Queueing Delay Guarantees in Bandwidth Packing

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1 Queueing Delay Guarantees in Bandwidth Packing ERIK ROLLAND 1, ALI AMIRI 2 and REZA BARKHI 3 1 Departent of Accounting & Manageent Inforation Systes, Fisher College of Business The Ohio State University, Colubus, OH E-ail: rolland.1@osu.edu 2 Departent of Manageent, College of Business Adinistration Oklahoa State University, Stillwater, OK E-ail: aairi@okway.okstate.edu 3 Departent of Accounting & Inforation Systes, Paplin College of Business Virginia Polytechnic and State University, Blacksburg, VA E-ail: reza@vt.edu Draft date: Septeber 29, 1998 This paper is to appear in Coputers and Operations Research, Abstract. This paper proposes a new forulation for the bandwidth packing proble, assuring axiu service delay in telecounications networks. The bandwidth packing proble is one of selecting calls, fro a list of requests, to be routed in the network. We liit the axiu queueing delay, while axiizing revenues generated fro the routed calls. An efficient Lagrangean relaxation based heuristic procedure for finding bounds and solutions to the proble is deonstrated, and coputational results fro a variety of proble instances are reported. We show that the procedure is both efficient and effective in finding good solutions. Key words: Bandwidth packing, call routing, queueing delay, telecounications networks, Lagrangean relaxation, subgradient search, heuristics.

2 Stateent of Scope and Purpose The bandwidth packing proble is one of selecting and routing calls in a telecounications network. The selection is norally perfored as to axiize the revenues fro the calls routed. However, this ay cause serious queueing delays in the network, possibly causing lost profitability and lost custoer satisfaction for the network owners. The scope of this paper is to propose a atheatical forulation that addresses the bandwidth packing proble one that axiizes revenues but also at the sae tie liits the axiu queueing delays in the network. In addition, we propose a Lagrangian-based solution procedure that produces both lower bounds and high quality solutions for the bandwidth packing proble. 1

3 Ali Airi is an Assistant Professor of Manageent Sciences and Inforation Systes at Oklahoa State University. He received the BS degree in Business Adinistration in 1985 fro the IHEC, Tunisia, the MBA in 1988 and the Ph.D. in Manageent Sciences and Inforation Systes in 1992 fro Ohio State University. His research interests include data counication and coputer network design and analysis, databases, cobinatorial and discrete optiization and general OR/MS odeling and analysis. He has published in Coputer Counications, Coputers & Operations Research, European Journal of Operational Research, and Naval Research Logistics. Reza Barkhi is an Assistant Professor of MIS in the Departent of Accounting and Inforation Systes, Paplin College of Business, at Virginia Polytechnic Institute & State University. His current research interests are in the areas of collaborative technologies and proble solving, and topological design of telecounication networks. Dr. Barkhi has published in journals such as Location Science, European Journal of Operational Research, Group Decision and Negotiation, and Decision Support Systes. He received a BS in CIS fro the College of Engineering, and an MBA, an MA, and a Ph.D. fro the College of Business all fro The Ohio State University. Erik Rolland is an Assistant Professor of MIS in the Departent of Accounting and Manageent Inforation Systes, Fisher College of Business, at The Ohio State University. His research interests include anageent and design of telecounications systes, cobinatorial odeling and analysis, and strategic MIS. He has published in journals such as Coputers & Operations Research, European Journal of Operational Research, Transportation Science, and Annals of Operations Research. He received a BS in CIS fro the College of Engineering, and an MA and a Ph.D. fro the College of Business all fro The Ohio State University. 1

