WAVELENGTH-DIVISION-MULTIPLEXING (WDM)

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1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 6, JUNE Logical Topology Design for Dynamic Traffic Grooming in WDM Optical Networks Chunsheng Xin, Member, IEEE, Bin Wang, Member, IEEE, Xiaojun Cao, Member, IEEE, and Jikai Li, Member, IEEE Abstract Traffic grooming in optical networks refers to consolidation of subwavelength client connections onto lightpaths. Depending on whether client connections are given in advance or randomly arrive/depart, traffic grooming is classified as static and dynamic. Dynamic traffic grooming has been traditionally performed through establishing/releasing lightpaths online. In this paper, the authors propose an alternate approach to design a static logical topology aprioriand then route randomly arriving client connections on it to avoid frequent lightpath setup/teardown. Two problems are considered: 1 minimize resource usage constrained by traffic blocking requirements and 2 maximize performance constrained by given resources. These are formulated as integer linear-programming ILP problems. The numerical results show that the resource usage dramatically decreases when the blocking requirement is relaxed, and the grooming performance slowly increases when given more resources. In addition, the number of ports at client nodes has more profound impact on traffic grooming than the number of wavelengths. Index Terms Dynamic traffic grooming, logical topology design, wavelength-division-multiplexing WDM optical networks. I. INTRODUCTION WAVELENGTH-DIVISION-MULTIPLEXING WDM optical networks offer huge transmission bandwidth and have become the transport infrastructure to connect networks such as Internet-Protocol IP networks and asynchronoustransfer-mode ATM networks, which are called as clients of WDM optical networks. Today s WDM optical networks use circuit switching and set up lightpaths between client nodes to transport client traffic. A lightpath is a logical link between a pair of source and destination client nodes sd-pair and is set up along a physical fiber route between the sd-pair, occupying one dedicated wavelength on each traversed link. The set of lightpaths among all client-node pairs forms a logical topology. There have been extensive studies on WDM optical networks assuming traffic demands from clients at the granularity of wavelength e.g., see [1], [2], and references therein. In prac- Manuscript received October 9, 2005; revised February 8, The work of B. Wang was supported in part by the U.S. Department of Energy Early Career Award DE-FG02-03ER The work of X. Cao was supported in part by NSF CAREER Award CNS C. Xin is with the Department of Computer Science, Norfolk State University, Norfolk, VA USA cxin@nsu.edu. B. Wang is with the Department of Computer Science and Engineering, Wright State University, Dayton, OH USA bwang@cs. wright.edu. X. Cao is with the Department of Networking, Security, and Systems Administration, Rochester Institute of Technology, Rochester, NY USA xiaojun.cao@rit.edu. J. Li is with the Department of Computer Science, The College of New Jersey, Ewing, NJ USA jli@tcnj.edu. Digital Object Identifier /JLT Fig. 1. Traffic-grooming illustration. tice, client-traffic demands are more likely at subwavelength granularity with heterogeneous bandwidth requirements and are usually consolidated onto lightpaths, as illustrated in Fig. 1, which is called traffic grooming in literature [3], [4]. In traffic grooming, the client traffic is assumed circuit traffic e.g., see [4] [10], such as ATM virtual path/circuits VP/VCs, IP flows [11], and label switched paths LSPs carrying IP packets [12]. We refer to them as client calls or client connections. Traffic grooming in mesh WDM optical networks can be classified as static and dynamic, depending on whether client calls are given in advance or randomly arrive/depart. Static traffic grooming can be formulated as an integer linear programming ILP problem to determine a set of lightpaths to accommodate a given set of client calls [5], [6]. Dynamic traffic grooming has been traditionally conducted by establishing/releasing lightpaths online to accommodate incoming client calls e.g., see [7] [10]. However, this approach requires the capability of dynamic lightpath provisioning, which may not be available in a large number of WDM optical networks for a while due to the cost and complexity of deploying dynamic provisioning technologies such as the generalized multiprotocol label switching GMPLS [13]. In this paper, we adopt an alternate approach to conduct a dynamic traffic grooming: Design a static logical topology apriori based on estimated traffic loads, and route each dynamically arriving client call on the established logical topology. There is no further lightpath setup/teardown after the initial establishment of the designed logical topology. The major benefits of this approach are the elimination of the large overhead of frequent lightpaths setup/teardown and elimination of packet delay in the traditional approach when invoking a new lightpath setup. Another benefit is the capability to conduct dynamic traffic grooming for optical networks without dynamic-lightpathprovisioning capability, where the traditional approach is not applicable /$ IEEE

