Construct QoS End-to-end Virtual Path With Lagrangean Relaxation Method

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(1)Int. Computer Symposium, Dec. 15-17, 2004, Taipei, Taiwan.. Construct QoS End-to-end Virtual Path With Lagrangean Relaxation Method Chang-Jiu Chen, Ming-Su Liu National Chiao Tung University, Taiwan,ROC {cjchen,msliu}@csie.nctu.edu.tw established connections may cause big load for connections that request to establish virtual circuits later. Cheng and Lin[5] took a user-optimization approach and considered a fairness issue by minimizing the maximum individual end-to-end packet delay in virtual network, but they didn’t consider the system’s perspective. In this paper, we attempt to jointly consider both system and user perspectives, and keep maximum tolerance to the later connections.. Abstract- In this paper, we have improved a QoS routing problem. We give an approach to minimize the congested link utilization while to satisfy individual connection’s packet delay. We use a Lagrangean Relaxation based approach augmented with an efficient primal heuristic algorithm, called Lagrangean Relaxation Heuristic (LRH). With the aid of generated Lagrangean multipliers and lower bound indexes, the primal heuristic algorithm of LRH achieves a near-optimal upper-bound solution. A performance study delineated that the performance trade-off between accuracy and convergence speed can be manipulated via adjusting the Unimproved Count (UC) parameter in the algorithm. We have drawn comparisons of accuracy and computation time between LRH and the Linear Programming Relaxation (LPR)-based method, under three networks named NSFNET, PACBELL, and GTE and three random networks. Experimental results demonstrated that the LRH is superior to the other approach, namely the LPR method, in both accuracy and computational time complexity, particularly for larger size networks Keywords: QoS, Routing Problem, Lagrangean Relaxation, LRH, LPR. 2. Problem Model and Formulation We will describe our problem, and model it in this section. Furthermore, we also formulate our problem into non-linear integer programming form. Problem Description We construct the network into load balance model subject to end-to-end packet delay constraints for each individual user. This model has two advantages. 1. This model can reduce packets delay implicitly. 2. This model reserves the maximum flexibility to the later connections. The problem has also been shown to be NP-complete which means no polynomial time algorithm for it unless P=NP. For the sake of obtaining sub-optimal solutions, Lagrangean relaxation is applied to the formulation to decompose the problem into several tractable subproblems in next section. The candidate path set does not need to be prepared in advance and the best paths are generated while solving the subproblems in our approach. A heuristic algorithm based on the solving procedure of the Lagrangean relaxation will be developed to obtain a primal feasible solution in the next two sections.. 1. Introduction To ensure reliable and high-quality network services, routing and capacity assignment policies should be carefully designed. Traditional quasi-static routing algorithms attempt to optimize a certain aggregate measure, e.g. to minimize the average endto-end packet delay [1, 2]. However, this kind of performance measures may not be consistent with the service objectives and may result in fairness problems. The routing problem in virtual circuit networks has been a traditional research topic in computer networks and has attracted even more attention since the emergence of the Asynchronous Transfer Mode (ATM) technology. However, most previous researches on virtual circuit routing considers the objective function of minimizing the average end-toend packet delay [1, 3, 4], which address a systemoptimization perspective without taking individual users into account. And also these researches do not consider the later connections, that is these current. Network Model and Definition A virtual circuit communications network is modeled as a graph where the processors are represented by nodes and the communication channels are represented by arcs. Let V = {1, 2, 3, ..........., N} be the set of nodes in the graph and let L denote the set of communication links in the network. Let W be the set of origin-destination (O-D) pairs (commodities) in the network. For each O-D pair w ∈ W , the arrival of new traffic is modeled as a Poisson process with rate. rw. (packet/sec). To reduce the problem’s complexity, we. assume that each O-D pair w, the overall traffic is transmitted over one path in the set capacity is. Cl. Pw .. For each link. l∈L ,. the. packets/sec.. For each O-D pair. w ∈W. , let. xp. be 1 when. p ∈ Pw. is used to transmit packets for O-D pair w and 0 otherwise. In. 338.