4 1. Introduction The reliability and response ties of telecounications networks are ajor factors affecting perceived quality of telecounications services. Users have coe to expect 100% reliability, and virtually iediate response ties. Telecounications copanies ust not only satisfy their custoers by providing reliable and responsive systes, but they also have a coitent to stakeholders to axiize profits. Maxiizing profits translates into decisions related to iproving the utilization of the network capacity. Decisions that affect capacity utilization involve deciding which calls on a list of requests, called a call table, should be routed on the network. Subsequently, a path for each call to be routed ust also be deterined. This path should be selected fro all possible paths in the network. The coplete network topology, as well as the call table, the revenues, and the traffic requireents are given. This proble is typically referred to as the bandwidth packing proble (BWP). Versions of this proble have been studied by Airi, Rolland & Barkhi [1], Anderson et al. [2], Laguna & Glover [13], Cox et al. [4], Parker & Ryan [15], and Park et al. [14]. The objective of the BWP has in these past research efforts been to axiize the total revenues fro calls that are routed without consideration to quality of service to users. Route, or path, selection influences response tie experienced by network users and has a ajor effect on the utilization of network resources (e.g. node buffers and counications links). A good routing policy would allow new users to use the network without significant deterioration of the quality of service to existing users. Lack of a good routing policy ay require unnecessary capacity expansions to the network. In anaging the network, one has to ake tradeoffs between revenue axiization and response tie to users. If the only consideration is revenue axiization, then network users ay experience significant delays, and the quality of 1

5 service will suffer. The odel developed in this paper incorporates response tie by including a non-linear constraint that liits the axiu queueing delay in the network to a anageent specified upper level. A version of the path assignent proble that considers only revenue axiization has been previously addressed in [2], [13], [4], [15], and [14]. A wide variety of solution procedures have been proposed: tabu search [2], [13], genetic algoriths [4], colun generation [15], as well as integer prograing [14]. The authors in [1] addressed the issue of iniizing queueing delay in the network. They include a cost ter associated with this queueing delay in the objective function of their odel. This ter is coputed by ultiplying total link queueing delay by a unit delay cost. The ain justification for using this ter was to control the delay in the network and therefore response tie to users. It ay be difficult to assign a weight to this unit delay cost, and further the nature of the solution to the proble ay change adversely with this value. A better way to control response tie to users through queueing delays is to ipose an upper liit on the link queueing delay in the network. Since the upper liit (or bound) for the delay ay not be exactly known, the network designer or anager can start with an estiate of this bound (e.g., the delay bound to obtain 60% average utilization of link capacity). With an increase in this bound, total profit increases. By deciding on the level of tradeoffs between total profit and quality of service to users, the network designer or anager can decide what delay bound to ipose depending on the strategy and priorities of the organization operating the network. Motivated by the iportant applications for path assignent in call routing, custoer satisfaction (i.e., reasonable response ties) and the coplexity of the proble, we present a forulation that seeks to axiize total revenues of calls to be routed while guaranteeing a certain level of quality of service to users. We develop a procedure that generates feasible solutions as well as bounds for this proble. 2

6 The reainder of this paper is organized as follows. In section 2, a atheatical forulation of the BWP proble is presented. A Lagrangean relaxation of the proble obtained by dualizing a subset of the constraints is presented in section 3. A heuristic solution procedure is developed in the following section. Coputational results are reported in section 5. The conclusions are suarized in the last section. 2. A Matheatical Proble Forulation We introduce the following notation necessary for developing a atheatical odel for the BWP: N E M d r O() D() Q δ the set of nodes in the network the set of undirected links (or arcs) in the network the set of calls. Each call is represented by a counicating node pair the deand of call M (e.g., the deand for network resources: bandwidth) the revenue fro call M (e.g., onetary units) the source node for call M the destination node for call M the capacity of link (i,j) (bandwidth) the upper liit on the queueing delay (network independent delay surrogate) The bandwidth packing proble is defined as follows: Given a graph G=(N, E) and a set of call requests (a call table) M, we seek to axiize the profits fro the routed calls, while assuring that the queuing delay does not exceed a pre-specified acceptable liit. Further, we cannot exceed the capacities on the counication links. A graphical representation of a siple network structure with two calls is provided in Figure 1. The dashed line shows a call being routed fro node 3 to node 5 via interediate nodes 8 and 7. The thick solid line shows a call being routed fro node 3