2 2268 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 6, JUNE 2006 The classic logical-topology-design problem has been studied extensively in the literature e.g., see [14] [20], which designs an optimal logical topology with regard to some performance metrics e.g., minimizing the maximum link carried load in [15] to carry given packet traffic loads. Static traffic grooming can be seen as another type of logical topology design, which determines static lightpaths and allocates dedicated bandwidth for each given static client call. In this paper, we study logical topology design from a different perspective, which results in different design objectives and formulations. The client traffic in our work is dynamic circuit traffic, which is different from both classic logical topology design and static traffic grooming. For such traffic, the major performance metric is the blocking probability, which plays a fundamental role in our studies on logical topology design. Specifically, we study two related problems. 1 Minimize resource usage in the physical topology while constrained by traffic blocking probability, e.g., requiring the call blocking probability between an sd-pair to be lower than a give value. 2 Maximize call accepting probability or revenue while constrained by the physical topology and resources such as the number of wavelengths or ports at each client node. The first problem can be used by service providers that buy or lease network resource from other providers to construct their own networks, as well as for network planning of optical networks without dynamic provisioning capability, to satisfy a preset network performance threshold while utilizing a minimum amount of resource. The second problem is used to maximize the revenue or grooming performance on an existing optical network. We adopt ILP models to formulate both problems. In order to make the formulation tractable, we only consider single-hop traffic grooming to avoid nonlinear formulation which is computationally expensive of path blocking-probability computation. The single-hop grooming, which aggregates the traffic between an sd-pair onto the lightpaths directly connecting this sd-pair [9], is desirable when the electronic forwarding at intermediate client nodes is more expensive than optical bypassing. For instance, the packetover-wavelength POW approach of [11] uses a single-hop grooming to aggregate IP flows. The rest of this paper is organized as follows. Section II introduces a call-blocking model. Sections III and IV formulate the two logical-topology-design problems and develop corresponding heuristics. Then, we present our analysis and numerical results in Section V. Finally, Section VI concludes the paper. II. CALL-BLOCKING MODEL We introduce a call-blocking model to compute the call blocking probability on lightpaths between an sd-pair. We first discuss call blocking on one lightpath. We assume that the lightpath capacity is an integral number of basic bandwidth units, and the call data rates are in arbitrary number of basic bandwidth units, up to the lightpath capacity. Call arrival is assumed to follow the Poisson process with arbitrarily distributed holding times. Based on their quality-of-service QoS requirements, e.g., bandwidth requirement, calls are classified into classes. We list notations to be used for this section. L lightpath bandwidth an integer; S set of call classes; b s data rate of class-s s S calls. We assume b s L; ρ total offered load to the lightpath; probability that an arriving call is a class-s call; p s ρ s = p s ρ. Offered load of class-s calls; B s blocking probability of class-s calls on the lightpath. Lemma 1: Denote qj 0 j L as the probability that exactly j bandwidth units are used in the lightpath. Then, qj satisfies the following recursive relationship: j qj = b s ρ s qj b s 1 s S where qj =0for j<0, and L j=0 qj =1. The lightpath blocking probability for class-s calls can be computed as B s = L j=l b s +1 qj. 2 Proof: See [9]. Denote ρ = {ρ s s S} and B = {B s s S}.Weusea function B = fρ to denote the computation in 1 and 2. If an sd-pair has more than one lightpath in between, we assume that a call offered to this sd-pair sequentially attempts all lightpaths, until it is carried by one lightpath or blocked after trying all lightpaths. The ρ is redefined as the total offered load to this sd-pair, and, correspondingly, ρ s = p s ρ is the offered load of class-s calls to this sd-pair. Let B m = {B s m s S} denote the joint call-blocking probabilities of first m lightpaths between this sd-pair. Then, we have B 0 =1={1,...,1} B 1 = B = fρ. When m 2, the B m is obtained as follows. Let the operator denote the scalar multiplication of two vectors, i.e., ρ B = {ρ s B s s S}, and as the scalar division of two vectors, i.e., ρ B = {ρ s /B s s S}. The offered loads to the mth lightpath is the blocked traffic by the first m 1 lightpaths, which is ρ B m 1. With such offered loads, the call-blocking probabilities on the mth lightpath is computed as fρ B m 1 by Lemma 1, and the blocked traffic amount on the mth lightpath is equal to the offered loads multiplied by the blocking probabilities, i.e., ρ B m 1 fρ B m 1. 1 Since the calls sequentially try the m lightpaths, the joint blocking probabilities of all m lightpaths are equal to the blocked traffic after trying all m lightpaths, i.e., the traffic blocked by 1 Note that f produces a vector.

