Topological Design of MPLS Networks

Transcription

1 6 number 6 capacity 6 capacity 6 fixed 6 capacity-dependent Topological Design of MPLS Networks M. Pióro Department of Communication Systems, Lund University Box 118, S Lund, Sweden A. Mysáek Warsaw University of Technology, Poland A. Jüttner, J. Harmatos, Á. Szentesi Ericsson Traffic Laboratory, Budapest, Hungary Abstract- The paper addresses the IP/MPLS network cost optimization problem consisting in selecting localizations of nodes and links, combined with links dimensioning. As MPLS allows for flexibility in routing of demands flows through the network, the considered problem is similar to the classical NP-hard topological design problems dealt with in the past, mostly in the context of the road traffic. We assume that the network nodes are composed of two disjoint sets: the set of access nodes (Label Edge Routers) and the set of transit nodes (Label Switching Routers). The access (end) nodes only generate demands and do not transit end-to-end demands flows, while the transit nodes do not generate demands but do transit the flows. The selection of the actually installed transit nodes and the links interconnecting (all) network nodes is the major subject of optimization. As the considered problem is hard, we discuss and propose branch-and-bound based solution methods. The effectiveness of the methods is illustrated by means of a numerical study. I. INTRODUCTION In the context of the telecommunications network planning, topological design refers to the problem of finding nodes and links locations, and corresponding routing of demands that minimize the resulting network cost, taking into account the cost incurred by installing nodes and links. The optimization problem we address in this paper, called the Transit Node and Link Localization Problem (TNLLP), is a (rather general) instance of the backbone network topology optimization problem. Our problem can be informally characterized as follows: For a given set of access nodes and the demand between each pair of access nodes find and locations of the actually installed transit nodes (in these nodes no traffic is originated, they only switch the traffic streams between access node pairs) of links connecting access nodes to transit nodes of links interconnecting transit nodes such that the demand is realized at a minimum total network cost. The network cost is composed of installation cost of each link and each transit node cost of each link. The work presented in the paper has been carried out in a cooperation between Warsaw Technical University and Ericsson Traffic Laboratory. The considered problem can be studied for MPLS-capable IP networks, with the access nodes representing ingress/egress Label Edge Routers (LER), and the transit nodes - Label Switching Routers (LSR). Since the evolving IP/MPLS networks will comprise large numbers of LERs and LSRs to be placed in many possible sites, the problem of optimal node and link location becomes important, especially for economical network extension. Of course, many other applications of the problem, also for other networking technologies, can be thought of. Although similar problems have been considered in the past, especially in the context of the road traffic (cf. [1] for references and discussion), it seems that there is still a lot of work to be done to find efficient solution methods applicable for large networks of realistic sizes. Since a problem very similar to TNLLP is known to be NP-complete [2], one cannot expect to find time efficient algorithms for exact solving of the latter problem. Roughly speaking, there are three basic approaches to TNNLP: branch-and-bound [3], [4], [5], cut-sa-turation heuristics [6], and greedy heuristics [7], [8]. The first approach, although exact, is in general too time-consuming. The cut-saturation methods do not treat the problem directly, so very inaccurate solutions can be yielded. Finally, the greedy algorithms, although fast, get stacked in local minima. Hence, in the paper we have chosen to study heuristic algorithms based on the branch-and-bound approach. Our numerical investigations show that the proposed approach is feasible for solving practical problem instances. The paper is a continuation of [1]. It is organized as follows. In Section II we give a formal statement of the considered problem. In Section III we present a branch-and-bound approach to the problem and its simplified variants. Heuristic methods for TNLLP are discussed in Section IV. Efficiency of the introduced methods is illustrated in Section V using numerical examples. We conclude the paper in Section VI. II. PROBLEM FORMULATION Consider two disjoint sets of nodes: access nodes labeled with w=1,2,...,w, and transit nodes labeled with v=1,2,...,v. The set of (undirected) links, labeled with e=1,2,...,e, is partitioned into

