

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ATM is basically connection oriented. All information coming from dierent trac sources such as data, voice, video etc. is transmitted by means of short xed-length cells comprising a 5 byte header and 48 byte payload, at standard transmission rates of 150 and 600 Mbps. A common assumption is that the buers in switches and multiplexers will be relatively small (in the order of 102 cells), so that the delays will be relatively short. An ATM network behaves like a hybrid of a circuit-switched and a packet network and was called in earlier days a type of fast packet network. The other feature of ATM networks is the potentially stringent grade of service (GOS) standard, such as a cell loss ratio of 1 in 109 , where cell loss could occur through buer over ow or resource saturation. The ATM layer also de nes virtual paths and virtual channels: a virtual channel is associated with a collection of ATM cells having a common identi er in the cell header (virtual channel identi er), and a virtual path describes unidirectional transport of cells belonging to virtual channels having a common virtual path identi er in the cell header. In other words, a transmission path may comprise several virtual paths, and each virtual path may contain several virtual channels. The implications for us are that the capacity for calls (channels) may be much less than the standard transmission rates of 150 or 600 Mbps, and also that the models of interest can apply at the connection level, at the virtual channel level within a virtual path, or at the virtual path level. The connection oriented property of the ATM technique suggests that we are holding some amount of resource for the life of a connection, and that there is some link with circuit-switching. In general, calls will not be constant bit rate (CBR) and, even if they are, by the time they have passed through various switches and multiplexers, perhaps some of which are on the customer's premises, they can incur jitter and so be distorted, thus looking like variable bit rate (VBR) calls. Because the target cell loss probabilities are so low, an appropriate mathematical technique is the theory of Large Deviations. It has been shown by a number of authors [15], that there is a concept of equivalent bandwidth associated with calls, a number somewhere between the mean and the peak bandwidth requirement of the connection, k say. Now, if the capacity (of a link or a buer) has size C , then n connections of this call type can be accepted at the resource and the GOS standard will be maintained provided that n
k  C . The equivalent bandwidth will depend on the GOS standard, and can in general also depend on the mix of other trac on a link. However, in many instances it is possible to use conservative equivalent bandwidths which are independent of the trac mix, in which case we are free to add equivalent bandwidths. It should be stressed that the equivalent bandwidth approach applies to VBR calls, and not just CBR calls. Note that this equivalent bandwidth approach uses statistical multiplexing, which we can sidestep by allocating all calls their peak capacity. It is helpful to consider the decomposition rst suggested by Hui [32]. On a long time scale (for example seconds), call requests arrive and calls or service demands have a holding time on this scale. However, for VBR sources, information is transmitted in bursts of information, perhaps on a millisecond time scale (consider for example speech bursts). These bursts are then converted into cells, which are transmitted at the line rate, so we are looking at a microsecond scale. The equivalent bandwidth approach essentially says if we allocate buers (in switches, multiplexers etc.) at the cell scale, only large enough to 2

cater for simultaneous arrivals of cells when the overall arrival rate is less than the service rate, and if our CAC is at a burst level (making the probability of the number of bursts exceeding the capacity very small) then at the call level our ATM network behaves like a multi-slot circuit-switched network, where cell level losses are guaranteed to meet the de ned cell loss ratio. This framework allows us to model call and cell level losses: calls have an equivalent bandwidth k and the situation is isomorphic to a circuit-switched network. If we wished to equalize blocking then we would apply trunk reservation values inversely related to k. More generally, there is also some cost associated with accepting a call, since it could cause resource saturation. Depending on the loss function associated with achieved cell loss, we can then maximize the net expected reward. The remainder of this monograph is organized as follows. GOS formulations are addressed in Chapter 2. Some general comments are given and key questions are discussed. Chapter 3 deals with a particular aspect of equivalent bandwidth mentioned above, namely how to derive certain linear approximations. One of the problems with multi-rate models is how to determine the blocking probabilities of the dierent trac streams; this is considered in Chapter 4. It is well-known that for Poisson arrivals in the absence of controls the distribution of busy circuits has a product form, which is insensitive to the holding time ratios, from which the blocking probabilities can be calculated. However, faster methods are often needed and these are discussed along with some behavior peculiar to multislot systems. The last part of this chapter deals with a comparison of the complexity and accuracy of dierent approximation methods if multi-slot peaked trac sources are considered. In the absence of controls, large bandwidth calls can experience a much higher blocking probability than low bandwidth calls. Ways of controlling this are discussed in Chapter 5, which particularly concentrates on the use of trunk reservation as a protection mechanism. The chapter looks at ways of calculating call blocking probabilities when trunk reservation is used. A general rule to equalize blocking for calls with dierent bandwidth requirements is given and the in uence of the call mean holding time ratio is studied. Chapter 6 examines call and burst blocking for ATM systems. A model which allows consideration of call and burst blocking simultaneously is presented. The problem of calculating end-to-end blocking in multi-rate networks becomes even more complicated. Therefore, Chapter 7 starts to examine some of the issues concerning multi-service networks. In Chapter 8 control and dimensioning aspects of such networks are given. Questions on trac integration and separation are discussed and rules and criteria for the design of virtual paths in ATM networks are proposed. The design of optimal control policies for CAC is discussed, too. Some concluding remarks are provided in Chapter 9.

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2. Grade of service formulations 2.1 General comments and some key questions For the design and dimensioning of integrated networks with multi-rate trac it is important to have the GOS constraints well formulated. The nal decisions about these matters are, of course, not taken in a project like the current COST 242. What can be done here is to list and discuss certain reasonable GOS structures, where the actual parameter values can be determined later. In ATM networks there are essentially two dierent types of GOS constraints to consider. First, there are probabilities of cell losses (or delays) in already accepted calls. Such constraints are formulated in terms of very small cell loss probabilities e.g. 10?5 or 10?9 . The other type of GOS constraint, the one we shall discuss here, is concerning the probabilities of rejecting new calls. Such constraints are formulated in terms of blocking probabilities with magnitudes from 10?1 to 10?3 and a typical value could be 10?2 . When we talk about calls here we essentially refer to on-demand services, though for example virtual connections and other types of semi-permanent reservations of capacity also need similar rules for their acceptance principles. Concerning the burst scale losses, they are traditionally regarded as cell losses in the GOS sense, since a situation with too many bursts going on simultaneously normally means that all ongoing calls lose a fraction of their cells, which will thus be included in the cell loss probabilities. However, another arrangement, suggested in [30], is to reject all the cells in a burst when there is not enough space on the link for the whole burst. This means that cells from other calls will not be lost due to this burst. Such a rejection of a whole burst will be done with probabilities in the range of 10?2 and is thus more of a call blocking type of constraint. Still, we start our discussions as if everything in the trac mixtures were on-demand calls. We assume that we have a network, or at rst simply a link, to which mixtures of calls with dierent (equivalent) bandwidths arrive for example as Poisson processes during some busy hour-like key period. Then they are characterized by their bandwidths and their trac intensities. The structure of the GOS we shall formulate is also connected with the arrangements and strategies we choose for the integrated trac streams in the network. Some key questions depending on the GOS formulations put in [(92)018] and [(92)065] are e.g.:

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 Should we have total or partial integration of dierent trac classes?  Should we use strategies to equalize the individual blocking probabilities or is a

control of the average blocking probability sucient?  If the link is ineciently used due to the combination of low blocking constraints and on-demand calls with very high bandwidths, could we consider a higher blocking probability e.g. 10?1 for such calls, or even an arrangement with pre-booking? These types of questions must later be answered in relation to the whole network and its structure but we start with GOS formulations for a single link.

2.2 Grade of service For a single link we could have the following types of multi-rate GOS formulations:

a) GOS in terms of average blocking In this context the average blocking probability is de ned by the sum of individual blocking probabilities weighted with the individual products of bandwidth and trac intensity. With a GOS constraint as a certain value of the average blocking probability, it is possible to avoid an equalization strategy. Thus we get an unplanned unfairness depending on the mixture and its bandwidths. In practice, the calls with highest bandwidths in the mixture will get all the essential blocking and their individual blocking will be clearly above the average value. The advantage here is the simplicity in the arrangement, which could perhaps be used if this average blocking value is low, e.g. 10?3 and the ratio between the highest and the mean bandwidths is not too large. It could be reasonable to add an extra constraint on the highest individual blocking probability at an essentially higher value, e.g. 10?2 , so that it is normally still not the dimensioning criterion.

b) Equal blocking probabilities for calls with dierent bandwidths This is the simplest GOS formulation to understand since it means complete fairness. In practice however, this means that we need an equalization strategy to ful ll such a constraint. An example of such a strategy which will be discussed in a later chapter is trunk reservation, where all types of calls are blocked at states where at least one of these types would be blocked. Another strategy is partial sharing.

c) Planned unfairness due to dierent eciency in the utilization of the link for dierent magnitudes of bandwidths This is not one speci c GOS formulation, but a whole class of dierent possibilities (see also [(92)018]). Essentially one could say that since we might be dealing with calls of 6

bandwidths from e.g. 40 kbps to 40 Mbps on link capacities which are multiples of 150 Mbps it might be reasonable to treat these very dierent magnitudes dierently in the GOS formulation. Let us for simplicity divide the bandwidths into three clusters, small bandwidths below 0.4 Mbps, normal (in ATM) bandwidths between 0.4 and 4 Mbps, and (very) high bandwidths above 4 Mbps. Of course the actual values of these limits are a bit arbitrary and they could later be modi ed. Now, the calls with small bandwidths can usually be carried very eciently on the link and it is thus reasonable for those to have a low blocking probability, e.g. 10?3 . However, with such a low average blocking it is perhaps not so important to have complete fairness within this cluster of bandwidths. An extra constraint checking that no individual class of calls in the cluster gets a higher blocking probability than a certain higher value, e.g. 10?2 , would do. For the second cluster with normal bandwidths, 10?3 might be a too expensive average blocking probability to have as GOS. The upper limit for the small bandwidths, 10?2 , is a more reasonable value here. With this higher average blocking probability, it is however more important to have fairness within this cluster of bandwidths. Since an integration between these clusters with dierent magnitudes of bandwidths does not give signi cant savings compared to integration within these clusters and since they have dierent rules it will be practical to dimension each of them separately. For the cluster with the highest bandwidths, above 4 Mbps in our case, with typical values around one tenth of the link capacity, not even blocking probabilities in the magnitude 10?2 would be compatible with an acceptable utilization of the link. The users of such services are hardly prepared to pay too much extra just for a low loss probability. Thus even higher blocking probabilities, perhaps in the magnitude 10?1 , should be considered here. Then also equalization would be needed. Another possibility for this last cluster of bandwidths is of course some kind of pre-booking. These calls are usually not of real on-demand type anyway. Everyone considering this particular suggestion can, of course, imagine a number of variants in the same spirit, but with other limits between clusters, with other levels of the blocking probabilities, or with other combinations with or without equalization within clusters. We shall not list all these variants here. Each of these GOS variants for one link must of course have its extension in GOS for the whole network. In a simple network with no alternative routing it is obvious that, without any special arrangements, calls using several links in series will get a higher end-to-end blocking than those which only use one link. Even though it is the end-to-end blocking that counts, it is for administrative reasons easier to actually choose the link blocking values with respect to the number of links certain calls might use. The alternative, to consider dierent types of reservations on each link and to account for the fact that certain calls can be blocked at several links and should thus have lower blocking probabilities on each link than those which only use one link, is more complicated. Thus we do not suggest equalized end-to-end blocking, though it is of course also a possibility. If the network with multi-rate trac also has over ow possibilities, then the GOS formulation in connection with alternative routing must be combined with the particular 7

multi-rate GOS constraint we have chosen for a single link. Several combinations are possible. If, for example, the over ow network has the constraint that each link in a nal route should have a blocking probability of say 10?2 for all the trac oered to it, either direct or over ow, and the multi-rate trac has the complete fairness constraint within the mixtures of each trac class, then a reasonable arrangement would be to use the equalization strategy already on the rst choice links and thus keep the same proportions in the mixtures in the over ow streams, which is a great practical advantage. As we have seen, the choice of arrangements, strategies and dimensioning methods are very much dependent on the structure of the GOS formulations and it is thus important to reach some consensus on this point.

8

3. Equivalent bandwidth estimations for VBR calls In ATM networks, VBR trac sources are of special interest since they can oer essential savings in link capacity due to statistical multiplexing. The use of statistical multiplexing is also of prime importance to obtain an ecient utilization of the network links. However, if no peak bit rate allocation is performed, the peak demand of all active sources may exceed the available link capacity. This eect, which may cause cell loss, should occur only with a small probability. Therefore, for CAC the problem is to determine the amount of bandwidth which should be allocated to an arriving call to meet the desired GOS quantities on a link or on a virtual path (VP). Since the CAC algorithm must handle connection requests in real-time, a simple and useful way to determine this amount of bandwidth for arriving calls is required. In principle, it lies somewhere between the mean and the peak bit rate of the trac source. In the literature several proposals exist to calculate this bandwidth amount which is usually called the equivalent or eective bandwidth or sometimes the equivalent capacity. The idea behind this concept is that the acceptance region for CAC is approximately linearly constrained and thus the CAC algorithm in an ATM network can be thought of as operating like a circuit switched environment with a linear resource allocation strategy. In [71] the equivalent bandwidth is calculated using an estimation of the bandwidth variance. Another approach based on the mean and peak bit rate was presented in [15]. In [16] the burst length was also taken into account for the determination of the equivalent bandwidth. Estimates for the equivalent bandwidth of so-called ON/OFF sources have been proposed in [26]. Further equivalent bandwidth estimations can be found e.g. in [25], [32] or [38]. Here, the analysis is based on the overall cell loss rate derived by measuring the proportion of time the resource, i.e. the transmission link, is saturated. In contrast, a GOS measurement based on the proportion of the lost cells is taken into account in [5] where dierent GOS requirements are considered too. In the following we will outline the analysis method presented in [15] and the extensions to that given in [(92)036]. In this approach the equivalent bandwidth essentially depends on two source characteristics, namely the mean rate m and the variance of the rate 2, as well as the link rate C and the overall cell loss rate Ploss . Therefore, it is independent of the calls in progress. The next section deals with the derivation of complete equivalent bandwidth formulae whereas the section afterwards summarizes the results and gives a brief outlook. 9

3.1 The complete equivalent bandwidth formulae If a superposition of a number of VBR sources is oered to an ATM multiplexer the variations of the resulting cell stream can be decomposed into two components on dierent time scales [15]. On the one hand, there is a cell scale component accounting for short term variations due to the discrete nature of the ATM cells and their asynchronous arrivals. On the other hand, a burst scale component re ects longer term uctuations due to the bandwidth variation of the trac sources. For the further derivation of an equivalent bandwidth estimation for given GOS requirements, the buer associated with the link is assumed to be large enough to take care of cell scale variations (about 102 cells). However, the buer is assumed to be too small to accommodate a burst, which means that we have a pure loss system in the burst scale sense. This assumption about the buer size is reasonable, because very large buers must be used to take care of burst scale variations (see e.g. [15]). Furthermore, a negative side eect of large buer sizes is that they lead to longer cell delays. This can be an important factor for delay sensitive sources. Using this assumption about the buer size and with a GOS de ned by the xed overall cell loss rate Ploss = 10?9, an empirical expression for the equivalent bandwidth k, i.e. the resource allocation value for a trac source, is suggested in [15]:

k = 1:2m + 602=C

(Ploss = 10?9 ):

(3.1)

If instead of the variance 2 only the peak rate h of the source is known, or if the source is of an ON/OFF type, the approximation 2 = m(h ? m) is used, which yields

k = 1:2m + 60m(h ? m)=C

(Ploss = 10?9 ):

(3.2)

According to [15], this expression for k holds with acceptable accuracy in an interesting part of the parameter space, i.e.

h=k  2; m=h  0:05 and 15  C=h  1000:

(3.3)

The values of the coecients 1.2 and 60 are, of course, essentially dependent on the overall cell loss rate Ploss = 10?9 . Until now, 10?9 has been almost the sole candidate for the cell loss rate. However, for certain sources such as voice calls, this value is far more strict than required. Concerning the cell loss rate, dierent levels of GOS should therefore be oered, e.g. on dierent VP's. Thus, an equivalent bandwidth expression as a function of a given cell loss rate Ploss is required. In the following we present a generalization of this equivalent bandwidth formula. The extension was derived in [(92)036]. General expressions for the coecients mentioned above are given in terms of Ploss . Furthermore, the corresponding equivalent bandwidth 10

formulae are extended to the whole parameter space so that we have complete equivalent bandwidth formulae for various cell loss rates. At the end of this chapter a short summary including a ow chart for the assignment of an equivalent bandwidth to a trac source is given. First, we start by deriving an expression for the overall cell loss rate Ploss if homogeneous trac sources are considered. In doing so we use the method of equivalent bursts, which means that the total trac stream is substituted with a superposition from an in nite number of sources generating bursts. The bursts are of an ON/OFF type with xed length and height, arriving according to a Poisson process. Thus, the overall cell loss rate is

Ploss = E(X ? n j XE(Xn))
P(X  n) ;

(3.4)

where X is the number of simultaneous bursts in progress, n = c=h is the link to burst rate ratio and h = 2=m is the bandwidth of a standard burst. As mentioned before, m and 2 denote the mean and variance of the rate of a single real source. Using the fact that X  Po(a), where a = mr=h with r equal to the number of real sources integrated to achieve the cell loss rate Ploss , the expression for Ploss can, after some eort, be written as n Ploss = (n ?n a)2
an! e?a :

(3.5)

With equation (3.5) as a starting point we shall look for the equivalent bandwidth of the real source. By applying Stirlings formula for n! and taking natural logarithms we get

p

ln(Ploss 2) = n ln a + (n ? a) ? 2 ln(n ? a) ? (n ? 1) ln n ? 0:5 ln n:

(3.6)

Introducing the equivalent bandwidth k = C=r and the variable q = 2=mC the quantities n and a can be expressed as

n = q?1

(3.7)

a = (qy)?1;

(3.8)

where y = k=m. Now, after some rearranging of the logarithmic expression above we get

p

ln y ? (1 ? y?1) + 2q ln(1 ? y?1) = 1:5q ln q ? q ln(Ploss 2): 11

(3.9)

From this equation it is easy to see that y is a function of the variables q and Ploss , i.e.

y = y(q; Ploss):

(3.10)

In Table 3.1, which is taken from [(92)036], some results for the y-values are given. To generate this table, Ploss was xed to certain values of interest and then q was varied.

q 6:0
10?3 8:0
10?3 1:0
10?2 1:5
10?2 2:0
10?2

10?9 1.557 1.684 1.807 2.115 2.432

10?8 1.500 1.611 1.719 1.984 2.253

Ploss 10?7 1.442 1.537 1.629 1.854 2.079

10?6 1.380 1.461 1.539 1.725 1.908

10?5 1.317 1.383 1.446 1.596 1.740

Table 3.1: Results of the function y = y(q; Ploss). By extending this table over a broader range of q-values, we can draw graphs of the type y(q) for the xed loss rates Ploss considered in Table 3.1. These graphs are found to be almost linear over a wide range of q-values including the region of interest, say q = 10?3 to 10?2 . Thus, the almost linear parts of these graphs can be approximated by

y = a(Ploss ) + b(Ploss )
q

(3.11)

for the dierent loss rates Ploss = 10?k , k = 5; : : : ; 9. Using the de nitions of y and q above, we get the equivalent bandwidth

k = a
m + b
2=C:

(3.12)

To nd the coecients a and b, a straight line has to be adjusted to the almost linear part of each graph y(q). There is, of course, some uncertainty involved in drawing these lines. However, the coecients presented in Table 3.2 can be found. The next step is to nd empirical functions for a and b in terms of Ploss . It is easy to check that the simple empirically found equations below are well matched to the results given in Table 3.2:

Ploss a = 1 ? 2 log 100

(3.13) 12

Ploss 10?9 10?8 10?7 10?6 10?5 a 1.18 1.16 1.14 1.12 1.10 b 63 56 48 41 33 Table 3.2: Coecients a and b as functions of the cell loss probability Ploss .

b=a = ?6 log Ploss :

(3.14)

So, through equations (3.13) and (3.14) we have general expressions for the coecients for the equivalent bandwidth formulae of equation (3.12). As was pointed out in the introduction, the equivalent bandwidth formula holds with acceptable accuracy in a relevant part of the parameter space for the case where the overall cell loss rate was set to Ploss = 10?9 . When leaving that region, the formula either underestimates or overestimates the bandwidth requirement of the source. From the network point of view, the most serious case is of course when an underestimation takes place. To deal with these circumstances we start by rewriting equation (3.12) using 2 = m(h ? m), i.e.

k = a
m + b
m(h ? m)=C:

(3.15)

By introducing the quantities

x = m=h

(3.16)

log Ploss z = ?2 C=h

(3.17)

and considering (3.13) and (3.14), equation (3.12) can be written as

k = x(1 + 3z(1 ? x)): ah

(3.18)

In the case of homogeneous ON/OFF sources the binomial distribution can be used to nd exactly how many sources can be integrated on a link if the parameters m, h, C and Ploss are given. This was done in [15]. Numerical comparisons between equation (3.18) and the left graph on page 151 in [15] reveal that equation (3.18) underestimates the equivalent bandwidth when C=h is small, i.e. when z is great. To compensate for this underestimation, it can be found empirically that the expression

k = x(1 + 3z2(1 ? x)) ah

(3.19) 13

should be used instead of equation (3.18) when z > 1. To study the improvement given by (3.19) compared with (3.18) in the case of z > 1, the maximum number of sources which can be integrated on a link is calculated. In [(92)036] this was done for the two formulae given in (3.18) and (3.19) and for the binomial distribution in [15]. The results are shown in Table 3.3 using the following expressions for N1, N2 and N3:

N1 = Number of sources integrated using the binomial distribution

(3.20)

N2 = ax(1 +C=h 3z(1 ? x))

(3.21)

N3 = ax(1 + C=h 3z2(1 ? x))

(3.22)

Ploss 10?9 c=h N1 N2 N3 N1 10 83 133 79 136 8 43 88 42 74 5 9 36 10 18

10?7 N2 170 113 47

N3 N1 128 235 69 137 18 38

10?5 N2 219 154 65

N3 219 128 35

Table 3.3: Comparison of N1, N2 and N3 for x = 0:01. The sources used for generating the table have a great peak rate and they are also extremely bursty (h=m = 100). With the column for N1 regarded as the correct solution it can be stated that (3.18) clearly yields an underestimation of the equivalent bandwidth while (3.19) is quite good but gives a slight overestimation when Ploss is increasing. From equation (3.18) (respectively (3.19)) it follows that in case of CBR sources, i.e. when x = 1, we get k = ah. This amount of bandwidth is, of course, the maximum a source should be assigned. However, it might happen that the right-hand side of (3.18) (respectively (3.19)) becomes greater than 1, which yields k > ah. To protect against such an undesired overestimation of the equivalent bandwidth, the right-hand side of (3.18) (respectively (3.19)) is required to be less than or equal to 1. For equation (3.18) this leads to the demand

x  31z

(3.23)

14

and for equation (3.19) to the demand

x  31z2 :

(3.24)

Thus, if in the case where z  1 the demand (3.23) is violated or, in the case where z > 1 the demand (3.24) is violated, the equivalent bandwidth k = ah is simply assigned to the source.