3 XIN et al.: LOGICAL TOPOLOGY DESIGN FOR DYNAMIC TRAFFIC GROOMING IN WDM OPTICAL NETWORKS 2269 the mth lightpath divided by the offered loads to this sd-pair, i.e., ρ. Therefore, we have B m = ρ B m 1 f ρ B m 1 ρ = B m 1 f ρ B m 1. 3 Let B ρ, mm 0 be the average blocking probability for all classes of calls when the offered load to the sd-pair is ρ, and there are m lightpaths between this sd-pair. Then, we have B ρ, m = s S ρ s B m s s S ρ s = s S p s B m s. 4 The B ρ, m can be calculated within OmL S time, where indicates taking the cardinality. III. TOPOLOGY DESIGN TO MINIMIZE RESOURCE USAGE 2 A. IIPFormulation We use the multicommodity flow model [22, Ch. 6] to formulate logical topology design for dynamic traffic grooming. The objective is to use a minimum resource for carrying the given offered loads while ensuring that the call-blocking probabilities be less than the given thresholds. Without loss of generality, this paper addresses the average blocking probability for all classes of calls instead of the blocking probability for each class of calls. Nevertheless, the formulations and heuristics developed in this paper can be adjusted with minor changes to address the latter case. As can be seen from Section II, the blocking-probability computation is nonlinear. If we directly formulate the blocking probability, we need to use the nonlinear-programming formulation, which, however, requires an intensive computation. Instead, we formulate the problem as an ILP problem in this paper, which is computationally more tractable than nonlinear programming. In order to apply the ILP formulation, we need to transform the nonlinear computation of blocking probability into linear operation. Our approach is to precompute the blocking probability. Specifically, we introduce a parameter M = min{m Bρ n,m < min ε, ε n for all sd-pair n} to denote the maximum number of lightpaths that could be set up between each sd-pair M can be reduced based on the problem size. For sd-pair n and its offered load ρ n, we can precompute the call blocking probability when there is one, two,...,orm lightpaths between this sd-pair, as Bρ n, 1, Bρ n, 2,..., Bρ n,m, and store them in a table as the input parameter to the formulation. As can be seen later on, the nonlinear computation of the blocking probability is then eliminated, and we can formulate the problem as an ILP problem. As illustrated in Fig. 2, we consider three types of resources: client-node ports CNPs e.g., linecards of routers, wavelengths, and optical-node ports ONPs e.g., optical transmitters and receivers. Each lightpath initiates from an outgoing 2 A preliminary version of this section has been presented in IEEE ICC 2005 [21]. Fig. 2. CNP and ONP traversed by lightpaths. CNP at the source client node, goes through an incoming ONP and an outgoing ONP at every intermediate optical node, and terminates at an incoming CNP of the destination client node. We develop formulations to minimize the number of CNPs, wavelengths, and ONPs, respectively, to set up the lightpaths constituting the logical topology. We introduce an artificial parameter W to denote the number of wavelengths on each fiber, which is used by the formulations to obtain the real number of wavelengths needed to set up the logical topology. Following are the parameters we need in the formulations. M number to denote the maximum number of lightpaths that could be set up between an sd-pair; W number of wavelengths on each fiber; W set of wavelengths i.e., {1,...,W}; J set of client nodes; N set of sd-pairs; K set of unidirectional links in the physical topology; T set of routes between all sd-pairs in the physical topology, which are used to set up lightpaths; ρ n offered load of sd-pair n n N, including all classes of calls; Bρ n,m call blocking probability between sd-pair n when there are m lightpaths in between [computed by 4]; upper bound of call blocking probability be- ε n ε A D tween sd-pair n n N; upper bound of the average call blocking probability for the entire network i.e., among all sdpairs; route-sd-pair incidence matrix, A tn =1 if the end nodes of route t is sd-pair n; otherwise, A tn =0; route-link incidence matrix, D tl =1if route t contains link l; otherwise, D tl =0; H, H client-node-sd-pair incidence matrices, Hjn = 1 H jn =1if the client node j is the source destination node of sd-pair n; otherwise, H jn =0 H jn =0. Let F m = m 0 m M and Q w = w 1 w W be two auxiliary parameters. In addition, we define a binary-matrix variable E, where E mn =1indicates that sd-pair n needs m lightpaths in the logical topology. 1 Minimize Number of CNPs: In addition to minimizing the total number of CNPs, sometimes service providers may also want to balance the CNPs distribution at client nodes, i.e., prevent a client node having so many CNPs, because a client node with so many CNPs is expensive. Let γ be a variable denoting the largest number of CNPs needed among all nodes.

