Network Dimensioning and Performance of Multi-service ... - CiteSeerX

Network Dimensioning and Performance of Multi-service, Multi-Rate Loss Networks with Dynamic Routing* D. Medhi and S. Guptany

Department of Computer Networking Computer Science Telecommunications University of Missouri{Kansas City 5100 Rockhill Road Kansas City, MO 64110 USA Tel: +1 816 235-2006 E-mail: [email protected] Fax: +1 816 235-5159 This revision: March 1997 Second revision: October 1996 First revision: December 1995 Original version: December 1994

Abstract In this paper, we address a multi-service, multi-rate loss network environment with dynamic routing. In this setting, we consider multiple trac load periods (multi-hour) during the day, and by observing network dynamics, we present a network dimensioning model that consists of two-steps: a bandwidth estimation estimation step followed by a multi-commodity ow model for multiple services and trac loads. For network operations, we discuss a probabilistic admission control policy and three multi-service routing schemes. We have used a 10-node network with multiple asymmetric trac data sets (partially extracted from an actual network) for our study. It was found that the capacity obtained using the analytic network dimensioning model provides a good estimate of network capacity required for meeting the grade-of-service goal for each service type in each trac load period; this observation is based on a simulated network environment that uses the proposed admission control and the dynamic routing schemes. Our observation suggests that it may not be not necessary for the dimensioning model to explicitly incorporate admission control policy, but admission control is needed for network operation to provide desirable gradeof-service.

Keywords: multi-rate loss networks; dynamic routing; network dimensioning; multi-commodity network ow optimization model; admission control; network performance.

31 pages (including tables and gures) * Research supported in part by the University of Missouri Research Board under grant K-340605 and by National Science Foundation under grants NCR-9506652 and CDA-9422092. y Now with Retix, Santa Monica, California.

1. Introduction

In this paper, we consider network dimensioning and performance of multi-service, multi-rate loss networks with dynamic routing. We assume that each service type requires a xed peak rate allocation of bandwidth for the duration of a call while dierent services may have dierent peak rates. This is, for example, the case for class-A (constant bit rate, CBR) service in B-ISDN [10, p. 135]. Thus, this scenario is similar to multi-rate circuit switching but with exible slot assignment, and the network operates as a loss network, i.e, a blocked call is cleared. Here, the grade-of-service (GoS) [61, p. 149] refers to call blocking for each trac (service) type. For example, 1% blocking GoS is typical for public voice networks. We will use the term basic bandwidth unit (BBU) used by [42] to refer to a base rate, and assume that all services require some multiple of this base rate. For example, if we consider 64Kbps to be the basic bandwidth unit, then a 64Kbps peak rate voice call requires one BBU, while a 384Kbps peak rate video call requires six BBUs. Also, we will use the term, service unit (SU), to refer to the number of BBUs required for a connection of a given service type. We assume that the call arrival of a call of each service type follows Poisson arrival, and the call holding time is exponentially distributed for which an SU is allocated for the duration of the call. We start with a discussion for a single-link system (no routing). In this case for a single service type, 10 erlangs of oered load at 1% blocking GoS requires 17.44 SUs (by Inverse ErlangB formula [34]); if this service happens to be, say, a 384 Kbps video service, then it requires 17:44  6 = 104:64 BBUs of capacity if a BBU assumed to be is 64 Kbps. Thus, the product of SU and BBU requirement per connection translates to capacity requirements. It has already been noted [41] for mixed bandwidth trac in a single-link system that if there is no control, then the trac type with lower bandwidth requirement per call has less blocking than the trac type with higher bandwidth requirement per call. For example, consider two trac streams | one requiring one BBU per connection (type one) and the other requiring six BBUs per connection (type two). When 18 erlangs of trac type one and 3 erlangs of trac type two are oered together to a link with a capacity of 54 BBUs, the blocking for type one is 1.0% and for type two is 7.9% under no control (obtained using a two-dimensional birth-and-death model); this shows the disparity in service grades. Thus, proper admission control schemes are required so that equitable grade-ofservice can be provided to all service types if that is the goal of network provisioning (obviously, dierent GoSs for dierent services types is also possible). There has been considerable work on admission control and performance of a single-link system where multiple services (multi-rate) are oered [12, 35, 36, 39, 41, 42, 44, 50, 56]. Now, consider a network with dynamic routing. Here we consider dynamic routing with at most two links to connect a call [19], [33], i.e., a call can be routed between the origin and the destination switching nodes either on the direct link (if available), or can use at most one via 2

node for alternate routing. Some examples of such dynamic routing schemes are dynamic nonhierarchical routing (DNHR) [5], dynamically controlled routing (DCR) [11], dynamic alternate routing (DAR) [18], real-time network routing (RTNR) [8]. In general, the dierences between these various routing schemes are when (how often) and how (what information about the network is taken into account, whether centralized or distributed etc.) the routing is computed and done, and whether crankback [5] is allowed. There have been extensive studies and analyses of various forms of dynamic routing for the single service case such as voice over the last couple of decades (e.g. [1, 13, 20, 37, 43, 51, 54, 58, 60, 69]); in recent years, there have been increasing number of works for multiple services loss networks also [7, 13, 15, 16, 17, 21, 23, 29, 32, 40, 45, 63, 67]. For example, in [67], Wang and Saadawi present performance models for symmetric heterogeneous networks where services have dierent holding times but for the single rate scenario. In [7], Ash et al. present RTNR in the context of multiple services where a minimum guaranteed bandwidth allocation is done for each service type, and some results on performance are presented. Girard and Bouley [21] discuss a two-moment model for performance in multi-service networks. In [13], Chung et al. present an analytical performance model for state-dependent routing with single-rate and extends to multi-rate trac; however, no admission control schemes are addressed in this work. In [45], a performance model is also provided for multi-rate case; however, admission control is not discussed. Dziong and Mason [15] present a performance model for admission control and routing. In [17], Gersht et al. have presented a detailed description on allocation, control and routing in Asynchronous Transfer Mode-based (ATM) networks, although no numerical results have been provided. As discussed in some of the work mentioned here, and also from the numerical example discussed in the previous paragraph in the case of a single-link system, it appears that some form of admission control may be appropriate in a network environment as well to provide desirable GoS for each service class. There have been several works on network dimensioning for dynamic routing networks for the single-service case [5, 19, 24, 25, 38, 62]; recently, multi-service networks have been addressed by several researchers [3, 4, 6, 23, 46, 53, 59]. Most researchers have approached dimensioning for single-service networks by iterating between solving a set of non-linear equations and a solution of linear programs (e.g. see [5, 19, 22]). For multi-service networks, Ash et al. [3, 4] discuss network sizing for multi-service networks by putting a limit on the number of calls of each service type and state that an iterative scheme is used between solving a set of non-linear equations and a solution of linear programs to arrive at network dimension without providing detailed mathematical formulation and description; a dimensioning model is presented for multi-service networks by Ash and Chang [6] where the multi-rate issue is not explicitly addressed; while in [46], we have considered multi-hour dimensioning with explicit incorporation for multi-rate but grade-of-service is not considered in the model. Pioro et al. [59] propose that for the multi-service case a network 3