7 1 to node 4 via interediate nodes 7 and 8. The input paraeters that need to be known in order to successfully solve this routing proble include the network topology, the capacity of the links, and the traffic requireents and revenues for all the calls. In a typical telecounications network the topology graphs are often sparse, necessitating the use of shared resources. This is exeplified in Figure 1, where both calls use one coon resource: link (7,8) Figure 1. An Exaple Network Topology We assue that all nodes in the graph (Figure 1) have infinite buffers to store essages waiting for transission on the links. Further, the arrival process of essages to the network follows a Poisson distribution, whereas the essage lengths follow an exponential distribution. Also, the propagation delays in the links are negligible 1. Note that we only consider a single class (or type) of service for each counicating node pair. 1 The packet travel tie is assued to be negligible. 4

8 Even though the list of calls is known in advance, the traffic requireent for each call ay typically be bursty. For exaple, both video and data transission exhibit variable bit rates, for which queueing delays can be approxiated by using an M/M/1 odel. The validity of this approxiation is supported by experiental evidence: it has been shown that the optial routing is insensitive to the shape of the delay versus link load curve, and is only affected by the asyptotic value of the link capacity [9]. Given the above assuptions, the telecounications network is odeled as a network of independent M/M/1 queues ([11], [12]). In this network, links are treated as servers with service rates proportional to the link capacities. The custoers are essages whose waiting areas are the network nodes. Using the notation described above and the decision variables defined below, the queueing delay in link (i,j) is d X Q M M d X and the average link queueing delay in the network is given by 1 d X M. E Q d X ( i, j) E M The nuber of links in a network ( E ) is constant. Therefore the queueing delay requireent can be represented by constraint (5) in the forulation of the odel below. The decision variables are: Y X = 1 if call is routed 0 otherwise = 1 if call is routed through a path that uses link (i,j) 0 otherwise 5

9 W = 1 0 if call is routed through a path that uses link ( i, in the direction of i to j otherwise j) Proble P: Z P = Max M r Y (1) subject to: j N W - j N W ji = Y Y 0 if i = O( ) if i = D( ) otherwise i N and M (2) W + W ji X (i,j) E and M (3) M ( i, j) E d X Q M d Q (i,j) E (4) M X d X δ (5) X (0,1) (i,j) E and M (6) Y (0,1) (i,j) E (7) W (0,1) (i,j) E and M (8) The objective function (1) represents total revenues of routed calls. Constraint set (2) contains the flow conservation equations, which define a route for each call represented by a counicating node pair. Constraints in set (3) links together the X and W variables. The proble can be correctly forulated with either X or W variables only. However, both variable sets are useful in the Lagrangean relaxation developed in the next section. The capacity constraints on the links are considered by constraint set (4). Constraint (5) enforces the upper liit on the queueing delay in the 6

10 network. The integrality conditions on X constraint sets (6)-(8), respectively.,y and W variables, are enforced in 3. Proble Relaxation. Proble P is a cobinatorial optiization proble with a nonlinear constraint (5). The proble studied in Anderson et al. [2], Laguna & Glover [13], and Park et al. [14] is a special case of proble P and is known to be NP-coplete [7]. Proble P is a nonlinearly constrained (0,1) integer prograing proble, and it is difficult to solve this proble to optiality with standard ixed-integer prograing tools. Hence, we propose a coposite upper and lower bounding procedure based on a Lagrangean relaxation of the proble. Consider the Lagrangean relaxation of proble P obtained by dualizing constraint set (3) using nonnegative ultipliers α for all (i,j) E and M, respectively, and relaxing constraint (5) using a nonnegative ultiplier ψ: Proble L: Z L = Max M r Y -ψ ( i, j) E Q M Subject to (2), (4), (6), (7) and (8). d M X d X + M ( i, j) E α (X - W - W ji ) +ψδ (9) Proble L can now be decoposed into two subprobles as follows: Proble L1: Max M r Y - M ( i, j) E α (W + W ji ) (10) Subject to (2), (7) and (8). Proble L2: 7