(2) Int. Computer Symposium, Dec. 15-17, 2004, Taipei, Taiwan.. a virtual circuit network, all of the packets in a session are transmitted over exactly one path from the origin to the destination. Thus. ∑x. p. =1 .. For each path p and link. p∈Pw. l∈L,. let. δ pl. link l is on path p and 0 otherwise. Then, the aggregate flow. g l , is. ∑ ∑x rδ p w. pl. .. p∈Pw w∈W. In the network, there is a buffer for each outbound link. Using Kleinrock’s independence assumption [6], the arrival of packets to each buffer is a Poisson process where the rate is the aggregate flow over the outbound link. It is assumed that the transmission time for each packet is exponential distributed with mean. Cl. ywl. is defined as. ∑xδ p. pl. p∈Pw. and fl denotes the estimate of the aggregate flow.. Decision variables:. denote the indicator function which is 1 if. over link l, denote as. variables are introduced:. −1. α. :percentage of capacity usage on maximum congested link. xp : 1 if path p is selected, 0 otherwise. ywl : 1 if source-destination pair w uses link l, 0 otherwise.. min α. Problem PII Subject to:. . Thus, each buffer is modeled as. an M/M/1 queue, as considered in [3, 7, 8].. Problem Formulation The following notations are used in the formulation. Input values:. NF. : the set of nodes in the network. L : the set of communication links in the communication network. W : the set of source-destination (SD) pairs. Wn : the set of SD pairs where node n is the source node. rw :(packets/sec.):the arrival rate of new traffic of each OD pair w ∈ W ,which is modeled of Poisson process for illustration purpose. Cl : (packets/sec.),the capacity of each link l∈L. Pw : a given set of of simple directed paths from the origin to the destination of O-D pair w ∈ W . gl : the aggregate flow over link l, which is equal to .. ∑∑. Redundant constraints associated with these auxiliary variables (3) ,(4),(7) and (9) are added. Lagrangian relaxation is a general solution strategy for solving mathematical programs that permits us to decompose original problems into several subproblems such that we can exploit their special embedded structures. We use Lagrangean relaxation to the heterogeneous Minmax end to end delay problem and decompose the original problem into several subproblems in next section.. x p rwδ pl. 3.Lagrangean relaxation and problem decomposition. p∈ Pw w∈W. We first dualize Constraints (1), (2), (3) and (4) to Problem PII to obtain the following Lagrangean relaxation problem. Problem (Dual_P): y Z ( ρ ) = m in{[1 − ∑ v c ]α + [ ∑ s ( ∑ − D )] + c − f. δpl. : 1 if path p uses link l; 0 otherwise. Dw : the maximum allowable end to end delay for O-D pair w ∈W . Decision variables: α :percentage of capacity usage on maximum congested link. xp : 1 if path p is selected, 0 otherwise. The formulation is modeled as the following integer linear programming problem.. Problem P. wl. dual. l. l. w. l∈ L. w ∈W. ∑ ∑t. wl. w. l∈ L. l. ( ∑ x p δ pl − y w l ) +. w ∈W l ∈ L. p ∈ Pw. l. ∑ [u. l. ( g l − f l ) + v l f l ]}. − (∗ ). l∈ L. subject to constraints (5), (6), (7), (8), and (9) .. min α. Subject to:. Reorganizes formulation (*), dual (P) becomes => Z dual ( ρ )= min{[1 − ∑ vl cl ]α + [ ∑ ∑ ∑ (t wl + ul rw ) x pδ pl ] l∈L. w∈W l∈L p∈Pw. ∑ sw ywl +[ ∑ ( w∈W − ∑ t wl ywl + (vl − ul ) f l ) − ∑ sw Dw ]} cl − f l l∈ L w∈W w∈W subject to Constraints (5), (6), (7), (8), and (9) and vectorρ= (s,t,u,v) is the non-negative Lagrangean multiplier.. The objective function is to minimize the largest utilization on the most congested link. For the purpose of applying Lagrangean relaxation method, we transform the above problem formulation into an equivalent formulation PII. In PII, two auxiliary. 339.

