End-to-end Blocking Probability Approximations ... - Semantic Scholar

End-to-end Blocking Probability Approximations for Resource Management in Multirate Loss Networks1 Tibor Cinkler, Gabor Fodor, Laszlo Ast, Sandor Racz fcinkler,fodor,ast,[email protected] Department of Telecommunications & Telematics, Technical University of Budapest, Sztoczek u. 2., H-1111 Budapest, Hungary Abstract

Eective trac control in multirate networks is vital, because a small increase in the carried trac may signi cantly increase the operator's revenue and at the same time enhance the user's satisfaction. Therefore evaluation of end-to-end blocking probabilities and carried trac when combining dierent routing and link allocation strategies is important. A broad spectrum of dierent routing and link allocation procedures is available in the literature; unfortunately most of the analytical models are either constrained by the network topology, or of high computational complexity. This paper (1) presents analytical and simulation results on a variety of combinations of routing and link allocation algorithms and (2) proposes an optimization model in which the revenue or carried trac in multi-rate loss networks is optimized with respect to load sharing parameters or an optimal partitioning between bandwidth classes is found. With respect to routing we consider non-alternate, sequential alternate, alternate routing with load sharing and adaptive least loaded routing. As for link allocation, we compare the performance of complete sharing, complete partitioning and trunk reservation policies. The analytical methods compute route blocking probabilities and network revenue either by the reduced load and link independence assumption or by a Whitt type approximation, while link blocking probabilities are chosen among a large set of well known methods. The paper points out the trade o between accuracy and speed in the above techniques, and presents simulation results con rming the analytical ones.

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This work has been supported by Ellemtel Telecommunications System Laboratories

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1 Introduction Optimization of resource management for maximal revenue in traditional telephone networks has a long tradition and a number of useful theoretical results exist ([10], [19]). In broadband networks where heterogeneous services are integrated the dimensioning and management become a more dicult task ([5], [4], [17]). We have to adopt new assumptions, apply approximations and develop mathematical models. We assume that on the call level an ATM network can be modelled as a multi-rate circuit switched network. The physical network is assumed to be given with its topology, link capacities and trac demands between certain pairs of nodes (can be based on estimation, measurement and/or prediction). We want to maximise the total carried trac. A reasonable model is to de ne a revenue function W which is the sum of all carried tracs (call rate multiplied by the holding time and bandwidth demand for each call) in the network. The most dicult term to calculate is the carried trac between node pairs (or more concretely the blocking probability of each call attempt). The subject of this paper is twofold. First, we present an analytical model and a simulation system pointing out their weaknesses and strenghts proposing application areas. Second, we present results of performance analyses using both techniques, showing the impact of dierent resource sharing and routing techniques on the total carried trac. In section 2 we describe methods for calculating/approximating the end-to-end blocking probabilities for dierent trac conditions, resource allocation and routing strategies, mentioning the topological impact. In section 3 we present our models, followed by performance analysis results in section 4 for dierent resource sharing and routing techniques. Section 5 draws conclusions.

2 On the End-to-End Blocking Probability Estimation As it was mentioned above in order to evaluate the performance of dierent resource allocation techniques, routing strategies and mathematical approximations end-to-end blocking probabilities have to be estimated. For that purpose we use either simulation or analytical evaluation, or both. In some cases the analytical method is satisfactory, for others we need to apply simulation because of the inaccuracy or unavailability of analytical formulae. For a single link with multirate trac the exact formula exists along with a wide wariety of different approximations ([1], [4], [5]). The choise depends on the trade-o between accuracy and speed. Furthermore we need to estimate the blocking probabilities of calls estabilished on a sequence of links, moreover on a set of paths if routing is present.