11 Max M ( i, j) E α X - ψ ( i, j) E Q M d M X d X (11) Subject to (4) and (6). Proble L1 can be further decoposed into M subprobles (one for each call) as follows: Max r Y - ( i, j) E α (W + W ji ) (12) subject to: j N W - j N W ji Y = Y 0 if i = O( ) if i = D( ) otherwise i N (13) Y (0,1) (i,j) E and M (14) W (0,1) (i,j) E and M (15) Proble L2 includes a non-linear ter in the objective function. In essence, this optiization proble resebles a non-linear ulti-diensional knapsack proble. It is difficult to obtain an optial solution procedure or a heuristic that would solve this proble well. To be able to solve proble L2, we decopose it into E sub-probles (one for each link) in the following anner: Max M α X - ψ ( i, j) E Q M d M X d X (16) subject to: M d X Q (17) X (0,1) (i,j) E and M (18) 8

12 Proble L does not satisfy the integrality property, since the relaxation of L does not necessarily have an integer solution. Hence, the relaxation of proble P can theoretically give a lower bound which is at least as good as, and possibly better than the relaxation of P. Each subproble of proble L1 can be solved by solving the shortest path proble fro O() to D() using the nonnegative ultipliers α as the cost of the links (i.e., link distances). If the revenue fro the call is greater than the cost of that shortest path, then the call is routed through that path. if not, the call is not routed and we set Y = 0 and W = 0 (i,j) E. Each subproble of proble L2 corresponding to a link (i,j) is equivalent to a single constraint (0,1) knapsack proble with a nonlinear objective function. We relax the integrality constraints and solve the continuous version of this proble using the following greedy type procedure. Procedure Greedy: Step 1: Reorder the X variables by sorting the in nonincreasing order of α /d ; Re-indexed the variables in this order, and Let =0. Step 2: Let =+1 and set X X 0 if α > 0 and X 0 > 0 = 0 otherwise where X 0 = in{ 1, 1 d [(Q - S) - (ψd Q α ) 1/2 ]} and S = d k X k k< Step 3: If = M stop; if X. < 1 then stop and set X k =0 for k=+1,..., M. Otherwise go to step 2. 9

13 4. The Solution Procedure Feasible solutions as well as lower bounds for the optial solution of proble P, can be obtained by using the relaxation presented above. As for all relaxation procedures, the success of the approach depends heavily on the ability to generate good Lagrangean ultipliers [8]. Theoretically, let ZL(α,ψ) be the value of the Lagrangean function with a ultiplier vector (α,ψ), then the best bound using this relaxation is derived by calculating Z L (α,ψ ) = Min ( α, ψ) {Z L (α,ψ) }. In practice, a good but not necessarily optial set of ultipliers is often derived using iterative ethods such as subgradient optiization ethod and various ultiplier adjustent ethods known as ascent (descent) ethods [14]. We use the subgradient optiization ethod to search for good ultipliers. The subgradient ethod is a odified version of the gradient ethod in which subgradients replace gradients [8]. Since this ethod is well understood, we do not provide its ipleentation details; we suarize, however, the particulars of our algorith in appendix A. We now outline a heuristic procedure to solve proble P. The below stated procedure (Procedure Exaine) attepts to generate a feasible solution to proble P at every iteration of the subgradient optiization algorith using inforation provided by the solution to proble L1. The best feasible solution is retained when the subgradient algorith is terinated. Note that in the solution to proble L1 every call is either routed through the links in the network or not routed at all. However, there ay be soe links with loads higher than their available capacities. This siple heuristic attepts to route calls through the network without exceeding link capacities. Thus, the heuristic guarantees to generate a feasible solution at every iteration of the subgradient optiization procedure. The coplete heuristic is stated below. Procedure Exaine: 10