(3) Int. Computer Symposium, Dec. 15-17, 2004, Taipei, Taiwan.. an example, consider the case that fl =0. The objective function is minimized by assigned ywl * (0) to 1 if. Problem (Dual_P) can be decomposed into following three independent subproblems (S1, S2 and S3) by separating the decision variables α, x, y. Therefore, we have Zdual=ZS1+ ZS2 +ZS3 − swDw , where ZS1(v)= min  1 − . ∑ l∈ L. subject to Constraint (8), ZS2(t,u)=. (. w∈W. points of fl as those points where.  v l c l α . min [ ∑ ∑. ∑ (t. wl. ∑( l∈L. w wl. w∈W. cl − fl. sw for each − twl ) = 0 Cl − fl. i. + ul rw ) x pδ pl ]. break points. We observe that when f l ≤ fl ≤ fl. i +1. the. *. value, the value of ywl ( f l ) remains constant for all. subject to Constraints (5) and (6) , and. ∑s y. (. w. These break points are sorted and denoted as n 1 2 fl , fl ,.............. fl . Note that there are at most |W|. w∈W l∈L p∈Pw. ZS3(s,t,u,v)= min{. and to 0 otherwise. We define a set of break sw − twl ) ≤ 0 cl. ∑. w ∈W − ∑twl ywl + (vl −ul ) fl )}. (. w∈W. subject to Constraints (7) and (9).. and is 0 otherwise. Therefore, within an sw − t wl ) ≤ 0 cl − f l. interval, [ fl i , fl i +1 ) , the objective is only a function of fl, and the minimum point within the interval can be found analytically. By examining at most |W| +1 intervals, we can find the global minimum point by comparing those local minimum points. When examining an interval, we first determine i * ywl ( fl ) within the interval for each w. We denote. Solving Subproblem 1 Subproblem S1 is a problem for decision variable α. Variableαis set to 1 if the corresponding cost 1 − v c is. ∑. *. . Within the above internal, ywl ( f l ) is 1 if. l l. l∈L. negative; otherwiseαis set to 0. Subproblem 1 runs on O(L) computation time.. ∑s. *. w. i. ywl ( fl ). as al and. ∑t. i. wl. * ywl ( fl ) as bl. Note that al. w∈W. w∈W. and bl are non-negative. Within the interval, the objective function can then be expressed as: al . A typical curve of the Z sub 3 _ l = − bl + (vl − ul ) f l Cl − f l. Solving Subproblem 2. objective function vs. fl within the interval f l i ≤ fl ≤ fl i +1 is shown in Figue 1. The curve of the objective function vs. fl is shown in Fig. 2. The local minimum point is either at. Subproblem S2 is a problem for decision variable x. It consists of |Wn| independent problems. Each one is an edge-disjoint-path problem rooted at the given source node and destined to all destination nodes for the SD pairs with non-zero traffic demand. To solve the problem, one can view the input network as a graph. This graph contains (L) arcs and (N) nodes. We set each arc l have Cl capacity (it means that the transmission time for each packet is exponentially distributed with mean Cl) and non-negative arc weight, (twl+ulrw) . In such graph, the subproblem is a minimum cost flow problem to send minimum cost flow from the source node to all its destination nodes with specified traffic demands. We use traditional minimum cost flow algorithm such as successive shortest path algorithm [9] to solve the problem.. the boundary *. fl = Cl. point,. fl. i. or. fl. i +1. , or at point. .. al ,((ul − vl ) ≠ 0) ul − vl. Subgradient Optimization Procedure From the weak Lagrangian duality theorem, Zdual(ρ) is a lower bound of the Problem (P) for any non-negative Lagrangean multiplier vector ρ = (s, t, u ,v) ≥ 0. Naturally, one wants to determine the largest lower bound byZlower_bound = max Zdual (ρ) ρ≥0 The subgradient method can be applied to solve (11). The solution to Problem (Dual_P) at iteration k of the subgradient optimization procedure is given below. In subgradient solution procedure, the Lagrangian multiplier vector ρ is updated by ρ k +1 = ρ k + θ k bk where b is a subgradient of Zdual(ρ) with vector size. Solving Subproblem 3 Subproblem S3 is a problem for decision variable y. It consists of |L| independent problems. For each link l ∈ L :. ∑ sw ywl − ∑ t wl y wl + (vl − ul ) f l ] min[ w∈W cl − f l w∈W. subject to (5) and (8). For different values of fl, the value of ywl for minimum objective function, denoted as ywl * ( fl ) may be different. As. |W+LW+L+L|. The step size. θk = 340. λk (UB − Z dual ( ρ )) bk. 2. θk. is determined by.