4 2270 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 6, JUNE 2006 We can minimize the total number of CNPs or balance CNP distribution among all nodes as follows: Minimize 2 M F m E mn, or γ. 5 Remark: Because one and only one of E 0,n,...,E M,n is equal to one [see 6], M F me mn is the number of lightpaths between sd-pair n, which is equal to half of the total number of CNPs. Each of the two objectives is minimized separately to focus on a different performance metric. Subject to n N, n Nwith ρ n > 0, M E mn =1 6 M E mn Bρn,m ε n. 7 Remark: It can be seen that M E Bρ mn n,m represents the blocking probability of sd-pair n. If the number of lightpaths between sd-pair n is i, i.e., E in =1, then, by 6, all E mn =0 for m i, and M E Bρ mn n,m= E in Bρn,i= Bρ n,i. Therefore, 7 constrains that the call blocking probability of sd-pair n is not higher than ε n.note that B is a constant precomputed by 4 ρ n M E mn Bρn,m ε ρ n. 8 Remark: Since M E Bρ mn n,m is the blocking probability of sd-pair n see the preceding remark, ρ M n E Bρ mn n,m/ ρ n represents the average call blocking probability among all sd-pairs. Thus, 8 constrains that the average call blocking probability is not larger than ε j J, j J, M H jn F m E mn γ 9a M H jn F m E mn γ. 9b Remark: This constraint is used only when the objective is to minimize γ. From the remark of 5, M F me mn is the number of lightpaths between sd-pair n. Then, H jn M F me mn denotes the number of lightpaths originating from client node j. Thus, this constraint ensures that the number of lightpaths originating from and terminating at client node j is not larger than γ. 2 Minimize Number of Wavelengths or ONPs: a No wavelength conversion: Let C denote a binarymatrix variable for wavelength assignment, where C wt =1 indicates that the wavelength w on all links of route t is reserved for route t to set up a lightpath. Let ζ denote the largest wavelength number 3 of the assigned wavelengths on every 3 Wavelengths are numbered as 1, 2,...,W. link. We can minimize the number of ONPs or wavelengths as follows: Minimize 2 w,t,l C wt D tl +2 M F m E mn, or ζ. 10 Remark: l C wtd tl is the number of optical links traversed by a lightpath that occupies wavelength w on route t. Thus, excluding the first and last ONPs of all lightpaths, the total number of ONPs used for lightpaths is 2 w,t,l C wtd tl,as illustrated in Fig. 2. On the other hand, the number of first and last ONPs of all lightpaths is equal to twice of the number of lightpaths, i.e., 2 M F me mn [see the remark for 5]. Therefore, the summation of these two terms is the total number of ONPs used to set up the logical topology. Subject to 6 8 and the following constraints: w Wand l K, C wt D tl Remark: Any wavelength on a link can be reserved for at most one route used to set up a lightpath on that route M n N, C wt A tn. 12 F m E mn t T t T w W Remark: The left side is the number of lightpaths needed by sd-pair n [see the remark for 5] to meet the blockingprobability constraint. The right side is the number of wavelengths that are reserved to set up lightpaths for sd-pair n. w Wand t T, Q w C wt ζ. 13 Remark: This constraint is used only when the objective is to minimize ζ. a Full wavelength conversion: We redefine the wavelength-assignment matrix C as a three-dimensional binary matrix, where C l wt =1indicates that the wavelength w on link l is reserved for route t to set up a lightpath. Let G t denote a variable indicating the number of lightpaths to be established on route t. The logical topology design under full wavelength conversion can be formulated as follows: Minimize 2 w,t,l C l wtd tl +2 M F m E mn, or ζ. Subject to 6 8 and the following constraints: w Wand l K, CwtD l tl 1 14 n N, M t T and l K, t T F m E mn t T D tl G t G t A tn W w=1 C l wt 15a 15b w Wand t T and l K, Q w C l wt ζ. 16

5 XIN et al.: LOGICAL TOPOLOGY DESIGN FOR DYNAMIC TRAFFIC GROOMING IN WDM OPTICAL NETWORKS 2271 Remark: Equations perform a similar functionality as in the formulation without wavelength conversion. a Sparse wavelength conversion: The sparse wavelength conversion means that only some nodes in the network are capable of wavelength conversion [23]. For sparse wavelength conversion, in addition to the constraints in the full wavelength conversion, we need to put another constraint to formulate the wavelength assignment. Let R be a set and t, i, j Riff the node between two adjacent links i and j on route t does not have a wavelength conversion, where link i is the upstream link of j. The additional constraint is t, i, j R, C i wt C j wt. 17 Remark: If the wavelength w on the link i is reserved for route t, and there is no wavelength conversion from i to the downstream link j on route t, i.e., t, i, j R, then wavelength w on link j must also be reserved for route t. Algorithm 1 Logical topology design to minimize used resource 1 Φ=. /*empty set*/ for n Ndo Find m such that Bρ n,m ε n. if Bρ n,m > ε then Φ=Φ {n} /* add sd-pair n to Φ */ end if U n = m. end for 2 if ρ n Bρ n,u n / ρ n ε then The current logical topology