Virtual Path Control for ATM Networks with Call Level Quality of Service Guarantees

Transcription

1 Virtual Path Control for ATM Networs with Call Level Quality of Service Guarantees Nios G. Aneroussis and Aurel A. Lazar Department of Electrical Engineering and Center for Telecommunications Research Columbia University, New Yor, NY Abstract We provide an algorithm for the VP distribution problem that satisfies nodal constraints on the processing load and blocing constraints for each Source-Destination pair. The algorithm maximizes the networ revenue under the above set of constraints and is independent of the number of traffic classes in the networ, the method of representation of networing resources, the admission control policy used in every lin and VP, and the networ routing scheme. We apply the algorithm to the Xunet ATM testbed and study several of its performance characteristics.. Introduction [ANE95a] presented a comprehensive management architecture that allows the networ manager to configure VP connection services under quality of service constraints and evaluate the resulting networ performance. The objective of this paper is to provide a supporting algorithmic framewor to the networ manager for formulating a bandwidth allocation policy for Virtual Paths (also referred to as the VP distribution policy), that guarantees Quality of Service (QOS) both at the cell and the call level. QOS at the cell level can be guaranteed through the concept of the Schedulable Region [HYM9]. At the call level, QOS is guaranteed by bounding the blocing probability of the VC service for every Source-Destination pair in the networ and the average connection setup time. User VC Service User Figure : The Networ Model Public Networ User VP Networ According to our model, users mae requests for the VC Service at the boundaries of the networ (Figure ). The networ operator establishes VPs within its administrative domain. This VP networ is transparent to the users and serves the purpose of carrying user calls through the public networ at reduced call processing costs. A separate routing mechanism determines how calls are to be routed through the VP networ. The VP distribution problem is a networ control problem with the following formulation: given the networ topology, the capacities of networ lins, the capacities of the signalling processors, and the matrix of offered networ load, calculate the routes and capacities of Virtual Paths in the networ such that the following requirements are satisfied:. The sum of VP capacities on each lin does not exceed its capacity. 2. The signalling load on each signalling processor is below a predefined upper bound. 3. The call blocing rate of each Source-Destination (SD) pair is below a predefined upper bound (also referred to as the SD blocing constraint). 4. The networ revenue is maximized under the above constraints. The requirement described in constraint 2) above is justified by our experiments with the Xunet III ATM testbed [ANE95b]. These experiments evaluated the relation between the transport and the signalling networ. When using wideband (video) calls, a small call arrival rate was sufficient to saturate the transport networ. If, however, we were to produce the same capacity demand using narrowband (voice) calls, we would have to apply much higher call arrival rates (the capacity of one video call in our experiments was roughly equivalent to 70 voice calls). In this case, however, there is a danger that the capacity of the signalling networ is exceeded. When this happens, only a small percentage of the total number of calls can be established (the remaining calls are rejected due to signalling message losses, excessive call setup times, etc.) and the transport networ operates well below its total capacity. In other words, even if the total capacity demand is the same, a small call arrival rate with high capacity demands puts pressure on the transport networ, whereas greater call arrival rates with small capacity demands shift the pressure to the signalling system. In reality, the signalling system can act as a bottlenec for the transport networ. Congestion in the signalling system can be relieved in two ways: by blocing the excessive call setup requests at their source, or by employing VPs to relieve the most loaded signalling nodes. If the first approach is followed, calls might be bloced even if there is