for each service is optimized separately and then is superimposed. Girard and Gardouh [22] have recently addressed integrated network dimensioning for probabilistic choice of alternate paths for dynamic routing. The work presented here is quite dierent from the work that has been done so far. Speci cally, we present a network dimensioning model and then, using dierent dynamic routing schemes and an admission control policy, show how the model provides a good estimate on network capacity to meet desired GoS goal as observed through network simulation. While various works discussed above have been often concerned with just one issue, or sometimes two (among dimensioning, routing performance, and admission control), none have shown results on their interrelation together, especially in the context of multi-rate loss networks with multi-hour trac. Speci cally, our work here on network dimensioning is dierent from other works in that a mathematical framework is presented where we rst consider a virtual bandwidth estimation problem and second consider an integrated multi-hour, multi-service, multi-commodity network ow (MCNF) model | this is done with multi-rate in mind. For the second part (i.e., MCNF), we have been greatly bene ted by pioneering work on modeling for synthesis of non-simultaneous as well as simultaneous multicommodity network ow problems by Gomory, Hu, Minoux and others (the reader is directed to [5, 26, 27, 31, 57] for more information). We have extended their ideas to address the problem in hand. For dynamic routing schemes presented here, we have extended the concepts from the existing routing schemes for single-service networks, and for admission control we have extended a probabilistic admission policy for use in the network context that we have discussed previously for a single-link system [50]. For our study, we have considered a ten-node network based on realistic data. For this network example, we have estimated network capacity using our analytic multiservice network dimensioning model for a given GoS (which considers dynamic ow, but no call admission control policy); this capacity was then used in a call-by-call network simulation where admission control and dynamic routing discussed here were implemented. Through this simulation, we observe that GoSs for services are met quite well, thus indicating that the network dimensioning model generates a good capacity estimate, although call admission control is not considered in the dimensioning model. Furthermore, we observe that the notion of service reservation is important in multi-rate, multi-service dynamic routing environment beyond the notion of trunk reservation already known for (single-rate, single-service) dynamic routing networks. The rest of the paper is organized as follows. In the next section, we present the network dimensioning model. In section 3, a probabilistic admission control policy and various routing schemes are discussed. In section 4, we rst present network dimensioning results for a test network data extracted from an actual network, and then present network simulation results to discuss the eectiveness of the network design when various routing and admission rules (based on an admission policy) are used. In section 5, we provide a summary with a discussion on future work. 4

2. Network Dimensioning Model

The aim of network dimensioning here is to minimize total network capacity cost to carry oered trac at a speci ed GoS goal. In our case, we consider this scenario given that we have multiple trac types each of which have dierent peak rate bandwidth requirements per connection, and that oered trac load matrices are given for dierent times during the day for each service, and the call routing is dynamic and uses at most two links to connect a call. This problem is inherently non-linear between routing, blocking, and capacity as noted, for example, in the case of single-service voice networks [5]. Non-linearity can be further complicated by the fact that the over ow trac has peakedness. Our aim here is to develop a simple, yet reasonable approximation that can capture the essence of network behavior, yet that can provide a good capacity estimate. Thus, our approach draws mainly from the following observations to see if a reasonable approximation can be obtained: (a) rst, in a dynamic routing network, the direct and alternate paths need to route pair-wise oered trac that is not blocked, thus, an estimate for service unit requirements on a pair-wise basis (without referring to which paths trac is actually routed) is desirable; (b) secondly, in a multi-service scenario, the requirement on the bandwidth coupled with oered load needs to be re ected in a proper way by taking into account the bandwidth requirements per call for each service; (c) thirdly, most of the carried trac in such network is routed on the direct link in case of low blocking GoS requirement (such as 1% blocking), and the rest is carried on two-link paths; (d), overall, we need to take into account multiple trac hours to address non-coincidence of busy hour trac. These observations lead us to consider the network dimensioning through two stages: 1) pair-wise demand estimation, and 2) routing/sizing. The rst three observations (a), (b) and (c) together lead us to the development of a simple estimate for service unit required for each trac pair and service type for given oered trac and GoS requirements. First, by oered load (trac) we refer to the load in terms of trac type without considering the bandwidth requirement per call. That is, oered load of a erlangs is given by a = = where  is the average Poisson arrival rate and 1= is the mean call holding time. The bandwidth requirement of a call w, instead of being considered as a part of oered load, is considered outside the estimation procedure in the dimensioning method explicitly [see model-(3) and Remark-A below for further explanation and illustration]. Thus, in the estimation phase we are interested in estimating the service units (not estimated bandwidth) required for a trac pair for a particular service type; we speci cally refer to it as virtual service units (VSU) on a pair-wise basis since this service units can be carried over multiple paths due to routing. We thus address the dimensioning problem in two decoupled stages: in the rst stage, we estimate a virtual service unit requirement for each service based on given oered load and the 5

GoS goal to account for non-linear behavior as well as the observations (a), (b) and (c); in the second stage, this VSU demand is used to determine link sizing through a linear multi-commodity

ow problem which tries to capture observations (b), (c) and (d) since the estimated demand is required to be routed in the dynamic routing network using the direct link and two-link alternate paths. (This approach bene ts from our work on the trac network design part for a uni ed survivable network design model for single-service networks [7, 47].) Thus, the non-linearity is addressed through the approximation of the estimated virtual bandwidth rst and then through the solution of linear multi-service, multi-hour, multi-commodity network ow problem. We rst start with the virtual bandwidth estimation step. For a given oered load and the GoS goal, the virtual service units requirement is approximated based on an approximation discussed for single-service networks in [7, 47, 62] which have been found to work well in practice. The load to be carried in a dynamic call routing environment can be carried in two ways: on a direct link or on two links in case of alternate routing. If we estimate a blocking bd of the load to be carried on the direct path, then the over ow trac can use two links to complete the requirements. This over ow trac shares capacity with other trac pairs. Conceptually, although it uses two links, the over ow can be visualized as being carried on a shared virtual link (as an incremental amount on a large group). Thus the total requirement to be carried over the direct link and alternate two link paths is the total virtual service units required for the oered load for a particular service type to be carried. Let B be the acceptable goal of blocking level. Let E (c; a) be the well known Erlang-B blocking formula (e.g., see [19]) for service units c and oered load a de ned by: c E (c; a) = Pc a (=cak!=k!) : (1) k=0 We denote the inverse Erlang-B loss formula to compute the number of service units necessary to carry the oered load a at a particular blocking level, say, b by E ?1 (a; b). Let the average desirable occupancy (utilization) of a network link be . Let the carried load be denoted by a0 = a(1 ? B ). Then the virtual service unit demand, v , for load to be carried is approximated by the following formula: v(a; B) = E ?1 (a0 ; bd) + a0 bd =: (2) The rst part is for the trac estimated to be carried on the direct link and the second part is for any over ow that is estimated to be carried on the shared virtual link which is assumed to be a large group. It may be noted that we use a simple expression above for the over ow term. This is due to our observation [item (c)] that in a dynamic routing network engineered for low blocking (such as 1% blocking), most of the trac, but not all, are direct routed (this observation is made from network simulation; see, for example, [47, 51]). This idea of simple splitting is used before [62] for single-service networks. Further, any alternate routed (over ow) amount may 6