3.2 Concluding remarks To summarize how to assign an equivalent bandwidth to a given VBR source according to the proposals of the last section, the ow chart in Figure 3.1 could be useful. With the help of this ow chart, an equivalent bandwidth can be assigned to all kinds of sources, if their peak rates h and mean rates m are given. The cell loss rate Ploss can be varied and the whole parameter space is covered. It can be assumed that for the vast majority of sources the two successive questions in the ow chart will be answered by: Yes, Yes. Otherwise, the statistical multiplexing gains which can be achieved would be poor. Given: m, h, C, Ploss

Compute: x = m/h , z =

k=ah

No

1 x< 2 3z

2 log Ploss -2 log Ploss , a =1C/h 100

No

z > < : m 0 N > > : m1 P p~(m ? mi)mi ii : 0 < m  M i=1 1

gcd means the greatest common divisor

21

(4.14)

λ2

λ2 λ1

λ1

0

λ2 λ1

1

2

µ1

2 µ1 µ2

λ1

λ1

3 3 µ1

3 2

λ2

µ2

4 4 µ1

2 µ2

5 5 µ1

5 2

µ2

Figure 4.3: State space reduction for the recursive solution (M = 5, m1 = 1, m2 = 2). After normalization, we get the state probabilities

p(m) = p~(m)


M X m=0

p~(m)

!?1

:

(4.15)

The blocking probability Bi for class-i calls can be calculated as

Bi =

M X m=M ?mi +1

p(m):

(4.16)

For the example in Figure 4.3, the resulting blocking probabilities are

B1 = p(5) and B2 = p(4) + p(5):

(4.17)

It should be mentioned, that regarding the blocking probability calculation the recursive solution discussed in this subsection is of an exact nature; see [34], [59]. Furthermore it has been shown in [59] that the solution delivers exact results also for general holding time distribution functions. To illustrate the results from the last two sections, we use an example considering four trac classes. The numerical results are provided in the Figures 4.4 and 4.5. Since the trac regulation is done at the call level while the link occupancy by dierent call service classes is measured at the bit (or cell) level, we will rst introduce some notation concerning oered trac, carried trac and link utilization at the call and bit rate level (cf. [(92)019]). While at the call level the normalized oered trac Ai and the normalized carried trac Yi of class i are given by

Ai = i i

and Yi = Ai
(1 ? Bi);

22

(4.18)

we obtain, at the bit rate level, the normalized oered bit rate trac i as

i = i CCi = Ai
CCi i

(4.19)

and the normalized carried bit rate trac i of class-i calls as

i = i CCi
(1 ? Bi) = i
(1 ? Bi): i

(4.20)

Note that i also represents the link utilization by class-i calls. Finally, the total link utilization  is given by N  C X  = i = i Ci
(1 ? Bi): i=1 i i=1 N X

(4.21)

According to [(92)019], the following parameters are chosen for the examples. The capacity of the transmission link is assumed to be 150 Mbps and the bandwidth requirements of the calls of the four classes are C1 = 2 Mbps, C2 = 4 Mbps, C3 = 10 Mbps and C4 = 20 Mbps. Calls from the dierent trac classes are assumed to oer the same load, i.e. 1 = 2 = 3 = 4. In Figure 4.4 the blocking probabilities are drawn as functions of the oered load . It is obvious that the blocking probabilities for calls with higher bandwidth requirements are always higher than for calls which require less bandwidth. This behavior can be illustrated simply by equation (4.5) in case of the product form solution or by equation (4.16) for the recursive algorithm. In these equations the sets of states where calls with lower bit rate demand are blocked are always subsets of those sets of blocking states which belong to calls that require more bandwidth. Thus, a kind of unfairness can be observed since low bandwidth calls are favored. This unfairness also appears if we look at the link utilization shared among the dierent trac classes (cf. Figure 4.5). Remember that the oered bit rate trac i is the same for all classes. The unfair bandwidth sharing is easy to understand due to equation (4.20). In Figure 4.4 the blocking probabilities are monotonic functions of the oered bit rate trac. However, the blocking probabilities for trac classes with low bandwidth requirements can be non-monotonic. This behavior depends strongly on the composition of the trac mixture. An illustration of this eect is given in Figure 4.6. For this example, the capacity of the transmission link is again C = 150 Mbps and the two trac classes have a bandwidth requirement of C1 = 2 Mbps and C2 = 40 Mbps. The oered bit rate trac is assumed to be the same for both classes (1 = 2). Looking at Figure 4.6, an interference between the two curves can be observed and the GOS for class-1 calls is improved at some particular ranges even though the oered trac increases. This behavior was also noticed in [33] and [(86)068]. 23

blocking probability

10E0 class 4 class 3

10E-1

class 2 class 1 10E-2

10E-3

10E-4 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

link utilization

Figure 4.4: Call blocking probabilities in a multi-service system.

0.4 class 1 class 2 0.3

class 3 class 4

0.2

0.1

0.0 0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

Figure 4.5: Link utilization sharing in a multi-service system. 24

blocking probability

The explanation for this (perhaps) surprising eect is the following: If the oered load increases, class-1 calls can take over an imaginary block of 40 Mbps of the transmission link (i.e. the bandwidth need of a class-2 call). At this point, the blocking probability of class-2 calls increases whereas the class-1 calls experience a better GOS since this imaginary block can be used by them exclusively. 10E0

10E-1

10E-2

10E-3 class 1 class 2 10E-4 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

Figure 4.6: Oscillation of call blocking in multi-service systems. If the oered trac increases further, the blocking probability of class-1 calls grows rapidly until another imaginary block is taken over. Thus oscillations of the blocking probability of class-1 calls occur. This eect becomes stronger, as the class-2 bandwidth comes close to the link capacity and the larger the bandwidth dierence becomes between the two classes. These oscillations must be taken into account in network dimensioning routines. In the next section we describe a simple way to get approximate values for the average call blocking probability and the highest individual call blocking probability of a trac mixture. Using these results, formulae for estimating the individual call blocking probabilities of the dierent trac classes are also given. The approximate results are compared to the exact values.

4.2 Simple approximations In the last section two simple algorithms for computing the exact blocking probabilities are shown. Thus it might look unnecessary to consider the prospect of an approximation. 25

However, an exact solution might not be practical in case of the product form solution if a large number of dierent trac classes is considered. If the granularity of the trac mixture is high, say call bandwidths of some kbps and a transmission link with a capacity of some hundreds of Mbps, a solution using the two presented algorithms might not be practical, either. A rst-shot approximation could also give a clearer view of the importance of certain key parameters, which can be useful in network planning. In the following we describe an approximate solution to get the average blocking pro . Using these results, an bability B  and the highest individual blocking probability Bmax  approximation of the individual blocking probabilities Bi is given. The methods have been derived in [51], [(92)018], [(92)065] and [(93)008]. The blocking probabilities are denoted with the superscript "", since the results are of approximate nature. The superposition of the N trac classes oered to the link can be characterized by its mean m and variance v:

m=a=

N X i=1

ai =

N  X i Ci  i=1 i

v =

N X i=1

aiCi =

N  X i 2 Ci :  i=1 i

(4.22)

For the following equivalent model we determine the mean bandwidth C of the trac mixture by

C = v=m:

(4.23)

Considering the special case with all Ci equal to C we have a simple and exact distribution for the states of the trac process. An equivalent approximation can be derived by assuming that other trac mixtures with the same value of the mean C behave in a similar way. However, certain interesting quantities need a special interpretation. If Cmin = minfCig = 1 and thus C is greater than 1, then a quantity like E( CC ; Ca )

(4.24)

should represent the probability of C bandwidth unit states rather than just one. E(:) denotes Erlang's loss formula. Now, let

Cmax = maxfCig

and

n = C ? Cmax

(4.25)

i.e. n is the last state where no type of call is blocked. It is expected, that Cmax is essentially smaller than n. For the derivation of the approximation we shall, for simplicity, assume that the smallest bandwidth, say Ci, is one bandwidth unit and that the other Cj 's (j 6= i) are multiples of that. However, the results shall later be generalized to hold also for non-integer values. 26

Now, the probability of the state n is approximated by

pn = E( n C? y ; Ca )=C

(4.26)

where y = (C ? 1)=2. Here y is a correction term with the choice above as the most natural one, though an alternative like y = C ? 1 should also be tested with various numerical examples. The probability of a state C steps above n, i.e. pn+C , should approximately have an extra factor a=C . This factor is equal to the oered load. Thus, the last blocking states are approximated by

pn+k

=

pn


 a  Ck C

for k = 1; 2; : : : ; Cmax:

(4.27)

Now, set b = (a=C )1=C and let Ci = i. Thus, the approximate blocking probability Bi for class-i calls is

Bi = pn

CX max j =Cmax +1?i

b j = pn (bCmax+1?i ? bCmax+1)=(1 ? b):

(4.28)

The mean blocking probability is then given by CX CX max a B  max 1 i i  C +1?i ? a bCmax+1 )=(1 ? b):   B = a = p ( ib max i n a i=1 i=1 a

(4.29)

With  = 1 ? b and b?i  1 + i we have approximately CX max i=1

aib?i 

CX max i=1

 )  ab?C ai(1 + i) = a(1 + C

(4.30)

and the result for the mean blocking probability will be

B  = pn (bCmax+1?C ? bCmax+1)=(1 ? b):

(4.31)

Using the same approach, the largest individual blocking probability is  = p (b ? bCmax+1 )=(1 ? b): Bmax n

(4.32) 27

If we use the approximation (bx ? by )=(1 ? b)  (y ? x)b(x+y?1)=2; then we have

 a (2Cmax+1?C)=2C     B = pn C C

(4.33)

 a (Cmax+1)=2C  = p C : Bmax n max C

(4.34)

and

The ratio between the highest individual and the mean blocking probability may also be of interest:

 (Cmax?C)=2C  Bmax C max C : B  = C a

(4.35)

This is an expression suitable for generalization and very easy to use e.g. for checking   can be a condition like Bmax  10B . Finally, the blocking probabilities B  and Bmax expressed in terms of Erlang's loss formula, i.e.  B  = E( C ? CC + 1 ; Ca )

(4.36)

 (Cmax?C)=2C  = B 
Cmax C Bmax : C a

(4.37)

and

For the derivation of these two formulae we assumed, for simplicity, that the smallest bandwidth of a call is one bandwidth unit and the others are integer multiples of that. However, the results can easily be generalized in the following way:  B  = E( C ? CC+ Cmin ; Ca )

(4.38)

and

B

max

 (Cmax?C)=2C C max C   : = B
C a

(4.39) 28

Here, the parameter Cmin denotes the minimum required bandwidth of the dierent trac classes. To illustrate the accuracy of these approximate formulae, results for several trac mixtures are compared with the exact values which have been derived using the product form solution or the recursive algorithm. We start this numerical study with some typical cases for which the approximations are originally intended, i.e. reasonable mixtures and rather small blocking probabilities, as suggested in [(92)065]. The considered trac mixtures are given in Table 4.1. Mixture 1 2 3 4 5 6 7 A 90 6 60 12 30 20 6 30 30 6 160 8 200 4 96 24 4.8 C 1 5 1 5 1 3 5 1 2 5 1 10 1 10 1 4 10 i

i

Mixture 8 9 10 A 40 20 10 8 5 4 40 40 10 8 4 24 12 16 6 4.8 4 6 2.4 C 1 2 4 5 8 10 1 2 4 5 10 1 2 3 4 5 6 8 10 i

i

Table 4.1: Trac mixtures to study the approximation accuracy. Note that e.g. mixtures 2 and 3, though dierent, have the same values of all the key parameters in the formulae, i.e. a = 120, C = 3 Mbps and Cmax = 5 Mbps. Thus, the same value of the transmission link capacity C should make those cases equivalent with respect to the approximation formulae and it is expected that their real blocking probabilities are actually similar. Other mixtures which are equivalent in this sense are (5, 7, 9) and (8, 10) respectively. The rst ten cases focus on the average and highest individual blocking probability. The capacity of the transmission link is assumed to be C = 150 Mbps for the mixtures 1 to 4 and C = 300 Mbps for the mixtures 5 to 10, i.e. a load factor of 0.8 for all the cases and the blocking should be rather low. The results from [(92)065] are depicted in the rst part of Table 4.2. Note rst that the values for equivalent cases are very close to each other with respect to both quantities. Thus, also other equivalent cases ought to be close to these values as expected in the approximation approach. We can also notice that the approximations for  are close to the corresponding exact values in every case, i.e. the accuracy is B  and Bmax  approximation can in some cases have errors which are slightly greater, good. The Bmax since it contains two factors of almost independent approximation. Now, knowing that the approximative formulae are accurate for parameter sets with low blocking probabilities, it is also important to check this out for cases where the blocking is higher. Therefore, in [(92)065] the same ten mixtures with a transmission link capacity reduced successively by 10% and 20% are studied. This means that cases 11 to 14 and 21 29

Case 1 App 2 3 App 4 App 5 7 9 App 8 10 App 6 App

B  1:0E ? 2 0:9E ? 2 2:1E ? 2 2:1E ? 2 2:2E ? 2 1:5E ? 2 1:5E ? 2 1:0E ? 2 1:0E ? 2 1:0E ? 2 1:0E ? 2 1:5E ? 2 1:5E ? 2 1:6E ? 2 0:3E ? 2 0:3E ? 2

 Bmax 2:6E ? 2 2:8E ? 2 3:6E ? 2 3:6E ? 2 4:0E ? 2 3:2E ? 2 3:4E ? 2 2:7E ? 2 2:7E ? 2 2:7E ? 2 2:9E ? 2 3:3E ? 2 3:3E ? 2 3:6E ? 2 1:5E ? 2 1:5E ? 2

Case 11 App 12 13 App 14 App 15 17 19 App 18 20 App 16 App

B  3:8E ? 2 3:9E ? 2 5:8E ? 2 5:8E ? 2 6:1E ? 2 4:8E ? 2 5:2E ? 2 3:9E ? 2 3:9E ? 2 3:9E ? 2 4:1E ? 2 4:9E ? 2 4:9E ? 2 5:2E ? 2 2:3E ? 2 2:2E ? 2

Case 21 App 22 23 App 24 App 25 27 29 App 28 30 App 26 App

B  9:7E ? 2 1:0E ? 1 1:2E ? 1 1:2E ? 1 1:3E ? 1 1:1E ? 1 1:1E ? 1 9:8E ? 2 9:9E ? 2 9:9E ? 2 1:0E ? 1 1:1E ? 1 1:1E ? 1 1:2E ? 1 7:7E ? 2 8:1E ? 2

Table 4.2: Accuracy of the approximate blocking probabilities. to 24 are mixtures 1 to 4 with C = 135 Mbps and C = 120 Mbps respectively, and cases 15 to 20 and 25 to 30 are mixtures 5 to 10 with C = 270 Mbps and C = 240 Mbps. The results derived in [(92)065] for these cases are given in the last two parts of Table 4.2. Here, only the average blocking probability B  is considered. When looking at the equivalent cases we can see that they still have very similar values for these quantities, also in a region with higher blocking probabilities, and it is reasonable to conclude that this will hold for other equivalent cases, too. The approximate values are not quite as accurate as for the cases with low blocking, but the relative error is still in the magnitude of typically 5%. Thus, the formulae can be used also for parameter sets with higher blocking probabilities, though they were not originally meant for this particular situation. Of course, approximation formulae of this simple type will have unacceptable errors at least in some untypical parts of the parameter space, so limitations are required. The examples above show, however, that they are rather good in large regions covering the 30

essential cases for the intended application. In the remainder of this section, the results above are extended to get approximate values for the individual blocking probabilities. This extension was presented in [(93)008]. The individual blocking probabilities Bi in this type of approximation are essentially proportional to the number of states which imply blocking for the particular type of calls. However, when the normalized oered bit rate trac is slightly below 1:0, then the last states have decreasing blocking probabilities, and thus the average value for the last Cmax  < k=Cmax. states is greater than that of the last k states with k < Cmax, i.e. Bk=Bmax The essential eect following the approximation approach is that the individual blocking probabilities Bk are proportional to

k(C=a)k=2C = k(1 + )k  k + k2

(4.40)

where   = (C=a)1=2C ? 1  (C=a ? 1)=2C:

(4.41)

Thus, the individual blocking probabilities can be estimated by

i(1 + i) Bi = i + i2 = N   (1 + (C =C )2)) ; P j + j 2 C (1 + C B

(4.42)

j =1

i.e. Bi is also dependent on the variance C2 of the mixture, at least when  has a signi cant value above 0. This is of course a little complication. However, when the normalized oered bit rate trac is rather close to 1:0, i.e.  is rather small, one can simplify the approximation by  Bi=B   i=C:

(4.43)

Since all the individual blocking probabilities are approximated, it is important that all these expressions add up to B , i.e. N X i=1

aiBi = aB :

(4.44)

This holds for equations (4.43) and (4.42). For more general cases with non-integer bandwidths, i is replaced by Ci in the equations (4.43) and (4.42), respectively. In [(93)008] the mixtures 5 and 7 of Table 4.1 are used to illustrate the accuracy of these quantities. It is shown, that in case of these parameter sets the approximate values are close to the real ones. 31

4.3 Approximations for peaked trac sources In this section dierent methods for approximating the blocking probability of multi-slot peaked trac streams at a call level are compared as presented in [(92)037]. As in the single slot case these methods become essential if alternative routing is applied within the network. Multi-slot is a special case of multi-rate. With multi-slot the bandwidths can be expressed as integers (times a basic bandwidth or slot) while for multi-rate the bandwidths are reals. For networks without alternative routing the blocking probabilities in the network generally satisfy a product form solution. For networks with alternative routing a product form solution no longer exists. Due to the complexity involved, exact evaluation methods have not been found in the literature. Decomposition methods with two moment trac approximations have been successfully applied to the single-slot case. The reduced load approximation form can also be applied in the multi-slot case, a description of the algorithm for evaluation of blocking probabilities in multi-slot networks is given in Chapter 7 and [(93)020]. The success of this method depends on the availability of an ecient multi-slot trac performance evaluation algorithm on a single link on the basis of a two moment trac approximation. This is the subject of this section.

4.3.1 System description We consider approximations which estimate call blocking probabilities Bi of N individual independent trac streams which are oered to a single link. The individual trac streams (call arrival processes) are described by their mean intensity mi, peakedness zi and bandwidth Ci (assumed to be an integer value). The admission control mechanism to the link is based on the decision whether the sum of the existing call bandwidths plus the bandwidth of the requested call exceeds the bandwidth of the link C (assumed to be an integer value). In that case the call is rejected (complete sharing).

4.3.2 Justi cation of the system assumptions An essential element of the single link models we will consider is that we assume that dierent trac streams arrive independently. The independence of trac streams is assumed due to the complexity of correlated models in large networks. By nature, the trac oered to the network originates from independent sources and is therefore independent when it enters the network. Only with the use of excessive non-hierarchical routing can some streams become correlated. We consider two-moment trac approximations. This is because of the substantial eect of peakedness on the blocking probability which is not taken into account in a one-moment model. This description also closely matches singleslot methods and can be measured in practice. It is essential that a bandwidth can be assigned to trac streams. This is the case for ATM networks on the basis of peak rate or equivalent bandwidth assignment. We consider stationary state measures. The time 32

scale over which the blocking probability is measured is larger than the occurring holding times. Complete sharing (i.e. no selective circuit reservation) has been assumed because of its simplicity both in modeling and in practical application in equipments. Considering the reduced costs of transmission, e.g. for investment and operation, it might well be more cost-eective to allow for complete sharing and not to apply a partial sharing policy. In addition to that, partial sharing would only be applied if it would increase the eciency and therefore would save capacity. The relevance of partial sharing becomes important when the network is suciently loaded. Multi-slot models as opposed to multi-rate have been considered. Since initial clustering of services in ATM networks reduces the complexity of the network equipment (CAC and UPC mechanisms) this is likely to be part of the rst implementations.

4.3.3 Approximation methods The approximation methods are based upon the rst two moments of the trac distribution, i.e. the mean mi and variance vi or peakedness zi = vi=mi (in case zi = 1, A is used instead of mi). These methods have been divided into four classes based upon the use of basic formulae which also determine the computational complexity of the method. The basic formulae and the approximations can be obtained from Table 4.3.

a) Erlang methods These methods convert the multi-slot trac streams into one composite trac stream. The blocking probability of the composite stream is evaluated with use of Erlang type formulae. This blocking probability is used together with a transformation to arrive at the blocking probabilities of the individual multi-slot streams. To arrive at the composite stream the following observation is made: Consider the random variable S , describing the number of slots a stream would occupy on an in nite trunk-group. The mean of the number of slots is equal to miCi and the variance is viCi2. The composite stream is a single-slot stream with mean and variance equal to the sum of the means and variances of S . The blocking probability of the composite stream is calculated using Erlangs loss formula. Three methods result from three dierent transformations to the individual blocking probabilities. The computational complexity of Erlangs loss formula is proportional to the link capacity C . Since the Erlang formulae are applied on a composite stream the computational complexity is best approximated by O(C + N ). Three methods have been examined:

 Method W:

The rst method, W, is based on the observation that the blocking probability of a trac stream will be higher for streams with higher peakedness and higher rate. Therefore the blocking probability will be computed by multiplying the blocking probability of the composite stream by a factor ziCi weighted with the average of all ziCi's. 33

Method

Formulae

Complexity

Erlang

 C ) = E (A; [C=C ]) BERL (A; C;

O(C )

(single-stream, Poisson) ERT2

 

+[C=C ])  C ) = E (CE;C BERT (m; v; C; (A ;C  )

O(C )

(single-stream, peaked) Product form (multi-stream, Poisson) Kaufman/Roberts (line occupancy) Kaufman/Roberts

N;C ?Ci ) BPRODi (A; C; C ) = 1 ? G(

G( N;C )

[ CCi ] j G( i;C ) = P Aj!i G( i?1;C?j
Ci ) j =0 N QKAUF` = P Ci`Ai QKAUF(`?Ci) i=1 C P QKAUF` = 1 `=0 C PKAUFi = P QKAUF` `=C ?Ci +1

(time congestion)

i = 1; : : :; N O(C N )

` = 0; : : :; C

`

Ci ] N C m [P P i i = (1 ? z ?1 )j ?1
Q

Delbrouck

QDEL`

(line occupancy)

` = 0; : : :; C

Delbrouck

PDELi =

i=1 `zi j =1 C P

PC

`=0

`=C ?Ci +i

i

DEL(`?jCi )

O(C
N )

O(C 2)

QDEL` = 1

QDEL`

(time congestion)

Table 4.3: Basic multi-slot formulae.

 Method WE:

The second method is an adjusted method, the transformation from method W is replaced with a technique from approximating single-slot peaked trac [20]. The peakedness of the multi-slot streams has been taken equal to ziCi. This method will be referred to as WE.  Method WR: The third method is another re nement of method W with a technique for approximation of single-slot peaked trac [62]. This method will be referred to as WR. 2

ERT=Equivalent Random Theory; C  and A are solutions from the Molyna-Nyquist equations.

34

According to these methods, the approximate blocking probabilities can be derived with the following equations. N N X X Bcomp = BERT ( miCi; viCi2; 1; C ) i=1 i=1 B Pcomp = Zcomp + (1 ? Zcomp )
Bcomp comp ziCi
B BiW = P comp N ( ziCi)=N i=1 z BiWE = Z iCi
Bcomp comp BiWR = 1 ? (1 ?zziCCi )
P
Pcomp i i

comp

(4.45) (4.46) (4.47)

(4.48)

b) Kaufman and Roberts methods These methods are based on the expression of Kaufman [34] and Roberts [59] for the distribution of the number of lines occupied (cf. Section 4.1.2). The computational complexity of this formula is O(C
N ). Four methods have been examined:

 Method K:

The rst method, referred to as K, considers the peaked streams to be Poisson. This corresponds to a one-moment method.  Method KE: The second method, KE, is a re nement of method K. It multiplies the time congestion computed by means of this method with the peakedness of the trac stream.  Method S: A third method, S, is based upon a technique for single slot approximations [65]. The method is based upon adding trac streams, one for each trac class, so that the composite trac in each class has peakedness 1. Therefore, we add trac with mean vi ? mi and variance 0. At the same time, the capacity of the link is increased with (vi ? mi)
Ci. From the resulting system the time congestions, Btot, can be evaluated using the results from Kaufman and Roberts. The congestion of the original streams can be found from the following relation: (4.49) Btot = mv i
Bi + vi ?v mi
0 i i Thus Bi = zi
Btot. 35

 Method E:

A fourth method, E, is based on the assumption that the dependencies of Ci and zi can be decomposed. This approximation uses two correction terms for the blocking probability of a composite stream. The rst correction is the eect of the rates which are measured by assuming Poisson trac and applying the method of Kaufman and Roberts. The second term is the eect of the peakedness which is measured by assuming single-slot trac and applying a transformation [49].

The formulae for numerical evaluation, according to these four methods, are given below.

BiK = PKAUFi (m1; : : :; mN ; C1; : : : ; CN ; C ) BiKE = zi
PKAUFi BiS = zi
PKAUFi (v1; : : :; vN ; C1; : : :; CN ; C + BiE = B (0) + (Bi(1) ? B (0)) + (Bi(2) ? B (0)) N X B (0) = BERL( miCi; 1; C )

N X i=1

(4.50) (4.51)

Ci
(vi ? mi))

i=1

Bi(1) = PKAUFi (m1; : : : ; mN ; C1; : : :; CN ; C ) zi (2) B
(2B + (1 ? B ) z ) Bi = 1+B N N X X with B = BERT ( miCi; miCizi; 1; C ) i=1

i=1

(4.52) (4.53) (4.54) (4.55) (4.56) (4.57)

c) Delbrouck methods These methods use a new expression for the line occupancy distribution. This distribution was rst introduced in [17] and has been proven to be equivalent to a process of batch arrivals with stochastic batch sizes [18]. The distribution is determined by the rst and second moments and the rates of the multi-slot trac streams. The time congestion is calculated by summation from the line occupation distribution. The computational complexity of the Delbrouck formula is O(C 2) if N  C . The dierent methods apply dierent transformations from time congestions to call congestions. Four methods have been considered:

 Method D1: D1 is a method proposed by Delbrouck for renewal processes, it makes use of the

average number of simultaneously active calls of a certain class.  Method D2: D2 is also proposed by Delbrouck for non-renewal processes. 36

 Method DE: DE combines time-congestion of Delbrouck with the transformation of [20].  Method DR: DR combines time-congestion of Delbrouck with the transformation of [62].  Method R: R is due to [60], where it is proposed to estimate the blocking probability by the use of the time congestion formula of Delbrouck with adjusted mean, mi = mi + zi ? 1. m~ i = BiD1 = BiD2 BiDE BiDR BiR

= = = =

C X

C ?j

(1 ? zi?1)[ Ci ]QDELj

j =0 mi ? m~ i mi mi ? m~ i mi + (zi ? 1)
m~ i zi
PDELi zi
P 1 ? (1 ? zi)
PDELi DELi PDELi (m1; : : : ; mN ; v1; : : : ; vN ; C1; : : : ; CN ;

(4.58) (4.59) (4.60) (4.61) (4.62)

C)

(4.63)

d) Product form methods Under the assumption of Poisson arrivals, the system state distribution described by the number of active calls of each trac class, satis es a product form (cf. Section 4.1.1). The recursive formulae of Kaufman and Roberts avoids the evaluation over all trac states (cf. Section 4.1.2). If, however, we have a general multi-rate system (not multislot) applying the recursive method is of no use. The product form solution remains valid for partial sharing policies, whereas the recursive solution does not. The computational complexity of the product form is proportional to the summation over all possible trac states, i.e. O(C N ).