meets the blocking probabilities requirements, with U n lightpaths between sd-pair n. Assign wavelengths to those lightpaths using a wavelength-assignment heuristic such as first-fit, and the algorithm terminates. end if 3 Find n Φ such that ρ n = max{ρ i i Φ}. Remove n from Φ. Find m such that Bρ n,m ε. U n = m. Goto Step 2. B. Heuristic For large networks, the ILP optimization may not be computationally viable. Hence, we develop a heuristic illustrated in Algorithm 1. In Step 3, the sd-pair is picked such that the calculated average blocking probability would be likely reduced more than picking other sd-pairs, as the offered load ρ n is a weight in such a computation. The algorithm is guaranteed to terminate, since at each iteration, one sd-pair is removed from Φ in Step 3. The running time is primarily on calculating the blocking probabilities Bρ n,m. Denote N = N and S = S, where indicates taking the cardinality. Equation 3 in Section II indicates that Bρ n, 1,..., Bρ n,m can be incrementally computed within OmLS time. In Algorithm 1, m M, where M is defined in Section III-A. Thus, Step 1 takes ONMLS time, and Step 3 takes ON + MLS time with at most N iterations. Thus, the total running time in the worst case is ON 2 + NMLS. IV. TOPOLOGY DESIGN TO MAXIMIZE THE PERFORMANCE AND REVENUE In this section, we assume that ONPs have been deployed for every wavelength of every link and use call accepting probability, i.e., 1 call blocking probability, as the major grooming performance metric. The parameters M, W, W, J, N, K, T, ρ n, Bρn,m, ε n, A, D, H, H, and variables E, C defined in Section III-A will be used. Note that the parameter W is not an artificial parameter any more, but the real number of wavelengths per fiber. The parameter ε is not used because it is the optimization objective for this problem. We also need the following parameters: θ j number of outgoing and incoming CNPs at client node j outgoing and incoming CNPs appear in pairs; δ n revenue or price to set up a lightpath for sd-pair n. We can maximize call accepting probability or revenue as follows: Maximize 1 ρ M ρ n 1 E mn Bρ n,m n or M δ n ρ n 1 E mn Bρ n,m. Remark: 1 M E mn Bρ n,m is the call accepting probability of sd-pair n [see the remark for 7]. Thus, the first objective is the average call accepting probability. The second is the revenue that resulted from accepted calls. Subject to: 1 equations 6 and 7 to guarantee the call accepting probability for each sd-pair within a certain level, while maximizing the average call accepting probability or revenue; 2 equation 9 changed to the following because different nodes may have different number of CNPs: j J, j J, M H jn E mn F m θ j 18a M H jn E mn F m θ j 18b 3 wavelength-assignment constraints [11 and 12 for no wavelength conversion, 14 and 15 for full wavelength conversion, and 14, 15, and 17 for sparse wavelength conversion]. We further develop a heuristic to solve this problem for large networks, as illustrated in Algorithm 2, where U n n N denotes the number of lightpaths to be set up between sd-pair n, and P n n N denotes the new call accepting probability or revenue if an additional lightpath is added between sd-pair n. The running time of the algorithm is dominated by two parts:

6 2272 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 6, JUNE 2006 computation of Bρ n,m and sorting of {P n }. Suppose that Algorithm 2 needs to compute Bρ 1, 1,..., Bρ 1,m 1 for sd-pair 1,..., Bρ N, 1,..., Bρ N,m N for sd-pair N. These computed Bρ n,m for each fixed m and n can be stored to eliminate repeated computations. From 3 in Section II, we know that the time to compute Bρ n, 1,..., Bρ n,m is OmL S. Thus, the total time to compute all B by Algorithm 2 is OL S N i=1 m i. The maximum number of lightpaths that can be set up is limited by the total number of CNPs, i.e., j J θ j.letγ= j J θ j.wehaveol S N i=1 m i OL S Γ. On the other hand, the sorting of {P n } takes O N log N, and the maximum number of iterations is less than j J θ j =Γ. Thus, the total running time of the algorithm is OL S Γ+Γ N log N. Fig. 3. A 14-node NSFNET. TABLE I MINIMIZE NUMBER OF CNPs Algorithm 2 Logical topology design to maximize call accepting probability or revenue 1 for n Ndo Find m such that Bρ n,m ε n. U n = m. Set up U n lightpaths using a routing and wavelength assignment RWA algorithm. If there is no enough wavelengths, or no enough ports to set up such lightpaths, terminate with failure. end for 2 for n Ndo if maximize grooming performance then P n = αρ n 1 Bρn,U n +1 + α else if maximize revenue then P n =δ n ρ n 1 Bρn,U n +1 + i n,i N i n,i N ρ i 1 Bρi,U i δ i ρ i 1 Bρi,U i. end if end for 3 Sort P 1,P 2,...,P N in the descending order. Denote the sorted series as P j1,p j2,...,p j N. for k =1to N do If both the source and destination nodes of sd-pair j k have a free port, then try to set up an additional lightpath for sd-pair j k. If success, let U jk = U jk +1 and jump out of the loop to Step 2. end for 4 Terminate with success. The average call accepting probability is α ρ n1 Bρ n,u