2 two subsets: the set of access links connecting access nodes to transit nodes, and the set of transit links interconnecting transit nodes. There are no links between the access nodes. The demand imposed on the network is realized by means of flows allocated to admissible paths. The individual demands are labeled with d=1,2,...,d; with each demand d there is associated its origin access node s d and its destination access node q d, demand volume h d, and a set of admissible paths labeled with j=1,2,...,m d. Each such path s at node s d, then traverses a non-empty subset of transit nodes, and ends at node q d. The volume h d of demand d is split into flows x dj allocated to the demand s paths. The paths are defined by the incidence coefficients a edj : a edj =1 if link e belongs to path j of demand d, and a edj =0 otherwise. The incidence of links and transit nodes is given by the incidence coefficients b ev : b ev =1 if link e is incident with transit node v, and b ev =0 otherwise. The access nodes are fixed (installed) and cost nothing. A transit node v can be provided or not. If provided, it costs l v. Also any link (access or transit) can be provided or not. In the former case it costs c e y e +k e (where y e is the capacity of link e), otherwise it costs nothing. Of course, if a link is provided then its end nodes must be provided as well. The network design problem considered in this paper is called the Transit Nodes and Links Localization Problem and can be formally stated as the following Mixed-Integer Program (MIP). TNLLP1 (link-path formulation) indices d=1,2,...,d demands j=1,2,...,m d paths for realizing flows of demand d v=1,2,...,v transit nodes e=1,2,...,e links constants h d volume of demand d a edj =1 iff link e belongs to path j of demand d c e cost of capacity unit of link e k e fixed cost of installing link e Y e upper bound for the capacity of link e b ev =1 iff link e is incident with transit node v l v fixed cost of installing transit node v G v upper bound for the degree of transit node v variables x dj flow realizing demand d allocated to path j (non-negative continuous variable) y e capacity of link e (non-negative continuous variable) 1 e =1 if link e is provided, 0 otherwise (binary variable) v =1 if node v is provided, 0 otherwise (binary variable) objective minimize C = e c e y e + e k e 1 e + v l v v (2.1) constraints j x dj = h d d=1,2,...,d (2.2) d j a edj x dj = y e e=1,2,...,e (2.3) y e Y e 1 e e=1,2,...,e (2.4) e b ev 1 e G v v v=1,2,...,v. (2.5) In TNLLP1 constraints (2.2) assure that the demands are realized, and constraints (2.3) and (2.4) that the capacity of a non-provided link is equal to 0. Constraint (2.5) assures that if a link is provided then also its end nodes are. The objective is to minimize the cost of all links and of transit nodes. The set of admissible paths for each demand d may be defined as the set of all paths between s d and t d, or it may be somehow limited (e.g. to at most two-hop paths). In the former case the values of m d can be very large, leading to an excessive amount of flow variables in the link-path problem formulation given above. If this is the case we can use the node-link problem formulation instead. The node-link formulation of TNLLP uses link flows instead of path flows and requires the flows to be directed. Hence with each link e we associate two directed arcs traversing the link in two opposite directions. Note that also the demands are directed. TNLLP2 (node-link formulation) indices: v=1,2,...,v transit nodes w=1,2,...,w access nodes e=1,2,...,e links t=1,2,...,t directed transit arcs (between transit nodes) f=1,2,...,f directed access arcs (between access and transit nodes) constants: h ww1 demand originating at access node w and destined for access node w1 H w = w1 h ww1 total demand originating at access node w b ev = 1 if node v is incident with link e, 0 otherwise b tv = -1 if transit arc t is incoming to transit node v = 1 if transit arc t is outgoing from transit node v b fv = -1 if access link f is incoming to transit node v = 1 if access link f is outgoing from transit node v b fw = -1 if access link f is incoming to access node w = 1 if access link f is outgoing from access node w a et = 1 if transit arc t is realized on link e, 0 otherwise a ef = 1 if access arc f is realized on link e, 0 otherwise variables: x tw flow realizing all demands originating at access node w on transit arc t (non-negative continuous variable) x fw flow realizing all demands originating at access node w on access arc f (non-negative continuous variable) y e capacity of link e 1 e =1 if link e is installed, 0 otherwise (binary variable) v =1 if node v is installed, 0 otherwise (binary variable) objective: minimize (2.1) constraints: t a et w x tw + f a ef w x fw = y e e=1,2,...,e (2.6) f b fw x fw = H w w=1,2,...,w (2.7) f b fw1 x fw = h ww1 w=1,2,...,w, w1=1,2,...,w (2.8) t b tv x tw + f b fv x fw = 0 v=1,2,...,v, w=1,2,...,w (2.9)