14 (1): Order the calls by sorting the in non-increasing order of the optial values of the objective functions of their corresponding sub-probles of proble L1. Start with the first call. (2): If the call can be routed using the path deterined in the solution of its corresponding sub-proble of proble L1 without exceeding the available link capacities and the queueing delay upper liit, then route the call and update the available link capacities. (3): If all calls have been exained then Stop. Otherwise proceed with the next call and go to step Coputational Results We coded the above solution procedures in Pascal, and perfored a nuber of coputational experients using a 7610 VAX coputer. 5.1 Test Proble Generation. To evaluate the effectiveness of the proposed procedures, we randoly generated various networks and call tables containing inforation about calls to be routed on those networks. We generated the networks as follows: First, the generator locates the specified nuber of nodes on a 100x100 grid. Each node has a degree equal to 2, 3 or 4 with probability of 0.6, 0.3 and 0.1, respectively. We repeat the following procedure for each node i Ν: Deterine node i s closest neighbor (in ters of Euclidean distance) with unsatisfied degree requireent, label this node j. Add arc (i,j) and repeat this until node i s degree requireent is satisfied or all the nodes with unsatisfied degree requireents have been considered. If the degree requireents are not et for node i, then connect node i to its closest neighbors to which it is not already connected until the degree requireent of node i is satisfied. At the end check if the network is connected; if not, add links necessary to ake it connected. 11

15 The paraeters that define the network (node degrees and their associated probability of occurrence) are chosen in order to generate realistic telecounications networks, where there are soe redundant counication links. A typical graph resulting fro the network generation using the paraeters stated above is depicted in Figure 2a. Figure 2b shows the sae nodes, but each node has a degree equal to n-1. The latter case (2b) is not realistic in telecounications networks, since full duplication of counications links is prohibitively expensive. Figure 2a. A telecounications network. Figure 2b. A fully connected network We generated five sets of networks with 20, 25, 30, 35 and 40 nodes respectively. The average nubers of links for each of the networks are shown in Table 1. Table 1. Average nuber of links in the networks Nuber of nodes Avg. nuber of links in the network in the network

16 Each link in the network is randoly assigned a capacity equal to 48, 96, 192, or 500 with equal probabilities. These capacity choices approxiately correspond to real line choices: a T3 line, OC2, OC4, and OC9, respectively (approxiations in Mbps). This provides realistic variations in the capacity liits to test the robustness of our algorith. The test proble generator also produced the call tables. The call table contains inforation about the origin and destination nodes for all counicating node pairs, as well as the revenues and counication deand for those calls. For each call, the generator randoly deterines an origin and a destination node. The traffic requireent, or deand (d ), as well as the revenue (r ) for each call are generated randoly fro two unifor distributions between 20 and 40 and between 10 and 50, respectively 2. As depicted in Table 2, we generated 5 sets of networks with paraeters as described above, to capture a wide range of proble structures. For each network set (20, 25, 30, 35 and 40 nodes), we generated five proble groups, where only the nuber of calls differed. For each proble, we used rando call tables to create 10 proble instances (each line in Table 2 reports averages fro these 10 proble instances). The nuber of calls (P) varies between 40 and 80 percent of all possible calls. For exaple, in a 40 node network, using a percentage of 60 (P=60), there are 936 calls (or 40x39x0.6) 3. Let DM, a delay ultiplier, be the upper liit on the average link utilization 4. This delay ultiplier was fixed at 60% in order to enable us to study the effects of variations in nuber of calls. To study the iplications of queuing delay, we generated a new set of probles (Table 3). The five proble groups are each based on one corresponding proble instance fro Table 2 Various other liits on the traffic requireents were used in separate experients to investigate the sensitivity of our proposed algorith to these paraeters. We found that the algorith was not sensitive to any specific paraeter liits in the range [5, 100]. 3 For reasons of coparison, we ention that Anderson et al. [2] reported results of coputational experients conducted using networks ranging in size fro 14 nodes and 35 calls to networks with 192 nodes and 20 calls. Park et al. [14] report results for probles with up to 30 nodes and 75 arcs, and with a axiu of 75 calls. 4 The upper liit on the total queueing delay, δ, in the network is equal to E *DM/(1-DM). 13