2.1 Fixed Routing In the case of xed routing (no alternative routes are present), exact steady state probabilities can be given modelling the network by the state-space, see e.g. [17]. However, in case of large capacities the model quickly becomes untractable. A quite simple upper bound estimation on route blocking is given by Whitt ([22], Corollary 1:2). It 2

was utilised by Ast et al. [1], [7] to yield a fast but rough estimation. Assuming that blocking occures independently from link to link and that the oered trac to a given link is Poissonian but thinned by the blocking on other links along the route we can apply the so called reduced load approximation [18]. A performance analysis for dierent link level approximations is given e.g. in [4]. The reduced load approximation leads to a set of nonlinear equations which can be solved by repeated substitution ([22], [18]), by Lagrangian multipliers or by Newton's method [9]. We have preferred the rst one on the basis of our experiences and because of its simplicity and speed [9]. These models are referred to as link decomposition methods. In the case of networks where the correlation between link occupancy states is high due to a large proportion of common trac a more re ned model is needed. There are re ned models that are able to take link trac correlation into account for circuit switched networks employing xed routing ([6]) but are correct for typical topologies only ([9]). When a route is longer (i.e. it consists of a large number of links) the link decomposition methods can be inaccurate and the re ned models become too complex. In adittion, in case of overload analytical models can fail to give accurate results. In these cases we revert to simulation. It can be time consuming but its accuracy depends on the con dence level, not on the trac and topological conditions. To estimate the topological dependence of the link blocking correlation we can follow the this idea: If the network has a fully connected topology the links block independently (single link paths). The worst case is when all nodes wish to communicate along a single path. In this case link blockings would be independent only if all nodes communicated with their neighbours only, which is not a typical situation.

2.2 For a Set of Alternative Routes In general, modern telecommunications networks do not employ xed routing, but oer a set of alternative paths for every node pair. Which path to use is determined by the routing strategy. Loss networks can be modelled as a multidimensional Markov process, where the dimension is determined by the number of routes. For the case of alternative routing the number of available routes can be quite large, and a product form solution does not exist. The equilibrium state probabilities can be obtained by solving a system of linear equalities. Since the number of states grows exponencially with the number of routes, this approach can not be used for networks of practical interest. When more alternative routes are available for one O-D pair the blocking probability can be obtained by decomposing the trac into single-route tracs and calculating their route-blocking probabilities. For load sharing this decomposition is quite easy: the load oered to a single route is the product of the load sharing parameters and the load oered to the chosen destination node. There appeared numerous analytical results to estimate end-to-end blocking probabilities for some special routing algorithms. For example [11] gives a polinomial approximation for end-to-end blocking probabilities for some special cases; [13] gives analitical model for the over ow trac, useful for dimensioning. Analytical models for some routing strategies are available: [10] or [3] where approximation for state dependent routing (least loaded routing (LLR)) is given. These can be dicult to implement or are not suitable for bigger networks. For many routing techniques no practical analytical model is available. 3

In these cases simulation is suggested for the evaluation of the end-to-end blocking probabilities.

3 Our Models Our models can deal with a number of dierent resource allocation techniques (sharing and/or partitioning techniques: S/P)

 CS (Complete Sharing): The calls of all bandwidth classes have access to a common media (link capacity) all together.

 CP (Complete Partitioning): The media is logically separated (partitioned) between classes of calls. Every partition can be modelled as a CS case [7].

 POL (Partial OverLap, Partial Sharing): It is combination of CS and CP techniques. Every class of calls has its own partition, and an additional common part is available for over ow [12].

 TR (a special case of Trunk Reservation): A treshold dM is set to be equal to the largest

call-bandwidth-demand. If the amount of free capacity on the link is less than dM all calls are rejected, if it is grater all calls are accepted. This technique is preferable from the aspect of fairness between calls of dierent bandwidth demands [5], [16], [2].

With respect to routing techniques our models support:

 NAR (Non-Alternative - Fixed Routing): One route per Originator-Destination node (O-D) pair, no alternatives.

 LS (Load Sharing): Statistical assignment of calls to routes from a prede ned set according to xed LS parameters.

 SPR (Shortest Path Routing): Finds the shortest path in the network where the links are weighted e.g. by the reciprocal of their free capacities. It is a type of adaptiv least loaded routing.

 AR (Alternative Routing): An ordered set of alternative routes is given. If the attempt on the rst route fails, the second one is tried, end so on.