 Method H:

The product form solution can be used as a generalization of a method published in [23]. The method transforms a peaked trac stream into a Poisson stream by dividing the mean by the peakedness and the capacity by the same peakedness. This relation is very similar to the Erlang formula for multi-slot single stream trac, where the capacity is divided by the rate. The generalization for multi-slot streams is arrived at by dividing the mean by the peakedness of the individual trac streams and multiplying the rate with the peakedness (similar to reducing the capacity). Because of the division by the peakedness the rates become non integer and therefore evaluation of the blocking probability must make use of the product form. 37

e) Other methods Another interesting method to approximate the blocking probabilities of multi-slot trac has been described in [60]. It uses an approximation of the line occupancy distribution with a truncated Pascal distribution which is tted to the multi-slot trac with the rst two moments of a composite trac stream for instance as in the Erlang methods described above. This method has not been evaluated.

4.3.4 Numerical results In order to be able to evaluate dierent approximation methods, a reference con guration has been de ned (cf. Figure 4.7). In this reference con guration we model peaked trac as over ow trac from Poisson trac oered to a link. A number of representative trac loads have been examined. These trac loads have been selected so as to model important parameters on which the blocking probability depends. Reference values have been obtained from simulation. In the basic con guration, two trac streams are oered to a single link. The trac stream with the higher rate is referred to as broadband trac, whereas the stream with the lower rate is referred to as narrowband trac. Trac stream one is assumed to have a Poisson arrival process with peakedness equal to 1. Stream 1 m 1 , z 1 , C1

Trunk Group

Stream 2

Size C

m 2 , z 2 , C2

Figure 4.7: Reference con guration. The parameters which determine and denote the trac load are:

m1 m2 v2 z2 C1 C2 C

the mean trac intensity of stream 1 (1=1) the mean trac intensity of stream 2 (2=2) the variance of trac intensity of stream 2 the peakedness of trac stream 2 (v2=m2) the bandwidth requirement of stream 1 the bandwidth requirement of stream 2 the capacity of the trunk group 38

Relevant parameter combinations are given by:

m1C1 or m2C2 m1C1 + m2C2 [m1C1=(m1C1 + m2C2)]
100% (m1C1 + m2C2)=C C1=C2

capacity demand of stream 1 or 2 total capacity demand of oered trac percentage capacity demand of stream 1 the (oered) load ratio of the trunk group rate ratio

For all considered trac situations the capacity C has been taken equal to 100 (lines) and the load ratio has been set to 1, the so-called critical load. The mix parameter has been varied and also the peakedness of stream two. All considered situations have been repeated for 5 dierent rate ratios, namely C1=C2 = 1=30; 2=6; 3=3; 6=2; 30=1. In the absence of reference material, a computation of the exact blocking probabilities Bi has been carried out using simulation. The results have been used to derive important dependency relations on important parameters or parameter combinations. They also serve as reference values to estimate the accuracy of the approximation methods. The peakedness of a multi-slot trac stream has little in uence on the blocking probability of other trac streams on the same link, see Figure 4.8. The blocking probability of other streams decreases slightly if the peakedness increases. However, the eect on the same trac stream is signi cant. The blocking probability increases if the peakedness increases. The blocking ratio between two multi-slot peaked trac streams sharing the same link can be expressed by a simple relation in the peakedness ratio, the rate ratio and the trac load, cf. equation (4.64). This formula has been derived empirically, using Figures 4.9 and 4.10. Therefore, its validity is more tentative than quantitative. It is valid for rates which are less than one tenth of the link capacity and is independent of the trac mixes. The dependency of the peakedness ratio is valid for low blocking thus low load conditions and is therefore not exact for the examined critical load.

B1 = z1
 C1  m1C1C+m2 C2 B2 z2 C2

(4.64)

The blocking ratio is linear in peakedness ratio and multiplicative in rate and load ratio where the rate ratio is the base and the inverse load is the power. This means that under critical load conditions the blocking ratio is proportional to the rate ratio, which has been proved in [57]. Under lower load conditions, the blocking ratio becomes more extreme, under higher loads the opposite is true. If the broadband trac is peaked, the blocking ratio is further enlarged by the peakedness ratio. If the narrowband trac is peaked, the ratio will be tempered a little.

39

B

1 0,8 0,6 0,4 0,2 0 0,01

0,1

1

10

100

C1:C2 B2; Z2=1

B2; Z2=2

B2; Z2=4

B1; Z2=1

B1; Z2=2

B1; Z2=4

Figure 4.8: Blocking probabilities as a function of the rate-ratio. B1/B2 100 10 Z2

1

1 2

0.1

4

0.01 0.001 0.01

0.1

1

10

100 C1/C2

Figure 4.9: Blocking probability for dierent peakedness factors.

4.3.5 Comparison results To compare the described approximate approaches, two measures of accuracy have been used. The absolute accuracy A is given by sP 2 A = (Bi n? Bi ) ; (4.65) 40

B1/B2 1,000 % S1

100

1% 10

10% 25%

1

50% 75%

0.1

90% 0.01 0.001 0.01

99% 0.1

1

10

100 C1/C2

Figure 4.10: Blocking probability for dierent trac mixes. whereas the relative error R is de ned by

sP  2 R = [(Bi ?nBi )=Bi] :

(4.66)

The variable Bi denotes the approximation for Bi and n is the number of experiments. The most accurate methods are Delbrouck methods followed by Kaufman and nally Erlang methods. The product form method F has an accuracy comparable to the Erlang methods but a higher computational complexity. Method E gives a good absolute accuracy but the relative accuracy is poor. The relative accuracy can be improved by applying the transformations proposed in [20], i.e. method WE, KE and DE, or with the transformations suggested in [62], i.e. method WR or DR. The transformation in [62] also improves the absolute error for the approximation methods while [20] worsens the absolute accuracy especially in case of high blocking probabilities. Methods which make use of the transformations in [20] give in some case unrealistic estimates (larger than 1) resulting in a poor absolute accuracy. Erlang methods do not explain non-monotonic behavior of the blocking probabilities. Therefore the relative accuracy is limited. The combination of Delbrouck's method with the transformation proposed in [62] is the most accurate approximation considered.

41

42

5. Trunk reservation and forming of call blocking Considering the model in Chapter 4, calls with higher bit rate demand Ci experience higher blocking probabilities in general than calls with lower bit rate demand Cj (Bi > Bj ). Therefore a more complex CAC algorithm is required to provide a fair access to the link for all trac classes. This chapter deals with a strategy for CAC which is called the trunk reservation mechanism. This mechanism is often discussed in the literature, e.g. in [33], [60], [69] and [(86)068]. The aim of trunk reservation is to in uence performance parameters of the system such as the blocking probability of dierent trac classes. To provide such a capability thresholds i are assigned to every trac class i. In addition to the trunk reservation mechanism there are several other access control strategies for in uencing the call blocking probabilities in multi-service systems discussed in the literature. Examples of such strategies are partial sharing or class limitation. For a comparison of several access control strategies see e.g. [47]. In this chapter we focus exclusively on trunk reservation, since it is the simplest and most eective method for in uencing call blocking. According to the CAC with trunk reservation, a call of type i will be blocked if CR < C ? i, and accepted otherwise. The parameter CR again denotes the available capacity of the transmission link upon arrival of a call of type i. In other words, trunk reservation is a mechanism of bandwidth prereservation for trac classes with higher bit rate requirements. This results in a reduction of call acceptance for lower bit rate call classes. The basic system environment for the model with trunk reservation is shown in Figure 5.1. λ 1 , µ1 , C 1

B1 Θ1 C Mbps ΘN

λ N , µN , C N

transmission link

BN

Figure 5.1: Basic link model for trunk reservation mode. Out of the number N of service classes we may want to in uence the GOS in such a way 43

that blocking probabilities of certain service classes should be dimensioned to be the same. This can be done by choosing appropriate values of the trunk reservation thresholds. The usage of trunk reservation in multi-service systems is often discussed in the literature. In Roberts [60] an approximate recursive solution for the state probabilities of the trac model with trunk reservation was proposed. It was stated how to choose the thresholds i to minimize the maximum blocking probability for all trac classes. An approximation of blocking probability for two trac classes where the trac of one class has a peakedness factor and trunk reservation is employed was derived in [49]. In [24] trunk reservation was investigated by an approximate iterative algorithm for the blocking probability evaluation. A general cost function for GOS management was introduced and an iterative algorithm for the minimization of this cost function was proposed. In the following, we present a simple rule for equalizing call blocking probabilities of arbitrary classes. Subsequently, two approximation methods for the call blocking probabilities are given. The rst one is based on the recursive solution in Section 4.1.2 and is quite accurate. For the second approximate algorithm, the simple solution of Section 4.2 is extended. This leads to less accurate results but the complexity remains quite low, even for models with a large number of trac classes.

5.1 Rule for equalizing call blocking As mentioned before, the trunk reservation mechanism can be used to in uence the call blocking probabilities for a given trac scenario. This can be done by setting the threshold variables i to certain values. In [69] the following simple and general rule for balancing blocking probabilities of calls from dierent classes was proposed: Rule: For any subset of trac classes !  f1; : : : ; N g the corresponding call blocking probabilities Bi (i 2 !) are equal, if all thresholds i for i 2 ! are set to b(C ? maxfCk j k 2 !g)c. This rule is more general than the one suggested in [60]. To motivate the rule we refer to Figure 4.3. The reason that the blocking probability for class-2 calls is higher than for class-1 calls is that a class-2 call is blocked when four basic bandwidth units are occupied. In that case a class-1 call could still be accepted. If 1 is set to b(10 Mbps ? maxf2 Mbps; 4 Mbpsg)c = 6 Mbps, which equals 3 basic bandwidth units, then the blocking probabilities B1 and B2 are equalized. As we will discuss later using numerical examples, a side eect of having trunk reservation for blocking probability equalization is that a fairer link utilization is obtained also (i.e. the class-i link utilization is proportional to the normalized oered bit rate trac of class-i calls).

44

5.2 Accurate approximations While considering the case of trunk reservation, we are leaving the area of state spaces which can be solved by a product form solution. In principle, at least for smaller state spaces, the state equation systems can be completely formulated and solved using e.g. an iterative algorithm. This possibility is however numerically intractable for more realistic parameter sets with larger state spaces. This motivates the necessity of an approximate solution. The recursive technique used here is again based on a mapping of the multi-dimensional state space into a one-dimensional state space, as discussed above. The use of this technique for this class of models was rst proposed in [60]. When employing trunk reservation, a call of class i is accepted only when at most mi basic bandwidth units are available upon arrival and not more than i Mbps are occupied. When calls from a class j will not be subject to trunk reservation, j is set to C ? Cj . This setting is reasonable, because class-j calls are blocked anyway if more than C ? Cj Mbps of the transmission link are occupied. Hence the call acceptance of class-j calls is not aected if j is set to this value. To derive a recursive solution for the state probabilities, the states of the trac model are de ned as in Section 4.1.2 by the number m of occupied basic bandwidth units. It should be stated here that this state space description is of an approximate nature. The state space from Figure 4.3 reduces with 1=4C = 3 to the state space shown in Figure 5.2 (2=4C is set to M = 3, i.e. class-2 calls are not subject to trunk reservation). λ2

λ2 λ1 0

λ1

λ1

1 µ1

2 2 µ1

µ2

λ2 λ1 3 3 µ1

3 2

λ2

µ2

4

5

4 µ1 2 µ2

5 2

µ2

Figure 5.2: Trunk reservation mode: example for the approximate state space. The dierence between the two state spaces in Figure 4.3 and Figure 5.2 is that the arrows between state 4 and 5 for class-1 arrivals and departures are not drawn. This is due to the fact that a class-1 arrival is blocked when 4 basic bandwidth units are occupied although the transmission link would be able to carry one more call of this class. As a consequence the blocking probability for class-2 calls is smaller, because some of the class-1 calls are prevented from entering the system. To emphasize the approximation we denote the state probabilities of the one-dimensional state space by p(m) and the resulting blocking probability obtained by the recursive solution by Bi. The unnormalized state probabilities can be approximately obtained using the following recursion algorithm (see [33], [60]) which is similar to the one given by

45

equation (4.14):

8 > 1 : m=0 > > < : m 0 > N > : m1 P p~(m ? mi)mi(m) ii : 0 < m  M i=1

(5.1)

where mi(m) is de ned by

8 > < m : m
4C  i : mi(m) = > i : 0 : m
4C > i

(5.2)

In this approximation approach, the results depend only on the oered trac Ai = i=i but not on the arrival and service rates themselves. However, this independence is not given if trunk reservation is employed and therefore the approximation accuracy should be compared in terms of these parameters. After a normalization of p~(m) we obtain the state probabilities p(m):

p(m)

=

p~(m)


M X m=0

p~(m)

!?1

:

(5.3)

The blocking probability Bi for calls of class i can be calculated by

Bi =

M X m=minfM ?mi;i =4C g+1

p (m):

(5.4)

In the example state space (cf. Figure 5.2) the blocking probabilities of class-1 and class-2 calls are the same according to the blocking probability equalization intention of the trunk reservation mechanism:

B1 = p(4) + p (5) = B2:

(5.5)

Now, numerical results taken from [69] will be presented to show two main eects:

 the approximation accuracy of the recursive algorithm and  the eciency of the trunk reservation mechanism and accordingly, the blocking probability and the GOS management issues. 46

blocking probability

The rst example deals with two trac classes with C1 = 2 Mbps and C2 = 20 Mbps which oer the same oered bit rate trac (1 = 2) to a transmission link of capacity C = 150 Mbps. The blocking probabilities and link utilizations for this set of parameters are shown in Figure 5.3 and Figure 5.4, respectively. When trunk reservation is not employed the blocking probability (see Figure 5.3) for class-2 calls is clearly higher than that of class-1 calls due to the higher bit rate demand of class-2 calls (C2 > C1). The trunk reservation thresholds 1 and 2 have been chosen according to the rule stated in Section 5.1 to 1 = 2 = 130 Mbps and therefore equalize the blocking probabilities of calls from both classes. One can observe that, at the call level, the blocking probability B1 increases much more than B2 decreases. This eect becomes more marked if the proportion between the bandwidths of the two trac classes C1 and C2 is chosen larger.

10E0 •

•

• • •

10E-1 • • •

10E-2

• 10E-3

without trunk reservation: class-1 class-2

•

with trunk reservation: class-1, class-2 simulation •

•• 10E-4 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

Figure 5.3: Blocking probability equalization by trunk reservation and approximation accuracy. Simulation results show that the accuracy of the recursive approximation is good for the whole range of trac intensities. We have observed this level of accuracy also for a large number of other parameter sets and dierent trac mixes (1 6= 2) whose results are not shown here. In general, it can be stated here that the approximate solution for the case of trunk reservation can be considered as sucient. The dierences between the exact blocking probabilities Bi and the approximate ones Bi are negligible for practical purposes. The transmission link utilization (cf. Figure 5.4) by class-1 calls 1 is higher than the utilization by class-2 calls 2 without trunk reservation. By employing trunk reservation 47

link utilization

1.0

0.8

without trunk reservation: total class-1 class-2

0.6

with trunk reservation: total class-1, class-2

equalization

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

Figure 5.4: Link utilization sharing with trunk reservation. the utilization from both classes is also equalized. The price to be paid for that equalization is that the total transmission link utilization  decreases with trunk reservation. Results for a more complex example are shown in Figures 5.5 and 5.6 respectively. The transmission link has a capacity of C = 150 Mbps and we consider four trac classes with bit rates C1 = 2 Mbps, C2 = 5 Mbps, C3 = 10 Mbps, C4 = 20 Mbps. Calls from the dierent trac classes are assumed to oer the same load (1 = 2 = 3 = 4) and we want to equalize the blocking probability and transmission link utilization of class-1 and class-3 calls. Figure 5.5 shows that a trunk reservation threshold 1 = 140 Mbps not only equalizes the blocking probabilities of class-1 and class-3 calls but also decreases the blocking probabilities of class-2 and class-4 calls. As expected from the results for the blocking probabilities the utilization of the transmission link by class-1 and class-3 calls (1; 3) is also equalized by means of the trunk reservation mechanism with 1 = 140 Mbps (see Figure 5.6). The transmission link utilization by class-2 and class-4 calls (2; 4) is increased as the blocking probability is decreased. The numerical results show that using the trunk reservation mechanism it is possible to equalize blocking probabilities from calls of dierent trac classes. Furthermore, the transmission link utilizations i can be set proportional to the normalized oered bit rate tracs i .

48

blocking probability

10E0

10E-1

4

4

3

1,3

2

2

1 10E-2

equalization TR

TR

10E-3

TR: without trunk reservation TR: with trunk reservation 10E-4 0.2

0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

link utilization

Figure 5.5: Blocking probability equalization in a multi-service system. 0.4 TR: without trunk reservation: TR: with trunk reservation:

equalization 1

0.3

2 2 1,3 0.2

3

4 4

0.1

TR

TR

0.0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

Figure 5.6: Link utilization sharing in a multi-service system. 49

5.3 Simple approximations After presenting very accurate approximations for the blocking probabilities for the model with trunk reservation, we now focus on a simpler method which is therefore less accurate. This simple approximation was suggested in Lindberger [51] and is based on the results derived in Section 4.2. In the sequel we assume that the blocking probability of all classes is equalized. Thus we have a common blocking probability B0. When trunk reservation is employed, the common blocking probability is in general also dependent on the ratio of the mean holding times of the dierent classes of calls. However, it would be very complicated to consider also the relationships between all the mean holding times when the number of trac classes is large. Thus we assume, for the approximation we describe, that the mean holding times are equal and the result is meant to cover cases when they are in the same order of magnitude. In this case, only the same two characteristics of a trac class as in the case without trunk reservation are needed. The probability of the last state before the blocking state is still pn , but for the blocking states we now have

pn+i = pn bi(aCmax + aCmax?1 + : : : + ai)=a:

(5.6)

The explanation of the last ratio in this equation is that the state n + i can now only be reached by arrivals from calls with bandwidth requirements greater than or equal to i. The common blocking probability is then

B 0

=

CX max i=1

pn+i

CX max 1  ? 1 = a pn (1 ? b) ai(b ? bi+1) i=1

(5.7)

and by again using the approximation (b ? b i+1 )=(1 ? b)  ib(i+1)=2 we get Cmax max a B  = CX (i+1)=2  X ia b(C+1)=2 = aCb  (C+1)=2: ia b i i 0  pn i=1 i=1

(5.8)

Thus, the common blocking probability B0 can be expressed approximately by

B0 = pn C (a=C )(C+1)=2C :

(5.9)

An estimate for the ratio between B0 and B  can easily be obtained using

B0 = (C=a)(Cmax?C)=C : B 

(5.10) 50

If B0 is expressed in terms of Erlang's loss formula, we get  C ? Cmax + 1 a   B0 = E ; C : C

(5.11)

For the derivation we assumed, for simplicity, the bandwidths of all classes to be integer multiples of the smallest bandwidth. The result can easily be generalized in the following way:  + Cmin ; a  : B0 = E C ? Cmax (5.12) C C Case 1 App 2 3 App 4 App 5 7 9 App 8 10 App 6 App

B0 1:3E ? 2 1:3E ? 2 2:4E ? 2 2:4E ? 2 2:6E ? 2 1:9E ? 2 1:9E ? 2 1:4E ? 2 1:3E ? 2 1:3E ? 2 1:4E ? 2 1:9E ? 2 1:9E ? 2 2:0E ? 2 0:6E ? 2 0:5E ? 2

Case 11 App 12 13 App 14 App 15 17 19 App 18 20 App 16 App

B0 4:7E ? 2 4:9E ? 2 6:4E ? 2 6:4E ? 2 6:8E ? 2 5:6E ? 2 5:9E ? 2 4:8E ? 2 4:8E ? 2 4:8E ? 2 5:1E ? 2 5:7E ? 2 5:7E ? 2 6:1E ? 2 3:2E ? 2 3:3E ? 2

Case 21 App 22 23 App 24 App 25 27 29 App 28 30 App 26 App

B0 1:1E ? 1 1:2E ? 1 1:3E ? 1 1:3E ? 1 1:4E ? 1 1:2E ? 1 1:3E ? 1 1:1E ? 1 1:1E ? 1 1:1E ? 1 1:2E ? 1 1:2E ? 1 1:2E ? 1 1:3E ? 1 9:6E ? 2 1:0E ? 1

Table 5.1: Accuracy of the simple approximation formulae. In Table 5.1 some results, which have been presented in [51], are given to illustrate the quality of this rough approximation. To do this, the approximate values are compared to very accurate ones which are derived using the recursive algorithm in Section 5.2. 51

Again, as in Section 4.2, the mixtures of Table 4.1 are used to generate the rst part of the table and the capacity of the transmission link is set to C = 150 Mbps for the mixtures 1 to 4 and C = 300 Mbps for the mixtures 5 to 10. Thus, the load factor is 0.8 for the rst ten cases. To obtain the values in the second and third part, the link capacity was reduced by 10% and 20%, respectively (cf. Section 5.2 for a detailed description of the cases 11 to 30). By looking at the equivalent cases in the rst part of the table, it can be stated that the values are very similar, just as in the case without trunk reservation. Furthermore, the approximate values for B0 are close to the corresponding exact values and thus the approximation is good. For the cases with reduced link capacity, i.e. part 2 and 3 of Table 5.1, the results from the simple formulae are usually a bit too high. With a relative error of typically 5%, however, they are still reasonably accurate.

5.4 Heavy trac approximation Recent developments in communication networks have led to much interest in systems where trac of widely diering characteristics is integrated together. Bean et al [6], [7] develop an analysis of single resource loss systems under the assumption of heavy trac. In this section we discuss the analysis with special emphasis on its practical implementation for solving real world examples that arise in the study of multi-service networks. The extension of this approach to networks is possible by means of the reduced load approximation (see Whitt [72]). The assumption of heavy trac also holds in this section, but there is good reason to expect results that are also accurate when the resource is near to critical loading. Formally we study a resource of integer capacity C oered a nite number of trac streams indexed in a set J . Calls of type j 2 J arrive as a Poisson stream of rate j and have exponential holding times of mean ?j 1; each such call requires integer capacity Cj , and is accepted if and only if the resulting free capacity of the resource is at least rj ; otherwise the call is lost. The parameters rj are referred to as trunk reservation parameters and provide an important mechanism for controlling the behavior of the system. Note that in this section we use, for the sake of simplicity, a dierent de nition of the trunk reservation thresholds than in the other parts of this book. This de nition is however equivalent to the rst one if rj is substituted by C ? j . All arrival streams and holding times are independent. In order to ensure irreducibility of all the stochastic processes involved, we assume, without loss of generality, that the capacities Cj ; j 2 J , have a greatest common divisor equal to 1. Of particular interest for loss systems is the determination of the blocking probability, i.e. the equilibrium call rejection probability, associated with each call type. For the model considered in this section it is possible in principle to determine blocking probabilities exactly, since they are functions of the equilibrium distribution of a Markov process. However, the state space for this process is typically so large as to make this determination impossible in practice. In Section 5.4.1 it is shown how, by studying the stochastic process 52

which merely describes the free capacity of the system at any time, good approximations for the blocking probabilities may be obtained. These approximations are asymptotically exact under the limiting scheme described there. In Section 5.4.2 we consider some numerical aspects of the determination of these approximations. Section 5.4.3 gives some numerical examples.