17 2, where P=60. Thus, for all these probles, we kept the nuber of calls constant. We varied the surrogate easure for link utilization, DM, fro 40% to 70%, in steps of 5% for each proble group. In order to achieve a reasonable level of confidence about the perforance of the solution procedure versus the proble structure, we generated 10 instances by randoly creating different call tables for each instance (that is, each line in Table 3 reports the average results fro 10 proble instances). 5.2 Analysis and Discussion. Tables 2 and 3 show the average perforance easures for different networks. The results of the experients are described by providing the nuber of nodes in the network ( N ), the percentage of the nuber of routed calls (P), the delay ultiplier (DM), the gap between the best feasible solution value and the upper bound expressed as a percentage of the upper bound, the total revenue, the average and axiu link utilization, and the CPU ties in seconds. The nubers describing the queueing delay ultiplier (DM) are unit-less. This surrogate easure can be applied to specific telecounications networks to calculate delays in actual tie-units. Table 2 shows the results for different percentages of the total nuber of available calls. As the nuber of calls increases, the average link utilization and total queueing delay do not change significantly because of the requireent of iniu level of response tie to users. However, when this nuber increases, total revenue of routed calls increases as a result of better selection of calls to be routed fro aong a larger set of available calls. Also, there is higher probability that ore calls will not be routed. The average gap between the feasible solution value and upper bound varies between 1.46% and 5.80% with a ean of 2.82%. The CPU tie (in seconds) varies between 33 for networks with 20 nodes and 846 for networks with 40 nodes. The effects of changes in delay ultiplier (DM) are reported in Table 3. As the delay ultiplier increases, a tradeoff between total revenue and response tie to users is 14

18 ade. When the delay ultiplier increases, total revenue of routed calls increases at the expense of quality of service (response tie) to users. The deterioration in response tie is reflected in the increase in the total queueing delay. For exaple, for the 40 node networks, the total queueing delay on average increases fro 39.2 when the delay ultiplier is 40 to when this ultiplier is 70. The deterioration in response tie is also indicated by the increase in average and axiu link utilization. For exaple, for the 40 node networks, the average and axiu link utilization increase fro 34.2% to 54.4% and 68.4% to 91.0%, respectively. The average gaps between feasible solution values and upper bounds vary between 0.90% and 5.68% with a ean of 2.79%. The CPU tie (in seconds) varies fro 46 for networks with 20 nodes to 642 for networks with 40 nodes. In suary, it is observed that as the delay ultiplier increases, the nuber of calls routed increases, thus revenue increases. However, this increase in revenue coes at the expense of increased delays. This pattern is expected beyond the tested range of the delay ultiplier (40-80%). Further, our experients have shown that as the average node degree goes up, keeping everything else constant, the delay will diinish because of additional routing capacity. The nuber of nodes in the graphs sees not to directly have any ipact on the solution quality, but it indirectly ipacts CPU requireents. The ain drivers of CPU requireents are the nuber of arcs and the nuber of calls. The nuber of arcs increases as the nuber of nodes increases. Due to the decoposition of the proble, the CPU requireents also increase as the nuber of calls increases. The total delay in the syste is related to the nuber of calls since it is the su of the products of the nuber of calls and the delay per call. 6. Conclusion We have provided a forulation and an efficient bounding procedure for the call routing proble with a iniu service quality threshold. Incorporating this threshold 15

19 is iportant in assuring a axiu acceptable delay for the incoing calls in telecounications networks. Our contributions are as follows: 1. We propose a new forulation for the call routing proble that considers all possible paths for routing the calls, and selects the best subsets. 2. We axiize the revenue generated fro selecting the calls, while we assure a iniu service quality with respect to axiu acceptable delay. 3. We propose efficient bounding and solution procedures for this proble, based on a two-level decoposition of the proble. In our coputational experients, we deonstrated that, on average, our heuristic produced consistently good solutions with average optiality bounds of approxiately 2.8% in less than five inutes of CPU tie. 16