2 bandwidth available in the networ. Therefore, the second approach is superior, but at the expense of a reduced networ throughput due to end-to-end bandwidth reservation. This paper is organized as follows: Section 2 presents an overview of related wor on VP distribution algorithms. Section 3 presents an algorithm for VP distribution together with the concepts that provide our quality of service framewor. Section 4 applies the algorithm to the Xunet ATM testbed and maes observations on its performance characteristics. Finally, Section 5 summarizes our conclusions and proposes directions for further study. 2. Notation and Review of Related Wor 2. Notation Before starting the review of the individual approaches to the VP distribution problem it is worthwhile to define a common problem setting based on which all approaches can be compared. For this purpose, we introduce the following canonical model: Topology Information: The networ topology is represented as a directional graph G(V,L), where V is the set of vertices (networ nodes) and L is the set of networ lins. W is the set of Source-Destination (SD) pairs: W { w ( u, v) u, v V }. The networ supports, 2,..., K traffic classes each with its own QOS requirements. P is the set of Virtual Paths. For each SD pair w, R w { r wi, i, 2,, N w } is a set of routes of cardinality N w. Each route consists of an arbitrary combination of VPs and lins between the source and the destination node. We define the logical graph G (V, E ), where the set of edges E is derived from the set of edges of the original graph G by adding one edge between every VP source and termination point. Capacity Information: The networing capacity for lin l is denoted by S l and is given by the Schedulable Region [HYM9]. Loading Information: The call arrival rate and the inverse of the holding time of calls for each SD pair w is denoted by λ w and µ w, respectively. The traffic intensity (in Erlangs) is denoted by ρ w. For each switching node v V, µ v denotes the processing capacity of the node signalling processor in requests per unit of time. Finally, λ v denotes the total arrival rate of call setup requests at node v. Control Policy: For each p in P, we denote as C p the networing capacity assigned to p. The networing capacity is in general given by the Contract Region [HYM93b]. For every lin l we must have that the sum of VP capacities travelling over that lin is less or equal to the lin capacity. At every outgoing lin and VP operates an admission controller. Encoded in the admission controller is the admission control policy, denoted by u. The admission controller accepts or rejects calls based on their traffic class and the number of currently active calls in the lin or VP. The state of the admission controller is denoted by x ( x, x 2,, x K ). For each set R w with N w > there exists a routing policy R w which specifies the schedule for finding available routes. For example, an incoming call might try the first route in the set, and if no capacity is available, the second route and so on. In order to guarantee that every signalling processor is operating normally, an overload control rejects the surplus of call requests whenever λ v > µ v. Constraints: The blocing constraint for each SD pair and traffic class is denoted by β w. The blocing constraint enforces the Quality of Service requirements at the call level. The blocing probability P w is the percentage of call attempts of class for the above SD pair that are denied service due to the unavailability of resources. We must always have that P w β w. At every signalling processor, the arrival rate of call setup requests must be less than the node s processing capacity, i.e., λ v < µ v. For every route r R w, the call setup time t r on route r must be bounded, i.e., t r < T r. 2.2 Taxonomy of Algorithms The VP capacity allocation problem in ATM networs can also be regarded in a more general context: given a networ topology, a portion of the capacity of the physical lins is reserved on an end-to-end fashion for creating logical lins connecting non-neighboring nodes. This problem rises in circuit switched networs, in the configuration of leased lines and Virtual Truns (VT), and also in the configuration of logical channels in SONET based networs. For this reason, we will review a variety of approaches for capacity allocation that are equivalent to our VP distribution problem. We provide the following taxonomy Synchronous vs. Asynchronous Synchronous algorithms update the VP capacity in real-time based on the observed demand for call establishment. In this context, the bandwidth of VPs can be expanded or contracted at call arrival or departure times. Such is the scheme pre-

3 sented in [SAT90, OHT92]. On the other hand, asynchronous algorithms maintain a fixed VP distribution for a time period T (also referred to as the update interval). These algorithms are called asynchronous, because the modifications of VP capacity are not driven by the call arrival or departure process associated with each VP. The VP distribution policy is computed at the beginning of the period, and remains fixed during that period. The decision on choosing the appropriate VP distribution policy is based on estimates of the offered load during the coming period. For this purpose, a load estimator is usually needed to predict the offered load Centralized vs. Decentralized Asynchronous algorithms can be further distinguished in two major categories: centralized and decentralized. Centralized algorithms are run in