5.4.1 Analysis of the model Let N (t) = (Nj (t); j 2 J ) where Nj (t) is the number of calls of type j in progress at time t. Then N (
) is a (vector) Markov process with state space S = fn 2 Z+jJ j : Pj2J Cj nj  C g (where Z+ is the set of non-negative integers). Let  = ((n); n 2 S ) denote its equilibrium distribution. For each time t, de ne also M (t) = C ? Pj2J Cj Nj (t). Then M (t) is the free capacity at time t and the process M (
) takes values in the state space M = f0; 1; : : : ;PC g. The equilibrium distribution  = ((mP ); m 2 M) of this process is given by  (m) = n2S(m) (n), where S (m) = fn 2 S : j2J Cj nj + m = C g. Note that a knowledge of the onedimensional distribution  is sucient for the determination of blocking probabilities. We now show how  may be determined, at least to a good approximation, without the very much more dicult evaluation of the higher-dimensional distribution . De ne X   (n)nj n 2S ( m ) (5.13) j (m) = X  ; m 2 M; j 2 J:  (n)

n2S(m)

Thus j (m) is the expected value of Nj (t) under the equilibrium distribution , conditioned on the event N (t) 2 S (m). De ne also  = (j (m); m 2 M; j 2 J ). Then ( ;  ) satis es the system of equations (5.14) to (5.17) in (; ), where  = (j (m); m 2 M; j 2 J ),  = ((m); m 2 M),   0 and   0, given by 2 3 X X (m) 4 j + j j (m)5 = j :rj +Cj m

X

j :rj m; m+Cj 2M

X m2M

X j 2J

j :m+Cj 2M

(m + Cj )j +

X

j :Cj m

(m ? Cj )j j (m ? Cj );

(m) = 1;

Cj j (m) + m = C;

m 2 M;

(5.14) (5.15)

m2M

(5.16) 53

and

gj (; ) = 0; j 2 J;

(5.17)

where, for each j , the function gj is de ned by

gj (; ) = j

X mCj +rj

(m) ? j

X m2M

(m)j (m):

(5.18)

To see these results, observe rstly that the equations (5.14) follow easily, for each m, by summing the global balance equations for the equilibrium distribution  of the Markov process N (
) over the states n 2 S (m) and using the de nition (5.13). The equation (5.15) is trivially satis ed by  =  . That  =  satis es the equations (5.16) follows immediately from (5.13), or from the interpretation of j (m) as an expected value. Finally, the equations (5.17) (with (; ) = ( ;  )) may formally be derived from the global balance equations for , or from the observation that, in equilibrium, the expected acceptance rate equals the expected departure rate for each call type j . Note that the equations (5.14) and (5.15) may be regarded as the equations determining the unique equilibrium distribution  of a Markov process on M with transition rates given, for each j 2 J , by 8 > < m ? Cj ; at rate j I (m  rj + Cj ); m!> (5.19) : m + Cj ; at rate j j (m)I (m + Cj 2 M) (where I denotes the indicator function). Thus, as usual, any one of the equations (5.14) may be omitted (being implied by the remainder). Similarly, if (; ) satis es the equations (5.14), then, by multiplying each of these equations by the corresponding value of m and summing over all m 2 M, it follows without too much diculty that Pj2J Cj gj (; ) = 0. Thus, any one of the equations (5.17) may also be omitted from the above system. In general, the equations (5.14) (5.17) are insucient to determine (;  ). We may however use them to determine (;  ) approximately by making appropriate assumptions, for each j , about the dependence of j (m) on m. To motivate our approximation scheme, we rst consider some asymptotic theory. Bean et al [7] show that, when C is large, and under the heavy trac condition

X j Cj > C; j 2J j

(5.20)

the process N (
)=C evolves approximately as a deterministic dynamical system x(
), and that the process M (
)=C eventually remains close to 0. The xed points of this dynamical 54

system are in one-to-one correspondence with the solutions of the system of equations (5.14), (5.15) and (5.17) modi ed by replacing M throughout by Z+ and, for each j 2 J , by replacing j (m) by a positive constant j (independent of m), where we additionally require that (in place of the equations (5.16))

X j 2J

Cj j = C:

(5.21)

For each solution (; ) (where  is interpreted as (j ; j 2 J )) of this modi ed system of equations, the corresponding xed point of the dynamical system is given by =C . Now suppose that the dynamical system x(
) possesses a unique xed point x to which all of its trajectories converge. Bean et al [7] show that the component  of the corresponding unique solution (; ) of the above modi ed system of equations then approximates the equilibrium distribution  of the process M (
). The intuitive explanation for this is that, since, in equilibrium, N (
)=C remains close to x and M (
)=C remains close to 0, for each j the constant j becomes a reasonable approximation to j (m). (Further, under these conditions, the process M (
) behaves approximately as a Markov process with transition rates given by the set of equations (5.19) with j replacing j (m).) Under the limiting scheme of Kelly [40], in which the arrival rates j and the capacity C are allowed to grow in proportion, and with the heavy trac condition (5.20) continuing to hold, the solution  of the modi ed system of equations remains constant and is the limit (under weak convergence) of the exact equilibrium distribution  . If the dynamical system x(
) does possess multiple xed points, then the component  of each of the corresponding solutions of the above modi ed system of equations represents an approximate quasi-equilibrium distribution of the process M (
), that is, a distribution which behaves as an equilibrium distribution over a sustained period of time. However, there is at present no evidence to suggest that such multiple xed points can occur. While the above approximation to  becomes exact under the limiting scheme described, it is nevertheless too crude for most practical applications. We therefore seek to improve it by retaining the state space M and, for each j , approximating j (m) as a linear function of m. Although this linear approximation is less than ideal from a theoretical viewpoint, it nevertheless appears to work well in practice. We also expect that it will continue to work reasonably well even when the heavy trac condition (5.20) is not satis ed, particularly if the resource is instead at or near to critical loading. Recalling that j (m) must also be always positive, we thus replace the equations (5.16) by

j (m) = aj (C ? m);

j 2 J; m 2 M

(5.22)

where

aj  0; for each j 2 J; and

X j 2J

Cj aj = 1: 55

(5.23)

The equilibrium distribution  is then approximated by the component  of the solution (a; ) (where a = (aj ; j 2 J )) of the system of equations (5.14), (5.15), (5.17), (5.22) and (5.23), provided that this solution is unique. Note that, after eliminating  and the redundant equations, the above system of equations consists of jMj + jJ j equations in jMj + jJ j unknowns. Multiple solutions, were they to occur, would again correspond to quasi-equilibrium distributions of the process M (
).

5.4.2 Discussion In the previous section we described a model for approximating  by treating j (m) as linear in m. In this section we consider the numerical determination of this approximation through the solution of the system of equations (5.14), (5.15), (5.17), (5.22) and (5.23). This problem may be regarded as that of nding the component a of the solution (a; ): for any given a in the (jJ j ? 1)-dimensional region de ned by the conditions (5.23), the equations (5.22) determine a value of  and then the equations (5.14) and (5.15) determine a distribution ; we must therefore choose a such that (; ) satis es the equations (5.17) (where again any one member of this set may be omitted). In the special case where jJ j = 2 a bi-section method can be used, but in general a modi ed version of the Newton-Raphson routine in which derivatives need not be supplied is highly eective. A further consideration arises in the solution of the equations (5.14) and (5.15) (for given ) at each iteration of any numerical procedure. For moderate values of C these may be solved by matrix inversion (omitting any one of the equations (5.14)). For large C this is impractical. In this case, we may make a further appeal to the asymptotic theory of Bean et al [7] which shows that, provided j (m)=C varies slowly with m (as is the case here), the tail of the distribution  determined by the equations (5.14) and (5.15) is approximately geometric with a parameter p which can be directly determined in the manner described below. We may therefore choose a threshold value m0 2 M and assume that

(m) = (m0)pm?m0 ;

for all m  m0:

(5.24)

The parameter p should be taken as the unique real root between 0 and 1 of the polynomial f : C ! C of degree 2^e (where e^ = maxj2J Cj ) given by

f (z) = ze^

X j 2J

(j + j j (m0)) ?

X j 2J

 ze^+Cj j + ze^?Cj j j (m0) :

(5.25)

The value of m0 should be chosen so that Pmm0 (m) is small. If m0 and p are determined in this manner, the set of equations (5.14) need only be considered for m = 0; 1; : : : ; m0 ?1. Substitution of the relation (5.24) into these equations and equation (5.15) yields m0 + 1 linear equations in (0); (1); : : :; (m0), so determining the entire distribution . An alternative approach to the problem caused by large C would be to assume that (m) = 0 for all m  m0. However, to achieve a similar degree of accuracy would require a much (m) would then need to be negligible. greater value of m as P 0

mm0

56

5.4.3 Examples In this section the algorithm of the previous section is used to calculate blocking probabilities for the call types given in Table 5.2 and the choice of system parameters given in Table 5.3, where the arrival rate of calls is in units of call per second rounded to two decimal places. We consider four choices of call characteristics, described by the equivalent bandwidth and mean call holding time, which might, for example, be associated with the applications shown. Application Call Type Digitized Voice I Interactive Video Retrieval II File Transfer III Distribution Video IV

Eective Bandwidth (Mbps) 0:04 0:50 2:00 2:00

Holding Time (s) 180 1200 60 1800

Table 5.2: Parameter values used to describe each call. Figure

Type 1 Call Type 5.7 II 5.8 III 5.9 I

Arrival Rate 0:24 2:68 32:38

Type 2 Call Type IV IV II

Arrival Rate 0:13 0:08 0:70

Capacity (Mbps) 622 622 622

Table 5.3: System parameter values. For each of the examples de ned in Table 5.3, we present the analytic results and compare them to simulations of the system, for which we also display the relevant 99% con dence intervals. In each example, we consider a single resource and two types of oered trac, and write s = r1 ? r2 while holding at least one of r1 or r2 equal to zero. We study how well our model approximates the two blocking probabilities as s varies. (The choice of an optimal value of s will, of course, depend on the cost structure adopted, and perhaps other criteria.) Figure 5.7 shows an example where the algorithm gives highly accurate estimates of the blocking probabilities for both types of call over a wide range of values for the trunk reservation parameters. In this example, the link is critically loaded since

X Cj j = C: j 2J j

(5.26)

This gure shows the typical behavior of the blocking probabilities at such a link. The relationship between the blocking probabilities of the call types is governed by the relative 57

0:15

Call Blocking Probabilities 0:05 0:10

IV

0:0

II

?10

?5 0 5 Trunk Reservation, s = r1 ? r2 (Mbps)

10

Figure 5.7: Analytical and simulation results with varying trunk reservation. values of C1 + r1 and C2 + r2. So, for example, when C1 + r1 = C2 + r2 the blocking probabilities are equal for both call types. The blocking probability for the calls of type 1 is almost zero when the trunk reservation parameters are such that s is large and negative and as s increases, the blocking probability decreases for calls of type 2 and increases for calls of type 1. Note that when there is no trunk reservation, that is s = 0, the calls with the higher equivalent bandwidth have a higher blocking probability than those of the other call type. In Figure 5.8 the link is again critically loaded and has two call types of identical equivalent bandwidth but very dierent holding times. We again nd that the algorithm provides accurate estimates for the blocking probabilities. Finally, in Figure 5.9, we consider a more extreme case where both the equivalent bandwidths and holding times dier considerably between call types. The link satis es the heavy trac condition (5.20), in fact

X Cj j = C  1:05; j 2J j

(5.27)

corresponding to 5% overload. In this case the results are somewhat less accurate despite capturing all the qualitative features. The results show that the algorithm described here can be used in a wide range of practical examples to give very accurate estimates for the blocking probabilities in multi-service networks. 58

Call Blocking Probabilities 0:05 0:10 0:15 0:20

III 0:0

IV

?40

?20 0 20 Trunk Reservation, s = r1 ? r2 (Mbps)

40

Call Blocking Probabilities 0:05 0:10 0:15

Figure 5.8: Analytical and simulation results with varying trunk reservation.

II

0:0

I

?0:8

?0:4 0:0 0:4 0:8 1:2 Trunk Reservation, s = r1 ? r2 (Mbps)

1:6

Figure 5.9: Analytical and simulation results with varying trunk reservation. 59

Further, the asymptotic theory of the preceding sections shows that any inaccuracies decrease as the system capacity increases. To improve these results, for example where call types vary widely or where the load is light, would require further investigation into the form of j (m) (see equations (5.22) and (5.23)).

5.5 In uence of holding time Since the call blocking probabilities depend also on the mean holding times and the holding time distribution type when trunk reservation is employed, the next two subsections deal with these two quantities. First, we focus on the in uence to the blocking probabilities if the mean holding times of the trac classes dier in order of magnitude. Afterwards we show that the trunk reservation mechanism is quite robust against the holding time distribution type.

5.5.1 In uence of the mean holding time In this section we are primarily interested in investigating the eect of varying the mean holding time ratio on the blocking probabilities experienced by two heterogeneous call types subject to trunk reservation on a single link of capacity C . Over the past few years, various models have been developed which calculate exact solutions or use approximate methods where numerical derivation of exact solutions is not possible, see e.g. [6], [17], [34], [48], [60] and [(92)018]. Our main concern is to concentrate on the case of service protection mechanism of trunk reservation.

a) Service protection of trunk reservation Suppose that a certain amount of capacity is reserved for exclusive access by a particular trac type. For example, in circuit-switched networks, fresh calls can be given priority over over ow calls by reserving a number of channels . In this case, either type of calls could occupy a channel if there are more than  channels free while only fresh calls have the chance of occupying the remaining  channels. The same idea applies in a multi-rate situation. One of the most useful applications of trunk reservation is to equalize the blocking probabilities of dierent trac types as high bit rate calls experience a worse GOS since they have less chance of occupying their required bandwidth. Consider the de nition of trunk reservation as given in Section 5.1, then the blocking probabilities for all call types are equalized if all thresholds i = b(C ? maxfCk j k 2 !g)c with ! = f1; : : :; N g, in other words all the calls will be blocked in the same states. In the case of trunk reservation, no product form solution exists, due to the fact that one-way transitions are present in the state space which destroy reversibility. Although in principle, the blocking probabilities could be calculated by solving the birth-death 60

equations, this becomes intractable as the number of call types and the link capacity grow. Consequently approximations are sought. In the rst part of this chapter, two approaches have been presented. However, these methods do not cater for dierent mean holding times. In the following, we will show the sensitivity of the blocking probabilities to changes in the mean holding time ratio of calls (in [28] this has already been shown for a single-slot case). A very recent method [6] derives these blocking probabilities approximately for truly heterogeneous trac scenarios, where these quantities are derived by focusing on the onedimensional equilibrium distribution of available capacity rather than the N -dimensional overall occupancy distribution. They derive asymptotic results when link capacity is large and consequently obtain approximations by solving for birth and death equations based on the expected number of calls of type i in progress. In the next section, exact results are obtained by solving for the birth-death equations when only two heterogeneous call types are present and they are subject to trunk reservation.

b) Numerical results Now, we shall show by means of exact results (solving for the state space probabilities using the algebraic package Maple) for a small link capacity, that the blocking probabilities of two dierent call types are sensitive to the mean holding time ratio of the calls. If we minimize the maximum blocking, i.e. equalizing the blocking probabilities, the blocking probabilities will still be sensitive to the mean holding time ratio. In the examples used, we assume that the total oered trac for both call types does not change and, as a result, only the arrival rate and the mean holding time of the call type with a larger bandwidth requirement changes. Figures 5.10 and 5.11 illustrate the graphs of call congestion for two call types as the mean holding time ratio 2=1 increases under two dierent trac scenarios when no trunk reservation is used and when trunk reservation equalizes the blocking probabilities. The data to represent the gures is summarized below. Note that throughout the examples given in this section, the capacities (link, call bandwidths and trunk reservation parameters) are in terms of any unit of capacity (e.g. circuits, channels, slots, Mbps). Figure 5.10: A heavily loaded link with capacity C = 30 being oered two call types of the following characteristics:

1 = 10 2 = 5=2

1 = 1

C1 = 1 C2 = 5

1 = 27 2 = 30

Where i , i , and Ci are the Poisson arrival rate, the mean holding time (exponentially distributed) and the bandwidth requirement of the call type i respectively. i is the trunk reservation parameter associated with the call type i. Blocking probabilities are equalized if 1 = 2 = 25. The total oered trac (normalized) is 35=30  1:17. When the mean holding time ratio 2=1 changes, 1 and the total oered trac of call type 2 are xed and consequently 2 varies. 61

Figure 5.11:

This is the same data as for Figure 5.10 but the link is oered less amount of call type 2, i.e. the link is lightly loaded now where 2 = 1=2 with a total normalized trac of 0.5.

1.0

call loss probabilities

0.8

0.6 call 2 (no TR)

0.4 equalized blocking 0.2 call 1 (no TR)

0 0.1

1.0 10.0 Mean Holding Time ratio(call2/call1)

100.0

Figure 5.10: Sensitivity to changes in mean holding time ratio (heavy load). The following observations can be made from Figures 5.10 and 5.11:

 The higher bandwidth call experiences a worse GOS (i.e. a higher blocking proba-

bility), where no trunk reservation is used (cf. Chapter 4).  Blocking probabilities are equalized if the rule proposed in Section 5.1 is employed.  The blocking probabilities are sensitive to the changes in the mean holding time ratio. This is a very important result as it has implications on taring and optimally allocating trunk reservation values amongst competing customers using the same resources in the network.  The sensitivity of the call blocking probabilities to the mean holding time ratio is more apparent when the link is lightly loaded. It is also true that the sensitivity is more apparent for the low bit rate call (call type 1) as its mean duration becomes smaller than that of the higher bit rate call. This will result in a better GOS being experienced by call type 1. This is because the call type 1 has an upper hand for two reasons; it requires less amount of capacity to set up and it also has a shorter duration.

Now, suppose that there are two competing call types and the call type 2 will always require more bandwidth. Furthermore, assume that the total oered trac of the two call types stays constant but the nature of the mean holding time of the call type 2 is not 62

0.05

0.04 call loss probabilities

call 2 (no TR)

0.03

equalized blocking

0.02

0.01 call 1 (no TR)

0 0.1

1.0 10.0 Mean Holding Time ratio (call2/call1)

100.0

Figure 5.11: Sensitivity to changes in mean holding time ratio (moderate load). known and could vary quite drastically. The question is: How could its GOS be maintained without restricting the GOS criterion for the call type 1 (i.e. assuming that call type 1 could experience any blocking probability)? Figure 5.12 shows that the trunk reservation values needed to guarantee a GOS of 1 percent for the call type 2 change as the mean holding time ratio varies (although this example illustrates that the ratio should be very large in order to see such an eect but this is only a small link size and the study should be repeated for larger link sizes). Figure 5.12 also shows the blocking probability of call type 1 for the corresponding trunk reservation value and mean holding time ratio as the call type 2 GOS is maintained. Obviously, the GOS of the call type 1 improves as more amount of capacity becomes available to calls of type 1. Data is the same for Figure 5.11 where 1 is also a variable. The important message is that the trunk reservation mechanism may not be such a robust mechanism if very dierent services with such varied characteristics are present.

c) Conclusions and further studies To conclude, preliminary results suggest that the blocking probabilities of dierent trac types are sensitive to changes in the mean holding time ratio even when the call blocking probabilities are equalized. This eect is more apparent when the system is lightly loaded and when the mean holding time of higher bit rate calls is larger than that of the lower bit rate calls. Furthermore, as the mean holding time ratio changes, the trunk reservation parameter varies quite drastically to maintain a nominal GOS of 1 percent, say, for a high bit rate call type. If dynamic allocation of trunk reservation is not practical, one solution to maintain the GOS for both call types is to allow call types with a speci c mean holding 63

20

0.5

0.4

0.3 10 0.2

GOS of call type 1

trunk reservation parameter

Maintaining GOS of call type 2 at 1%

0.1

0 0.001

0.010

0.100 1.000 10.000 mean holding time ratio (call 2 / call 1)

0 100.000

Figure 5.12: Sensitivity of GOS of call type 1 and its trunk reservation parameter to changes in mean holding time while maintaining GOS of call type 2. time ratio to share a link. The results obtained here require further investigation for a wide range of link sizes. This will be the subject of a further study once the method explained in [6] is implemented. The task of identifying appropriate trunk reservation values in general, whether to provide an equal GOS for dierent services or guarantee a preferential performance for speci c call types is non-trivial. This is a major study in itself. Even though we have shown that with trunk reservation the result for a mixture of two trac streams can, in cases where the mean holding time ratio is very great, dier signi cantly from the case with similar mean holding times, a simple method which does not include mean holding time relations could still be used in practical dimensioning for the following reasons:

 Trunk reservation is only recommended to be used in mixtures where the bandwidth

ratio between calls is less than or equal to a value of the order of 10.  Mean holding time ratios as large as, for example, 100 are very dicult to combine with the assumption of constant arrival intensities for both types of calls during common busy hour-like periods anyway.  The result will always be more extreme in cases with only two types of bandwidths. The methods are usually meant to be used in mixtures with a greater number of bandwidths. In such cases it is most likely that we will have examples of both smaller and larger mean holding times, both among the smaller and the larger bandwidths 64

in the mixture. This will make the result similar to the case with equal mean holding times.  If still all call types with great bandwidths should have mean holding times in another magnitude than the other call types, we should expect this magnitude to be the greater one, and thus the blocking probability would be smaller than what the simple method, independent of mean holding times, would give. That this could happen sometimes is not too serious.  Methods depending upon all of the mean holding time relations in a mixture with many classes of bandwidths are dicult to use in the practical dimensioning and optimization of the network anyway. All in all, trunk reservation is still the simplest and most eective method for equalization.

5.5.2 In uence of the holding time distribution type As pointed out in Section 4.1.2, the resulting probabilities for blocking calls of the dierent classes depend only on the mean holding time and not on the holding time distribution type itself, if trunk reservation is not employed. However, in the case of trunk reservation the holding time distribution type may have an in uence on the blocking probabilities.

1 + 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

cTH1 = cTH2 = 0

class 1

class 2

cTH1 = cTH2 = 1=3

cTH1 = cTH2 = 1

cTH1 = cTH2 = 4

class 1

class 2

class 1

class 2

class 1

class 2

1.28E-3 8.10E-3 2.68E-2 5.97E-2 1.05E-1 1.56E-1 2.08E-1 2.58E-1 3.04E-1 3.47E-1

1.34E-3 8.06E-3 2.69E-2 5.95E-2 1.05E-1 1.56E-1 2.09E-1 2.58E-1 3.05E-1 3.47E-1

1.28E-3 8.05E-3 2.68E-2 5.95E-2 1.05E-1 1.56E-1 2.08E-1 2.58E-1 3.04E-1 3.47E-1

1.32E-3 8.05E-3 2.68E-2 5.95E-2 1.05E-1 1.56E-1 2.09E-1 2.58E-1 3.04E-1 3.47E-1

1.27E-3 7.90E-3 2.66E-2 5.94E-2 1.04E-1 1.55E-1 2.07E-1 2.57E-1 3.03E-1 3.45E-1

1.29E-3 7.96E-3 2.65E-2 5.92E-2 1.04E-1 1.55E-1 2.07E-1 2.57E-1 3.03E-1 3.45E-1

1.27E-3 7.80E-3 2.56E-2 5.73E-2 1.00E-1 1.51E-1 2.03E-1 2.53E-1 2.99E-1 3.44E-1

1.28E-3 7.74E-3 2.57E-2 5.73E-2 1.00E-1 1.51E-1 2.03E-1 2.53E-1 2.99E-1 3.44E-1

Table 5.4: Illustration of robustness of trunk reservation against the holding time distribution type.

65

Using simulation studies, the authors in [69] have observed that the results for the system with trunk reservation are almost insensitive with respect to the holding time distribution type as can be seen in Table 5.4. The blocking probabilities of the two call classes are shown to depend on the coecient of variation cTH of the holding time distribution which varies from 0 (Deterministic holding distribution), 1/3 (Erlang-9), 1 (Poisson), to 4 (Hyperexponential). It can be observed that the blocking probabilities of the two call classes are almost the same, no matter what kind of holding distribution type is used. For practical purposes, the blocking probabilities can be regarded as equalized for these cases too. Furthermore, the equalized blocking probabilities for the four considered holding time distribution types are also very close to each other. Thus, the results derived with the presented approximation methods are quite accurate also for models with general holding time distributions.