3.1 Short Description of the Analitical Model For the purpose of optimization, comparison and performance evaluation of dierent con gurations and techniques an objective function is needed. As a natural choice the total network revenue is used. According to this approach connections accepted on route r generate revenue at a rate wr and the total expected network revenue is then

W=

X wrr = X wrr(1 ? Lr) r

r

(1)

where r is the carried trac on route r, r is the trac oered to route r and Lr is the end-to-end blocking probability for route r. 4

This formula is appropriate for both evaluation of the network performance and optimization. If wr is proportional to the bandwidth requirement of each call than W is the total carried trac. When the link capacities are partitioned in a network the carried trac can be increased optimizing these partitions. The partitioning can be seen as forming virtual private networks, separation of dierent groups of trac to ensure fairness/priorities or can be seen as the system of virtual paths (V P s) in an ATM network. For the purpose of optimization a constraint has to be added: The sum of partition capacities (CV P ) in any link may not excede the capacity (C ) of that link. The mathematical notation of the optimization problem is:

max W (C~V P ) subject to

X

for all

V Ps

CV P  C for all links

(2)

We can also extend the optimization problem to increase the revenue optimizing networks where alternative routing is applied. In the case of load sharing (the simplest quasy-alternative routing technique) the optimization problem can be formulated as:

max W (~a) subject to

X

for all routes r of a single source

ar = 1 for all sources

(3)

where ~a is the vector of all load sharing parameters. The oered trac to route r is r = ar
OD where OD is the total oered trac for the O-D pair r belongs to. Of course we can optimize W (C~V P ;~a) subject to both constraints 2 and 3 obtaining better results but dealing with higher computational eort and more local optima. For some other alternative routing techniques analitical results exist. For DAR it is quite simple [13], for LLR more dicult [3] to implement. For the optimization we have used the program CFSQP [20] because of its speed and ability to deal with linear and nonlinear equalities and inequalities as constraints. Furthermore it can optimize by numerical approximation of derivatives. Therefore it can be used to optimize partitions of networks that use arbitrary routing technique. In that case the revenue function can be evaluated by simulation. The mathematical formulation is the same as in 2. In order to guarantee fairness and certain GOS values we may have additional (nonlinear) constraints in the model. In the analytical model Whitt-like and reduced load approximations are used on route level. The Knapsack and Gaussian models are chosen from a set of available approximations to be used on link level. For more details see [1] and [7].

3.2 Short Description of the FSP Simulator The FSP has originally been developed for the simulation of ATM networks [8], but is equally well suited for multirate circuit switched networks, since it models the ATM network on the call level assuming constant bandwidth demands during the call. It has been developed in order to simulate routing and link allocation in networks with arbitrary topology, number of nodes, link sizes, and permanent V P /V C arrangements. It employs the conventional discrete event simulation (DES) paradigm to simulate virtual call requests and releases which manipulate the link capacities as state variables in the 5

program according to the actual routing and link allocation algorithm under study. The information model of FSP was proposed by [14], which de nes a nite set of object instances, where each object is associated with a well-de ned portion of the ATM network, such as a V P /V C switch, a transmission link or a V P link. Current version of the FSP does not take into account the statistical behaviour of the trac sources (generation of ATM cells) in detail, the eect of statistical multiplexing in the cell level resource allocation [15] is not considered. Thus the basic assumption regarding resource allocation at the virtual call level is that Bi basic bandwidth units (BBUs), representing the equivalent capacity of the bandwidth are enough to ensure an acceptable cell loss rate for trac type i. It allows performance analysis of dierent routing and link allocation algorithms. Virtual call requests are served by non alternate routing (NAR), shortest path routing (SPR), sequential alternate routing (AR) or alternate routing with load sharing (LSH). The FSP supports gathering statistical information on demand while the simulation is running. Statistics may include route and link blocking, route usage (i.e. how many times a given route has been used by a speci ed trac source) and V P link usage (i.e. on the average how big portion of the V P link capacity has been utilized). Some of these statistics will be used in Section 4.