one location (typically at the networ manager s site) and require the collection of up-todate information from all networ nodes, while decentralized algorithms are run in every networ switch and information is passed along between neighboring nodes. Such is the scheme of [SHI9] and the game-theoretic approach of [LAZ95] Form of the Cost Function Algorithms can be also categorized based on the cost (objective) function employed in the decision maing process for VP capacity assignment. The scheme of [SAT90, OHT92] does not use a cost function. For selecting the VP that will receive a bandwidth increase, [SHI9] uses a cost function based on the VP blocing probability. In its other variation, a linear combination of the carried traffic in every VP is employed. [LIN93] uses the total call blocing rate. [GER9] has selected multiple cost functions: the lin residual capacity and the amount of bandwidth demand that needs to be bloced overall among SD pairs. [GER94] defines a weighted sum of the total rejected bandwidth demand and the total load assigned to each lin. [HUI9] employs a linear combination of the reserved bandwidth on each lin, weighted by a cost factor. [KIM95] also uses a function of the reserved capacity on every lin in combination with a switching cost for every VP. The AT&T capacity design system [ASH9] uses a weighted combination of the residual lin capacities. [GOP9] uses the revenue for each iteration of the algorithm expressed as the difference of the expected blocing probabilities before and after increasing the capacity of a path times the expected load on the path. Essentially, the same cost function is used by [ARD94], slightly modified to incorporate the cost observed by other SD pairs when the capacity of every VP is increased, while the capacity of all others is held constant. Since the objective of the VP distribution problem is to achieve a satisfactory networ throughput, it is logical to include the call blocing rates in the cost function. This approach is taen by [SHI9], [LIN93], [GOP9] and [ARD94]. [GER9] and [GER94] incorporate the total rejected bandwidth demand in the cost function. The solution in this case maximizes the total carried capacity in the networ. This has some advantages because the call blocing rates (which are non-linear functions of the VP capacities) do not participate in the cost function, and the optimization problem becomes more tractable. However, in either case, in order to guarantee Quality of Service at the call level, the call blocing constraints must be introduced as constraints into the optimization problem. Only [ARD94] addresses this problem. In the other cases, even if the solution maximizes the networ throughput (or the carried capacity), the call blocing rates might be acceptable only for some SD pairs and violating the blocing constraints for others Trade-off between Signalling Costs and Capacity Allocation Most algorithms except [ASH9] and [KIM95] assume that one or more VPs are established between every SD pair, and thus the destination can be reached from the source in one hop only. In our opinion, this approach has two major flaws. Firstly, it is not scalable: a networ with hundreds of nodes will need a very large number of VPs, and consequently, the VP distribution tas will be overwhelming. Secondly, there is a substantial cost associated with loss of throughput due to the rigid capacity reservation for each VP. On the other extreme, a solution with no VPs at all implies increased call processing costs. We envision an intermediate solution that maes available to each SD pair a number of routes. Every route consists of any concatenation of physical lins and VPs. This scheme can maintain a natural balance between the processing costs and the cost due to the reservation of resources, since it does not require a single direct VP between every SD pair. Further, VPs can be used to carry traffic of different SD pairs. The scheme is similar to the AT&T real time networ routing system [ASH9]. This system achieves very low call blocing probabilities because it allows, in addition to a direct route, a large number of non-direct alternate routes to be followed (the equivalent of a call setup using 2 VPs and an intermediate node in an ATM environment) and has been shown to absorb well traffic load fluctuations. In this way, VPs can be used to reduce processing costs without maing significant throughput sacrifices. From all the capacity allocation algorithms reviewed, only [OHT92] investigates the trade-off between capacity utilization and signalling costs. The context is slightly different, because there is only one route available to each destination (using a VP), and the signalling cost is associated with the frequency of messages needed to expand or contract the capacity of the VP. These messages must travel along the route of the VP in the same way as the hop-by-hop call setup messages that would be used to establish a VC on the same route.