be split over multiple paths and carried on network links that have direct trac on their own. Thus, this incremental amount can be visualized as carried on a large group and is estimated by considering simply additional service unit estimate if network links were to maintain a certain occupancy/utilization . See Remark-B for additional explanation including the omission of the peakedness factor. Once VSU estimation is computed, we need to consider the actual dimensioning of the links in the network. This is where observations (b) and (c) together with (d) are considered through a multi-hour, multi-commodity ow model by explicitly addressing bandwidth requirement per call for each service type. Such optimization model helps us address the following issues: noncoincidence of busy hours of trac for dierent trac pairs and possibly for dierent service classes towards minimizing overall network bandwidth required through cost minimization. To take into consideration observation (c) again, an upper bound on each trac path is desirable since all trac is not typically routed on just one path. Also, considering all trac types in an integrated model has the bene t that the capacity can be minimized compared to considering each trac type separately and then summing the capacity at the end (see section 3.1). Based on the above discussion, the second step is then to route this total estimated quantity given that a call can be connected using at most two-links. Furthermore, we need to take into consideration dierent BBU requirements for each SU for dierent services for dierent trac node pairs in the network and at dierent time during the day. We present this in a multi-hour, multiservice, multi-commodity linear network ow model. We rst present the notation to be used for this purpose.

K L H S ws

Set of trac switching node-pairs Set of links Set of dierent load periods during the day Set of dierent types of services Peak rate bandwidth required for a single connection of service type s (in multiples of BBUs)

Pksh Set of candidate (at most two-link) paths for service s for node pair k 2 K in load period h2H y` Capacity required (in BBUs) on trac link (group) ` 2 L (variable) xsh km Amount of ow (in virtual service units) on path m for node pair k for service s in load period h (variable) 7

sh` Entries for arc-path incidence matrix for the network ; 1 if path m for node pair k, km service s uses link ` in load period h, 0 otherwise ash k Oered load (trac demand) in erlangs for node pair k, service type s in load period h; sh sh here, ash k = k =s , where 1=s is the mean call holding time for service type s and k is

the arrival rate { this is the average number of connection requests per unit service time for each class

Bksh Acceptable blocking goal for node pair k, service type s in load period h sh v(ash k ; Bk ) Virtual service units estimate for node pair k, service type s in load period h for a blocking goal Bksh (obtained from the rst stage) sh ush km Upper bound on ow, corresponding to path variable xkm

c` Unit BBU cost on trac link ` We do routing of estimated virtual service bandwidth units demand depending on the time of the day so that the total network capacity cost is minimized in the following model: X

minsh

subject to

fy`;xkm g `2L X

m2Pksh X

sh sh xsh km  v (ak ; Bk );

ws

X

X

k2K m2Pksh sh 0  xsh km  ukm ; s2S

c` y `

(3a)

s 2 S ; k 2 K; h 2 H

(3b)

sh` xsh  y` ; km km

h 2 H; ` 2 L

(3c)

m 2 Pksh ; s 2 S ; k 2 K; h 2 H

(3d)

y`  0;

` 2 L:

(3e)

Expression (3a) is to specify the total network capacity cost which is to be minimized. Constraint (3b) says that for each service, trac pair and load period, a given set of candidate (at most two-link) paths together is required to carry at least the virtual service bandwidth unit demand. Given ow on paths, the service units required for each service on each link and load period is shown to be multiplied by the BBUs needed per connection for each service, ws , and summed up for all services so as to determine the total number of BBUs needed on each link and in each load period; this amount is to be less than the actual BBUs needed on each link so that the request can be satis ed at any load periods during the day | this is done through constraints (3c). The constraint set (3d) is to specify the upper bound on service unit ow on each path variables and 8

to specify that path variables are non-negative, and (3e) is to specify that the link capacity variables are non-negative. We have used here the arc-path formulation [27] due to the imposition of maximum two links for connection of a call. Note that if #(L) = #(K), then the network is fullyinterconnected (here, #(
) denotes the cardinality of a set). Further note that there is no explicit routing cost in this network dimensioning model { this is in line with the discussion presented for single-service network; see, for example, [5, 19]. To recap, the dimensioning procedure, thus, involves a VSU demand estimation given by (2) which is then followed by solving the ow model (3). For broadband network services requiring other than peak rate allocation which may be characterized using three parameter trac descriptors (such as peak rate, mean rate and mean burst period), some form of equivalent bandwidth concept ([28]) may be used to determine the bandwidth weight ws towards sizing multi-service networks with dynamic call routing. This would then also require addressing cell loss probability as one of the quality-of-service parameters; this issue is considered elsewhere (see, for example, [48, 49]).

Remark-A: It is well known regarding inverse Erlang-B formula that E ?1 (wa; b)  wE ?1 (a; b) for w > 1:

(4)

Thus, it is easy to see that the following holds:

v(wa; B)  w v(a; B) for w > 1:

(5)

This implies that if bandwidth requirement per call is considered to be a part of oered load (i.e., `wa' on the left hand side of (5)), instead of considering explicitly as in model (3), then the total bandwidth requirements can be underestimated. As for illustration, consider the simple case of inverse Erlang-B formula (without routing) as in (4). If 15 erl of oered load for a service type that requires 6 BBU per call is to be engineered for 1% blocking, then SU requirement (without rounding) is 23.64 thus requiring a total 23:64  6 = 141:84 BBUs { this corresponds to the right hand side of (4); however, if bandwidth requirement per call is considered as a part of the load for the service type, i.e., if 15  6 = 90 erl is considered as oered load, then for 1% blocking, BBU requirement is 106.30 for the left hand side of (4), thus underestimating the actual BBU requirement of 141.84 by a signi cant amount (about 33%) for this service type. This also shows that a call of w BBUs/call is not equivalent to w calls of 1 BBU/call from call blocking perspective. Thus, the explicit introduction of bandwidth requirement per call for each service type is required in the dimensioning model. In model (3), v (a; B ) is used with demand in (3b) while ws is used with link bandwidth; the equivalency with using w v (a; B ) as demand is shown in appendix. 9

Remark-B: The expression (2) deserves further explanation in light of model (3). It may seem

that a multiplier two would be appropriate in the second term in (2) since the over ow estimate is to be carried on two-link alternate routes. This is not needed to avoid double counting in model (3) through additional VSU demand in input and through the link summation done in (3c) for any two-link over ow. Also, note that we use a instead of a0 in (2); this is since multi-commodity

ow models as in (3) is for carried trac and to satisfy ow conservation. Hence, the blocking GoS factor needs to be accounted up front in (2) so that estimate of demand to be carried is considered in model (3). Also, note that expression (2) does not directly address the multi-rate aspect since (2) is used for each service type separately; the multi-rate aspect is captured through the bandwidth parameter ws used in model (3). This can be considered as a limitation of our approach; on the other hand, this simpli cation also allows us a simple way to incorporate service dependent candidate routes (see Remark-C also). Finally, it is well known that over ow trac has peakedness [19]. This is ignored here due to the over ow being typically a small percentage which is envisioned to be carried on an incremental basis on a large group (due to alternate routing). Note that Wilkinson's equivalent random method (ERM) [14] is not applicable here since ERM is for given over ow and peakedness to determine load and the size of the group the over ow occurred from, whereas here we are given oered load and GoS, to determine VSU demand.