66

6. Call and burst blocking in ATM systems In B-ISDNs several dierent types of trac will be transmitted on the same medium. Since the trac types dier not only in their peak and mean bandwidth requirements but also in their bandwidth variances, the CAC has to be designed very carefully. To guarantee a certain GOS for accepted calls, CAC procedures using peak or equivalent bandwidth allocation or trunk reservation have been proposed. In this chapter we do not focus on a particular CAC algorithm but investigate the blocking behavior at the call and the burst level of trac mixes from CBR and VBR sources to derive a basis for the discussion of CAC methods for such trac mixes. Therefore, we outline the analysis of the trac model presented in [30]. This model is an extension of the one presented in Chapter 4 which additionally allows for the investigation of the blocking probabilities occurring at the burst level. Again, a broadband system transmission link with a xed transmission speed C measured in Mbps is considered. The input trac consists of the superposition of the trac of N dierent trac classes. Some trac classes contain CBR and others VBR trac. The CBR calls are considered only at the call level whereas the VBR calls are looked at at the call and at the burst level. The arrival trac of service class i is assumed to follow a Poisson process with rate i . The holding time THi of a class-i call is assumed to be negative exponentially distributed with mean 1=i . CBR calls of class i generate a constant bit rate Ci during their holding time. VBR calls (class j ) are assumed to behave like ON/OFF sources, i.e. they generate a constant bit rate Cj in times of activity (ON state) and nothing during times of silence (OFF state). The distribution of the time in the ON or OFF state is negative exponential with mean 1= j and 1= j respectively. If an ON/OFF call is accepted by the CAC it starts with an active or a silent phase. Calls which start in the ON or OFF state arrive with rates jON and jOFF respectively. These rates can be calculated from the overall arrival rate j and the probability, that a class-j source is in either the ON or the OFF state: j jON = j
1= 1= j + 1= j

and

j jOFF = j
1= 1= : j + 1= j

(6.1)

At the arrival of a VBR call of class j an equivalent bandwidth C~j is reserved at the call level and released if the call has been completed. This bandwidth could be the mean bit rate of the source or the equivalent bandwidth derived with the algorithm in Chapter 3. 67

The basic system environment is shown in Figure 6.1. λ 1 , µ1 , C 1

Bi B1

C Mbps

λ i , µi , C i

1γ

j

1β

j

λ j , µj , β j , γ j , C j λ N , µN , β N , γ N , C N

transmission link

BN Bj

Figure 6.1: Basic link model for CBR and VBR calls. The connection acceptance control and blocking behavior is basically dierent for CBR and VBR calls: 1. CBR calls (class-i say) are accepted when their required bandwidth Ci is available at the call and burst level, i.e.  the sum of the bandwidths of already accepted CBR and the equivalent bandwidths C~j of already accepted VBR calls at the call level must be less than or equal to C ? Ci and  the sum of the bandwidths of already accepted CBR and the sum of the peak bandwidths of currently active VBR calls at the burst level must be less than or equal to C ? Ci. 2. VBR calls (e.g. class-j ) are accepted when their equivalent bandwidth C~j is available at the call level, i.e. when the sum of the bandwidths of already accepted CBR calls and the sum of the equivalent bandwidths of already accepted VBR calls is less than or equal to C ? C~j . Calls of class i or j are blocked at the call level with probability Bic and Bjc respectively, and do not in uence the system any longer. VBR calls can also be blocked at the burst level (with probability Bjb), since the sum of the bandwidths from the actual number of VBR calls being in the ON state and the bandwidths from the accepted CBR calls can exceed the link capacity. If burst blocking occurs, the aected burst is not transmitted and the VBR source remains in the OFF state. The normalized oered bit rate trac from class-i calls i is de ned by

i = i CCi i

(6.2)

68

for CBR calls and by ~ j = j CCj j

(6.3)

for VBR calls of class j . The overall oered bit rate trac  is therefore

=

N X k=1

k :

(6.4)

The bandwidth utilization by class-k calls is denoted by k and the total bandwidth utilization by . In the next section, we present analytical approaches to derive approximate state probabilities and related performance measures such as blocking probabilities and link utilization of the model described above (cf. [30]). The analysis in Section 6.1 deals only with VBR sources, whereas CBR and VBR sources are considered in Section 6.2. Results derived by an exact analysis or by simulation respectively, are presented to illustrate the accuracy of the approximation.

6.1 Approximations for ON/OFF-sources To illustrate the structure of the state space for the model with only VBR sources, we show in Figure 6.2, taken from [30], a simple example state space with only one VBR trac class. The system states (~n1; n1; : : : ; n~ N ; nN ) are de ned by the number n~ i of accepted calls of class-i and by the number ni of class-i calls which are currently in the ON state. Thus, the multi-dimensional state space has twice as many dimensions as trac classes. In our example, a transmission link with C = 12 Mbps serves one class of VBR sources with a bandwidth requirement of C1 = 4 Mbps while being in the ON state. The arrival rate is 1 and the mean holding time is 1=1. The ratio of mean active and silent times is assumed to be 3 1 = 1. An equivalent bandwidth of C~1 = 3 Mbps, which is equal to the mean bit rate, is used for CAC. The structure of the state space contains an exception between states (3,3) and (4,3). In state (3,3) an arriving call which wants to start in the ON state is accepted at the call level but blocked at the burst level. Therefore it has to start in the OFF state. This causes a change of the rate 1OFF to the rate 1ON + 1OFF .

6.1.1 Blocking probabilities For state spaces like the one described above, a product form or recursive solution does not exist. In principle, an exact computation, at least for smaller state spaces, can be 69

3,3 λ1 ON 3µ1 λ1 OFF

2,2 λ1 ON 2µ1 1,1 λ1 ON µ1 0,0

λ1 OFF µ1

γ1 β1 1,0

λ1 OFF µ1 λ1 ON µ1 λ1 OFF 2µ1

γ1 2β1

µ1

λ1 ON

2µ1 λ1 OFF

2,1 2γ1 β1

2µ1

λ1 ON

2,0

µ1 λ1 OFF 3µ1

λ1 ON + λ1 OFF µ1

γ1 3β1

λ1 ON

3µ1 λ1 OFF

3,2

2µ1

2γ1 2β1

λ1 ON

2µ1 λ1 OFF

3,1

3µ1

3γ1 β1

λ1 ON

2γ1 3β1 4,2 3γ1 2β1 4,1

β µ 1 4γ1 1

λ1 OFF

3,0

4,3

4µ1

4,0

Figure 6.2: Example state space (N = 1, C = 12 Mbps, C1 = 4 Mbps, C~1 = 3 Mbps). performed by solving the system of state equations, e.g. with an iterative algorithm. For realistic parameter sets, which lead to larger state spaces, this possibility is however numerically intractable. Thus, we rst focus on the calculation of the blocking probabilities and present in Section 6.1.2 a simple algorithm to derive approximate state probabilities. If the state probabilities p(~n1 ; n1; : : : ; n~ N ; nN ) are known, the call and burst blocking probabilities can be derived as follows. The call blocking probability Bic for class-i calls is given by

Bic =

X (~n1 ;n1 ;:::;n~N ;nN )2Sic

p(~n1; n1; : : :; n~ N ; nN ):

(6.5)

In the set Sic are those states, in which the available capacity at the call level is less than the equivalent bandwidth of an arriving class-i call:

X Sic = f(~n1; n1; : : : ; n~ i; ni; : : :; n~ N ; nN ) j (~ni + 1)
C~i + n~j C~j > C g: N

j=1 j6=i

(6.6)

For the example in Figure 6.2 the call blocking probability B1c is

B1c = p(4; 0) + p(4; 1) + p(4; 2) + p(4; 3): 70

(6.7)

Burst blocking for class-i calls can occur for two reasons. Firstly, an already accepted call in the OFF state wants to switch to the ON state but the required bandwidth is not available at the switching time instant. Secondly, an arriving call can not start in the ON state due to a lack of available bandwidth. The resulting burst blocking probability Bib can not be calculated by adding the corresponding state probabilities, because the rates for changes from OFF to ON states are not the same for all states. In addition, arrivals of new calls can cause a higher bandwidth utilization at the burst level. Due to the unequal rates that cause higher bandwidth utilization, the state probabilities must be weighted with the corresponding rates. The set Sib of states where burst blocking for class-i calls can occur is

Sib = f(~n1; n1; : : :; n~i; ni; : : : ; n~ N ; nN ) j (ni + 1)
Ci +

N X j=1 j6=i

nj Cj > C g:

(6.8)

For the example in Figure 6.2 we obtain

S1b = f(3; 3); (4; 3)g:

(6.9)

If S is the set of all existing states, the burst blocking probability Bib for class-i calls can be derived using the weighted state probabilities: P p(~n)

(~n ? n ) + P p(~n)
 i

~n2Sib

i

i

iON

~n2Sib =Sic

Bib = P p(~n)

(~n ? n ) + P p(~n)
 : iON i i i c ~n2S

~n2S=Si

(6.10)

The variable ~n is an abbreviation for the state vector (~n1; n1; : : :; n~ N ; nN ). The rst part of the sum in the numerator of equation (6.10) represents the weighted state probabilities for state transitions, where an accepted call switches from the OFF to the ON state. The weighted state probabilities of the state transitions for arriving calls, which want to start in the ON state, are represented by the second part of this sum. The denominator in equation (6.10) plays the role of a normalizing constant and represents the same weighted state probabilities, where all states are considered, instead of only the burst blocking states. The link utilization i of class-i calls can be computed by

i =

X (~n1 ;n1 ;:::;n~N ;nN )2S

p(~n1; n1; : : :; n~N ; nN )
ni
CCi

(6.11)

if the state probabilities are known. Finally, the total link utilization  is given by N C! X X (6.12) = p(~n1; n1; : : :; n~ N ; nN )
ni Ci : i=1

(~n1 ;n1 ;:::;n~N ;nN )2S

71

In the next section we describe an approximate calculation of the state probabilities, from which an estimation for the blocking probabilities and the link utilization can be obtained with the equations introduced above.

6.1.2 State probabilities Looking at the model with only VBR sources we have, in principle, two dierent processes. The rst one is given by the arrival and the departure of calls and the second one is the switching of accepted calls between ON and OFF states. State transitions of the rst process are independent of the present state of the second process. Because of this independence, we can calculate the exact state probabilities for macro states, which represent the number of accepted calls of each class, by the product form solution described in Chapter 4.1.1. In Figure 6.2, the macro state with two accepted calls is shaded. Knowing the macro state probability, the probability distribution for the states within the macro state must be determined. This is done by the approximate assumption that there is always enough available bandwidth at the burst level. Since the capacity at the burst level is limited, the state space is truncated at the capacity limit and the corresponding arrival rates are changed. In Figure 6.2 state (4,4) does not exist and therefore the transition rate from state (3,3) to (4,3) is changed from iON to iON + iOFF . By neglecting this eect, we can get an approximate solution for the state probabilities by evaluating the Markov chain in each macro state. The unnormalized state probabilities of the macro states Q~ (~n1; : : :; n~N ) are given by

8 QN ( = )n~i N P > < i=1 i n~i!i : i=1 n~ iC~i  C Q~ (~n1; : : :; n~ N ) = > : N :0 : P n~ iC~i > C

(6.13)

i=1

After normalization, we obtain the macro state probabilities Q(~n1; : : : ; n~ N ):

0bC=C~1c bC=C~N c 1?1 X X Q(~n1; : : :; n~ N ) = Q~ (~n1; : : : ; n~ N )
@


Q~ (~n1; : : :; n~ N )A : n~1 =0

n~N =0

(6.14)

An approximate distribution of the local state probabilities within the macro states can be obtained also by a product form solution. The unnormalized local state probabilities q~(~n1; n1; : : : ; n~ N ; nN ) within the macro state (~n1; : : : ; n~ N ) are given by

8 QN N n P > < i=1 (~nin~?in! i )!
( i= nii!) i : i=1 niCi  C q~(~n1; n1 : : : ; n~ N ; nN ) = > N :0 : P niCi > C i=1

72

(6.15)

and by normalization we obtain the local state probabilities q(~n1; n1; : : : ; n~ N ; nN ):

1?1 0minfn~1 ;bC=C1cg minfn~N ;bC=CN cg X X q~(~n1; : : : ; nN )A : (6.16)


q(~n1; : : : ; nN ) = q~(~n1; : : : ; nN )
@ n1 =0

nN =0

For the shaded macro state of the example state space in Figure 6.2 the local state probabilities are

q(2; 0) = 161

6 q(2; 1) = 16

9 q(2; 2) = 16

:

(6.17)

Finally, the approximate state probabilities p(~n1; n1; : : :; n~ N ; nN ) are computed by multiplying the local state probabilities with the probabilities of the corresponding macro states:

p(~n1; n1; : : :; n~N ; nN ) = q(~n1; n1; : : :; n~ N ; nN )
Q(~n1; : : :; n~ N ):

(6.18)

Thus, the approximate calculation of the state probabilities consists of a product form solution in two phases. In the rst one, the exact probabilities of the macro states are derived. In the second phase, an approximate calculation of the probability distribution in the macro states is done. The nal state probabilities are then obtained by multiplying the corresponding local state and macro state probabilities. Using these state probabilities, an approximate solution for the blocking probabilities and the link utilization can be obtained with the equations in Section 6.1.1. It should be noted, that the call blocking probabilities are exact, because of the accurate calculation of the macro state probabilities. To compare the approximate solutions with exact values, we make use of a numerical example presented in [30]. It deals with one VBR trac class with C1 = 20 Mbps and a transmission link of capacity C = 150 Mbps. The mean times in the ON and OFF states are assumed to be equal, i.e. 1 = 1, and the equivalent bandwidth is chosen as the mean bandwidth C~1 = 10 Mbps. For the computation of the exact solution the complete system of state equations was solved by an iterative algorithm. The resulting blocking probabilities are depicted in Figure 6.3. In case of the exact solution the results for the burst blocking probabilities are dependent on the ratio of the rates 1 and 1 which represents the mean number of active phases within a call duration. Figure 6.3 shows the call and burst blocking probabilities as functions of the oered bit rate trac1  for three dierent ratios 1=1 . This ratio is chosen as 1 (case 1), 5 (case 2), and 50 (case 3). Note that 1=1 = 1 means that the VBR sources 1

Numerical tests have shown that the blocking probabilities depend only on the ratio of 1 and 1 but not on their individual values although the corresponding state space does not allow for a product form solution.

73

blocking probability

10E0 exact calculation: call level: case 1-3 burst level: case 1 case 2 case 3

10E-1

approximation: burst level 10E-2 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

total offered bit-rate traffic

Figure 6.3: Approximation accuracy of blocking probabilities. behave like CBR sources since there is only one active phase per call in equilibrium. It can be seen that the curves of the exact call blocking probability are the same for each of the cases. This is due to the fact that the blocking process at the call level is independent of the burst level process. If we look at the burst blocking probability, we can observe a slight dierence between the curves of the exact solution and the curve for the approximate solution, which is the same for all three cases and constitutes an upper bound. The dierence between the curves decreases as the ratio 1=1 increases. This eect is due to the fact that most state changes take place inside the macro states if the ratio is large and therefore the exact probability distribution within the macro states is closer to the one calculated approximately. The ratio 1=1 = 1 can be seen as a worst case of the approximation, because this ratio represents a CBR source. For realistic parameter sets the ratio 1=1 is of the order of 102 and so the approximation provides values which are very close to the exact ones.

6.2 Approximations for CBR and ON/OFF-sources If CBR and VBR sources are considered, the approximate blocking probabilities and the link utilization can also be computed with the results of Section 6.1.1 and 6.1.2. To make use of these, the CBR sources are considered as VBR sources with time in the ON state tending to in nity and time in the OFF state tending to zero. Another way to derive the performance measures from this model is through an approximate analysis by 74

inspecting the complete state space. This analytical method, described in the following two subsections, is also taken from [30]. The advantage of this way of evaluation against the one mentioned before is a reduction in the numerical complexity.

6.2.1 Blocking probabilities Again, we will rst focus on the computation of the blocking probabilities and the link utilization if the state probabilities are known. Without loss of generality, we assume that trac classes 1 to k consist of CBR sources and the other N ? k classes consist of VBR sources. The system states (~n1; : : :; n~ N ; nk+1; : : :; nN ) are then de ned by the number n~ i of accepted class-i calls and by the number ni of active calls of class-i. For the CBR classes no dimension ni for the number of calls in ON states exists, because they are always active. To get a simpler notation of the following equations, we use the mean bandwidth C~i for the CBR sources too. Thus, we get C~i = Ci for the trac classes i = 1; : : : ; k. The call blocking probability Bic for the CBR calls of class-i are given by the sum of the state probabilities, where either more than C ? Ci Mbps of the transmission link capacity is reserved at the call level, or the available bandwidth at the burst level is less than Ci. If we denote the sets of this states by Sic for i = 1; : : : ; k, we get:

X Sic = f(~n1; : : : ; n~ N ; nk+1 ; : : :; nN ) j (~ni + 1)
C~i + n~ j C~j > C g [ N

j=1 j6=i

f(~n1; : : : ; n~ N ; nk+1 ; : : :; nN ) j (~ni + 1)
C~i +

N k X X n~ j C~j + nj Cj > C g: (6.19) j=1 j6=i

j =k+1

We can compute the call blocking probabilities Bic for the trac classes i = 1; : : : ; k by X p(~n1; : : :; n~ N ; nk+1; : : : ; nN ): (6.20) Bic = (~n1 ;:::;n~N ;nk+1 ;:::;nN )2Si

For the N ? k VBR classes there exist blocking probabilities Bic at the call level and Bib at the burst level. The calculation of these probabilities can be done as shown in Section 6.1.1. At the call level we get X Bic = p(~n1; : : : ; n~ N ; nk+1 ; : : :; nN ) (6.21) (~n1 ;:::;n~N ;nk+1 ;:::;nN )2Sic

for classes i = k + 1; : : : ; N with

X Sic = f(~n1; : : : ; n~ N ; nk+1 ; : : :; nN ) j (~ni + 1)
C~i + n~ j C~j > C g: N

j=1 j6=i

75

(6.22)

The sets Sib of states where a burst blocking for calls of class-i can appear are given by

Sib = f(~n1; : : :; n~ N ; nk+1; : : :; nN ) j (ni + 1)
Ci +

N X j=k+1 j6=i

nj Cj +

k X n~j C~j > C g (6.23)

j =1

for trac classes i = k + 1; : : : ; N . For VBR calls of class-i the burst blocking probability Bib can be calculated with the sets Sic and Sib by using equation (6.10). The parameter ~n is now an abbreviation for the system states (~n1; : : :; n~ N ; nk+1; : : :; nN ) and the rates iON are derived from the equations (6.1). The link utilization i for classes with CBR sources, i.e. i = 1; : : : ; k, is given by

i =

X (~n1 ;:::;n~N ;nk+1 ;:::;nN )2S

p(~n1 ; : : :; n~ N ; nk+1; : : :; nN )
n~ i CCi

(6.24)

and for VBR classes i = k + 1; : : :; N by

i =

X (~n1 ;:::;n~N ;nk+1 ;:::;nN )2S

p(~n1 ; : : :; n~ N ; nk+1; : : :; nN )
ni CCi

(6.25)

To compute the total link utilization , the individual utilizations i of all N trac classes must be summed to give:

=

N X i=1

i:

(6.26)

Since the state probabilities must be known for the presented computation of these performance measures, the next section deals with an algorithm to derive approximate values for the state probabilities.

6.2.2 State probabilities An approximate calculation of the state probabilities can be carried out as described in Section 6.1.2. First, the probabilities of the macro states Q(~n1; : : : ; n~ N ), which represent the number of accepted calls of each class, are computed approximately. This can be done by a product form solution (cf. equations (6.13) and (6.14)). The calculation is approximate since the process at the call level is not independent of the burst level process.

76

The probability distributions within the macro states can also be derived by a product form solution. For the unnormalized local state probability q~(~n1; : : :; n~ N ; nk+1; : : :; nN ) within the macro state (~n1; : : :; n~ N ) we get

8 Q N N ni Pk P > > < i=k+1 (~nin~?in! i )!
( i= nii!) : i=1 n~ iC~i + i=k+1 niCi  C : q~(~n1; : : : ; nN ) = > N k P P ~ > niCi > C n~ iCi + : :0 i=1

(6.27)

i=k+1

Using normalization as in equation (6.16), we arrive at the approximate local state probabilities q(~n1; : : :; n~ N ; nk+1; : : : ; nN ). The dimension of the local state space within the macro states is equal to the number of VBR classes. Finally, the approximate state probabilities p(~n1; : : :; n~N ; nk+1; : : : ; nN ) are given by multiplying the local state probabilities with the probabilities of the corresponding macro states:

p(~n1; : : : ; n~ N ; nk+1 ; : : :; nN ) = q(~n1; : : :; n~N ; nk+1; : : : ; nN )
Q(~n1; : : :; n~ N ): (6.28) Now, the performance measures for the model with CBR and VBR input trac can be derived approximately with the equations shown in Section 6.2.1. In contrast to the model with only VBR sources, the results for the call blocking probabilities are not exact, because the acceptance of arriving CBR calls is not independent of the utilization at the burst level. Some numerical results are presented in the following. In Figures 6.4 and 6.5 the in uence of the choice of the equivalent bandwidth on the blocking behavior if one CBR and one VBR class is taken into account is depicted. These gures are taken from [30]. The authors considered a transmission link with capacity C = 600 Mbps while CBR calls require C1 = 2 Mbps and VBR calls have a peak bit rate of C2 = 20 Mbps. The mean times in the ON and OFF states are assumed to be equal ( 2 = 2) and the oered bit rate from both trac classes is the same (1 = 2). The mean bit rate is chosen as equivalent bandwidth C~2 = 10 Mbps for the curves in Figure 6.4. It should be noted that an iterative algorithm, as used above for only one trac class, is not appropriate if more trac classes are considered. This is due to the huge number of states that must be taken into account. As mentioned in Chapter 4, calls with higher bandwidth requirements experience higher blocking probabilities if only CBR calls are considered. Looking at mixes of CBR and VBR trac classes, it turns out (cf. Figure 6.4) that CBR calls of lower bit rate requirements than VBR calls can be blocked at the call level with higher probability depending on the oered bit rate trac. Moreover, Figure 6.4 shows that the burst blocking probability of the VBR calls is unacceptably high (> 10?2 ). Therefore, in Figure 6.5, a higher equivalent bandwidth of C~2 = 17:5 Mbps is used for CAC of VBR calls. 77

blocking probability

10E0 call level: class 1 class 2 burst level: class 2

10E-1 10E-2

• • •

10E-3 10E-4

• ••

10E-5

•• •

• • •

••

• ••

•

•

•

• •

•

••

••

•

•

10E-6 •

simulation: 10E-7 0.2

0.4

0.6

0.8

1.0

total offered bit-rate traffic

blocking probability

Figure 6.4: Blocking probabilities for mean bit rate as equivalent bandwidth. 10E0 call level: class 1 class 2 burst level: class 2

10E-1 10E-2

• • •

•

•

• •

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

10E-3

•

•

•

•

• 10E-4

•

•

•

10E-5

•

•

•

•

•

•

10E-6

simulation:

•

10E-7 0.2

0.4

0.6

0.8

1.0

total offered bit-rate traffic

Figure 6.5: Blocking probabilities for increased equivalent bandwidth. 78

It can be observed that increasing the equivalent bandwidth leads to lower burst blocking probabilities for VBR calls (< 10?4 ). In contrast, the call blocking probabilities for both trac classes are higher compared with the numerical results from Figure 6.4. Thus, it can be concluded that the burst blocking probabilities are strongly dependent on the estimation of the required bit rate for the VBR calls. It should be noted that the simulation results are quite close to the approximate results obtained with the algorithm from Section 6.2. The investigation of larger numbers of trac classes does not imply any numerical problems and the requirements on memory eort and run time are not increased considerably if a suitable implementation is used.

79

80

7. Extensions to multi-service networks From the conceptual point of view, the problem of calculating end-to-end blocking in a multi-service network can be handled in a nice and simple way, since the product form solution in Section 4.1.1 still applies, see for example [39]. Therefore, we have to consider one factor for each link on each route. The only dierence with respect to a completely shared link is that the set of allowed states is now constrained by several linear capacity constraints, but still the set is coordinate convex. However, from the practical point of view, the problem of calculating end-to-end blocking probabilities when analyzing multi-service networks becomes even more complicated, because the evaluation using the product form solution becomes numerical intractable. Another problem is the use of trunk reservation in such a network, which can not be analyzed by the use of a product form solution. Ideally methods such as reduced load approximations or xed point models are used. These models should be built upon a suitably ecient and accurate algorithm to calculate the blocking probabilities for dierent call types on a single link. In a single-slot circuitswitched network, the appropriate xed point model uses Erlang's loss formula (or an extended version of it) for the calculation of single link blockings. A number of authors [(92)018], [48] have also used Erlang's loss formula for the multi-slot case in approximate methods but these do not allow for unequal mean holding times when trunk reservation is used. Let us assume that, for the time being, only xed routing is used as even this case has not yet been suciently analyzed when trunk reservation is used. Fixed point methods based on one or two moment models have been reported for networks without use of trunk reservation and allowing for unequal mean holding times ([14], [55] and [(93)020]). In Section 5.5 and [3] it is shown, that the blocking probabilities of heterogeneous call types are sensitive to the mean holding time ratio when trunk reservation is used. In this chapter, two xed point models are derived to calculate end-to-end blocking in multi-rate networks. The rst one is based on Kelly's xed point model [37] which has been extended to the multi-rate case. We allow VBR calls to have dierent equivalent bandwidths on dierent links of their path. The second xed point model is based on a more accurate calculation of single link blocking probabilities using the algorithm in Section 4.1.2. What we have also allowed for in the latter model is the calculation of an approximate mean holding time for a typical channel on a link in the network which forms an integral part of the approximation technique. However, this xed point model may not be suitable for the optimization of large networks. But the idea used could equally apply 81

to any xed point method which uses a fast model to calculate blocking on a single link when trunk reservation is used and mean holding times are not explicitly modeled.