4 Thus, this wor models the cost for allocating capacity to VPs and we believe that this must be fully taen into account in a VP distribution algorithm. 3. The VP Distribution Problem By observing the limitations of previous approaches to the VP distribution problem, we propose an algorithm that satisfies the following basic requirements: supports any number of traffic classes. explicitly guarantees QOS at the call level by introducing hard bounds on the call blocing rates for each SD pair and bounds on the time to establish calls through the signalling system. As will be shown later in Section 3.4, a sufficient condition to guarantee bounded call setup times is to bound the call arrival rates at every signalling processor. can wor with any combination of admission control and routing scheme (i.e., static priority or adaptive routing). In addition, the algorithm has the following desirable properties: is independent of the abstraction used to describe the networing capacity of lins and VPs. is independent of the admission control policy used at every lin and VP. The algorithm tries to maximize the networ revenue (considered as the weighted sum of the throughputs for each traffic class and SD pair multiplied by the respective call holding times), while satisfying the node processing and SD blocing constraints. The algorithm is presented in Section 3.. Sections and 3.3 provide a methodology for computing the quantities needed by the algorithm for a simple static priority routing scheme. 3. An Algorithm for VP Distribution Initialization: Begin with all VPs at zero capacity. Traffic between all SD pairs follows a hop-by-hop call setup procedure, except for the SD pairs which are served only by routes comprised of VPs (in that case all traffic is bloced). Compute the blocing probabilities for all SD pairs for the given call arrival rates. Step : Find the SD pairs for which the blocing constraints are not satisfied. If none are found, proceed to step 2. Else consider every VP whose capacity can be increased as a Bandwidth Increase Candidate (BIC). The capacity of a VP is increased by removing the necessary resources from all the lins on the path of the VP. If the BIC set is empty then exit (i.e., there is no spare capacity available in the networ). Else, compute the blocing drift D b for the current vector of VP capacities (also referred to as the current solution) from: D b max( 0, Pw β w ), w W w,. (EQ ) This quantity represents how far we are from satisfying the blocing constraints. For each BIC create an alternative solution by assigning one unit of capacity to the BIC while holding all other VPs to their current capacity. For each alternative solution, compute the blocing drift. If a larger number is obtained for all alternatives, then there appears to be no way of satisfying the blocing constraints for the given networ load; exit. Else, select the alternative with the lowest blocing drift and mae it the current solution. Step 2: Chec for nodes that violate signalling capacity constraints. If none are found, and there are no blocing constraint violations, proceed to step 3, else return to step. Else (there exist capacity violations), compute the capacity drift D c from: D c max ( 0, λ v µ v ), v V. (EQ 2) v Similarly, this quantity represents our distance from satisfying signalling capacity constraints. Every VP whose capacity can be increased by one unit is considered a BIC and an alternative solution is constructed in the same way as above by adding capacity to the BIC while holding all other VPs at their present capacity. If the BIC set is empty then exit. For each alternative solution, compute the capacity drift. If all alternatives have a higher capacity drift from the current solution, there appears to be no way of satisfying the signalling capacity constraints; exit. Else, select the alternative with the lowest capacity drift, mae it the current solution and go to step. Step 3: Now we have satisfied all constraints. Every VP whose capacity can be increased by one unit is considered a BIC. First compute the throughput for each SD pair: γ w ( P w ) λ w The networ revenue is then given by: J γ w Fw. (EQ 3), (EQ 4) w where F w denotes the revenue obtained by accepting one call of SD pair w and class. For each BIC, create an alternative solution and compute the resulting networ revenue. Drop the alternatives for which blocing or signalling capacity constraints are violated. If there is no remaining alternative that produced a higher revenue, then STOP. Else select the alternative with the highest revenue increase and repeat step 3. Intuitively, the algorithm wors as follows: In steps and 2 capacity is added to VPs until a solution is reached that satisfies the problem s constraints. The objective used in each of these steps is representative of the corresponding set of constraints that need to be satisfied. Step 3 attempts to further improve the solution by adding capacity to the VPs that promise a higher increase in revenue, while satisfying all