Remark-C: Non-coincidence of busy hour trac due to trac pairs as well as service type is

captured by model (3). Further, service dependent path variables are explicitly de ned to provide any restriction on a via path for a particular service type. This is possible, for example, if say a certain node in the network has not been upgraded to provide a new emerging service yet (which is deployed in the rest of the network) and thus this node may not be able to do alternate routing for other trac pairs; another situation where a node may not be allowed for alternate routed trac is if the node call processing capacity is reaching a certain threshold and thereby possibly making a high bandwidth call not desirable for alternate routing through this node; these types of feedback from network elements are desirable for proper network dimensioning. Finally, the bound on paths is dictated by the observed behavior from simulation of dynamic routing networks that under normal network operating conditions, a small percentage of trac is always alternate routed for low blocking GoS. Numerical results to re ect these issues will be discussed in the section on results.

Remark-D: As it can be seen from our discussion in this section so far, we have made several sim-

pli cations to arrive at the dimensioning model. Due to these simpli cations, a notable drawback of our model is that the model is not appropriate for computing bandwidth estimate just for a single link or when the multi-hour trac is not present. Furthermore, this model neither consider a small routing cost in the objective functions (sometimes desirable to avoid highly degenerate solutions) nor incorporate any speci c call admission policies that may be active during network operation. Our simulation results discussed later address the issue of an admission control policy which is presented in the next section. 10

3. Routing and Control

From the analytic network dimensioning model presented in the previous section, we can obtain the network capacity requirement; our approach presented is an approximation model as already discussed. To see whether the network capacity designed through such an approach meets the performance requirement in a multi-service dynamic routing network, we have computed the network capacity for asymmetric trac (extracted partly from an actual network which is discussed later) and then simulation of network using dynamic routing schemes and admission control. In this section, we provide the description of dynamic routing schemes as well as a probabilistic admission control policy used for a connection request considered in network simulation. The role of admission control here is to check whether a newly arrived call should be admitted to the network; if it passes this level, then for this call, a dynamic routing scheme is employed to check if there is capacity in the network to connect the call; if there is not enough capacity, then the call is blocked and rejected from the network. The implication of the admission control is discussed further in the section on results.

3.1 Routing Methods We present below three dynamic routing schemes which have their roots in existing dynamic routing schemes for single-service networks already discussed in introduction. 3.1.1 Dynamic Routing { 1 (DR1)

For each switching pair, an arriving call for dynamic routing scheme, DR1, for each service type between two switching nodes that passes the admission control rst tries the direct trac link. If there is capacity on the direct link to serve the bandwidth requirement for this call, the call is connected. If there is no free capacity on the direct link or the direct link does not exist, then the call rst tries through the rst alternate via node given in the routing table computed periodically (see below); if it cannot nd any available trunks (subject to trunk reservation [1, 43, 68], also known as state protection [19]) on this alternate route then the call is crankbacked ([5]) and tried via the next alternate via node as given in the routing table. If the call cannot nd any available capacity after trying all the alternate routes given in the routing table, then the call is blocked. (See [51] for more discussion on this routing scheme). Like dynamic non-hierarchical routing (DNHR) [5] and trunk status map routing (TSMR) [2] for single-service networks, this routing has the crankback feature; while DNHR used an o-line computed routing (with some real-time network management added routes in case of overload [9]) and TSMR uses DNHR with some added routes computed regularly, the routing used here updates entire routing table at a regular interval, somewhat similar to DCR [11]. Note that the routing we use attempts various alternate routes in the order given in the routing table using crankback, if needed, while DCR 11

uses probabilistic values to pick the alternate route from the routing table; additionally, DCR does not have crankback. Also, note that DR1 is aimed for multiple services, and thus, the routing table computation takes into account the peak rate bandwidth required per connection by calls of a particular service type. This is described next. Consider the trac link (i; j ) for the node pair k := [i; j ] with end nodes i and j . Let

t(i;j) := Total number of BBUs on link (i; j ) o(i;j) := Number of BBUs on link (i; j ) that are presently allocated to active calls of all types r(i;j) := Number of BBUs on link (i; j ) reserved (trunk reservation) for its own direct trac Then, the available capacity for pair [i; j ] via node v (= 6 i; j ) at the instant of computation of

the routing table is given by:

z[vi;j] = min f t(i;v) ? o(i;v) ? r(i;v) ; t(v;j) ? o(v;j) ? r(v;j) g:

(6)

If z[vi;j ] < ws , then via node v may not be able to accommodate a new call on this via trac path for trac type s for pair [i; j ] and as such may not be put in the candidate list. However, it is possible that some amount of bandwidth on such a path may free up since the last routing table update of available capacity due to completion of existing calls, making such a path possibly acceptable for alternate routing. Thus, for each service type s, we consider the candidate list n



o

V i;js; s = v zvi;j  maxf1; swsg ; ( [

)

]

[

]

(7)

where s satis es 0  s  1. If s = 0 for all service types s, then the candidate routing list is the same for each service type (chosen with at least one unit of BBU available). If s = 1 for all s, then the candidate list for each service type contains the alternate paths which have at least ws units of BBU available (at the time of computation) to connect a call for that service type. If s is between 0 and 1, then it means that a path is considered in the candidate list even if it does not have enough bandwidth to connect a call for that service type at the time of routing table update, but it may be close enough such that it merits consideration in case any existing calls are completed. The candidate list is sorted in descending order (of available capacity as given by z[vi;j ] in (6)) to determine the routing table for service type s for trac pair [i; j ] subject to the condition given in (7). Note that although the routing rule is the same for each service type for a speci c trac pair, the routing table can be dierent for dierent service types depending on the value of s. As noted earlier, this updating is done periodically. This computation can be done either in a centralized manner where each switch sends its information regularly to a central processor, or in a distributed manner where the busy capacity information is exchanged between switches using the signalling network. 12

3.1.2 Dynamic Routing { 2 (DR2)

The second routing scheme, DR2, is inspired by RTNR [8]. In this routing, a trac pair has an alternate via node available (\stored" via node) at any time. Once a call passes admission control, it rst tries the direct link. If there is capacity available, the call is connected. If there is no capacity on the direct link or the direct link does not exist, then the stored via node is tried for alternate routing; at the same time a process is spawned to compute a new via node. If the call is completed using the stored via node, then the newly computed via node becomes the stored via node for the next call that arrives for the same trac pair and the service type. If the call cannot be completed via the originally stored via node due to non availability of enough capacity, then the call waits for the completion of the process to compute the newly computed via node and then tries to route it using this via node. The idea behind this concept of \one call old" routing is to minimize the call set up time for most calls [8]. The computation of the new via node has similarities to the computation of the periodic routing table in DR1 (note the dierences with DAR [18]). In this case, it uses the rule given in (6) [along with (7), where s = 1 for all s] in a distributed manner to determine the available via nodes and picks the via node with the maximum available capacity to be the newly computed via node; note that this is done on per call basis. Thus, a switching node pair requests all its possible via nodes to send availability of capacity for computing best via path. Note that DR2 is conceptually similar to RTNR [8]; however, instead of computing sets of via nodes based on network status bit map as in RTNR, DR2 uses the information on free capacity on each link for computation of route. 3.1.3 Dynamic Routing { 3 (DR3)