n, and the revenue is δ nρ n 1 Bρ n,u n. V. N UMERICAL RESULTS We use the National Science Foundation Network NSFNET illustrated in Fig. 3 as the sample network. Each optical node is attached with one client node generating calls following the Poisson process with exponentially distributed holding times. In case of sparse wavelength conversion, nodes 4, 8, 9, 11, and 14 are assumed having a wavelength conversion. TABLE II MINIMIZE γ We use the first fit for wavelength assignment in heuristics. The offered load of each sd-pair ρ n is randomly distributed between 0 and 0.4 Erlangs. Each sd-pair uses one shortest route in the optical network to set up lightpaths. In the blocking-probability computation, the lightpath capacity L is assumed 100. The number of call classes S is also assumed 100; the data rate of class-s s S= {1,...,100} calls is assumed s i.e., b s = s; and the call data-rate distribution is assumed uniform i.e., p s =1/ S. In our experiments, the ILP models are quite efficient for medium-size networks such as the 14-node NSFNET. A. Minimize Resource Usage We assume blocking requirements ε n =0.2 for all n and ε = 0.03 moderate blocking. Table I illustrates the ILP results of minimizing the number of CNPs. The left column bold and italic indicates the ILP objective, and the right column shows γ, which is the largest number of CNPs among all nodes when optimizing the left column. Table II shows the ILP results of minimizing γ. Table III presents numerical results of minimizing the number of wavelengths and ONPs. Similar to Tables I and II, the bold and italic column is the ILP-optimization objective, and other columns are obtained when optimizing the bold column. We can see that the heuristic obtains a performance close to ILP models and may even obtain a better performance for nonoptimizing objective. For example, when minimizing ζ, the number of ONPs obtained by ILP is 1612 this number is suboptimal since it is not the optimizing objective, which is more than the number of ONPs obtained by the heuristic We have also found that when minimizing the number of wavelengths or ONPs, the number of CNPs is suboptimal. This is because when minimizing the number of wavelengths or ONPs, multiple short lightpaths on short routes may be set up in favor of one long lightpath on a long route, even though

7 XIN et al.: LOGICAL TOPOLOGY DESIGN FOR DYNAMIC TRAFFIC GROOMING IN WDM OPTICAL NETWORKS 2273 TABLE III MINIMIZE NUMBER OF ONPS Fig. 5. Minimize per-node CNPs. Fig. 4. Minimize number of CNPs. the latter can have the same contribution to reduce the traffic blocking probability as the former. An interesting observation in Table III is that the wavelength conversion exhibits no benefit on ILP results. Similar observations were also reported in [24], where only RWA was considered. To verify this observation, we have conducted 100 experiments using different parameters, and found that there is a little difference on ILP results for no conversion and full conversion. One possible reason is that wavelength conversion primarily reduces blocking due to the delay of the wavelength status updating in dynamic wavelength assignment, and thus not much useful in static wavelength assignment. Next, we examine the required resources to satisfy different blocking-probability constraints. Fig. 4 illustrates the minimum number of CNPs [obtained by 5] required to construct a logical topology to satisfy the varying blocking-probability constraints ε. When we require very light blocking probability ε =0.001 for incoming traffic, the logical topology needs a very large number of CNPs. If the blocking-probability requirement is relaxed, e.g., ε =0.005, which is still low, the number of CNPs dramatically decreases from 748 to 630. When the blocking-probability requirement is further relaxed, the number of CNPs further dramatically decreases, until ε is relaxed beyond 0.1, and then becomes flat. We can see that the heuristic performs very well and can obtain the same number of CNPs as by ILP in most cases. In fact, there is no difference between ILP and the heuristic when the blocking-probability constraint ε is higher than Fig. 5 plots the minimized γ largest number of CNPs among all nodes as a function of the blocking-probability constraints. Similar to the results in Fig. 4, γ also dramatically decreases when the blocking-probability requirement is relaxed. Once again, the heuristic performs close to the ILP model. Although there is a difference of 1 or 2 CNPs when ε is higher than 0.005, the ILP needs more total number of CNPs than the heuristic. Fig. 6. Fig. 7. Minimize number of wavelengths. Minimize number of ONPs. Figs. 6 and 7 show the results when minimizing the number of wavelengths and ONPs, respectively, assuming no wavelength conversion. Both the number of wavelengths and number of ONPs dramatically decrease when the blocking-probability requirement is relaxed from to 0.1 and then become flat when ε is relaxed beyond 0.1. In Fig. 6, there is a noticeable difference between ILP and the heuristic on the number of wavelengths needed to set up the logical topology. This is due to the benefit of optimal wavelength assignment by ILP against the first-fit wavelength-assignment algorithm used by the heuristic. In Fig. 7, there is a slight difference on the total number of ONPs obtained by the heuristic and ILP when the blockingprobability constraint ε is higher than 0.005, but it is illegible due to the large scale of Y -axis. Therefore, we explicitly draw the difference on the right Y -axis. The difference of the number of ONPs in Fig. 7 when ε =0.001 is because the heuristic needs much more lightpaths than ILP to satisfy the