5 constraints. Modeling Assumptions VPs are established hop-by-hop using a signalling protocol or a management function activated by a central networ management facility. Assuming non-blocing switches with output buffering, at every node along the route of the VP, the necessary networing capacity must be secured from the outgoing lin that the VP is traversing. The networing capacity of the output lins is described by the Schedulable Region (SR) and of the Virtual Paths by the Contract Region (CR). Informally, the Schedulable Region is a surface in a dimensional space (where is the number of traffic classes), that describes the allowable combinations of calls from each traffic class that can be accepted on the lin and be guaranteed Quality of Service. An analogous definition is given for the Contract Region (CR) of Virtual Paths. The Contract Region is a subregion of the SR reserved for exclusive use by a VP. Since the CR is by itself a region, the VP can carry calls of different traffic classes. The remaining region for a physical lin available for admitting VCs is obtained by subtracting from the SR the CRs of all VPs traversing this lin. The calculus of regions for this operation is described in [HYM93b]. We also mae the following modeling assumptions: the call arrival process for all SD pairs is Poisson, and each call has an exponentially distributed holding time. All logical lins (VPs or physical lins) bloc independently, and the overflow traffic is also Poisson. The Poisson model is widely used to model the call arrival process in current telephone networs, and we believe it will be adequate for modeling the call-level behavior in broadband networs as well. There has been no study so far to contradict this assumption, i.e., that other types of services lie video and data would be better described at the call level using a different model.. Representation of Capacity Regions The schedulable region is represented as an arbitrarily shaped region in the K-dimensional space. The Contract Regions however are represented as hyperplanes. This simplifies the implementation of the VP admission controller. Every hyperplanar region can be uniquely represented by a tuple (m,..., m K ), where m is the number of calls of class that can be admitted into the system when the number of calls from every other traffic class is 0. In this way the admission control test for VPs is given by: x (EQ 5) m xk m K The advantage of this representation is that higher accuracy can be achieved during addition and subtraction operations (which are performed in the K-dimensional space) while maintaining simplicity for the admission control test, at least for VPs..2 Selection of the Alternative Solution Set An important aspect in the execution of the algorithm is the construction of the alternative solution set. When adding capacity to a VP, the shape of the new hyperplanar region has to be specified as well. In order to determine what is the most appropriate shape, the following technique is employed: For each BIC, K alternative solutions are constructed. The CR of the BIC for each alternative is derived from the original one by expanding into one of the K possible directions by a certain amount (which can differ between classes). Thus, the alternative solution set consists of possibly more than one alternative for each BIC. The algorithm will then compute the objective function for each alternative, and will select the VP and the CR shape that provides the best improvement in the objective function. In this way, after several iterations of the algorithm, CRs can tae an arbitrary hyperplanar shape..3 The Route Selection Policy We assume a static priority route selection scheme, similar to dynamic hierarchical routing in circuit switched networs. This choice was made to simplify the analysis. The VP distribution algorithm however is not tied to a particular routing scheme. Other schemes such as adaptive routing (where the route for a call is dynamically determined based on real-time utilization information) can be applied as well. According to the static priority routing model, the route set R w consists of N w routes. An incoming call first attempts the first route in the set. If the call is bloced, the second route is attempted, and so on. The call is bloced if no available capacity was found on all the routes in the set. As mentioned previously, every route consists of one or more logical lins. Every logical lin is either a physical lin or a VP. A E D Figure 2: Example of route selection Figure 2 shows an example of a route selection policy. For the SD pair (A,B), the first attempted route is comprised of the direct VP from A to B. The second route consists of two VPs and the intermediate node D. The third route does not contain any VPs, In that case, the call request goes through the intermediate nodes E and F. 3.3 Finding the Blocing Probabilities Let us now focus on a single logical lin. We denote by λ l and µ l the arrival rate and the inverse of the holding time of class calls arriving on lin l. u l ( x), x Sl denotes the state dependent admission control policy for class calls [HYM93a]. We also define some set macros that will be used C F B