4 Performance Analysis - some examples The performance analysis has been carried out on three test networks: 4F four node fully connected, 5R ve node ring and 5N ve node network - 4F extended with one node and two links (looks like an open envelope). In cases of 4F and 5R there have been two bandwidth classes, a narrowband class with bandwidth demand 1 BBU, and a wideband class with bandwidth demand 10 BBUs. In case of complete partitioning O-D pairs of dierent classes are separated into logical networks. Our trac streams are symmetric in the sense that we have used all possible O-D pairs. For each O-D pair the two possible routes are utilized in the ring network and the (one-link) direct and the two possible two-link alternatives in the four-node fully connected network. Between these O-D pairs we have both a narrow and a wideband trac demand. In the case of network 5N three bandwidth classes have been used: 1, 10 and 120, the link capacities were 1500 BBUs. For every O-D pair 3 routes have been de ned. However, beyond the topological symmetry neither the physical resources, nor the oered trac values are completely symmetric in order to ensure nontrivial and reasonable solutions. By this we mean we have load/resource sharing at the (locally) optimal points without excessive blocking on any of the routes.

4.1 On Resource Sharing/Partitioning Techniques For speed reasons Whitt-like approximation was used with normal approximation on the link level (see [7]). The results were also compared with a more re ned evaluation of the revenue function at the end of the optimization phases using the reduced load approximation with the Kaufman-Roberts formula on the link level. The error was less than 1% even in the worst case, therefore it was justi ed to use 6

the faster approximations in the optimization process. In case of trunk reservation the approximation given by Tran-Gia and Hubner (see [16]) was utilized with the reduced load assumption. Since in revenue optimization problems dierent local optima can be present (see [19] and [1]), the best results were chosen given from several random starting points, unless noted otherwise. The results are summarized in Table 1 and Figures 1-16. The total oered bandwidth was 1390 in the four-node fully connected case and 890 working with the ve-node ring. After the revenue (or carried bandwidth due to the special revenue factors) optimization some quantities of our interest were evaluated. The revenue is the highest in the case of complete sharing and the lowest when CP is applied. The TR performs quite well, in our example very close to CS case, but it should be noted that its performance depends on the proportion of the threshold dM to the link capacity, and on the proportion of call arrival rates for trac classes with dierent bandwidth demands. If the link capacity is not much larger than dM but the calls of smaller bandwidth demand dominate the total carried trac can be very poor. Fig's 9-12 validate the well-known characteristics of the CS and CP allocation schemes [21]. The lack of link allocation policy, i.e. CS is unfair in the sense that it leads to heavy blocking of the broadband service class already at low trac load conditions while allowing practically zero blocking to narrow band services. The need for trac segregation has already been recognised e.g. in [7], [12], [5] and algorithms to nd the optimal partitioning in terms of VP link capacities with respect to virtual call blocking probability and carried trac are presented in these reports. These algorithms consist of de ning an objective function, such as the carried trac or route blocking and applying an optimisation method with respect to this objective function assuming the CP policy. Under this scheme (CP) the wide band service class' performance is improved at the expense of introducing blocking to narrow band service classes. Note that this observation is valid only for medium and high trac loads; for low load the CP results in even higher blocking probability than in the CS case. This can be explained by the low link utilisation of the available capacity at low loads: the CP rule restricts all service classes to their "own" VP links and leaves other VP links under-utilised when capacity there -possibly- is available. At higher loads, however, the CP successfully protects the broadband classes from the attacks of the intensive narrow band classes. The important conclusion here is that CP cannot be merely based on the oered load: service classes with identical oered load but diering intensity and bandwidth demand suer signi cantly dierent blocking probabilities in case of equal partitioning. To conclude we nd that the CS policy is adequate when trac is tolerable by the network, while trac segregation is bene cial at medium and especially at high loads. To combine the advantages of the CP and CS policies and to achieve fair distribution of blocking independent of the oered load we apply the partial overlap (POL) and trunk reservation (TR) allocation rules (Fig's 7-8, 13-16). We note that the POL policy provides for a more ecient allocation of available bandwidth than the CS/CP rules. (Fig's 1-6, 9-12) This improvement is most impressive under low load values, and practically vanishes as the network becomes congested. The interesting phenomenon here is that the narrow band service class is not seriously eected by the fact the broadband service class is multiplexed 7

into its links. The medium service class, in contrast, suers a notably higher blocking compared with the CP case.