20 Appendix A Given an initial ultiplier vector (α 0, ψ 0 ) (set to the zero vector in this study), a sequence of ultipliers is generated by updating the vector at the iteration k using the forula (α k+1, ψ k+1 ) = (α k, ψ k ) + t k (W k - X k ), where (α k+1, ψ k+1 ) and (α k, ψ k ) are the ultiplier vectors at iterations k+1 and k respectively, (W k,x k ) is part of the optial solution to the Lagrangean Proble L with ultiplier vector (α k, ψ k ) and t k is a positive scalar step size. Inf k=0 It is well known that li sup Z L (α k, ψ k ) converges to Z L (α*,ψ*) if t k 0 and t k [16]. Since in general these conditions are very difficult to satisfy, the subgradient optiization ethod is always used as a heuristic. In this study, we used the following step size that has been found to be satisfactory in practice (Bazaara and Goode [3]): t k = λk( ZL(α k, ψ k ) - Zf) / W k - X k 2, where Zf is the value of the best feasible solution found so far and λ k is a scalar satisfying 0 λ k 2. This scalar is set to 2 at the beginning of the algorith and is halved whenever the bound does not iprove in 20 consecutive iterations. The algorith is terinated after a specified nuber of iterations (set equal to 500 in this study) unless an optial solution is reached before that point. The algorith is also terinated if the gap between the best upper bound and the best feasible solution found is less than 0.01% of the best upper bound, or the best upper bound does not iprove in 100 consecutive iterations by at least 0.01%. 17

21 References [1] Airi, A., E. Rolland and R. Barkhi, "Bandwidth Packing with Queuing Delay Costs: Bounding and Heuristic Solution Procedures", European Journal of Operational Research, forthcoing, [2] Anderson, C.A., K. Fraughnaugh, M. Parker and J. Ryan, Path assignent for call routing: An application of tabu search, Annals of Operations Research 41 (1993) [3] Bazaraa, M. S. and J.J. Goode, A Survey of Various Tactics for Generating Lagrangean Multipliers in the Context of Lagrangean Duality, European Journal of Operational Research 3 (1979) [4] Cox, L.A., L. Davis and Y. Qui, Dynaic anticipatory routing in circuit-switched telecounications networks, in Handbook of Genetic Algoriths, L. Davis (ed.), Van Nostrand/Reinhold, New York, [5] Fisher, M.L., Lagrangean Relaxation Methods for Solving Integer Prograing, Manageent Science 27 (1981) [6] Ford, L.R. Jr. and D.R. Fulkerson, Flows in Networks, Princeton University Press, Princeton, N.J., [7] Garey, M.R. and D.S. Johnson, Coputers and Intractability: A Guide to the Theory of NP-Copleteness (1979) 215. [8] Gavish, B., On Obtaining the 'Best' Multipliers for a Lagrangean Relaxation for Integer Prograing, Coputers and Operations Research 5 (1978) [9] Gerla, M., The design of store-and-forward (S/F) networks for coputer counications, Ph.D. Dissertation, Coputer Science Dept., Univ. of California, Los Angeles (1973). [10] Held, M., P. Wolfe and H.P. Crowder, Validation of subgradient optiization, Matheatical Prograing 5 (1974) [11] Kleinrock, L., Counications nets: stochastic essage flow and delay, New York, Dover, [12] Kleinrock, L., Queueing systes, Volues 1 & 2, Wiley-Interscience, NewYork, 1975, and [13] Laguna, M. and F. Glover, Bandwidth Packing: A Tabu Search Approach, Manageent Science 39 (1993)

22 [14] Park, K, S. Kang and S. Park, An Integer Prograing Approach to the Bandwidth Packing Proble, Manageent Science 42 (1996) [15] Parker, M. and J. Ryan, A Colun Generation Algorith for Bandwidth Packing, Telecounications Systes 2 (1995) [16] Poljack, B. T., A General Method of Solving Extreu Probles, Soviet Math. Doklady 8 (1967)

23 Table 2. Effect of Changes in Nuber of Calls Percent Total Total % Link Utlization N P DM Gap Revenue Delay Average Maxiu CPU * Average: * All CPU ties are easured in seconds on a VAX 7610 coputer 20

24 Table 3. The Effects of Changes in the Delay Multiplier Percent Total Total % Link Utlization N P DM Gap Revenue Delay Average Maxiu CPU * Average: * All CPU ties are easured in seconds on a VAX 7610 coputer 21

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