6 in the sequel: Lins(r): The set of (logical) lins composing route r. Nodes(r): The set of nodes on route r. First(r,l): The set of lins of route r prior to lin l. Routes(l): The set of routes traveling over lin l. Out(v): The set of outgoing lins from node v. Let π l ( x) be the equilibrium probabilities of the corresponding Marov chain. The global balance equations can be written as: π l ( x) q l ( x, y) y S l π ( l y ) y S l where q l ( x, y) u l ( x)λl x µ l y π l ( y)q l ( y, x), (EQ 6), (EQ 7) and e is the elementary vector with a at position and zeros everywhere. After solving the above system of equations, we can compute the blocing probability of the lin for each class as: P l. (EQ 8) x S l x S l : u l ( x) 0 Let us now focus on the route level. The probability that a call is bloced on route r wi is: P wi ( P l ) l Lins( r wi ) Therefore the call load offered to route r wi is:. (EQ 9) The blocing probability for SD pair w is given by: j and thus, he total load offered to lin l is: r Routes( l) q First( r, l) Finally, the load on the lin due to route r is: S l if y x + e if y x e u l ( x)πl ( x) π l ( x) λ wi λ w i i. (EQ 0) λ w P wj < i N w j λ l P w N w λ r P wj, (EQ ) ( P q ). (EQ 2) λ l, r λ r. (EQ 3) q First( r, l) It is usually realistic to assume that the time between the arrival of the call setup message on a lin (when the resources are actually reserved) and the time the resources are freed because the call setup request was not successful is negligible compared to the holding time of an accepted call. This is true for example when the signalling protocol supports timeouts during call setups. For example, in the Xunet ATM testbed, a call setup request is timed-out after 2 seconds if no reply has been received, an interval significantly smaller than the average holding time of a call. In this case, we have that: µ l, (EQ 4) r Routes( l) where µ µ r, for all routes r that belong to SD pair w. w The P l can be determined using a fixed point approximation. We can rewrite (EQ 8) as a function of the offered load, and the Capacity Region of the lin in the following form: The above set of equations defines a continuous mapping from the compact set [0,] L into itself, and thus, by Brouwer s fixed point theorem, there exists a solution. In [KEL86] it is proven that the solution is unique, and the loss probabilities calculated by this solution converge to the exact loss probabilities. Starting with an arbitrary initial value in [0,] for the P l, the λ l and µ l can be evaluated from (EQ 2) and (EQ 4), respectively. Using the obtained values, the P l can be determined from (EQ 8). The new values are then used in the next step. The procedure stops when the improvement to the solution has satisfied a convergence criterion. In our experiments, the procedure stops when the improvement for every P has dropped below 0-6 l. Note that, the above algorithm can be easily parallelized. At every step, a separate processor can compute the P for every lin only with the nowledge of the P l in the previous l computation step. 3.4 The Call Setup Times ( P q ) ---- λ l, r λ µ l r P l F λ l -----, S. (EQ 5) l, u l, l,, E' µ l At equilibrium, the arrival rate of signalling messages at a node v has two components. The first component is due to the messages used to setup calls. It is approximated by the sum of the call arrival rates at the outgoing logical lins. The second reflects the contribution of the messages used to tear down the established connections: λ v 2 λ l + l Out( v) 2 λ r ( P r ) l Out ( v) r Routes( l). (EQ 6)

7 The above formula depends on the nature of the signalling protocol. We have assumed a single pass call setup and teardown. Every action involves a request travelling in the forward direction and a reply in the reverse direction. For this reason, every arrival rate in the above equation is multiplied by 2 to tae into account the reply messages. By modeling the signalling processor as an M/M/ queue with service rate g v, the queueing and processing delay for a signalling message is given by: Tq v (EQ 7) g v λ v The M/M/ model has been used before in the literature to model the behavior of signalling processors [MUR89]. If we select an upper bound for the arrival rate of signalling messages µ v, the call setup time on route r is always bounded: t g v r T p l v Nodes( r) l Lins( r) (EQ 8) where T p l is the propagation delay on lin l. The above equation verifies that the call setup time increases with the number of intermediate nodes that process signalling messages, and so, by using VPs, significant savings in call setup time can be achieved. 