(4) Int. Computer Symposium, Dec. 15-17, 2004, Taipei, Taiwan.. not feasible, the following heuristic is applied to find a feasible solution. Proposed Lagrangean-based Heuristic Algorithm. UB is an upper bound obtained from a heuristic solution described in the next section and λk is a constant in a range from 0 to 2. The details of this procedure see below:. Based on the solution obtained from solving Lagrangean relaxation in each iteration. The traffic demand is then routed onto the network with cost assigned as the same as they assigned in the Lagrangean relaxation. Traffic demands are then routed onto the network for each SD pair sequentially (from short delay required connections to long delay required ones) by applying Dijkstra’s shortest path algorithm. The capacity of those arcs used by the above accepted paths are updated by subtracting the flow of this connection from the capacity of this link. If the utilization of a link will become greater than the best known lower bound (LB)×|Cl| when a further virtual circuit setups on it, the weight of this arc is replaced by multiply a constant term on its weight as a penalty for avoiding further setup paths on this link. The process continues until all of the traffic demands are satisfied or the network cannot accommodate the traffic request. A feasible solution is obtained in the former case.. Evaluation of the Feasible Schedule. Summary of Lagrangean Relaxation Method The algorithms are described below: LRM denotes the Lagrangean relaxation method.. 4. Lagrangean-based Algorithm. From the weak Lagrangian duality theorem, Zdual(ρ) is a lower bound of the Problem (P) for any non-negative Lagrangean multiplier vector ρ = (s, t, u ,v) ≥ 0. Naturally, one wants to determine the largest lower bound by. Z lower_bound = max Z dual ( ρ ) ρ ≥0. (11). Heuristic. Since the Lagrangean relaxation is obtained by the relaxation of some constraints from the problem formulation, the solution to the dual problem might be infeasible for the original primal problem resulting from dissatisfaction of those relaxed constraints. However, such solution can still be used as a base to develop efficient heuristic algorithms to seek feasible solutions and obtain upper bounds for the original problem. In practice, in each iteration of the subgradient solving procedure, the solution of Lagrangean relaxation is used to obtain a lower bound of the primal problem. In addition, we verify the feasibility of the solution in the constraints of primal problem. If the solution is feasible, it is used to calculate an upper bound of the primal problem (Actually it is an optimal solution.). If the solution is. 5. Experimental Results Performance Comparisons We first carried out numerical computation of the lower and upper bound values of α , the maximum aggregate arrival rates of a link divided by the capacity of each link, Cl , using our LRM-based method and a Linear Programming Relaxation (LPR)-based method. In the computation, we considered three widely used networks. They are: NSFNET with 14 nodes and 42 links; PACBELL with 15 nodes and 42 links; and GTE with 11 nodes and 46 links. In the LRM-based heuristic algorithm, we adopted a penalty term of 2. In addition, if the Lagrangean lower. 341.