7.1 Fixed point model without trunk reservation First consider the following notation which is used throughout the model de nition.

kr kr akjr kj Cj Bkj Lkr j ?j

Poisson arrival rate for call type k on route r mean holding time for call type k on route r equivalent bandwidth of call type k on link j of route r trunk reservation threshold associated with call type k on link j total capacity on link j blocking probability seen by call type k on link j approximation for loss probability of call type k on route r approximate mean holding time for a typical channel on link j set of all call types which use link j

It is very important to assume that links block independently, see [37]. The blocking probabilities as a solution to the xed point equations ((7.1) and (7.2) solved by repeated substitution) are asymptotically correct as link capacity and oered trac are increased together, cf. [39]. The link blocking probabilities are then calculated using Erlang's loss formula:

Bj = E(j ; Cj )

(7.1)

where E(.) is Erlang's loss function. j is the link j reduced load or eective oered trac after the thinning eect in the network. j is calculated by

j = (1 ? Bj )?1

X X k2?j r;j 2r

Y kr kr akjr (1 ? Bi)akjr : i2r

(7.2)

The stream blocking for call type k on route r is then approximated by

Y kr = 1 ? (1 ? Bj )akjr :

(7.3)

j 2r

Note that the term akjr has been used as in ATM networks calls may require dierent bandwidth on dierent links of their route. 82

7.2 Fixed point model with trunk reservation The xed point model presented here is based on the single link model of Kaufman and Roberts, see Section 4.1.2. The xed point equations (7.4) to (7.7) are solved by repeated substitution until convergence is reached. The end-to-end blocking probabilities for call type k on route r are then given by Lkr . The calculation of Bkj is done using the recursive method of Section 4.1.2

Bkj = Ek (1j ; : : : ; Nj ; 1j ; : : :; Nj ; 1j ; : : :; Nj ; Cj ) kj = j (1 ? Bkj )?1

X r

akjr kr (1 ? Lkr )

Y 1 ? Lkr = (1 ? Bkj ) P P a  (1 ? L ) kjr kr kr kr j = r Pk P a  (1 ? L ) r k

(7.5) (7.6)

j

kjr kr

(7.4)

(7.7)

kr

where kj is the reduced load or eective oered trac of call type k on link j after the thinning eect in the network. j is the approximate mean holding time for a typical channel in link j and Ek (:) is the solution for the single link model. The idea behind the approximation is that the stream of rate kr is thinned by a factor 1 ? Bki at link i before being oered to link j . If this thinning could be assumed to be independent both from link to link and over all routes containing link j , then the type-k trac oered to link j would be Poisson with a rate given by kj . The approximation for the proportion of calls of type k requesting route r that are lost is Lkr .

7.3 Numerical results Consider the case without trunk reservation rst. The model given in section 7.1 was tested and compared with the results obtained in [14] for the star network of Figure 7.1. In [14], the stream blocking probabilities are calculated using three dierent methods: Kelly's xed point model, Knapsack xed point model (based on Kaufman [34] and Roberts [60] algorithm) and the Pascal xed point model. It is concluded [14] that Kelly's approximation gives the least accurate results especially at light and moderate loads and we know from [39] that the approximation performs best under heavy loads and for large networks with homogeneous trac. Our results based on Kelly's xed point model match those in [14]. Kelly's xed point model is used without normalizing the dierent bandwidth requirements in the model as Erlang's loss formula can only cope with homogeneous trac characteristics. A xed point model is also constructed using Kaufman and Roberts 83

1

90 5

120

4

2 100

110

3

Figure 7.1: A multi-rate star network (taken from [14]). algorithm referred to as K&R's model. The simulation results in [14] together with the results given by the two xed point models for the star network are summarized in Tables 7.1 and 7.2 for moderate and heavy trac scenarios respectively. The blockings are expressed in terms of percentages. Two call types are used with a1 = 1 and a2 = 5. route

fresh oered trac

simulation 95% con dence intervals

152 153 154 253 254 354

10 10 10 10 10 10

(0.34, 0.35) (0.29, 0.30) (0.28, 0.29) (0.07, 0.07) (0.06, 0.06) (0.01, 0.01)

1

2 2 2 2 2 2 2

1

2

(2.29, 2.30) (1.96, 1.98) (1.90, 1.92) (0.49, 0.50) (0.43, 0.44) (0.08, 0.08)

Kelly's model

1

0.006 0.006 0.006 0.000 0.000 0.000

2

0.032 0.032 0.032 0.000 0.000 0.000

K&R's model

1

0.355 0.303 0.293 0.074 0.065 0.012

2

2.351 1.993 1.924 0.527 0.458 0.093

Table 7.1: Blocking probabilities for moderate trac load (without trunk reservation). route

fresh oered trac

simulation 95% con dence intervals

152 153 154 253 254 354

15 15 15 15 15 15

(5.7, 5.7) (4.6, 4.6) (4.1, 4.2) (2.4, 2.4) (1.9, 1.9) (0.8, 0.8)

1

2 3 3 3 3 3 3

1

2

(28.0, 28.1) (23.3, 23.4) (21.4, 21.4) (13.2, 13.2) (10.9, 11.0) (4.8, 4.8)

Kelly's model

1

4.86 3.91 3.76 1.30 1.16 0.16

2

22.06 18.06 17.46 6.34 5.65 0.81

K&R's model

1

5.74 4.65 4.20 2.49 2.02 0.89

2

28.56 23.76 21.57 13.83 11.37 5.40

Table 7.2: Blocking probabilities for heavy trac load (without trunk reservation).

84

A 210

210 E

B

180

180 150 D

C

Figure 7.2: A multi-rate ring network. As these tables show, Kelly's approximation improves as the network load increases but still underestimates the blocking probabilities by a factor of two in some cases and in one case (route 3 5 4) by a factor of six. The approximation is fairly poor under light and moderate trac pro les. In [14], it is shown that the two approximation methods of the Knapsack and Pascal are signi cantly more accurate with the Pascal approximation having the same computational eort as the Kelly's approximation (or the Knapsack). We shall now consider a ring network where trunk reservation is used (cf. Figure 7.2). The numbers on the links represent the link capacity. The results of the xed point model given in Section 7.1 are compared with simulation results. Two heterogeneous call types with bandwidths a1 = 1 and a2 = 2 are modeled. The oered tracs and routes are summarized in Tables 7.3 to 7.5. The blocking probabilities are expressed in terms of percentages. In all four tables, the oered trac is such that the network is critically loaded, i.e. load factor 1.0 for all links. The results obtained are in good agreement with the simulation results. In Tables 7.3 to 7.5 trunk reservation is used to equalize the blocking probabilities. The ratio of the mean holding time of the two call types changes from 0.01 in Table 7.3 to 1.0 in Table 7.4 and nally the ratio in Table 7.5 is 100.0. The rst observation is that the results from the xed point model agrees well in the majority of cases with the simulation results. The xed point model tends to underestimate the blocking probabilities slightly when the mean call duration of call type 1 is less than that of call type 2. The blocking probabilities are slightly overestimated for the opposite case. Secondly, it is apparent that the end-to-end blocking probabilities change as the mean holding time ratio changes even though the total oered trac is xed. As an example, consider route A-B-C. Two call types originate at node A and are destined for node C with the total oered trac of 90 Erlangs (other trac is also present on links A-B and B-C). The equalized blocking probability changes from 39.8% (Table 7.5) to 34% (Table 7.4) and then to 19.6% (7.3), i.e. by almost a factor of two. The blocking probabilities improve as low bit rate trac has a shorter call duration giving more chances for the higher bit rate calls to set up. The same argument applies for other routes such as E-A where the improvement in the blocking probabilities is in the same order.

85

route

fresh oered trac

simulation 95% con dence intervals

AB ABC BCD CD DE DEA EA

60 30 30 30 30 30 60

(17.2, 23.7) (22.9, 29.9) (23.4, 30.2) (16.9, 23.4) (19.9, 27.4) (33.9, 41.4) (14.9, 20.8)

1

2 2 2 2 1 2 2 2

1

2

(10.9, 18.7) (16.5, 26.4) (18.2, 29.0) (9.4, 19.8) (10.4, 19.0) (31.5, 43.6) (11.3, 20.4)

1

K&R's model

13.9 19.6 23.9 18.6 14.0 23.9 11.5

2

13.9 19.6 23.9 18.6 14.0 23.9 11.5

Table 7.3: Heavy trac with trunk reservation (1=2 = 0:01).

route

fresh oered trac

simulation 95% con dence intervals

AB ABC BCD CD DE DEA EA

60 30 30 30 30 30 60

(17.8, 21.6) (28.5, 32.9) (34.3, 39.9) (25.0, 29.6) (22.5, 26.5) (36.3, 40.4) (19.1, 21.9)

1

2 2 2 2 1 2 2 2

1

2

(17.5, 24.2) (29.5, 34.4) (35.4, 41.0) (22.6, 29.5) (20.0, 25.3) (36.7, 43.8) (18.1, 23.1)

1

K&R's model

20.5 34.0 36.7 23.7 22.2 37.3 19.4

2

20.5 34.0 36.7 23.7 22.2 37.3 19.4

Table 7.4: Heavy trac with trunk reservation (1=2 = 1:0).

route

fresh oered trac

simulation 95% con dence intervals

AB ABC BCD CD DE DEA EA

60 30 30 30 30 30 60

(21.3, 26.3) (32.3, 40.1) (33.3, 39.0) (24.6, 30.0) (24.4, 29.8) (37.6, 45.5) (16.3, 20.7)

1

2 2 2 2 1 2 2 2

1

2

(20.6, 24.3) (31.9, 35.4) (38.4, 40.8) (28.5, 32.0) (23.8, 26.3) (38.4, 42.1) (20.8, 25.5)

1

K&R's model

27.1 39.8 42.0 29.8 25.2 44.3 25.5

2

27.1 39.8 42.0 29.8 25.2 44.3 25.5

Table 7.5: Heavy trac with trunk reservation (1=2 = 100:0).

86

Therefore, end-to-end blocking probabilities are sensitive to the mean holding time ratio when trunk reservation is used. If the ratio is orders of magnitude dierent and the aim is not just to equalize the blocking probabilities but also to provide preferential GOS for a trac type, it is worth using a xed point model which approximately models the mean holding time explicitly. This obviously becomes unnecessary if a fast and simple to implement heterogeneous single link model is developed.

7.4 Conclusions and further studies Kelly's xed point model has been extended to the multi-rate environment without normalizing the bandwidth requirements. When trunk reservation is not used, this model underestimates the end-to-end blocking probabilities even when the network is heavily loaded. Chung & Ross [14] have shown that the Pascal xed point model gives more accurate results with the same computational eort as Kelly's approximation. They also give asymptotic proofs on all three xed point models studied. Fixed point models based on algorithms such as K&R are viable for small or medium networks. In order to cater for alternative routing, xed point models based on two moments have been suggested [55], [(93)020]. When trunk reservation is present and the mean holding time ratio of the calls is quite large, a more appropriate xed point model should be used. A xed point approximation model for dierent call types is derived to allow for trunk reservation and unequal mean holding times based on the single link model presented in Section 4.1.2. The performance of this model is good compared with the simulation model for the studied ring network.

87

88

8. Towards dimensioning and control of multi-service networks Models and methods presented in the previous chapters mainly concern blocking evaluation and dimensioning of networks or network elements, i.e. links, if well known and speci ed tracs are oered. In the remainder of this book we focus on some aspects of dimensioning and control of multi-service networks. Therefore, we start with investigations concerning the integration and separation of trac having dierent characteristics and service requirements. These considerations result in guidelines for the trac clustering process. A second section addresses the design of virtual paths in multi-service networks, which plays a central role in the design of ATM networks. If we assign a kind of worth to calls, then the determination of an optimal control policy for ATM networks, in particular for CAC, is another dicult task. This problem is discussed in Section 8.3 where methods to determine such an optimal control policy, with respect to achieving a maximum reward, are described.

8.1 Integration and separation of trac Trac engineering has concentrated a signi cant eort to the extension of multi-rate models in order to make them applicable to ATM networks. In particular, the concept of equivalent bandwidth has been introduced (cf. Chapter 3). However, the eorts have partially eclipsed some special features of ATM networks, i.e.:

 the complexity of GOS evaluation due to trac processes at the burst and cell levels  the uncertainty and variability of operational conditions involved in ATM networks  the strict relationships existing between dimensioning, network control and management aspects.

It can be noted that operational conditions foreseeable in ATM networks seem rather dierent from the ones discussed in many studies on multi-rate networks. In multi-rate networks, trac can only dier in bandwidth requirements and service time; a greater heterogeneity is envisaged in ATM networks. 89

With such considerations in mind, this section focuses on criteria to integrate or separate trac in ATM environments. First, we review suggestions found in the literature. In [61] three main VBR trac classes are distinguished. Class 1 is represented by trac with well known statistical characteristics and low peak bit rates whereas class 2 contains ON/OFF-trac of low peak bit rate with unknown statistical characteristics. ON/OFFsources with a high peak bit rate compared to the systems capacity are assigned to class 3. The integration of trac belonging to class 3 with trac of the other two classes does not result in considerable multiplexing gains, while separation of such trac permits absorption of burst congestion by buering, leading to a better dimensioning. On the other hand, classes 1 and 2 can be advantageously integrated. In this case, cell priority levels and trunk reservation can be used to meet desired GOS objectives. In [(91)SE] three trac clusters are envisaged in relation to CBR services, short burst services (SBS) and long burst services (LBS). Statistical multiplexing is only performed within the clusters. The capacities of the CBR and SBS clusters are xed according to their dimensioning, whereas the capacity of the LBS clusters is dynamically assigned. A relation of the clustering process to diculties arising with policing sources which are multiplexed according to their equivalent bandwidth is discussed in [27]. In Chapter 2, further criteria for the clustering of trac with dierent bandwidth and GOS requirements have been suggested.

8.1.1 Problem formulation The question \How to integrate or separate trac?" cannot be answered without considering resource dimensioning, clustering alternatives and management policies. Moreover, any reasonable formulation of this problem should consider trac variabilities and abnormal network condition. To reduce model complexity, trac variability can be expressed by a set A of possible trac realizations Ai, i.e. A = fAi g, 1  i  I . Each trac realization Ai can be characterized by a probability or weight wi. If the network carries J dierent trac types, every trac realization Ai is described by a set of matrices Aji , 1  j  J , each representing point-to-point trac of type j in the realization i. From the dimensioning point of view, a possible criterion to derive a nal resource dimensioning vector R = frk g, 1  k  K , from the separate dimensioning Ri = frkig, where rki is the size of network element k for reference trac i, could be based on average case (f1) or worst case (f2) evaluation, i.e.:

8 I > wirki < f1(rki ) = iP =1 : rk = > : f2(rki ) = 1max r ki iI

(8.1)

Under such assumptions the following notation can be set:

90

A Aji D W B Bji

B Bij Ri R C

: : : : : : : : : : : : :

reference trac set, A = fAig type-j trac matrix belonging to the reference trac i vector of call bandwidth for trac type j , D = fdj g trac weight vector, W = fwig call blocking probability matrix, B = fBji g call blocking probability for type-j calls, trac realization i matrix for maximum call blocking probabilities, B = fBij g maximum call blocking probability for type-j calls, trac realization i resource dimensioning relevant to the i-th trac realization, Ri = frkig nal resource dimensioning vector, R = frk g vector of unitary capacity costs, C = fck g available management policy set trac clustering alternatives.

MPS TC The MPS can include dierent call access policies, namely, complete sharing (CS), trunk reservation (TR), partial sharing (PS) and complete partitioning (CP). Assuming that the network is dimensioned according to a call blocking constraint, the problem can be formulated by the following optimization problem:

Given Find Minimizing

Subject to

A, B , D, W , MPS, TC R, MPS , TC Zd (R; MPS; TC) + Zc (R; MPS; TC) with Zd = C
RT , Zc = g(R; MPS; TC) where either R = f1(Ri) or R = f2(Ri) BB

This problem cannot be solved globally due to its complexity. To de ne a proper solution strategy, it is advisable to make some preliminary numerical evaluations considering the dimensioning of a two-link network structure. Our purpose is to derive guidelines to structure the optimization procedure following a suitable step-by-step solution approach, by which trac and dimensioning methods are merged with proper design rules.

91

8.1.2 Investigation of integration policies a) Capacity saving Looking at two trac classes, high bit rate trac experiences a higher loss probability than low bit rate trac if a CS policy is considered. Moreover, low bit rate trac losses can show an oscillatory behavior (cf. Chapter 4), so, to assure a stable design, we use in the following an upper bound for the low bit rate trac loss, which is monotonic. The bandwidth ratio is very signi cant in evaluating capacity savings due to service integration [50]. If two trac classes have the same loss requirements, then capacity saving s by CS and TR policies, with respect to CP policy, are most signi cant for low load and a low proportion of high bit rate trac. Generally, with TR slightly greater savings can be achieved. In conclusion, the TR policy allows savings in quite a large set of oered trac scenarios, if the loss requirements are the same for all trac classes. Moreover, it is well known that, at least in case of Poisson arrival trac, the TR parameters for loss equalization depend only on the bandwidth of the calls but not on the trac composition or the trac load itself.

b) Eect of trac instability To analyze the eect of trac instability and unpredictable variability of working conditions, the trac matrix A and the loss matrix B have been considered in relation to the two-node network depicted in Figure 8.1. L2 L1 : dedicated to narrowband traffic A2 A1 A2

L2 : shared between A1 and A2 TR-equalization policy applied

L1

Figure 8.1: A two-node network with over ow. Overload conditions are identi ed with uniform and selective 100% overloads whereas a higher loss is allowed for the increased trac. The links in Figure 8.1 have been dimensioned for dierent trac mixes using the three management policies CP, TR (set for equalization) and PS. The total size of both links is reported in Table 8.1. To derive a nal dimensioning from the results for each working condition, the functions f1 and f2 given in equation (8.1) have been used in conjunction with a probability vector W . As shown in Table 8.1, savings with TR can reduce signi cantly if an instability of the oered trac is considered. Under overload conditions, the CP policy is better for worst case dimensioning due to the dierent GOS requirements of the trac classes. On the other hand, TR has a better performance for average case dimensioning. However, PS is preferable if the trac classes have dierent GOS requirements in normal conditions, see Table 8.2. This holds for both dimensioning criteria. 92

working condition Normal Overload 1 Overload 2 Overload 3

A

B

MPS W j = 1 j = 2 j = 1 j = 2 CP TR PS 0.7 0.1 0.1 0.1

10 20 10 20

100 0.001 0.001 100 0.1 0.001 200 0.001 0.1 200 0.1 0.1 f1 (average case) f2 (worst case)

338 358 398 418 354

312 447 412 399

345 356 483 438 345 370 418 447 483

d1 = 10; d2 = 1 Table 8.1: Dimensioning for equal GOS requirements in normal conditions. working condition Normal Overload 1 Overload 2 Overload 3

A

B

MPS W j = 1 j = 2 j = 1 j = 2 CP TR PS 0.7 0.1 0.1 0.1

10 20 10 20

100 0.01 0.001 100 0.5 0.001 200 0.01 0.05 200 0.5 0.05 f1 (average case) f2 (worst case)

308 238 401 312 311 401

312 447 371 436 344 447

299 244 371 323

304 371

d1 = 10; d2 = 1 Table 8.2: Dimensioning for unequal GOS requirements in normal conditions.

8.1.3 Trac clustering rules For the results in Section 8.1.2 only two trac classes have been considered. The extension to a larger number of trac classes is not straightforward. For such large scenarios with dierent trac volumes and characteristics (bandwidth, service time, loss requirement) system behavior becomes more complex to evaluate [13]. In particular, the determination of TR parameters becomes a dicult task if dierent GOS levels are considered, resulting in a less controllable network. For example, varying the reservation level of one trac class can produce unpredictable variations to the losses of the other trac classes [47]. This is another reason not to apply TR to distribute GOS dierently among the trac classes. Even if a pure TR policy can be bene cial in resource utilization, its applicability to heterogeneous trac scenarios can be constrained by dierent requirements and by the need of a robust dimensioning. In such situations, a policy like PS is advisable. Although the numerical examples above show that no simple and general conclusion can be drawn about the convenience of fully integrating heterogeneous trac, some qualitative 93

criteria can be derived:

 the interest in integration is potentially greater if limited trac volumes are involved  if GOS standards have a dierent nature and imply dierent management policies,

it is worthwhile to keep the relevant trac classes separated  if trac classes are of the same nature, the potential advantages of integration should be veri ed considering the cost of service provisioning, because it is not economically advisable to oer a GOS better than the required value to a high volume of trac. According to these items, a two step clustering process can be envisaged:

1. evaluation of the expected capacity savings achievable with integration policies belonging to the MPS 2. evaluation of complexity and of control and management eort required for implementation.

Considering the rst step, the following clustering rules can be suggested for trac scenarios where the GOS is characterized by the maximum call blocking probability:

 separate trac having bandwidth or holding time ratios larger than 1:10 [50]  avoid provisioning of high trac volumes with a GOS much better than required  verify convenience of trac clusters where the proportion of the low bit rate trac

is small (low capacity saving)  verify convenience of clusters with high trac volumes (low capacity saving)  verify convenience of clusters taking into account capacity savings achievable in all signi cant normal and abnormal operational conditions. Relating to the second step, it can be reasonably argued that control operations (e.g. call admission, routing and GOS monitoring) can be simpli ed in a network with homogeneous trac. More disputable is the impact of integration and separation on management operations like resource management, resource and service provisioning, etc. From this point of view, the optimal degree of integration can be found evaluating the greater complexity or simpli cation produced in management operations by the integration process. To reduce model complexity it is also necessary to adopt solutions which facilitate controllability and manageability of the network. A possible application of the suggested clustering rules is depicted in Figure 8.2.

94

Analysis of QOS standards and traffic characterisation

PHASE 1 Traffic / QOS standards

Y

Different stand. / charact.

Y

Overdimensioned Network ? N

N

Y

High volume of traffic parcels N

Traffic analysis

Different bandwidth

Y

Different holding time

Y

Y Critical Mix

Clustering rules

Clustering rules

Clustering rules

Traffic Clustering

Traffic Integration

Integration policy selection

Normal and overload conditions

Integration Policy Selection

Management Policy Set (CS, TR, CP, PS) Dimensioning rules and models

Analysis of control / management requirements

PHASE 2 Control / management requirements

Clustering and Integration Policies Modification

Figure 8.2: Clustering process example.

95

8.2 Design of virtual paths The design of virtual paths (VPs) plays a central role in ATM network architectures. VPs represent a logical network located at an intermediate level between call admission, trac enforcement and resource management [31]. They can be managed on a semipermanent or on-demand basis to optimize switching operations and to improve resource utilization and management exibility, cf. [8], [11], [15], [22] and [68]. VPs can be used in particular to:

 con gure temporary dedicated networks  dynamically assign services to network resources  execute more easily control and management functions. The optimization of VPs can involve the consideration of many objectives, e.g. throughput, reward, fairness, cost minimization as well as requirements and operating conditions. In the following, we assume that before entering the VP optimization phase, service clustering according to Section 8.1 has already been carried out. Therefore, we consider clusters with similar trac characteristics and GOS requirements. Under these assumptions, the basic problem of VP network optimization is the determination of the VP network structure and its dimensioning. These problems can be treated together or separately, and can be solved according to two complementary approaches, which will be presented in this section. The rst approach, cf. [2] and [(92)025], considers the VP network as a network infrastructure allowing low cost per unitary trac and high performance connections between node pairs, as a result of link economies of scale and simplicity of switching operations. In this simple approach, trac and switching operations at the call level are not directly considered. In a second approach, the VP network is optimized to balance the overall VP/VC switching and control operations required in an ATM network, cf. [10]. The call control costs, the trac integration policy and the GOS requirements are taken into account.

8.2.1 VP networks designed as a network infrastructure In [(92)025] and [12], the design of the VP network is decomposed in a process with three phases:

1. nd the optimal structure of the VP network 2. route VPs on the physical network 3. map demand on VPs, i.e. load, and dimension the VP network.