4. Experimental Results g v µ v T p l l Lins( r) v Nodes( r) We applied the VP distribution algorithm on the Xunet ATM testbed. The topology of the networ is shown in Figure Figure 3: Xunet Topology All the lins are bi-directional with a capacity of 45 Mbps. We assume two traffic classes: class corresponds to a video service with a pea rate of 4Mbps; class 2 to a voice service with a pea of 64Kbps. The admission control policy is Complete Sharing. We used a blocing constraint of 0.6 for class and 0.3 for class 2. The capacity of the Xunet signalling processors is 400 calls/min. Our static priority routing scheme provided a direct VP for each SD pair and a linonly route for the overflow traffic of the VP. 0 2 After applying the VP distribution algorithm, we observed that the initial solution (Phase ) does not satisfy the signalling capacity constraints at nodes, 4 and 7. In particular, the arrival rate at node 4 is 766 calls/min. The blocing constraints however are satisfied for all SD pairs. The algorithm then enters the st and 2nd steps (Phase 2), where it attempts to satisfy the above constraint violations by assigning capacity to VPs. When this happens, it enters the 3rd step (Phase 3) where it attempts to further increase the networ revenue. The blocing probabilities for each SD pair are shown in Table. The first number in every box corresponds to class, Table : SD call arrival rates and blocing probabilities for Complete Sharing SD 0 SD SD 2 SD 3 SD 4 SD 5 SD 6 SD 7 Nodes Arrivals.0 0 Phase Phase Phase Xunet Experiment and the number below to class 2. The last row of the table provides experimental results after installing the configuration obtained at the end of Phase 3 on the Xunet testbed. We used the management system of [ANE95a] to configure the VPs and the call generators and obtain the blocing probabilities. The measurements were taen over a period of 6 hours, during which a total of approximately 300,000 calls were generated. Due to limited networ access we were not able to run the experiment repeatedly in order to obtain a reliable confidence interval for the above measurements. The experimental results show that the actual blocing probabilities approximate very closely the figures predicted by the VP distribution algorithm. The obtained figures when compared to the prediction (Phase 3) appear in general to be lower for class and higher for class 2. In either case, they are well below the SD blocing constraints. The reader may have noticed that we use rather high numbers for the blocing constraints. The reason is that when verifying the VP distribution on the testbed, the experiment requires a smaller number of generated calls to provide accurate blocing measurements. If we were using blocing constraints of the order of, e.g., 0-3, we would need to run the experiment for significantly longer times to obtain accurate measurements. Table 2 shows the VP capacities in Mbps and the resulting networ revenue. The CR of every VP is represented as a 2- dimensional hyperplane, and the VP capacities are given as the points where the hyperplane meets the axes. Note the decrease in revenue at the end of Phase 2. This is due to the fact that during that phase, the algorithm attempts to satisfy the constraints rather than increase the revenue. The latter happens exclusively during Phase 3, hence the small revenue increase at the end of that phase. The reader will quicly

8 Table 2: VP capacities and revenue for Complete Sharing VP Revenue Nodes Phase Phase 2 Phase 3 observe that the VPs that carry the traffic of the most loaded SD pairs have been assigned more capacity. Also, SD #6 and #7 that are not assigned any VP capacity experience increased blocing. Overall, the algorithm required a total of 0 iterations to complete. 5. Conclusions and Further Study In this paper we addressed the VP distribution problem for ATM networs. VPs can be used to tune the fundamental trade-off between call throughput and overall performance of the signalling system. We provided an algorithm for the VP distribution problem that for the first time satisfies the combination of nodal constraints on the processing of signalling messages, and constraints on blocing for each Source-Destination pair. The algorithm maximizes the networ revenue under the above set of constraints and wors independently of the number of traffic classes in the networ, the admission control policy used in every lin, and the networ routing scheme. We provided a solution methodology for a static priority routing scheme. The methodology can be easily extended to networs with adaptive routing policies. We finally applied the algorithm to the Xunet ATM testbed and studied various performance characteristics. This wor was funded in part by NSF Grant CDA , and in part by a Grant from the AT&T Foundation. A more detailed version of this paper can be obtained from ftp:// ftp.ctr.columbia.edu/ctr-research/comet/public/papers/ 95/ANE95c.ps.gz. References [ANE95a] N.G. Aneroussis and A.A. Lazar, Managing Virtual Paths on Xunet III: Architecture, Experimental Platform and Performance, Proceedings of the 995 IFIP/IEEE International Symposium on Integrated Networ Management, Santa Barbara, CA, May -5, 995. 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