(5) Int. Computer Symposium, Dec. 15-17, 2004, Taipei, Taiwan.. bound remains unimproved for 50 iterations (UC=50), the step size coefficient (λk ) would be divided by two. The simulation was written in the C language and terminated at the end of 2000 iterations and operated on a PC running Windows XP with a 1.8 GHz CPU power. In the LPR-based method, by removing Constraints (6) and (7), the original Integer Linear Programming (ILP) problem is relaxed to a Linear Programming (LP) problem. Thus, the solution to the relaxed problem is a legitimate lower bound of the original ILP problem. To obtain an upper bound, we also develop a corresponding heuristic algorithm. The algorithm ranks all SD pairs in accordance with the desired packet delay. The next feasible path founded in the LP solution is then assigned to the SD pair with the smallest packet delay. There may be multiple feasible paths for an SD pair; we select the shortest path with the largest xp value in the algorithm. The path assignment process repeats until either the traffic demands of all SD pairs are satisfied (i.e., feasible), or there is no remaining resource (i.e., infeasible). In the simulation, the LP problem was solved using the CPLEX software, operating in the same PC environment previously described. Numerical results for the NSFNET, PACBELL, and GTE are summarized in Table II, III, and IV, respectively. The traffic demands (i.e., the traffic arrival rate) for all SD pairs are randomly determined with their mean value shown in the first column of the tables. Moreover, the Gap in the third column of the tables is computed as the ratio of the difference of the upper and lower bounds to the lower bound in percentage. As shown in Table II for NSFNET, the LPR-based method reaches a low guarantee of 20% gap, incurring high CPU computation time. Compared to it, the LRMbased method achieves ideal lower and upper bounds (gap< 5%) under all four traffic demand cases except case 1. The algorithm also improves the CPU computation time by one order of magnitude. We discover that, even though both methods achieve optimal lower bounds, the LRM-based heuristic algorithm arrives at much improved upper bounds due to the use of the Lagrangean multipliers derived upon seeking the Lagrangean relaxation solution. In Table III for PACBELL, the LPR-based method reaches a low guarantee of 29% gap. Compared to it, the LRM-based method again achieves ideal lower and upper bounds (<8%). The LRM-based algorithm also improves the CPU computation time by two order of magnitude. It is worth mentioning that in the case of the mean traffic demand being equal to 3.0, while the LPRbased method fails to obtain a feasible solution, the LRM-based method arrives at the optimal solution. Finally, in Table IV for the GTE network, the LRMbased method outperforms the LPR-based method in both the solution superiority and the computation time in all traffic cases. Specifically, the LPR-based method again reaches fairly low guarantee of 27% gap. The method produces a non-optimal solution. 342. but with an improved guarantee of 13% gap. This justifies the viability of the LRM-based method for providing efficient QoS routing method.. 5. Conclusions and Future Works In this paper, we have improved a QoS routing problem using a Lagrangean Relaxation based approach augmented with an efficient primal Heuristic algorithm, called LRH. With the aid of generated Lagrangean multipliers and lower bound indexes, the primal heuristic algorithm of LRH achieves a near-optimal upper-bound solution. Our method has three major characteristics. First, we start to consider user’s perspective and system’s perspective jointly. Second, in our routing procedure, the candidate path set does not need to be prepared in advance and the best paths are generated while solving the subproblems in our approach. Third, our method can both provide the upper bound and lower bound to the problem, and this distinguishing feature can help us to verify the performance of our solutions. Future Works We are able to reconfigure the virtual topology to adapt to changing traffic patterns. Some reconfiguration studies on virtual networks have been reported before [10, 11, 12]; however, these studies assumed that the new virtual topology was known a priori, and were concerned with the cost and sequence of branch-exchange operations to transform from the original virtual topology to the new virtual topology. We propose a methodology to obtain the new virtual topology, based on optimizing a given objective function, as well as minimizing the changes required to obtain the new virtual topology from the current virtual topology. This approach would result in the minimum number of switch retunings, thus minimizing the number of disrupted virtual paths. Consequently, this approach also.