DR3 is similar to DR2 except that the determination of the via node to attempt at the instant of a call arrival for some services is dierent than DR2. Usually, when a new service is introduced in the network, the amount of trac for such services is signi cantly lower than existing services. As such, the request for connection for such new services can arrive quite infrequently. This leads to the situation that the network state may have changed signi cantly since the via node was computed for the last call for this service type; i.e., the information may be outdated for a newly arrived call for the same service [30]. Thus, in such a situation it may be preferable to force computation of the via node on each call basis at the expense of possible increase in call set up time for new services. The computation of the via node is done as discussed for DR2. Thus, in DR3, some services use \one call old" routing for alternate routing while others use freshly computed via node for alternate routing.

13

3.2 Admission Control Our admission control policy is probabilistic in the sense that depending on the amount of free capacity on the direct link available (within a speci ed range) at the instant a call for a service type arrives, it is accepted (not connected yet) to the network with a certain probability. (We have considered such an acceptance policy in the context of performance analysis of a single-link system for heterogeneous trac [50].) If the call is not accepted, then it is blocked and cleared. If the call is accepted in the admission control phase, then it goes to the routing phase to determine if the call can be actually routed using DR1, DR2 or DR3. This probabilistic acceptance concept is somewhat similar to the code blocking concept used in overload situation for trac network management of telephone networks [66]; here we use it on a trac pair-wise basis and under normal network condition for admission control based on the availability of capacity at the time of arrival of a call. The admission control can be given for service s 2 S for trac pair [i; j ] by the following acceptance function:

s[i;j] =

(

ps[i;j] ; if L(i;j)  t(i;j) ? o(i;j) < U(i;j) 1;

otherwise.

(8)

where L(i;j ) and U(i;j ) are the lower and upper bounds given in BBUs, respectively, on free capacity for probabilistic acceptance, and 0  ps[i;j ]  1. If ps[i;j ] = 1; s 2 S for all trac pairs, then there is no admission control in the network; which is sometimes known as complete sharing [36]. Through our results on network performance we will show the implication of the probabilistic acceptance idea and of no control for normal network operating conditions.

4. Results

In this section, we rst present a numerical illustration through a three-node network example to show eects due to demands for dierent services at dierent load hours on network sizing for several methods [including model (30)]. We then consider a ten-node network data drawn from an actual network for network dimensioning; for this problem, we present network simulation results with admission control and dynamic routing discussed earlier to see how good is the estimation of network capacity due to the network dimensioning method as observed from network simulation with admission control and dynamic routing.

14

4.1 A Numerical Illustration In Remark-A in section 2, we have already discussed why the explicit use of bandwidth requirement per call for a service unit is required to avoid underestimation of pair-wise VSU demand. Thus, for the illustration here, we consider a three-node fully-connected network where the demands are assumed to be given already in VSUs, v (:; :) (Table 1) to illustrate network sizing. Here we consider two service types for three dierent load hours. For the rst service (s = 1), we use w1 = 1, and for the second service (s = 2), we use w2 = 6. For simplicity, we will use c` = 1 for all links for this example. Thus, in this case cost minimization in (3a) reduces to obtaining minimum capacity in the network to meet the given demand. If there is no restriction on upper bounds on the paths, i.e. setting ush kj = 1, then the minimum network capacity obtained by solving integrated model (3) considering both services together is 139 BBUs (no rounding); if the multi-hour multi-commodity ow problem is solved separately for each service [can be obtained by setting #(S ) = 1 and solve model (3) for each service independently] and then the capacity is superimposed afterward (e.g. as in [59]), then the total capacity required is 144.5, about 4% increase from the model-(3) solution. If now bounds on paths (3d) were set to 90% of the VSU demand for that pair, then solving model (3) for this three-node problem we obtain that the total minimum capacity required for the network is 146.8. If the dimensioning is done for each service separately and then the capacity is added afterward (again as in [59]) but this time with 90% upper bound on paths, then the network requires 151.3 units of capacity, about 3% more than the capacity determined by our model for 90% upper bound case. If a `simple heuristic' where 90% is put on the direct path, and the rest on the alternate path for each load hour and then the maximum link ow (for each link) is determined over the three load hours (without using model (3)), then the total capacity requirement would be 160.4 BBUs which is 9% more than obtained using model (3) with 90% for upper bound. Note that this heuristic is meant here speci cally for the three-node problem since there is only one alternate path. Finally, one may wonder if there is any gain from routing. Thus, if there is no alternate route, each link has to meet the demand for each pair directly; speci cally, in this case, each link needs to be engineered to meet the highest demand among all trac hours for that pair { this `no alternate routing' case results in capacity requirement of 160 BBUs in total, showing that we can gain by doing alternate routing. The results for each link for these various cases are summarized in Table 2. This example illustrates the need for upper bounding to re ect the observation that in a dynamic routing network with low blocking requirement a part of trac does not take a direct route, otherwise the capacity obtained may be lower than needed to meet the GoS requirements; on the other hand, the solution from solving for each services separately and then superimposing at the end shows that the capacity obtained may be more than what one needs as compared to solving the integrated model (3). 15

4.2 Network Sizing We rst discuss the network data for a ten-node test network. The data for this network is based on an actual public switched network spanning the continental US (courtesy: Sprint Corporation). Three dierent load periods of trac data are considered to re ect variations of trac during the day; they are for morning, early afternoon and late afternoon, i.e, #(H) = 3. The oered load for voice trac is extracted from this network. Some discussion about the data for voice trac can also be found in [47, 51]. For brevity, we refer to these three load periods as ld-1, ld-2, and ld-3, respectively. For the purpose of this work, we consider that a single voice call takes one BBU per connection. Since at this point we do not have any realistic trac data available to us for any other trac type, we consider a second (video) service type that requires six BBUs per connection. For convenience, we will, at times, refer to the voice service as service type 1 (s = 1), and the second service as video service with service type 2 (s = 2). The oered load for this second trac type (since not available) for dierent trac pairs in the network for various load periods of the day is generated using a uniform random number generator by picking a value between 0 and 5% of the oered load for the voice trac [i.e, a2kh = Uniform (0; 1)  0:05  a1kh ]. The total network trac for three load periods during the day for each service is listed in Table 3; this data set will be referred to as dataset-1 for brevity. Note that the second service is chosen to have signi cantly less oered load than the rst service; this is not unusual when a new service is introduced in the network. To summarize, for this network we have #(K) = 45, #(H) = 3, #(S ) = 2, ws=1 = 1; ws=2 = 6. For computation for network dimensioning, the unit link BBU cost of a link consists of two components: a port based cost and a distance based cost. Given the trend that the ber cost is comparatively low compared to port cost at the nodes, we have used 100 as the cost of each termination port and 0.1 to be the distance cost per mile. Thus, c` = 2  100 + 0:1  D` , where D` is the distance in miles. Note that the dimensioning model can take dierent GoS requirements for dierent services; for simplicity in computational work here, we consider the case of 1% blocking GoS for each service type, i.e., equitable blocking. Note that the model is not restricted to using only for 1% blocking GoS; however, the model is usually appropriate for low-blocking GoS operating conditions. The dimensioning model discussed in section 2 is implemented in C++ where the linear program in (3) is called as a subroutine and is solved using MINOS [55]; since the two trac streams we considered for computational work requires integral multiples of a BBU per connection, we have rounded up the capacity (solution y` to (3)) to an integral multiple of a BBU. There are three parameter sets involved in our dimensioning procedure: (a)  and bd in estimation model (2), (b) upper bound on the paths which can be speci ed as a percentage of virtual service units demand in model (3), and (c) the number of candidate paths to be considered for each trac pair 16