3 6 x 6 x 6 e 6 e 6 G y e M e 1 e e=1,2,...,e (2.10) e b ev 1 e G v v v=1,2,...,v. (2.11) In the formulation of the constraints in TNLLP2 only the following flow variables are used: tw for all pairs (t,w) such that t=1,2,...,t, w=1,2,...,w fw for all pairs (f,w) such that f=1,2,...,f, w=1,2,...,w and either access link f is outgoing from access node w (i.e. b fw =1) or link f is incoming to some other access node w1 with h ww1 >0 (b fw1 =-1 and h ww1 >0). In TNLLP2 each link e supports two directed arcs (either both access or both transit) whose numbers are specified by coefficients a ef and a et (respectively) equal to 1. Constraint (2.7) forces the total demand H w generated in access node w to flow out, constraint (2.8) - that the portion h ww1 of the flow originated at node w and destined for node w1 stays at w1, and constraint (2.9) - that no flow stays in a transit node. In order to illustrate the savings in the number of variables and constraints while using TNLLP2 instead of TNLLP1 consider a network with full demand matrix with W(W-1) entries. Then the number of flow variables K and the number of constraints M (constraints associated with links are not counted) are equal to: K=W(W-1)J, M=W(W-1) K=WV+W(W-1)V+WV(V-1), M=W+W(W-1)+WV=W(W+V) for TNLLP1 for TNLLP2. For example, putting W=100, V=10, J=100 (J is the number of paths for each demand in TNLLP1) we get K10 6 and M10 4 for TNLLP1, and K10 5 and M10 4 for TNLLP2 - a one order of magnitude gain. Note that in the optimal solution of TNLLP, the demands can be routed on single paths, so if a demand corresponds to an MPLS tunnel, the solution guarantees that the tunnel is realized on one path. This follows from the linearity of the capacitydependent cost used in the cost function: for each fixed configuration of nodes and links, the capacity dependent part of (2.1) is minimized by assigning the entire demand volume h d to one of its shortest paths with respect to link metrics c e. III. BRANCH AND BOUND APPROACH Let 1=(1 1,1 2,...,1 E ) and J=(J 1,J 2,...,J V) denote the vectors of links and nodes binary variables, respectively. Suppose that the set of link indices N={1,2,...,E} is partitioned into three disjoint subsets N u, N 0 and N 1, and -(N u,n 0,N 1 ) denotes the set of all binary E-vectors 1 such that en u implies 1 e =0, en 1 implies 1 e =1 (for en u the value of 1 e is either 0 or 1). Similarly, suppose that the set of transit nodes indices K={1,2,...,V} is partitioned into three disjoint subsets K u, K 0 and K 1, and let (K u,k 0,K 1 ) denote the set of all binary V-vectors J such that vk 0 implies J v=0, vk 1 implies J v=1 (for vk u the value of J v is either 0 or 1). Assume also that K 0 and N 0 are consistent in the sense that vk 0 and b ev =1 imply en 0 and consider a linear relaxation of TNLLP1 where the unspecified components of vectors 1 and J are continuous in [0,1]. The relaxed problem is as follows. TNLLP1 (linear relaxation) objective: minimize (2.1) constraints: (2.2)-(2.4), 1-(N u,n 0,N 1 ), (K u,k 0,K 1 ) and enu b ev 1 e g v v vk u (g v =G v - en0fn1 b ev ) (3.1) 0 1 e 1 en u (3.2) 0 v 1 vk u. (3.3) In the optimal solution of the above problem all constraints in (2.4) and in (3.1) are binding, so the problem becomes the simple uncapacitated flow task: objective: minimize e e y e + en1 k e + vk1 l v (3.4) where e = c e +(k e + vku b ev l v /g v )/Y for en u, e = c e for en 1, e = for en 0 (3.5) constraints: 1-(N u,n 0,N 1 ), (K u,k 0,K 1 ), (2.2)-(2.3), easily solved by allocating all demands to their shortest paths according to link metrics e. This allows for using the link-path formulation even for the case when all paths are admissible. Note that the quality of the lower bound calculated as the solution of the above problem is strictly related to the values of the constants Y e and G v : the lower the values the better the lower bound. Of course, the constants (being in fact the upper limits for the link capacities and the degrees of the nodes) cannot be decreased too much since then the original problem would be affected. Nevertheless, these limits can be deliberately decreased, even