8 2274 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 6, JUNE 2006 TABLE IV MAXIMIZE CALL ACCEPTING PROBABILITY AND REVENUE Fig. 9. Revenue versus wavelengths. Fig. 8. Revenue versus number ONPs. blocking-probability constraint, which can be seen from Fig. 4 the number of lightpaths is equal to half the number of CNPs. B. Maximize Performance or Revenue In this section, the number of wavelengths W is assumed to be 16. The number of both outgoing and incoming CNPs at every node is also assumed to be 16 i.e., θ j =16. The revenue to establish a lightpath for the sd-pair n, i.e., δ n,is set as the route length between sd-pair n, so that the longer the lightpath, the more the service provider charges to the client. We let ε n =0.2 for all n to ensure that the blocking probability for every sd-pair is below 0.2, while maximizing the overall accepting probability or revenue. As can be seen from the experiment results tabulated in Table IV, the wavelength conversion still does not bring much benefit in this scenario. Next, we change the number of per-node CNPs θ j W is fixed as 16 and examine the revenues. Fig. 8 plots the revenues as a function of θ j. When θ j is less than 13, there is no solution. The revenue increases when θ j is more than 20. However, the increment amount is very small and negligible as indicated by the flat curves in Fig. 8 when θ j increases beyond 20. We also examine the revenues as a function of the number of wavelengths W θ j is fixed as 16, as illustrated in Fig. 9. Similarly, when the number of wavelengths is larger than 20, the revenue does not change much. We can also see that the impact of the wavelength constraint on revenue is not as profound as the CNP constraint θ j. In particular, the revenue grows 7% when increasing the number of CNPs at the client nodes but less than 2% when increasing the number of wavelengths. The numerical results presented above are obtained based on a single set of traffic load. The simulations based on different sets of traffic loads indicate that although the specific numerical values e.g., revenue change with different sets of traffic loads, the growing trends and curve shapes are similar to those in Figs We have not used these values obtained on different Fig. 10. Fig. 11. Minimize ONPs. Maximize revenue. traffic loads to draw confidence intervals in Figs. 4 9, because different sets of traffic loads result in a large variance in confidence intervals, which obscures the real growing trends and curve shapes of the data. However, we present the performance difference between the ILP and heuristics with confidence intervals in Figs. 10 and 11 to observe the performance of heuristics under different sets of traffic loads. Because the performance difference between ILP and heuristics is small for any set of traffic load, the variance of confidence intervals is not large. Under different sets of traffic loads, one can see that our heuristics can also obtain a performance close to ILP. VI. CONCLUSION We have studied how to design a static logical topology for dynamic traffic grooming and formulated them as ILP problems with heuristics for large networks. The formulations are

9 XIN et al.: LOGICAL TOPOLOGY DESIGN FOR DYNAMIC TRAFFIC GROOMING IN WDM OPTICAL NETWORKS 2275 efficient for small- to medium-size networks, and the heuristics perform close to ILP in most of the cases. Our simulation results and analysis indicate that the resource usage dramatically decreases when the blocking requirement is relaxed, and the grooming performance slowly increases when given more resources. We have also found that the number of ports at client nodes has a more profound impact on traffic grooming than the number of wavelengths. [22] M. Gordran and M. Minoux, Graphs and Algorithms. Hoboken, NJ: Wiley, [23] S. Subramaniam, M. Azizoglu, and A. Somani, All-optical networks with sparse wavelength conversion, IEEE/ACM Trans. Netw., vol. 4, no. 4, pp , Aug [24] N. Wauters and P. Demeester, Design of the optical path layer in multiwavelength cross-connected networks, IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp , Jun REFERENCES [1] A. E. Ozdaglar and D. P. Bertsekas, Routing and wavelength assignment in optical networks, IEEE/ACM Trans. Netw.,vol.11,no.2,pp , Apr [2] X. Chu, B. Li, and Z. Zhang, A dynamic RWA algorithm in a wavelengthrouted all-optical network with wavelength converters, in Proc. IEEE Infocom, 2003, pp [3] E. Modiano and