4.2 Fairness of S/P Techniques In this section results of our fairness study are presented for dierent P/S techniques. The fairness between trac classes is studied comparing the carried trac for calls of two dierent bandwidth demands. 4F 5R (oered bw = 1390) (oered bw = 890) CS CP TR CS CP TR carried bw 1371.45 1349.95 1368.94 872.75 854.77 870.411 bw loss [%] 1.33 2.88 1.51 1.94 3.96 2.20 max class loss [%] 2.45 4.30 1.56 3.56 5.91 2.21 min class loss [%] 0.221 1.46 1.46 0.312 2.01 2.18 max/min 11.1 2.94 1.07 11.4 2.94 1.02 Table 1: Comparison of CP, CS & TR The experiences can be summarized as it follows (Table 1.):

 Tolerating the larger blocking probabilities of classes with larger bandwidth requirements (i.e. unfairness) complete sharing (CS) gives the best performance (largest carried bandwidth).

 Applying trunk reservation (TR) to equalize blocking probabilities of all trac classes on a link can give very good results. We obtained an almost ideal fairness factor because we have had the same dM for all links, and we had the same load from both trac classes for all route lenghts (number of hops). In a general case this policy can be less fair.

 At last, separation (CP) caused notable performance penalty in both cases, but it is the most

exible of the three investigated methods.2 Considering fairness, in our examples performance of the partition is between the two others, but much closer to TR. With additional constraints it could be noticeably improved.

4.3 Evaluation of Routing Techniques In this section we try to evaluate the impact of routing algorithms on call blocking and network revenue when combined with dierent link allocation strategies. We consider four dierent routing techniques: non alternate routing (NAR), shortest path routing (SPR) alternate routing (AR) and alternate routing with load sharing (LSH). As for link allocation, we consider CS, CP, POL and TR. In Figures 1, 5, 6 and 9-12 we observe that both narrow and wide band services perform worst under the NAR algorithm irrespective of the link allocation scheme applied when trac load is tolerable by Since applying CP it is possible to realize almost arbitrary fairness ratios or to ensure priorities with a simple separation of e.g. trac classes. 2

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the network. This can be explained by the fact that NAR does not make use of possibly free VP links on alternate routes, but blocks the call if the prede ned route is congested. However, if trac load becomes high, NAR blockings are smaller than that of AR, and especially than SPR. This is because NAR always uses only one route to build up a call and thus behaves "more greedy" with bandwidth than the two other algorithms, i.e. AR and SPR use more resources (links) in average to establish connections than NAR which clearly results in blockings of forthcoming calls, which have to be routed along one or several links of the already congested route. We also note that SPR and AR are often the best for all trac classes under each link allocation scheme when the trac load is tolerable by the network. Because of the weight function (reciprocal of free bandwidth), SPR spreads the incoming calls in the network, i.e. it eagerly seeks new routes instead of utilizing the already used but still not congested routes. It explains why this algorithm performs best under low load conditions. SPR obviously wastes more rapidly link capacity as trac load becomes higher then the AR, which chooses a new route only when it has to, i.e. when the route of higher priority becomes congested. That is why we experience that as soon as the SPR starts blocking, it indicates that available resources have been consumed up and it rapidly goes up to the "almost sure" blocking probability after a small further increase of the load.

5 Conclusion If we want to qualify the network con guartion and routing techniques or to optimize the partitioning we shall determine the end-to-end blocking probabilities. For this purpose we choose one of the available methods: either one of the analytical models (if it exists and is accurate and fast enough for our purposes) or simulation. Generally, analytical models will be preferred because of their speed. Simulation will be preferred when routes consist of a big number of links and link blocking probabilities are high, when link dependence is strong, or when complicated routing techniques are present.

6 Acknowledgements We are grateful to Sren Blaabjerg for fruitful consultations, for his ideas, help and support. We would like to thank Andre L. Tits for allowing us to use the CFSQP optimization routine.

References

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