if this leads to a slightly different problem. For example we could assume the following values of the quantities in (2.4) and (3.1): is an access link: Y e is the total demand generated and terminated at its access end node is a transit link: Y e is the total demand generated by all access nodes divided by 2 v is equal to K u FK 1 (K u FK 1-1)/2 + W/4. The lower bound can be in many cases further improved by limiting the number of paths using a link (P e is the maximal number of flows using link e): TNLLP1 (equivalent formulation) objective: minimize e d j c e h d a edj x dj + e k e 1 e + v l v v (3.6) constraints: (2.2), (2.4), (3.1) and d x dj = 1 d=1,2,...,d (3.7) d j a edj x dj P e 1 e e=1,2,...,e. (3.8)

4 The lower bound resulting from the relaxation of the binary variables requirement leads to the following simple flow allocation task: objective: minimize d j ( e ed a edj )x dj (3.9) where ed =c e h d +(k e + v b ev l v /g v )/P for all e and d (3.10) constraints: (3.7), (3.8). Of course, the above task is solved immediately by assigning flows to the cheapest paths (the cost of path j of demand d is equal to e ed a edj ). The above described lower bounds are used by the MIP solvers based on the LP relaxation (as [9]). However, such general solvers cannot exploit the fact that the lower bound is most effectively computed applying the shortest path algorithm to the link metrics defined by (3.5) or (3.10). Therefore we have implemented our own branch-and-bound algorithm, called BB. The recursive form of BB is as follows. procedure BB (N u,n 0,N 1 ) if N u = L and cost (N 0,N 1 ) < C best then C best := cost (N 0,N 1 ); N 0 best := N 0 ; N 1 best := N 1 end else if L(N 0,N 1 ) C best then return else choose en u ; BB (N u \{e},n 0 F{e},N 1 ); BB (N u \{e},n 0,N 1 F{e}) end end { procedure } {function cost (N 0,N 1 ) computes the cost (lower bound) of sub-network for 1 defined by N 0 and N 1 ; {L(N 0,N 1 )= if the sub-network defined is not connected} Although the nodes variables J are not considered explicitly in BB, it is assumed in the calculations of the cost and the lower bound that if all links incident with a node are not provided in the current solution, then the node is not provided either (costs nothing). BB can of course be used directly in the case when the locations of transit nodes are fixed and only the locations of links must be optimized. Because the number of links is usually much larger than the number of nodes, a good approximate solution to TNNLP can be obtained with a two-step procedure (called BB4 in Section V). First, only the transit node locations are found, using an appropriate branch-and-bound procedure (similar to BB) under assumption that the links variables 1 are continuous, i.e. by not branching on the link variables. In the second step, the node locations are fixed and the corresponding locations of links can be found by another application of BB (to links only) or by using one of the heuristic methods described in Section IV. A branch-and-bound based approach for locating links is described in [4] and [5]. The lower bound used there, however, assumes that there is a direct demand between end nodes of the links which is not necessarily the case in IP/MPLS networks. IV. HEURISTIC APPROACH TO LINKS LOCATION Heuristic algorithms for finding sub-optimal links locations (with fixed nodes locations) can be devised using the following idea of [8]. Suppose that 1 k is the current link variables vector and y k =(y 1k,y 2k,...,y Ek ) are the resulting link capacities. Let e1 be the index of the link maximizing the following quantity, over all links e with 1 ek =1:, e (1 k ) = y ek F e (1 k - 1 ek ) - (c e y e k + k e ), (4.1) where F e (1) denotes the cost of the shortest path (with respect to unit capacity costs c e :s) between the end nodes of link e in the sub-network corresponding to vector 1. If gain, e1 (1 k ) is nonpositive the algorithm terminates, otherwise the capacity of link e1is rerouted to a shortest path, the capacities on all the links along this shortest path are increased by y e1k, link e1 link is removed (1 e1 k+1 =0), and the procedure is repeated. Four heuristic algorithms (H1-H4) based on extensions of the above idea are described in Section 4.2 of [1]. V. NUMERICAL RESULTS Below we report results for three artificially generated network structures with 7, 14 and 28 nodes (referred to as N7, N14 and N28, respectively). The networks are characterized in Table I. All links are potentially available and the unit capacity costs c e are proportional to the geographical lengths of links. The fixed installation cost is given by k e =10nc