P. Lin, Traffic grooming in WDM networks, IEEE Commun. Mag., vol. 39, no. 6, pp , Jul [4] R. Dutta and G. N. Rouskas, Traffic grooming in WDM networks: Past and future, IEEE Netw., vol. 16, no. 6, pp , Nov./Dec [5] K. Zhu and B. Mukherjee, Traffic grooming in an optical WDM mesh network, IEEE J. Sel. Areas Commun., vol. 20, no. 1, pp , Jan [6] J. Q. Hu and B. Leida, Traffic grooming, routing, and wavelength assignment in optical WDM mesh networks, in Proc. IEEE Infocom, 2004, pp [7] H. Zhu, H. Zang, K. Zhu, and B. Mukherjee, A novel generic graph model for traffic grooming in heterogeneous WDM mesh networks, IEEE/ACM Trans. Netw., vol. 11, no. 2, pp , Apr [8] C. Ou, K. Zhu, H. Zang, L. Sahasrabuddhe, and B. Mukherjee, Traffic grooming for survivable WDM networks Shared protection, IEEE J. Sel. Areas Commun., vol. 21, no. 9, pp , Nov [9] C. Xin, C. Qiao, and S. Dixit, Traffic grooming in mesh WDM optical networks Performance analysis, IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp , Nov [10] W. Yao and B. Ramamurthy, A link bundled auxiliary graph model for constrained dynamic traffic grooming in WDM mesh networks, IEEE J. Sel. Areas Commun., vol. 23, no. 8, pp , Aug [11] J. Bannister, J. Touch, A. Willner, and S. Suryaputra, How many wavelengths do we really need? A study of the performance limits of packet over wavelengths, Opt. Netw. Mag., vol. 2, no. 2, pp. 1 12, Apr [12] E. C. Rosen, A. Viswanathan, and R. Callon, Jan Multi-Protocol Label Switching Architecture. IETF RFC [Online]. Available: [13] E. Mannie et al., Oct Generalized Multi-Protocol Label Switching GMPLS Architecture. IETF RFC [Online]. Available: [14] Z. Zhang and A. S. Acampora, A heuristic wavelength assignment algorithm for multihop WDM networks with wavelength routing and wavelength re-use, IEEE/ACM Trans. Netw., vol. 3, no. 3, pp , Jun [15] R. Ramaswami and K. Sivarajan, Design of logical topologies for wavelength-routed optical networks, IEEE J. Sel. Areas Commun., vol. 14, no. 5, pp , Jun [16] R. Krishnaswamy and K. Sivarajan, Design of logical topologies: A linear formulation for wavelength-routed optical networks with no wavelength changers, IEEE/ACM Trans. Netw., vol. 9, no. 2, pp , Apr [17] B. Mukherjee, D. Banerjee, S. Ramamurthy, and A. Mukherjee, Some principles for designing a wide-area WDM optical network, IEEE/ACM Trans. Netw., vol. 4, no. 5, pp , Oct [18] D. Banerjee and B. Mukherjee, Wavelength-routed optical networks: Linear formulation, resource budgeting tradeoffs, and a reconfiguration study, IEEE/ACM Trans. Netw., vol. 8, no. 5, pp , Oct [19] A. Gencata and B. Mukherjee, Virtual-topology adaptation for WDM mesh networks under dynamic traffic, IEEE/ACM Trans. Netw., vol. 11, no. 2, pp , Apr [20] A. Brzezinski and E. Modiano, Dynamic reconfiguration and routing algorithms for IP-over-WDM networks with stochastic traffic, in Proc. IEEE Infocom, 2005, pp [21] C. Xin and B. Wang, Logical topology design for dynamic traffic grooming in mesh WDM optical networks, in Proc. IEEE ICC, 2005, pp Chunsheng Xin S 02 M 03 received the Ph.D. degree in computer science from the State University of New York at Buffalo in From 2000 to 2002, he was a Research Co-Op with Nokia Research Center, Boston, MA. Since 2002, he has been an Assistant Professor with the Computer Science Department, Norfolk State University, Norfolk, VA. His research interests include optical networks, programmable radio wireless networks, and performance evaluation and modeling. Bin Wang M 00 received the Ph.D. degree in electrical engineering from the Ohio State University, Columbus, in In September 2000, he joined the Department of Computer Science and Engineering, Wright State University, Dayton, OH, where he is currently an Associate Professor. His research interests include wavelength-division multiplexing optical networks, wireless and mobile networks, network information security, stochastic modeling and queuing analysis of systems, simulation optimization, and network protocol development. Dr. Wang received the US Department of Energy Early Career Award in Xiaojun Cao M 02 received the B.S. degree from Tsinghua University, Beijing, China, in 1996, the M.S. degree from the Chinese Academy of Sciences, Beijing, in 1999, and the Ph.D. degree from the State University of New York at Buffalo in He is currently an Assistant Professor in the Department of Networking, Security, and Systems Administration, Rochester Institute of Technology, Rochester, NY. His research interests include modeling, analysis, and protocol/algorithm design of communication networks. Dr. Cao received the National Science Foundation CAREER award in Jikai Li S 02 M 04 received the Ph.D. degree in computer science from the State University of New York at Buffalo in He is currently an Assistant Professor in the Department of Computer Science, College of New Jersey, Ewing. His research interests include optical networks, optical burst switching networks, scheduling, traffic engineering, and high-speed network protocols.

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