in model (3). For VSU estimate, we have set  = 0:87 and bd is set to 0.1 for high loads (i.e., for a > 60; otherwise bd is set to 0.5) { this is henceforth called `rule-A' for brevity. In Table 4, P we show cost and capacity requirements (i.e., ` y` ) for four values of upper bound (100 means bounds are set to in nity while 90 means bounds on paths are set to 90% of the VSU estimate) and three values for the number of paths (2, 4 and 6). As expected, a lesser number of paths means the cost is higher; similarly, changing bound from 100 to 85 means the cost is higher. From preliminary network simulation we have observed that almost all of the trac is carried in three or four paths in a dynamic routing network under normal network operating conditions, and on average, often 90% of trac is direct routed. Thus, the capacity obtained when the candidate number of paths is set to six and bound is set to 90 is too stringent and does not provide desired GoS in all the load periods (as observed from network simulation); on the other hand, the capacity obtained when the number of path set to two and bound is set to 85 is higher than required. Finally, to contrast the choice of the bd and , dimensioning results based on the straight inverse blocking estimation of model (2) without the over ow quantity [obtained by setting bd = 1% and  = 1 in model (2)] followed by solving model (3) where the number of paths is set to four are shown in the last column in Table 4 { this shows the gross overestimation of capacity required using straightforward inverse blocking estimation for model (2). To summarize, the use of `rule-A' in model (2) coupled with upper bound set to 90% and the number of candidate paths to four in model (3) results in good capacity requirements towards meeting GoS goal | the capacity design from this setting is used in network simulation discussed in the next section which shows the eectiveness in meeting GoS goal. The cost of this network is found to be within 3% of the optimal cost as observed empirically (since network simulation based on another set of capacity [obtained from the dimensioning model when `rule-A' with bound set to 90 and the number of paths to six was used] shows that the GoS is not met). It takes less than three minutes of CPU time to solve the model for the ten-node network data on a DECstation 5000-200 running Ultrix, a 25 MIPS machine. The same set of parameter values are also used in dimensioning networks for two more trac data sets (labeled dataset-2 and dataset-3) for a ten-node network with signi cantly more video oered trac; trac for the datasets and the cost/capacity required are reported in Table 5. Simulation studies (see [65]) on these additional data sets have shown that the capacity obtained is a good estimate in meeting GoS goal, similar to what is observed for dataset-1 reported in the next section.

17

4.3 Network Performance Study The purpose of the network performance study is to see how well the capacity requirement obtained using the network dimensioning model for the 10-node test network is met in a dynamic routing network; what impact do the dierent routing schemes and admission control rules have, and any implications due to dierent call holding time for dierent services. To study these issues, we have developed a call-by-call network simulator where the routing schemes and the admission control policy discussed in section 3 are implemented; this is written in CSIM, a process-oriented simulation language [64]. In our simulation, the mean call holding time for voice (s = 1) is set to three minutes and is kept xed for all of our work. The routing update for DR1 is computed every fteen seconds using s = 0:5 for selection of possible paths as given (5); this value of s was chosen based on preliminary simulation so as to allow some routes to be listed in the routing table although they that may not have enough bandwidth at the time of routing table update. In DR3, voice calls use the \one call old" routing rule while video calls use forced computation of routing for every newly arriving call. The call set up time is assumed to be negligible and is not modelled in our simulation. For each case we considered, we ran ten independent replications, and all the results are based on computing 95% con dence intervals (shown by the short vertical lines in graphs). The mean call holding time for the second service is set to 7.5 minutes; this is 2.5 times more than that of voice service (we have also done a study when it is set to 15 minutes which is explicitly stated in the result that follows). In each case (replication, load period, routing, control), the rst six hours of simulation data are discarded to take into account simulation warm up period, and then the simulation is run for another twenty hours to collect statistics. (The longer warm up and statistics collection time, although not necessary for voice trac, is necessary for the video trac due to its relatively low oered load and higher call holding time.) Trunk reservation, r(i;j ) , is dynamically triggered based on pair-wise blocking in all the routing schemes; this is similar to [8]. Speci cally, if the pair-wise blocking is less than 1%, no capacity is reserved for the direct trac; if the pair-wise blocking is between 1 and 5%, then 5% of the capacity is reserved for direct trac; if the pair-wise blocking is between 5 and 15%, then 15% of the capacity is reserved for direct trac. It is worth mentioning that in RTNR [8], trunk reservation is done separately for each service to provide service grade; however, in our case we have used a shared trunk reservation approach for all services for a particular trac pair while letting admission control address service speci c GoS. Before we go into the discussion on results, we discuss the speci c admission control rules used in our study based on (8). If there is no admission control in the network (i.e., when ps[i;j ] = 1; s 2 S for all trac pairs in our case), it has already been observed for the single-link system case that the lower bandwidth call experiences lower blocking than higher bandwidth calls [41]; through our simulation, we have observed the same behavior for dynamic routing networks as well. Speci cally, 18