96

In the rst phase pairs of nodes that will be connected by VPs, the so-called VP terminator nodes are selected. This is done using a clustering algorithm. The clustering also de nes an initial load assignment for the VPs. The task of the second phase is to route the VPs on the physical network. This route selection is done by a randomized algorithm that is based on random choices among the set of all possible shortest paths connecting the given terminators. The objective is to distribute the routes in the network as evenly as possible to avoid overloading of links. The load of a link is interpreted as the total load of VPs that traverse it. It is proven in [(92)025] and [12], that this algorithm produces a VP route-distribution close to the best possible. Having found the VP terminators and the corresponding routes, a re ned loading and dimensioning of VPs is done in the third phase. This can be described as follows. For the convenience of the description we introduce the concept of logical links. A logical link between two nodes is either a VP connection or the non-VP fraction of a direct physical link between the two nodes (the non-VP fraction of a physical link is the part of the link capacity that is not reserved for VPs). We interpret logical links as directed links. Let us introduce the following notations:

L : set of all logical links in the network N : set of network nodes Lout (i) : set of nodes j such that (i; j ) 2 L Lin(j ) : set of nodes i such that (i; j ) 2 L Lpq : set of logical links that use the physical link (p; q) ij : trac demand from node i to node j Cpq : capacity of physical link (p; q) x(klij) : the part of capacity that is used for (i; j ) trac on logical link (k; l) ckl(ij) : the cost of routing one unit of (i; j ) trac on logical link (k; l). Now the capacity assignment can be solved by the following linear programming model: P c(ij)x(ij) Minimize i;j;k;l

Subject to

kl

kl

P x(ij)   ij il out

l2L (i)

P x(ij) = kl in

k2L (l)

P x(ij) lk out

k2L (l)

P P x(ij)  C pq kl i;j (k;l)2Lpq

8 i; j 2 N 8 i; j; l 2 N l 6= i; j

8 k; l 2 N

The rst constraint guarantees that all demands will be satis ed. The second constraint ensures ow conservation whereas the third constraint re ects the requirement that the 97

trac ow cannot exceed the physical link capacities. Having found the optimum for the x(klij)'s by linear programming, the VP capacities can be computed as X (8.2) Vkl = x(klij) i;j

where Vkl is the capacity of a VP connecting node k to node l. Completing the third phase, the result can be re ned by iteratively repeating the three phases, yielding a good sub-optimal solution.

8.2.2 Design of ATM networks integrating VPs and VCs In [10], the design of a VP network corresponding to one trac cluster is formulated as a cost minimization problem in a network with alternative routing meeting multi-service trac demands at the established end-to-end GOS at the call level (eventually varying for dierent services).

a) Problem formulation The formulation of the problem can be derived from the one formulated in Section 8.1.1 by introducing some simpli cations. We consider i = 1 reference trac set, j = 3 trac types, CS, TR, PS and CP policies and constant blocking constraints, i.e.

Bij  B

MPS  fCS; TR; PS; CPg;

Now, we can make the following formulation of the VP optimization problem:

Given Find Minimizing Subject to

A, B , D, MPS, F P , Q, MPS Z = Zd(P ; Q; MPS) + Zc (P ; Q; MPS) BB

where

A : trac matrix set Aj : trac matrix for trac type j D : vector for equivalent bandwidth of type-j calls, D = fdj g 98

F : maximum number of VPs B : blocking probability matrix B : maximum blocking probability matrix P : set of feasible VPs Q : set of VP capacities Zd : transmission and switching cost Zc : control cost.

b) Trac model For dimensioning requirements, the GOS has been expressed in terms of end-to-end call blocking. According to the models presented in Chapter 3, ATM links loaded by VBR sources are assigned to bandwidths which are linear functions of the number of carried calls of a given type. Each call requires an equivalent bandwidth, whose size depends mainly on the mean and the peak bit rate of the service as well as on the link capacity. Following this approach, the trac streams carried on a link and those over owing from a link (in mean and variance) are derived by extending the Hayward-like approximations to a multi-rate environment. This is done for the CS as well as for the TR policy.

c) Cost model The total cost of the network is decomposed in three components:

Z (T ) : transmission cost Z (C ) : switching cost Z (S ) : set-up cost. The transmission cost depends on the number of ATM links (or transmission paths) necessary to carry the trac between two nodes. For dimensioning purposes, it is useful to refer this cost to the required bandwidth amount which is expressed as a multiple of a certain bandwidth module. The transmission cost can now be expressed by the number of modules on the dierent transmission sections. Assuming for simplicity that the VP length is proportional to the number of transmission sections and VP switching is performed in each transmission node, then the total transmission cost is given as X Z (T ) = Nf tf ZT ; (8.3) f

where ZT is the transmission cost per section and per module, f is the VP index and the quantities tf and Nf represent the number of sections and modules of VPf , respectively. 99

The switching cost refers to the VP/VC switching operated in the network nodes. Assuming a linear relationship between cost and average switched bandwidth, the total switching cost is

Z (C ) =

X f

[ZC + (tf ? 1) ZP ]Df ;

(8.4)

where ZC and ZP are the switching costs per VC and VP respectively for one switched bandwidth unit and Df is the average bandwidth carried on VPf . We assume that one VC switching and (tf ? 1) VP switchings are performed in every VP. For each end-to-end trac relation, the set-up cost is computed proportional to the average call rate and to the number of VPs tested during the call set-up phase:

Z (S ) = ZS

X r

Lr Ur :

(8.5)

ZS are the set-up costs per call and per VP involved in the routing procedure, Lr is the average call rate for trac relation r and Ur de nes the average number of VPs involved in the call set-up phase.

d) Optimization procedure The heuristic procedure used to solve the VP network optimization problem is composed of three modules. This procedure is illustrated in Figure 8.3 and the modules are described in the following. Virtual Path Choice Module The object of this module is to optimize the VP structure. Consider a network with (L +1) nodes. The L VPs belonging to a minimum spanning tree (one-hop VPs) are immediately included in the set of optimal VPs. The problem is now to choose the residual F -L optimal VPs with a hop distance more than one. Their choice should be based on the end-to-end trac demands, the transmission network topology and the cost factors. In order to perform this function, two algorithms have been investigated. Algorithm A is based on a heuristic consideration: a certain VP is more suitable to be included in the optimal set where the more trac it can carry and the lower are its costs. So, all the VPs are ranked according to Kf = Af =Cf , where Af is the trac oered to VPf and Cf is the cost of one module on VPf . Finally, the F -L VPs with the highest rank are chosen. Algorithm B is based on a clustering algorithm similar to the one presented in Section 8.2.1. Numerical comparisons of the two algorithms for the 15 node network considered in [10] show that the performance of both algorithms is almost the same. 100

Point-to-point Demands

Service Characteristics

VP Cost / Efficiency Ratio

Network Topology

Cost factors (TX, SW, CTR)

Virtual Path Selection

Traffic Integr. Policy

Max VP Number

Routing Plan Generation

Routing Constraints

VP/VC Switching Optimization

End-to-end Blocking Constraints

VP Dimensioned Network

Figure 8.3: VP network optimization procedure. Routing Plan Generation Module This module derives the routing plan for each trac relation in the network. Obviously, the routes depend on the VPs chosen by the Virtual Path Choice Module and can also depend on the adopted service management policy (CP, CS, TR, PS). Basically, the routing structure can be hierarchical or non-hierarchical. In hierarchical networks, the routing plan follows some simple rules depending on a prede ned node hierarchy. According to the hierarchical principle, the shortest routes (from the hop point of view) over ow onto the longest routes. The last or nal route is composed of single hop VPs. The number of routes, which can become huge in case of trac relations having large hop distances, can be limited by an input parameter. An alternative solution is to adopt a non-hierarchical routing structure. In this case, any possible path between the origin and the destination node can be chosen. The trac distribution among the routes and the over ow order can be varied, too. In order to limit the number of routes and to choose the best ones, it is useful to adopt a cost structure. The natural assumption is that the cost of a route is the sum of the costs of the involved VPs. Thus, the problem reduces to that of nding the K shortest paths in a weighted graph. The logical over ow order is from the least to the most expensive routes. A more precise cost structure can be adopted taking into account the eciency of each VP from its utilization point of view.

101

VP/VC Switching Optimization Module The object of this module is to determine the capacity of the VPs, minimizing the global network cost and meeting the GOS constraints. The optimization is performed on the basis of the VPs and the routing structure obtained with the previous modules. Adopting the equivalent bandwidth approximation, the problem can be approached using the same framework as the design of multi-rate networks. A simple approach for the switching optimization in the case of a hierarchical routing scheme is presented in Figure 8.4, where Pratt's method [56] is adopted. According to this approach, the VPs are classi ed as nal or high usage. The high usage VPs are dimensioned to reach minimum cost, whereas the nal VPs (one-hop VPs) are engineered in order to meet the desired GOS constraints. In Figure 8.4, the following notation is used:

Af : trac oered to VPf Hf : marginal occupancy of high usage VPf . In the minimum cost network, according to the cost functions de ned above, the marginal occupancies of the high usage VPs satisfy the following optimization equation:

X ZT Htf = ZT b1 + (ZC ? ZP )(tf ? 1) + ZS tf ; f i2Sf i

(8.6)

where bi is the marginal capacity of the nal VPi and Sf is the set of nal VPs receiving the over ow of the high usage VPf . The blocking probabilities Bf of the nal VPs, are adjusted iteratively until the end-to-end GOS constraints are satis ed for each trac relation and each service. In the network performance analysis part, the end-to-end blocking probabilities are evaluated using the models presented in Chapter 4. In particular, the variances of the trac streams carried or over owing from a VP are computed extending a Hayward-like approximation as proposed in [?].

e) A case study The envisaged procedures have been applied to analyze the network shown in Figure 8.5. In this gure, the lines represent the ATM transmission links among the nodes. Assuming bidirectional VPs, the maximum number of VPs in the network is equal to 105. Three classes of service (T1, T2, T3) have been oered to the network having link capacities of 150 Mbps. The trac characteristics are shown in Table 8.3. For each service class an end-to-end trac matrix has been adopted. The total trac load on the network is equal to 2191 Erlangs subdivided in 1508 Erlangs for service T1, 351 Erlangs for service T2 and 332 Erlangs for service T3. 102

Initial values of Af, Bf, Hf

Dimensioning of high usage VPs according to Af, Hf

Dimensioning of final VPs according to Af, Bf

Network performance analysis: new values of Af, Bij

N

Convergence of Af ? Y

Adjustment of Bf according to GOS constraints

N GOS constraints met?

Y New values of Hf according to optimization equations

N

Minimum cost?

Y End

Figure 8.4: Example owchart for the optimization module.

103

1 4

11 3

12

2

5

7

14 8 6

13

10

15

9

Figure 8.5: Case study network. The adopted cost coecients in cost units are ZT = 1, ZC = 3, ZP = 0:9 and ZS = 100. To evaluate the impact on network costs of dependence on management policy and VP network structure, the following policies are compared with each other:

 CP: Each service class has its own VPs, completely independent of the ones used

by other classes. Also, the structure of the network is optimized service by service.  CS: All services share the same VP network.  TR: A trunk reservation threshold corresponding to the service with the greatest bandwidth is used for each VP for blocking probability equalization (cf. Chapter 5).  PS: The high usage VPs, if they exist, are individually assigned to the services while the nal VPs are shared among the services.

To describe the integration policy we use the code X1=X2 where X1 refers to the policy used on high usage (HU) VPs and X2 to the policy used on the nal choice (FC) VPs. As we are considering three classes of trac oered to the network, we have X1 = (x11; x12; x13) and X2 = (x21; x22; x23). The generic values of the xij 's have the following meaning: Service Peak Rate Mean Rate Call Duration T1 64 kbps 64 kbps 100 s T2 2 Mbps 2 Mbps 2s T3 10 Mbps 2 Mbps 200 s Table 8.3: Service class characteristics.

104

0 1 2 3

: : : :

trac j is not oered to the HU or FC VPs trac j is separated from other tracs trac j is integrated with some other trac using the CS policy trac j is integrated with some other trac using the TR policy.

For example the case (1,2,2/1,2,2) means that we are considering the complete integration of broadband tracs T2 and T3 and the separation of narrow band trac T1 on both high usage and nal choice networks. CS is represented by (2,2,2/2,2,2), TR by (3,3,3/3,3,3) and CP by (1,1,1/1,1,1). We also address other policies like (1,1,1/2,2,2), (0,1,1/2,2,2), (1,0,0/2,2,2). As to the network structure, the nal VP network has always 14
k VPs (1  k  3), depending on the adopted degree of integration. Similarly, the number of HU VPs is equal to p
k (1  p  90) depending on the integration policy. If the required end-to-end blocking probability B is set to 1% for all trac classes, the network costs for dierent integration policies X1=X2 and dierent VP network structures p are reported in Table 8.4. If the maximal blocking probability is chosen equal to 1% for T1 and 5% for T2 and T3 the results are shown in Table 8.5. Policy Code (1,0,0/1,0,0) (0,1,0/0,1,0) (0,0,1/0,0,1) (1,1,1/1,1,1) (1,1,1/1,1,1)* (1,2,2/1,2,2) (2,2,2/2,2,2) (1,1,1/2,2,2) (1,1,1/3,3,3) (0,1,1/2,2,2) (1,0,0/2,2,2)

10 3174 899 14093 18666 17821 18008 18243 19433 20645

VP Network Structure p 20 30 40 2951 2843 2764 898 915 938 13920 13956 14046 17769 17714 17748 17397 17206 17679 17395 17381 17807 17794 17862 18319 18525 18802 19553 19553 19675 20416 20311 20232

90 2579 1060 14554 18193 17610 18286 19868 20287 20046

Table 8.4: Total costs (B1 = B2 = B3 = 0:01). The network is such that with CS the total cost is equal to 18499 for the structures with the minimum number of VPs (14) and equal to 22124 for the maximal number of VPs (105). CS (2,2,2/2,2,2) is better than the CP policy (1,1,1/1,1,1) if we keep the same structure (p = 30) for each separated VP network. However, if we remove this constraint and optimize each VP network separately, we obtain a solution (1,1,1/1,1,1)* characterized by a dierent structure for dierent tracs leading to a cost lower than 105

Policy Code (1,0,0/1,0,0) (0,1,0/0,1,0) (0,0,1/0,0,1) (1,1,1/1,1,1) (1,1,1/1,1,1)* (1,2,2/1,2,2) (2,2,2/2,2,2) (1,1,1/2,2,2) (1,1,1/3,3,3) (0,1,1/2,2,2) (1,0,0/2,2,2)

10 3174 884 13656 17714 17546 17723 18204 19156 17206

VP Network Structure p 20 30 40 2951 2843 2764 882 897 918 13597 13652 13737 17430 17392 17419 17058 16916 17646 17114 17081 17503 17482 17545 18237 18403 18671 19384 19218 19353 16986 16883 16807

90 2579 1033 14209 17821 20941 18010 19743 19981 16626

Table 8.5: Total costs (B1 = 0:01; B2 = B3 = 0:05). in the case of the CS policy. In fact, the optimal structure for narrow-band trac T1 is achieved for p = 90 while for T3 trac the optimal structure is achieved for p = 20. It can be noted that, in the last case, the network cost is rather at with respect to p variations. A further improvement is given by the partial sharing policy (1,2,2/1,2,2), in which only T2 and T3 trac is integrated, and which constitutes the best considered policy (cost equal to 17206) as the other considered policies imply network costs greater than the optimal. Moreover, in the given network the TR and CS policies seem almost equivalent to each other. When considering Table 8.5 with required blocking probabilities B1 = 0:01 and B2 = B3 = 0:05, all the results are better than the results reported in Table 8.4. However in such a case, the best policy is the PS policy (1,0,0/2,2,2), whose cost is equal to 16626. The numerical results above show that when the bandwidth ratios between oered tracs are too high, trac integration may not be bene cial and other policies like those based on trac clustering and trac integration within the clusters can be preferable. Such a consideration holds even more when dierent GOS constraints or strict prescriptions during network overloads are present among the design objectives.

106

8.3 Control in multi-service networks In this section we are particularly concerned with control of multi-service networks and if we assign some notion of worth associated with calls, then we have enough information to at least pose sensible questions. Section 8.3.1 discusses the framework as it applies to networks. In Sections 8.3.2 and Section 8.3.3 some comments on reversible and adaptive controls are given. We brie y summarize some aspects of Markov Decision Theory in Section 8.3.4, which provided a nice theoretical model allowing us to say that optimal controls exists. Unfortunately we cannot calculate them for realistically sized networks. Therefore Section 8.3.5 looks at a single link and argues that trunk reservation has some appealing features, thus we perhaps should seek to maximize our understanding of this control before toying with more complex controls. Section 8.3.6 summarizes some key points.

8.3.1 Framework Consider a graph, where the vertices represent switches, and the edges represent logical circuits between them, edge j having integer capacity Cj . Calls of type r arrive at nodes destined for other nodes at rate r , and require Ajr units of resource j , which they seize for a holding time with mean 1=r . More generally we can allow alternative routing by having a number of possible matrices A. If a type-r call is accepted, we earn a reward at rate wr . We can mimic the eect of in ation by discounting at rate , where a reward w received in the future at time t is only worth we?t. Our goal is to maximize the expected reward, thus we have a multicommodity ow problem. We shall allow the A's to be positive numbers, re ecting the eective bandwidths. Thus in context of ATM networks, Ajr is the eective bandwidth of call r at resource j , which can vary on the dierent links of a route. It is usually convenient to reparameterize so that the eective bandwidths and capacities have greatest common divisor equal to 1, enabling us to work with integer quantities. We have chosen to work with rewards, with the objective of maximizing the expected rewards. We could equally well associate a cost with call rejection, where accepting a call incurs no cost and seek to minimize costs. The two formulations are equivalent provided that rewards (costs) and arrivals do not depend on the state of the network.

8.3.2 Reversible controls To make progress, we need to apply more structure. For example, with Poisson arrivals the probability of having xr calls in progress can be derived with the results in Chapter 4. The expected return g is then given by "X # g = E xr wr : (8.7) r

107

This models a reversible process which is insensitive to the type of the holding time process (see for example [9], [36] and [74]). Reversibility is an attractive feature giving a closed product form and some authors have looked at controls which stay within this framework. For example we may wish to maximize equation (8.7) subject to retaining reversibility. Reducing the arrival rates is such a control, which can be achieved practically by probabilistically thinning the arriving stream, or by looking at coordinate convex controls [63], which eectively truncate the state space. Note that controls such as trunk reservation discussed in Chapter 5 and below destroy reversibility, because they allow one-way transitions into certain regions of the state space.

8.3.3 Asymptotic results and adaptive control In fact, by taking expectations of the number of calls in progress for any dynamic routing scheme or control scheme, we can deduce a bound which links in reversibility. The optimal return under any non-anticipatory dynamic routing scheme is bounded by the solution to the following deterministic ow problem (cf. [43]):

Maximize

Px w

Subject to

0  xr  r =r Px A  C

r

r

r r

r jr

j

This is straightforward to prove by considering the associated process which does not lose any calls because of capacity constraints, and then taking expectations. The only restriction we make is to schemes which have no knowledge of the future. We can show that the bound is asymptotically approached and moreover is asymptotically achieved by an adaptive strategy which rejects a proportion of trac and routes the remainder directly (cf. [43]). Similarly, if we allow a number of alternative routes for type-r trac, then an asymptotically optimal control is to reject a certain proportion of the trac and use proportional routing to route the rest of the trac amongst its routes with probabilities derived from the linearPprogram. The interesting cases to consider are those when some links are overloaded ( r Ajr r =r > Cj for some j ), meaning we have to reject some trac, since otherwise we just accept all the trac in the limit. The quantities xr are the average ows under a given policy, or the carried trac r . In a practical situation, there might be constraints on the blocking probabilities achieved, in which case we might have additional constraints of the form

r  xr  (1 ? pr ) r

(8.8)

r

instead of xr  0, where pr is the maximum allowed blocking probability. This ts naturally into the framework given, by adding additional constraints into the linear program. 108

The conclusions concerning the asymptotic optimality of adaptive routing for Poisson arrivals although now there is the possibility of no feasible solution if P (1 ? pare)Aalsor unaltered, >C. r

r

j

jr r

It is instructive to consider the following dual problem: P s r + P C c Minimize r

Subject to

r r

j j

j

sr  wr ? P Ajr cj j

with sr  0 and cj  0 From standard results it follows, that under any policy the expected reward per unit of time is bounded above by the minimum of this problem. The dual problem also has a natural derivation obtained by adapting the proof of [41] for a dierent problem concerned with routing, supposing that a call arriving at the network is charged an amount sr when it is accepted, and in addition is charged an amount cj per unit of capacity it uses on link j . Using complementary slackness conditions [73], we have that a call will not be accepted on a route if the charges are too high, i.e. X (8.9) sr + Ajrcj > wr ; j

a link charge cj is zero if there is spare capacity on the link and a route charge sr is zero if calls are being lost (xr < r =r ). The notation used ties in with earlier work. The quantities cj can be interpreted as the implied cost of using a unit of capacity from link j , and sr is the surplus value of accepting a call of type r, see [42]. The notions of implied costs and surplus values have proved very useful in the analysis and design of circuit-switched networks (cf. [42], [44]). The asymptotically optimal routing described above is adaptive, since it is independent of the network's state. This result says that asymptotically we can use adaptive routing to maximize our gain, and moreover there is nothing to be gained by accepting more trac onto the network than this solution to the linear program. As just mentioned, the interesting cases are when some of the capacity constraints are binding, for some link j , since otherwise we just oer all the trac to the network in the limit. However, although these controls are asymptotically optimal, in practice the asymptotic limit is a coarse one. The proof relies on the convergence of the Erlang xed point to the deterministic limit and the asymptotic accuracy is weakest in the critically loaded case when trac and capacities coincide (cf. [41]), which is exactly what happens with this adaptive control when some link is overloaded. Practically speaking, the state-dependent controls which are optimal can give a markedly better return for realistic sizes of circuits, and moreover are more robust (see for example [43], [45]). Within our framework, there is a stationary optimal state-dependent control if we consider negative-exponential holding time processes and therefore have a nite state space and a 109

Semi-Markov Decision process. In other words, a process that is deterministic, does not depend on time and says whether to accept or reject a call of type r depending upon the network state. In general however, this control process will not be reversible. We now brie y describe a way of calculating such an optimal policy, and then discuss some of the problems associated with trying to apply such theory to networks. Introductions to Markov Decision Theory can be found in [64] and [67].

8.3.4 Markov Decision Theory and state-dependent controls For the moment, assume that we discount at rate  and let V denote the expected discounted reward earned over an in nite time horizon. Discounting at a positive rate ensures that the expected reward is bounded. Further, let (x, r) denote an arrival of type r in state x and, without loss of generality suppose that rewards are earned as a lump sum (wr =r ) if a type-r call is accepted. We can use uniformization to make the rate of transitions between states P constant, namely  = Pr Pr + , by adding ctitious events which occur at rate  ? r xr r , where  = max r xrr . In this case the dynamic programming recursions (optimality equations) for Vn; , the reward when n transitions remain, are [43]:

V0;(x)  0 Vn; (x) =

(8.10)

 P  P n?1; (x; r) + r xr Vn?1; (x ? er ) + ? r xr Vn?1; (x) r r (8.11) +

P V r

r

Vn; (x; r) = maxfwr + Vn;(x + er ); Vn;(x)g

(8.12)

where of course the constraints Pj Ajr xr  Cj are understood. In other words it is optimal to accept a call of type r if and only if

wr > W (x; r)  V (x) ? V (x + e ); r r

(8.13)

where we have suppressed the dependence on n and . This relation de nes a policy at each stage. It can be shown that Vn; converges to the limit V (see e.g. [64]), and hence the control policy used at each stage n converges to the optimal policy. This way of calculating V by successive approximations, known as value iteration, lends itself naturally to computer implementation. We have a nite state space and moreover the empty state x = 0 is accessible from every state. Hence the average reward g does not depend upon the initial state and is the limit of V as  ! 0. The quantities  (x) = lim!0fV(x) ? V (0)g are identi ed with relative values, determined up to a constant. The dierence  (x) ?  (y ) has a natural economic 110

interpretation as the amount we are prepared to pay for starting in state x rather than y . For  = 0 and large n we have Vn (x)  ng +  (x), which links the nite horizon gain with the in nite horizon average reward case. For practical purposes, the relation







 min x fVn(x) ? Vn?1 (x)g  g   max x fVn (x) ? Vn?1 (x)g



(8.14)

allows the average reward to be approximated to the desired accuracy. Value iteration has several advantages over other ways of calculating the optimal policies and reward. For example linear programming formulations can suer from complexity considerations. Relaxation factors can be used to speed up convergence of equations (8.10) to (8.12), cf. [67]. This provides a way of calculating the optimal control for small systems or isolated links. However, note that the state space is dependent on the number of calls of each type in progress. Thus in practice little can be said about the optimal control of networks even with a reduced state-space model where we look at links in isolation and assume that a multi-link call behaves as a sequence of independent single link calls. Moreover, nice monotonicity properties, which hold for the case where all the bandwidths are the same, disappear in the multi-service case [43]. Consider the case of several streams of Poisson trac, with common negative-exponential distributed call holding times but diering worths oered to a single link, each requiring unit resource. In [52] it is shown by induction, that the function Vn;a is concave, and hence that W is a non-decreasing function, which implies that optimal policies are monotonic increasing. If a call is rejected in state x, then it is optimal to reject this call in states bigger than x. Hence the optimal policy is a critical number policy or threshold policy, where a trunk reservation value of r is applied to stream r, with r  s if wr  ws. One can also show that Wn is a non-decreasing function of n and that there is a trunk reservation policy which is optimal in the nite horizon case for n suciently large and  small enough. However the analogy for networks, particularly for multi-service networks, does not hold in general, which means that the value functions are not multi-modular (see e.g. [46] and [66]). Indeed, in the broadband case optimal policies can fail to be self-monotonic, which will happen if we reject a type-k call in state x but accept it in state x + ek . An example for this is discussed below.