(6) Int. Computer Symposium, Dec. 15-17, 2004, Taipei, Taiwan.. minimizes the time it takes to complete the reconfiguration process. In the ideal situation, given a small change in the traffic matrix, we would prefer for the new virtual topology to be largely similar to the previous virtual topology, in terms of the constituent virtual paths and the routes for these virtual paths, i.e., we would prefer to minimize the changes needed to adapt from the existing virtual topology to the new topology. More formally, it would be preferable if a large number of the δ pl variables retain the same. values, then (12) become linear, i.e., if. ywl (1) − ywl (2) ≡ (1 − ywl (2)) ; then. λsd1. and. λsd2. , taken at two not-too-distant. corresponding. to. λsd2. which. matches. “closest” with the virtual topology corresponding to. λsd1. (based on our above definition of “closeness”).. Reconfiguration Algorithm 1).. We perform the following sequence of actions: F (1) and Generate formulations corresponding to traffic matrices. 2).. λ. and. F (2). λsd2. ,. respectively, based on the formulation in Section 3. Derive solutions S (1) and S (2) , corresponding to. F (1). values in. F (2) , respectively. Denote the variables’ S (1) as x p (1) and ywl (1) , and those in. S (2). x p (1). and. as. and. ywl (1) ,. value of the objective function for. OPT1 3).. 1 sd. respectively. Let the. S (1). OPT2 , respectively. F (2) to F '(2) ) by. and. S (2). be. and. Modify ( constraint. adding the new. α=OPT2. (10). This ensures that all the virtual topologies generated by would be optimal with regard to the objective. F '(2). function.. The new objective function for. Minimize: ∑∑ ywl (1) − ywl (2) w. F '(2). is. (12). l. Note that the mod operation,. function. If we assume that. ywl. x. ywl (1) = 0 ,. ywl (1) − ywl (2) ≡ ywl (2) . Hence, F '(2). may be. solved directly.. time instants. We assume that there is a certain amount of correlation between these two traffic matrices. Given a certain traffic, there may be many different virtual topologies, each of which has the same optimal value with regard to the objective function. But we will terminate after the first such optimal optimal solution is found. Our reconfiguration algorithm finds the virtual topology. else if. then. References. values in the two solutions, without compromising the quality of the solution (in terms of minimizing the congested link utilization). Let us consider the snapshot of two traffic matrices,. ywl (1) = 1 ,. , is a nonlinear. can only take on binary. 343. [1] B. Gavish and S.L. Hantler, “An Algorithm for Optimal Route Selection in SNA Networks”. IEEE Trans. on Communications COM-31, pp. 1154-1160, Oct. 1983. [2] L. Fratta and M. Cerla and L. Kleinrock, “The flow deviation method: An approach to tore-andforward communication network design”. Networks 3, pp. 97-133 1973. [3] F. Y. S. Lin and J. R. Yee, “A New Multiplier Adjustment Procedure for the Distributed Computation of Routing Assignments in Virtual Circuit Data Networks”. ORSA Journal on Computing, Vol. 4, No. 3, Summer 1992. [4] P. J. Courtois and P. Semal, “An Algorithm for the Optimization of Nonbifurcated Flows in Computer Communication Networks”. Performance Evaluation, pp. 139-152, 1981. [5]K. T. Cheng and F. Y. S. Lin, “Minmax End-toEnd Delay Routing and Capacity Assignment for Virtual Circuit Networks”. Proc. IEEE Globecom, pp. 2134-2138, 1995. [6]L. Kleinrock, “Queueing Systems”. WileyInterscience, New York, Vol.1 and Vol.2 19751976 [7] A. Shaikh, J. Rexford, and K. Shin, “Dynamics of quality-of-service routing with inaccurate linkstate information”. U. of MI Tech. rep. CSE-TR350-97, Nov. 1997. [8] B. Awerbuch et al., “Throughput- Competitive On-line Routing”. 34th Annual Symp. Foundations of Comp. Sci., Pala Alto, CA, Nov. 1993. [9]R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, “Network Flows: Theory, Algorithms, and Applications”. Prentice-Hall, 1993. [10] D. Bienstock and O. Gunluk, “A degree sequence problem related to network design”. Networks, vol. 24, no. 4, pp. 195–205, July 1994. [11]J. F. P. Labourdette, G.W. Hart, and A. S. Acampora, “Branch-exchange sequences for reconfiguration of lightwave networks”. IEEE Trans. Commun., vol. 42, pp. 2822–2832, Oct. 1994. [12] G. N. Rouskas and M. H. Ammar, “Dynamic reconfiguration in multihop WDM networks”. J. High Speed Networks, vol. 4, no. 3, pp. 221–238, 1995..

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