although the network is designed for 1% GoS goal for each service, the networkwide blocking for voice trac is observed to be less than 1% while the blocking for video trac is observed to be quite higher than 1% when there is no admission control for the trac loads and the network routing considered in this paper (see speci c graphs for the actual values). Thus, to provide equitable GoS to both services at 1% blocking, it is necessary that some admission control is introduced. We consider two simple instances (rules) of the admission control given by (8): AC-1 :: for video, p2[i;j ] = 1 in all cases; for voice, admit with probability p1[i;j ] = p1 (< 1) when L(i;j) := 0; U(i;j) := 6 AC-2 :: for video, p2[i;j ] = 1 in all cases; for voice, admit with probability p1[i;j ] = p1 (< 1) when L(i;j) := 3; U(i;j) := 9 The idea behind AC-1 is that when the number of free BBUs on a direct link falls below six (the number of BBUs required for an SU for s = 2), then admit voice call for this group with a certain probability (p1 ) while all video calls are admitted (without any restriction) so as to give some preference to video. The idea behind AC-2 is that if there are fewer than three BBUs (i.e., half of BBUs required for an SU for s = 2) of free capacity left, it may be unlikely to get six BBUs free soon for a video call if it arrives and thus, leave that open to voice with probability one; however in the range between three and nine BBUs of free capacity left, admit voice calls with a certain probability, p1 ; again, there is no restriction on acceptance of video calls. It should be noted that for simplicity, we have used the same voice acceptance probability value p1 for all trac pairs. The aim here is to provide a simple admission control by looking at local status for an arriving call, i.e. only the direct link, without looking at the status of the entire network. At the same time, it is activated when the available capacity is in a speci ed range providing some idea about the global status of the network. In Figure 1, we plot networkwide blocking for both voice and video services for load, ld-1, when AC-1 is used and the mean video call holding time is 7.5 minutes { this is shown for three values of p1 : 0.9, 0.95 and 1.0. Bear in mind that p1 = 1:0 refers to no admission control in the network (i.e. complete sharing) but does not mean that there is no trunk reservation; in fact, the dynamic trunk reservation is always active while no speci c reservation for each service is in place (when p1 = 1:0). The corresponding gures for ld-2 and ld-3 are plotted in Figure 2 and Figure 3, respectively. We observe that for ld-1, under no admission control both services have blocking less than 1%; however, for ld-2 and ld-3, video blocking under no control is in the 2% to 3% range (depending on routing schemes). On the other hand, if p1 = 0:9, we observe that often video blocking is lower than voice blocking. It can be seen that for p1 between 0.95 and 1.0, the cross-point (cross-band) of equal blocking for both services is attained. We further observe that this equal blocking point meets 1% GoS 19

for both services under DR1, DR2 and DR3 for all three load periods. From the graphs, we also observe that there is virtually no dierence among the three routing schemes as far as voice trac is concerned; however, for video trac, the average blocking is higher for DR2 and DR3 than for DR1 at all the three acceptance probability values, although there is no dierence in statistically sense at the 95% con dence interval. Note that there is virtually no dierence between DR2 and DR3 as far as the network blocking performance is concerned; however, their dierence lies in the way the connection set up time may be impacted. Recall that alternate routed call for DR3 and possibly for DR2 can have increased call set up time resulting from querying each switch for available link capacity. Post-processing of simulation results reveals that of the alternate routed calls for video service (which can all have increased connection set up time under DR3), about only 10% of calls (for p1 = 1, less for other values of p1 ) may be impacted with increased connection set up time under DR2. Thus, it is up to the network provider to determine whether it is desirable to impact increased call setup time using DR3 compared to DR2 without any perceived performance gain as far as blocking is concerned. Note nally that there is virtually no dierence in blocking for voice service for all three routing schemes in terms of performance. Using admission control AC-2 instead of AC-1 (keeping everything else the same), we observe similar behavior as with AC-1 (see Figure 4, Figure 5, Figure 6). In fact, there is virtually no dierence in performance behavior between admission control rules AC-1 and AC-2; the only dierence is that the cross-point of equal blocking with AC-2 is observed at a point for which the value of p1 is less than for the corresponding p1 when AC-1 is used. We have compared the routing schemes and the impact of admission control when the mean holding time for the video is set to 15 minutes instead of 7.5 minutes while mean call holding time for voice is kept xed at three minutes | this is done for AC-1 (accordingly, for this case, the length of simulation is doubled). We observe that for the no admission control case, there is no dierence due to dierent values of mean video holding time; however, under admission control (i.e., p1 < 1), dierent values of holding time do impact blocking somewhat and the gap is bigger as p1 decreases (shown only for DR1 in Figure 7). (This observation, although may not be intuitive, is similar to when a single-link system was tested using the analytical model given in [50] for Poisson arrival). Finally, to summarize the dierences among dierent scenarios due to dierent admission control parameter values and for dierent values of video call holding time, we have also added the case for AC-2 with mean video holding time of 7.5 minutes in Figure 7 for DR1 for ld-3 (not to clutter up, the con dence interval is not shown). We observe that the point of equal blocking is about 1% while the value of p1 for which this is attained is dierent for the three scenarios. Overall, our simulation studies suggest that the capacity estimate obtained using the dimensioning model (with the speci c choice of parameter set discussed earlier) results in a good estimate in terms of maintaining 1% blocking GoS for all the three routing schemes considered when an appropriate admission control scheme is applied under normal network operating conditions. 20

5. Discussion and Future Work

In this paper, we have presented a multi-hour network dimensioning model for multi-service, multi-rate loss network with dynamic routing. We present computational results on network cost and capacity requirements for dierent sets of trac data and provide guidelines on appropriate choices for the parameter values. We then discuss three routing schemes and a probabilistic admission control policy. From our simulation results, we can infer that the analytic multi-service network dimensioning model (that does neither consider admission control policy nor peakedness for over ow trac) can provide a good estimate on capacity requirements for the desired GoS goal under proper admission control and dynamic routing policies during network operation. This may seem rather surprising; our conjecture is that the estimation step and the multi-commodity ow together tend to average out various factors in network dynamics as far as the dimensioning is concerned. Note that a concrete result on how far our solution is from the actual optimal solution (this itself requires a methodology to estimate lower bound on the actual optimal cost) is desirable; our empirical observation based on the comparison of simulation on the capacity obtained with other parameter values (such as the number of paths set to six) indicates that we may not be too far from optimal. Regarding routing schemes DR1, DR2 and DR3, the acceptance value for p1 for equal blocking is about 0.96 that corresponds to 1% GoS for each service in all three load periods. It should be noted that while we kept the trunk reservation activated (when necessary) to protect direct trac shared by all services, we use admission control to provide \service reservation" or service protection; this is an alternative approach to the RTNR type environment where each service is explicitly set with a minimum allocated guaranteed service reservation [8]; a similar notion to ours has been independently stated in [16, 17]. As noted earlier, \one call old" routing with or without forced computation for newly arrived calls, as in DR2/DR3, is comparable to frequently updated routing scheme such as DR1 in terms of network performance (for additional work on network performance that considers also other routing schemes that have no crankback as well as trac with network overload, the reader is directed to [52]). However, the dierence is that call set up time delay due to query for link status information impacts DR3 more than DR2 while DR1 has no impact (since update of routing table is done frequently/periodically instead of on each call basis). The results on network performance leads to another interesting observation from simulation results: the network dimensioning model appears to be indierent to the actual dynamic routing scheme employed (as far as the three routing schemes presented here); furthermore, with a proper admission control rule the desirable property of equitable GoS for all services can be provided with any of the routing schemes without needing to add additional capacity. Of course, our observation is based on the limited network test data used here. It should be noted that the 21

dimensioning model is more general than just for 1% blocking GoS; we have used here 1% blocking GoS for computational work. Other possible related work may explore the following issues:

> Given that a good capacity estimate can be obtained using the dimensioning model, the

development of ecient computational algorithms to solve model (3) for large networks may be explored. We have also listed some limitations of our model in Remark-D of section 2. Thus, the development of an enhanced dimensioning model (that may consider, e.g., explicit development/incorporation of multi-rate blocking estimation as well as the fact that the set of paths used by dierent services may be dierent; small routing cost; call admission policies of services) may be explored and compared to the procedure presented here. Additionally, nonlinear link cost or discrete (modular) values for link capacity may also be addressed, which is not done here. Also, simulations show that GoS goals are observed on a service class basis; it remains to be seen how the individual pairwise per service GoS is met for the network and any enhancement needed in the dimensioning model to meet this requirement.