8.3.5 Single link models and trunk reservation For simplicity, consider just two trac streams with parameters wi, i , i and eective bandwidths ki, oered to a resource of size C . The bound on the rate of return is given by the solution to the linear program: 111

Maximize

w1x1 + w2x2

Subject to

0  xi  ii

x1k1 + x2k2  C from which it immediately follows that

 if  = k1 11 + k2 22  C then the bound w1 1 + w2 2 1

(8.15)

2

is obtained by accepting all trac  if  > C then the bound is !   1 1 w2 w1  + k C ? k1  ; 1

2

(8.16)

1

assuming that the indices are ordered such that w1=k1 > w2=k2, where we now reject a proportion of type 2 trac. We have already mentioned that this is an asymptotic bound, and we can easily calculate the optimal policy for Poisson arrivals and negative-exponential holding time processes provided that C 2=(k1k2) is not too large. Using the equations (8.10) to (8.12), we obtain the optimal control policy. For the parameters i = 1, i = 4, w1 = k1 = 1, w2 = k2 = 6 and C= 24, the rejection region for narrow-band calls is shown in Figure 8.6. It can be observed, that there are holes in the rejection region; we reject narrow-band calls in state (0,3) but accept them in state (1,3). In general, the optimal control approximates to trunk reservation. The appearance of holes is a consequence of boundary conditions, and we would expect these features to disappear as the resource increases in size. In fact, we can show that trunk reservation is asymptotically optimal. It is easy to see that in the lightly loaded case ( < C ) an optimal control is no control, namely to accept all calls. The critically loaded case ( = C ) is a little trickier, and we sidestep it for the moment. However we can see that trunk reservation is asymptotically optimal in the heavy trac case ( > C ), and we sketch a justi cation below. In [7] it is shown that in the heavy trac case the vector x = (x1=C; x2=C ) converges to a xed point of the equations

k1x1 + k2x2 = C

(8.17)

112

4

wide band calls

3 2 1 0 0

5

10

15

20

25

narrow band calls

Figure 8.6: Rejection region for narrow-band calls.

X x1 = 1 (1 ? B1) = 1 m

(8.18)

X x2 = 2 (1 ? B2) = 2 m

(8.19)

1

1 mC ?1

2

2 mC ?2

where xi is the number of type-i calls in progress. Type-i calls are subjected to a trunk reservation of i and i is the equilibrium distribution of the free-circuit process, which operates on a fast time-scale compared to the x process. Bi is the probability that a type-i call is blocked. Then in the case w1=k1 > w2=k2 we can make the return w11=1 + w22=2(1 ? B2) converge to the optimum (cf. equation (8.16)) by setting 1 = C ? k1 and choosing 2 approximately. Note that C ? 2 is expected to grow as log C . The above is based on modeling the free-circuit process on the integers j as having constant birth rates xj j . A more accurate approximation is to replace the constant xj by something that depends on the number of free circuits. For example, we can model the number of busy circuits (1 ? i) as a birth-death process with arrivals i subjected to trunk reservation, but variable departure rates ) ^i = n  = (1 ? Bi(1) +?Bi= (1 ? B ) 1

1

1

2

2

2

(8.20)

where n is the number of busy circuits. The attraction of this approach is that it gives a good one-dimensional approximation, which is much easier to solve and remains onedimensional regardless of the number of call-types present. Note that the solution is still not trivial, since the birth-death process is not skip-free, see Section 5.4.2. 113

Indeed, we can use either of the two one-dimensional approximations to obtain relative values. For example, in the average optimal case we can use Howard's equations [29] for a system where trunk reservation is applied. This approach is useful for a number of reasons:

 it enables us to use value iteration to determine the optimal trunk reservation pa-

rameter (see for example [43] and [64])  we can in principle use relative dierences as the basis of a multi-service dynamic routing scheme using analogies with the proposal in [53], or develop a more sophisticated divide and conquer approach to construct optimal polices or dynamic routing schemes along the lines suggested in [43].

The latter works by treating links as isolated resources and calculating a reduced oered trac that is oered to the link, which takes account of blocking on other links on the route. This could be calculated from a xed point approximation, or even from measurements. To calculate the relative values we assign a link based worth, that is we have to distribute the worth wr of the call amongst the links. A simple way to do this is for example to divide the worths equally. We can then calculate a set of relative values Wj for each link j and accept a call of type r which uses links 1; : : : ; m if

W1(n1) + : : : + Wm(nm ) < wr ;

(8.21)

where nj is the number of busy circuits on link j , and Wj (nj ) is a shorthand notation for Vj (nj ) ? Vj (nj + Ajr ) which depends on r through the eective bandwidth. This gives us a new policy. If this policy is now approximated by a trunk reservation policy, we can calculate new link-based relative values Wj and repeat the process. Details of this need to be explored. The fact that a simple closed form solution for Wj is unlikely to exist poses some problems.

8.3.6 Discussion The optimal control of multi-service networks is a complex problem, and for realistically sized networks the exactly optimal control is likely to be too complicated to implement, being critically dependent on the values of the oered trac and requiring precise knowledge of the network state. We know that as networks get large, simple controls become asymptotically optimal, but how large is large? One weakness of the theory is that we do not have good convergence results or bounds, so it is hard to answer this question quantitatively. We do know that if the resources are two or three orders of magnitude larger than the eective bandwidths of the calls, then the Erlang xed point provides a good approximation, which is used in assessing the ecacy of certain classes of control. However, in ATM networks we are unlikely to have this multiplier, which means that initially resources are small to medium. In the limit, sensible controls become asymptotically optimal. However, what we really want are controls that are robust. In practice we do not precisely know oered tracs, 114

mean holding times etc., and we want our controls to be robust against these assumptions. For example in large networks, probabilistically thinning arrival streams tends to an optimal control, but this is not robust, since the proportions depend on the trac and if we suddenly altered the trac, our return from the network could be far from ideal. Controls like trunk reservation are much more robust in this sense. Indeed, we would argue that the sensible question to concentrate on is how to set trunk reservation controls in a network, rather than nding improvements to trunk reservation. Here we are assuming that we want to do something more than just equalize link blockings. This is not a trivial task in a multi-service network, since the relative ordering of calls (in terms of worth) can change from link to link. Even if we assume the eective bandwidth is constant across all resources that a call uses and make the worth proportional to this, on some links there could be calls with a greater worth, and not on others. Thus perhaps a sensible starting place is to consider a very small number of trunk reservation parameters and attempt to say which calls should be at high-priority. The simple linear program which provides a bound on performance provides a starting point for examining this question and multi-slot xed point models enable the question to be tackled in more detail.

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9. Concluding remarks Since the nature of this document is an interim report of the research activities in the COST 242 project in the middle of the project lifetime, no nal or general conclusion will be or should be drawn. However, with regard to the forthcoming implementation of ATM networks, we would like to outline some ndings which we have gained from the studies and results so far. Models and methods presented in the rst part of this interim report mainly concern blocking evaluation and dimensioning of single network elements if well-known and speci ed trac is oered. To enable an ecient use of the transmission link capacity, it turns out to be helpful to introduce an equivalent bandwidth concept for VBR calls. By introducing this concept, multi-service networks like ATM networks behave at the call level like multi-slot circuit-switched networks, where cell level losses are guaranteed to meet a de ned cell loss ratio. Thus, a number of simple and easy-to-compute methods like the product form and the recursive solution exist for dimensioning single links to meet desired GOS constraints if no particular CAC method is implemented, i.e. no bandwidth reservation is performed. It proves, however, that such a complete sharing policy is in general an unfair CAC strategy, since calls which require only a small bandwidth amount are favored above the others in terms of low call blocking and high link utilization. To deal with these problems a more fair CAC strategy like trunk reservation is required. The trunk reservation algorithm is quite simple to implement and enables the network provider to adjust the blocking constraints individually for each trac class. One possibility is for example the equalization of the blockings for a number of trac classes. For the trunk reservation policy, the evaluation of the blocking probabilities is, however, not as simple. Therefore, we have presented a number of robust approximate methods which provide results which are suciently accurate for practical purposes. We have also presented an extended model which enables us to consider blocking on dierent time scales simultaneously, i.e. call and burst blocking. Such a model provides a helpful tool, since there are generally dierent GOS constraints on dierent time scales. An application for which this model can constitute a dimensioning tool is for example Fast Resource Management. The extension of such single link models to whole networks is, however, not straightforwarded due to the modi cations of the trac characteristics while passing through several network elements. This is caused for example by routing rules used for network 117

management and the intrinsic unfairness shown by multi-rate networks carrying heterogeneous trac. A rst step towards the evaluation of end-to-end blocking in multi-service networks by the use of xed point models was presented. However, such a model can only deal eciently with rather small networks. For larger, more realistic networks the computation eort becomes intractable and the accuracy is expected to be worse. The last part of this interim report deals with the problem of dimensioning multi-rate networks like ATM networks by introducing transmission and switching costs. A major diculty here is the problem of integrating or separating trac as well as controlling it. As a rst approach we have presented clustering and dimensioning rules and have shown that trunk reservation is an asymptotically optimal CAC strategy. These methods represent, however, only rst steps towards the dimensioning of multi-rate models and need to be investigated further. Overall, during the investigation and rst implementation phase of multi-rate systems like the ATM network, several new problems have arisen which have never occurred or been thought of in existing networks such as the narrow-band ISDN. This mainly concerns the evaluation of end-to-end measures. As one major part of the future research activities in the current COST 242 project we will concentrate on the development of network design strategies which can be applied to various trac scenarios, GOS requirements and cost functions. To do so, trac in multi-service systems like ATM networks must be characterized in a more speci c way to allow for more realistic modeling. Another research area related to the topic of this interim report is the investigation of trac integration in terms of GOS requirements. Trac classes with very dierent GOS requirements, like e.g. cell loss rates, should not generally be integrated on one link or virtual path.

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COST documents [(92)025] J. Chlamtac, A. Farago, T. Zhang, Optimizing the System of Virtual Paths in ATM Network Architecture, COST 242 Technical Document (025), 1992. [(91)SE] G. Gallassi, G. Rigolio, L. Verri, Key Role of Trac and Performance Issue in ATM: Open Approach to Trac Control, COST 224 Seminar, Paris, October 1991. [(92)019] F. Hubner, P. Tran-Gia, An Analysis of Multi-Service Systems with Trunk Reservation Mechanisms, COST 242 Technical Document (019), 1992. [(87)084] V.B. Iversen, A Simple Convolution Algorithm for the Exact Evaluation of Multi-Service Loss Systems with Heterogeneous Trac Flows and Access Control, COST 214 Technical Document (084), 1987. [(92)037] C.P.H.M. Lavrijsen, A Comparison of Approximation Methods for the blocking probability on connection level of Multi-slot trac peaked streams on a single link, COST 242 Technical Document (037), 1992. [(92)018] K. Lindberger, Some ideas on GOS and call scale link-by-link dimensioning, COST 242 Technical Document (018), 1992. [(92)065] K. Lindberger, The Use of Simple Methods for Integrated Call Scale Streams, COST 242 Technical Document (065), 1992. [(93)008] K. Lindberger, Some Additional Quantities in Approximate Multi Bit Rate Models, COST 242 Technical Document (008), 1993. [(93)020] A.H. Roosma, F.J.M. Panken, Trac Models for Multi-rate Networks with Alternate Routeing, COST 242 Technical Document (020), 1993. [(92)036] S.-E. Tidblom, Complete equivalent bandwidth formulae for various cell loss ratios, COST 242 Technical Document (036), 1992. [(86)068] J.T. Virtamo, Partial Sharing Access Control Policy in Switching Two Dierent Tracs in an Integrated Network, COST 214 Technical Document (068), 1986.

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Bibliography [1] J.M. Aein, A Multi-User-Class, Blocked-Calls-Cleared, Demand Access Model, IEEE Transactions on Communications, Vol. 26, No. 3, 1978, pp. 378-385. [2] M. Ammar, S. Cheung, C. Scoglio, Routing Multipoint Connections Using Virtual Paths in an ATM Network, IEEE INFOCOM 1993, pp. 98-105. [3] M. Azmoodeh, R.N. Macfadyen, Multi-Rate Call Congestion: Fixed Point Models and Trunk Reservation, Proceedings of IEEE U.K. Teletrac Symposium, Cambridge, March 1994. [4] G. Barberis, R. Brignolo, Capacity Allocation in a DAMA Satellite System, IEEE Transactions on Communications, Vol. 30, No. 7, 1982, pp. 1750-1757. [5] N.G. Bean, Eective bandwidths with dierent quality of service requirements, IBCN&S, Denmark, April 1993, paper 13.3. [6] N.G. Bean, R.J. Gibbens, S. Zachary, The performance of single resource loss systems in multiservice networks, to appear in Proceedings of the 14th ITC, Antibes 1994. [7] N.G. Bean, R.J. Gibbens, S. Zachary, Analysis of Large Single Resource Loss Systems under Heavy Trac, with Appplications to Integrated Networks, to appear in Advances in Applied Probability, March, 1995. [8] J.L. Burgin, Broadband ISDN Resource Management, ITC Specialist Seminar, Adelaide 1989. [9] D.Y. Burman, J.P. Lehoczky, Y. Lim, Insensitivity of blocking probabilities in a circuit-switched network, Journal of Applied Probability, Vol. 21, pp. 850-859. [10] E. Cavallero, U. Mocci, C. Scoglio, A. Tonietti, Optimization of virtual-path/virtualcircuit management in ATM networks, 5th International Planning Symposium Networks '92, Kobe 1992. [11] CCITT Draft Recommendation I.361, ATM Layer Speci cation for B-ISDN, SGXVIII, January 1990. [12] J. Chlamtac, A. Farago, T. Zhang, How to Establish and Utilize Virtual Paths in ATM Networks, IEEE International Conference on Communications, Geneva 1993, pp. 1368-1372. 121

[13] J. Choi, M. Choi, T. Chung, Y. Shin, K. Kim, Overall Design Requirements of Broadband ATM Networks, IEEE INFOCOM 1991, pp. 352-357. [14] S. Chung, K.W. Ross, Reduced Load Approximations for Multirate Loss Networks, IEEE Transactions on Communications, Vol. 41, No. 8, 1993, pp. 1222-1231. [15] COST 224 Final Report, J.W. Roberts (ed.), Performance Evaluation and Design of Multiservice Networks, Paris, October 1991. [16] M. Decina, L. Faglia, T. Toniatti, Bandwidth Allocation and Selective Discarding for Variable Bit Rate Video and Bursty Data Calls in ATM Networks, IEEE INFOCOM 1991, pp. 1386-1393. [17] L.E.N. Delbrouck, On the Steady-State Distribution in a Service Facility Carrying Mixtures of Trac with Dierent Peakedness Factors and Capacity Requirements, IEEE Transactions on Communications, Vol. 31, No. 11, 1983, pp. 1209-1211. [18] E.A. van Doorn, F.J.M. Panken, Blocking probabilities in a loss system with arrivals in geometrically distributed batches and heterogeneous service requirements, submitted for puplication 1992. [19] Z. Dziong, J.W. Roberts, Congestion Probabilities in a Circuit-Switched Integrated Services Network, Performance Evaluation 7, 1987, pp. 267-284. [20] A. Elldin, Basic Trac Theory, ITU Seminar on Trac Engineering and Network Planning, New Delhi, 1975, pp. 11-133. [21] O. Enomoto, H. Miyamoto, An Analysis of Mixtures of Multiple Bandwidth Trac on Time Division Switching Networks, Proceedings of the 7th ITC, Stockholm 1973, pp. 635.1-8. [22] J. Filipiak, Structured System Analysis Methodology for Design of an ATM Network Architecture, IEEE Journal on Selected Areas in Communications, October 1989. [23] A.A. Fredericks, Congestion in Blocking Systems - A Simple Approximation Technique, Bell Systems Technical Journal, Vol. 59, No. 6, 1980, pp. 805-828. [24] A. Gersht, K.J. Lee, Virtual-Circuit Load Control in Fast Packet-Switched Broadband Networks, Proceedings of Globecom 1988, pp. 7.3.1-7.3.7. [25] R.J. Gibbens, P.J. Hunt, Eective bandwidths for the multi-type UAS channel, Queueing Systems 9, 1991, pp. 17-28. [26] R. Guerin, H. Ahmadi, M. Naghshineh, Equivalent Capacity and Its Application to Bandwidth Allocation in High-Speed Networks, IEEE Journal on Selected Areas in Communications, Vol. 9, No. 7, 1991, pp. 968-981. [27] S. Gupta, K.W. Ross, M. El Zarki, On Routing in ATM Networks, IFIP Workshop TC6, Modelling and Performance Evaluation of ATM Technology, La Martinique, January 1993.

122

[28] S.J. Hindmarch, The Eect of Mean Holding Times on Determining the Trunk Reservation Parameter - A Markov Decision Approach, BT Internal Paper, RTM RT32/22/90, May 1990. [29] R.A. Howard, Dynamic Programming and Markov Processes, Wiley, New York, 1960. [30] F. Hubner, M. Ritter, Blocking in Multi-Service Broadband Systems with CBR and VBR Input Trac, 7th ITG/GI Conference, Aachen, September 1993, pp. 212-225. [31] D. Hughes, K. Wajda, Comparison of Virtual Path Bandwidth Assignment and Routing Methods, Cracow's Workshop on Network Management, Cracow, May 1993. [32] J.Y. Hui, Resource allocation for broadband networks, IEEE Journal on Selected Areas in Communications, Vol. 6, 1988, pp. 1598-1608. [33] S.A. Johnson, A performance analysis of integrated communication systems, British Telecom Technical Journal, Vol. 3, No. 4, October 1985. [34] J.S. Kaufman, Blocking in a Shared Resource Environment, IEEE Transactions on Communications, Vol. 29, No. 10, 1981, pp. 1474-1481. [35] J.S. Kaufman, Blocking with Retrials in a Completely Shared Resource Environment, Performance Evaluation 15, 1992, pp. 99-113. [36] F.P. Kelly, Reversibility and Stochastic Networks, Wiley, New York, 1979. [37] F.P. Kelly, Fixed Point Models of Loss Networks, Journ. Austral. Math. Soc., Series B31, 1989, pp. 204-218. [38] F.P. Kelly, Eective bandwidth at multi-class queues, Queueing Systems 9, 1991, pp. 5-16. [39] F.P. Kelly, Loss Networks (Special Invited Paper), The Annals of Applied Probability, Vol. 1, No. 3, 1991, pp. 319-378. [40] F.P. Kelly, Blocking Probabilities in Large Circuit Switched Networks, Advances in Applied Probability, Vol. 18, 1986, pp. 473-505. [41] F.P. Kelly, Bounds on the performance of dynamic routing shemes for highly connected networks, Research Report 92-34, Statistical Laboratory, Cambridge. [42] F.P. Kelly, Routing in circuit-switched networks: optimization, shadow prices and decentralization, Advances in Applied Probability, Vol. 20, 1988, pp. 112-144. [43] P.B. Key, Optimal control and trunk reservation in loss networks, Probability in the Engineering and Info Services, Vol. 4, 1990, pp. 203-242. [44] P.B. Key, Implied cost methodology and software tools for a fully connected network with DAR and trunk reservation, British Telecom Technological Journal, Vol. 6, 1988, pp. 52-65.

123

[45] P.B. Key, Some Control Issues in Telecommunication Networks, Probability, Statistics and Optimization: A Tribute to Peter Whittle, Wiley, 1994. [46] P.B. Key, Markov decission processes and optimal control in circuit-switched networks, Proccedings of the 5th UK Teletrac Symposium, 1988, IEEE publications. [47] R. Kleinewillinghofer-Kopp, E. Wollner, Comparison of Access Control Strategies for ISDN-Trac on Common Trunk Groups, Proceedings of the 12th ITC, Torino 1988, paper 5.4.A.2. [48] J.P. Labourdette, G.W. Hart, Blocking Probabilities in Multi-Trac Loss Systems: Insensitivity, Asymptodic Behaviour and Approximations, IEEE Transactions on Communications, Vol. 40, No. 8, 1992, pp. 1355-1367. [49] K. Lindberger, Simple Approximations of Over ow System Quantities for Additional Demands in the Optimization, Proceedings of the 10th ITC, Montreal 1983, paper 5.3-3. [50] K. Lindberger, Blocking for Multi-Slot Heterogeneous Trac Streams Oered to a Trunk Group with Reservation, 5th ITC Seminar, Lake Como 1987, North Holland 1988. [51] K. Lindberger, Dimensioning and Design Methods for Integrated ATM Networks, Proceedings of the 14th ITC, Antibes, 1994, pp. 897-906. [52] S.A. Lippman, Applying a New Device in the Optimization of Exponential Queueing Systems, Operations Research, Vol. 23, 1975, pp. 687-710. [53] T.J. Ott, K.R. Krishnan, State Dependent Routing of Telephone Trac and the Use of Seperatable Routing Schemes, Proceedings of the 11th ITC, Kyoto, 1985. [54] E. Pinsky, A. Conway, Performance Analysis of Sharing Policies for Broadband Networks, 7th ITC Seminar, Morristown 1990, paper 11.4. [55] M. Pioro, U. Korner, B. Wallstrorm, Design Methods and Routing Control in Integrated Services Networks with Alternative Routing, Proceedings of the 12th ITC, 1989. [56] C.W. Pratt, The Concept of Marginal Over ow in Alternate Routing, Proceedings of the 5th ITC, New York 1967, pp. 51-58. [57] M.I. Reiman, A critically loaded multi-class Erlang loss system, Queueing Systems 9, 1991, pp. 65-82. [58] M.I. Reiman, Optimal Trunk Reservation for a Critically Loaded Network, Proceedings of the 13th ITC, Copenhagen 1991. [59] J.W. Roberts, A Service System with Heterogeneous User Requirements - Application to Multi-Service Telecommunications Systems, Proceedings of Performance of Data Communication Systems and their Applications, G. Pujolle (ed.), North Holland, pp. 423-431. 124

[60] J.W. Roberts, Teletrac Models for the Telecom 1 Integrated Services Network, Proceedings of the 10th ITC, Montreal 1983, paper 1.1.2. [61] J.W. Roberts, Trac Control in the B-ISDN, Computer Networks and ISDN Systems, North Holland, 1993, pp. 1055-1064. [62] A.H. Roosma, On estimating call congestion and time congestion, Internal Document, PTT Research, Notitie 85 INF/130, Leidschendam, Netherlands, 1985. [63] K.W. Ross, D. Tsang, The Stochastic Knapsack Problem, IEEE Transactions on Communications, Vol. 37, No. 7, 1989, pp. 740-747. [64] S.M. Ross, Introduction to Stochastic Dynamic Programming, Academic Press, New York, 1983. [65] B. Sanders, W.H. Haemers, R. Wilcke, Simple Approximation Techniques for Congestion Functions for Smooth and Peaked Trac, Proceedings of the 10th ITC, Montreal 1983, No. 7, paper 4.4b.1. [66] S. Stidham, Optimal control of admission to a queueing system, IEEE Transactions on Communications, Vol. 30, 1985, pp. 705-713. [67] H.C. Tijms, Stochastic Modelling and Analysis, Wiley, Chichester, 1986. [68] I. Tokizawa, T. Kanada, K. Sato, A new Transport Architecture Based on Asynchronous Transfer Model Technique, ISSLS, Boston 1988. [69] P. Tran-Gia, F. Hubner, An Analysis of Trunk Reservation and Grade of Service Balancing Mechanisms in Multiservice Broadband Networks, IFIP Workshop TC6, Modelling and Performance Evaluation of ATM Technology, La Martinique, 1993. [70] J.T. Virtamo, Reciprocity of Blocking Probabilities in Multiservice Loss Systems, IEEE Transactions on Communications, Vol. 36, No. 10, 1988, pp. 1174-1175. [71] E. Wallmeier, A Connection Acceptance Algorithm for ATM Networks Based on Mean and Peak Bit Rates, Int. Journal on Digital and Analog Communication Systems, Vol. 3, 1990, pp. 144-153. [72] W. Whitt, Blocking when Service is Required from Several Facilities Simultaneously, AT&T Technical Journal, Vol. 64, 1985, pp. 1807-1856. [73] P. Whittle, Optimization under constraints, Wiley Interscience. [74] P. Whittle, Systems in Stochastic Equilibrium, Wiley Interscience, 1986.

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