e (n is a parameter) multiplied by 3 for the access links, and by 2 for the transit links. The fixed installation cost is l v =10k for all transit nodes (k is another parameter). TABLE I TEST NETWORKS PARAMETERS W V F T D min h d max h d d h d N N N We have applied an exact branch-and-bound algorithm with three different branching strategies. In all three cases we first branch on nodes (in effect fixing the nodes variables either to 0 or 1) and then on links. Of course, if a node is eliminated all the links incident with the node are eliminated and not used in the current branch-and-bound sub-tree. The first strategy (BB1) is based on excluding nodes and links first, and then, after checking the sub-tree, including them back. The second method (BB2) evaluates the lower bounds (LB) of both branches of the

5 current sub-tree to decide whether to exclude or include the node/link being the root of the sub-tree (the possibility with the smaller LB is considered first). The third strategy (BB3) uses a dynamic list of all the branch-and-bound tree nodes, sorted according to their lower bounds; in each iteration the sub-tree with the lowest LB is taken. All the strategies give exact solutions, but they differ in the way of approaching the optimal solution. BB1 has small memory requirements, proportional to the number of links and nodes. BB2 needs more memory, and in each iteration it calculates LB two times. BB3 requires a lot of memory for large problems; it sometimes visits more subtrees than BB1 and BB2, but frequently the first encountered complete solution is optimal. To deal with not only small networks we implemented a heuristic branch-and-bound based algorithm, called BB4. First BB4 looks for the best (in terms of LB) configuration of transit nodes, then for the obtained set of nodes it performs an exact branch-and-bound on all links between the selected nodes. TABLE II RESULTS FOR TNLLP (COST IN 10 6 ) Network n k EX BB4 Difference Rel. dif. [%] N N N N N N N N N N N N N N N N N TABLE III RUNNING TIME FOR TNLLP [S] (SUN SPARC ULTRA-4) Network n k BB1 BB2 BB3 BB4 CPLEX N N N N N N N N N N N N N N N N N Table II shows results for the exact methods and for BB4, with respect to the latter. Table III presents running times for the four strategies and for the MIP solver of [9]. The symbol - indicates that the exact methods are not able to find the optimal solution in a reasonable time. Optimal solutions generally used one transit node, except for N7 with n=4, k=5 and 6, and N14 with n=4, k=7, and n=5, k=8. BB4 selected more nodes (2) for two cases: N7, n=3, k=6, and N7, n=4, k=6. VI. CONCLUDING REMARKS We have formulated a topological design problem applicable for IP/MPLS networks. The problem, consisting mostly in optimal locating of links and nodes, is NP-hard, and hence heuristic approaches must be considered, especially for medium and large networks (with more than 15 nodes, say), for which the exact branch-and-bound algorithms become too slow. We have proposed an approximate branch-and-bound algorithm based on separate branching on nodes and then on links, and used it for a set of network examples composed of access and transit nodes. Numerical examples show that the proposed approximate branch-and-bound approach is quite efficient as for a simple method. Of course, the proposed solution is not perfect and there is still room for its improvement. REFERENCES [1] M.Pióro, A.Jüttner, Á.Szentesi, J.Harmatos, P. Gajowniczek, and A.Mysáek, Topological Design of Telecommunication Networks - Nodes and Localization under De-mand Constraints, submitted to 17 th International Teletraffic Congress, Salvador de Bahia, September [2] D.S.Johnson, J.K.Lenstra, and A.H.G.Rinnoy Kan, The Complexity of the Network Design Problem, Networks, Vol.8, 1978, pp [3] Boyce, D.E., Farhi, A., Weischedel, R.: Optimal network problem: a branch and bound algorithm, Environment Planning, Vol , pp [4] H.H.Hoang, A Computational Approach to the Selection of an Optimal Network, Management Science, Vol.19, 1973, pp [5] R.Dionne, and M.Florian, Exact and Approximate Algorithms for Optimal Network Design, Networks, Vol.9, 1979, pp [6] W.Chou, and D.L.Sapir, A Generalized Cut-Saturation Algorithm for Distributed Computer Communications Network Optimization, IEEE International Conference on Communications, ICC'82, vol.2, New York, 1982, pp.4c.2/1-6. [7] B.Yaged, Jr., Minimum Cost Routing for Static Network Models, Networks, Vol.1, 1971, pp [8] M.Minoux, Network Synthesis and Optimum Network Design Problems: Models, Solution Methods and Applications, Networks, Vol.19, 1989, pp [9] ILOG Cplex 6.5, User s Manual, ILOG, 1999.