> For network simulation, we have used one value of p1 for the entire network; it may be extended

to use the acceptance probability to be dierent for dierent trac pairs as we suggested in (8), and/or to possibly change p1 depending on network load, thus, providing a dynamic acceptance scheme.

Appendix The optimization model (3) can be written in following compact form using matrix notation: min cT y x;y subject to

E shxsh  vsh; s 2 S ; h 2 H X

s2S

ws
sh xsh  y; h 2 H

0  xsh  ush ; s 2 S ; h 2 H

y  0: This model is equivalent to the following model: min cT y z;y 22

subject to

E shzsh  wsvsh; s 2 S ; h 2 H X


sh zsh  y; h 2 H

s2S 0  zsh

 qsh ; s 2 S ; h 2 H y  0:

To see this, set zsh = ws xsh , and adjust the bound on ow variables using the transformation qsh = ws ush . We use the former model in the main body of this paper to provide the interpretation that the ow x refers to owing virtual service demands on dierent paths, and ws is used with link ow of virtual service units to obtain link bandwidth in BBUs which is to be minimized with appropriate unit link cost.

Acknowledgement We thank S. Rajendran for his work in the initial phase of the development of the network simulator. The trac data used in this work is based on data provided for another work [47] by Sprint Corporation and is greatly appreciated. We gratefully acknowledge I. Sukiman for identifying and xing a bug with one of the routing schemes in the simulation code. We thank the anonymous reviewers for constructive comments that helped improve the content and the presentation of the paper signi cantly. We also thank one anonymous reviewer for bringing to our attention references [16, 17].

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h=1 s=1 s=2 5 6 10

5 6 0

h=2 s=1 s=2 11 13 2

3 3 6

h=3 s=1 s=2 8 6 7

12 2 4

Table 1: Three-node trac demands in VSUs for each pair and load period

link (1,2) (1,3) (2,3) Total

(no upper bound) model-(3) superposition1 70.0 28.0 41.0 139.0

90% upper bound model-(3) superposition1

74.5 33.5 36.5 144.5

72.0 32.3 42.6 146.8

76.5 36.7 38.1 151.3

Simple Heuristic

No Alt. Routing

76.9 43.3 40.2 160.4

80 42 38 160

Table 2: Solution for the three-node problem by for dierent methods and bounds (1 { superposition of solution obtained for each service separately) load periods ld-1 ld-2 ld-3

Service Type, s = 1 Oered load (erlangs) 2684.80 2826.16 3224.08

Service Type, s = 2 Oered load (erlangs) 69.06 72.96 84.34

Table 3: Summary of network trac (dataset-1) for 10-node network 27

Upper Bound Rule 100 95 90 85

`rule-A' with #path=2

`rule-A' with #path=4

`rule-A' with #path=6

`Inverse blocking rule' with #path=4

1,350,825.3/4,460 1,399,346.2/4,634 1,453,596.6/4,835 1,508,417.4/5,036

1,340,922.4/4,443 1,385,103.6/4,626

1,305,266.4/4,286 1,340,638.7/4,424 1,388,949.4/4,627 1,439,247.5/4,835

1,738,486.7/5,715 1,796,859.2/5,960 1,858,822.9/6,210 1,917,779.0/6,452

1,432,769.1/4,815 1,477,632.5/4,996

Table 4: Cost and capacity for dierent parameter settings for 10-node network problem (dataset-1) load periods dataset-2 dataset-3

ld-1 ld-2 ld-3 ld-1 ld-2 ld-3

Oered load s=1/s=2 2,953.26/345.32 3,108.81/364.82 3,546.49/421.72 3,221.75/690.64 3,391.38/729.64 3,868.88/843.44

Cost/Capacity 2,332,348.8/7,800 3,404,342.4/11,304

Table 5: Network trac, cost and capacity for dataset-2 and dataset-3 ld-1, AC-1, s2ht=7.5 2

Networkwide Blocking (%)

DR1, s=1 DR2, s=1 DR3, s=1 DR1, s=2 DR2, s=2 DR3, s=2

1

0 0.9

0.95 Acceptance probability (p1)

1

Figure 1: Networkwide blocking for each service as the voice acceptance probability (p ) changes 1

under admission control AC-1, for load period ld-1, mean video call holding time = 7.5 minutes 28

ld-2, AC-1, s2ht=7.5 4 DR1, s=1 DR2, s=1 DR3, s=1 DR1, s=2 DR2, s=2 DR3, s=2

Networkwide Blocking (%)

3

2

1

0 0.9

0.95 Acceptance probability (p1)

1

Figure 2: Networkwide blocking for each service as the voice acceptance probability (p ) changes 1

under admission control AC-1, for load period ld-2, mean video call holding time = 7.5 minutes ld-3, AC-1, s2ht=7.5 3

Networkwide Blocking (%)

DR1, s=1 DR2, s=1 DR3, s=1 DR1, s=2 DR2, s=2 DR3, s=2

2

1

0 0.9

0.95 Acceptance probability (p1)

1

Figure 3: Networkwide blocking for each service as the voice acceptance probability (p ) changes 1

under admission control AC-1, for load period ld-3, mean video call holding time = 7.5 minutes 29

ld-1, AC-2, s2ht=7.5 DR1, s=1 DR2, s=1 DR3, s=1 DR1, s=2 DR2, s=2 DR3, s=2

Networkwide Blocking (%)

1.5

1

0.5

0 0.9

0.95 Acceptance probability (p1)

1

Figure 4: Networkwide blocking for each service as the voice acceptance probability (p ) changes 1

under admission control AC-2, for load period ld-1, mean video call holding time = 7.5 minutes ld-2, AC-2, s2ht=7.5 4 DR1, s=1 DR2, s=1 DR3, s=1 DR1, s=2 DR2, s=2 DR3, s=2

Networkwide Blocking (%)

3

2

1

0 0.9

0.95 Acceptance probability (p1)

1

Figure 5: Networkwide blocking for each service as the voice acceptance probability (p ) changes 1

under admission control AC-2, for load period ld-2, mean video call holding time = 7.5 minutes 30

ld-3, AC-2, s2ht=7.5 3

Networkwide Blocking (%)

DR1, s=1 DR2, s=1 DR3, s=1 DR1, s=2 DR2, s=2 DR3, s=2

2

1

0 0.9

0.95 Acceptance probability (p1)

1

Figure 6: Networkwide blocking for each service as the voice acceptance probability (p ) changes 1

under admission control AC-2, for load period ld-3, mean video call holding time = 7.5 minutes load: ld-3, routing scheme: DR1 4 s=1, ht=7.5, AC-1 s=1, ht=7.5, AC-2 s=1, ht=15, AC-1 s=2, ht=7.5, AC-1 s=2, ht=7.5, AC-2 s=2, ht=15, AC-1

Networkwide Blocking (%)

3

2

1

0 0.9

0.95 Acceptance probability (p1)

1

Figure 7: Networkwide blocking with routing scheme DR1 for load period ld-3 under combination of dierent values of admission control and video call holding time 31