Minimal Delay Traffic Grooming in WDM Optical Star Networks

Photon Netw Commun (2006) 11:323–330DOI 10.1007/s11107-005-7359-z

ORIGINAL ARTICLE

Minimal delay traffic grooming in WDM optical star networks

Hongsik Choi · Nikhil Grag · Hyeong-Ah Choi

Received: 10 May 2005 / Revised: 22 December 2005 / Accepted: 22 December 2005© Springer Science+Business Media, Inc. 2006

Abstract All-optical networks face the challenge of reduc-ing slower opto-electronic conversions by managing assign-ment of traffic streams to wavelengths in an intelligentmanner, while maximizing the bandwidth resources utiliza-tion. This challenge becomes harder in networks closer to theend users that have insufficient data to saturate single wave-lengths as well as traffic streams outnumbering the usablewavelengths. Traffic grooming has been proposed as a possi-ble solution in the network closer to the end users. However,it requires costly traffic analysis at access nodes. We studythe problem of traffic grooming that reduces the need to ana-lyze traffic, for a class of network architecture mostly used byMetropolitan Area Networks; the star network. We first provethat the problem is NP-hard, then provide an efficient greedyheuristics that can be used to intelligently groom traffics at theLANs to reduce latency at the access nodes. Simulation re-sults show that our greedy heuristics achieves a near-optimalsolution.

Keywords Traffic grooming · WDM star networks ·NP-complete · Greedy heuristics

H. Choi(B)Department of Computer Science, Virginia CommonwealthUniversity, Richmond, VA 23284, USAe-mail: [email protected]

N. Grag · Hyeong-Ah ChoiDepartment of Computer Science, George Washington University,Washington, DC 20052, USAe-mails: {nikhil, hchoi}@seas.gwu.edu

Introduction

WDM networks can be viewed as a Wide Area Network(WAN) that consists of a number of Metropolitan Area Net-works (MANs) that in turn consist of Local Area Networks(LANs). The local area network nodes are connected to theend users and currently require less data throughput capacityas individual users do not engage in high-speed connectionsthat fully utilize the fiber bandwidths. WANs are increas-ingly becoming all-optical and are capable of handling tera-bits of data. With the emergence of all-optical LANs usingthe WDM technology [1], the last-mile-delay in communica-tion is further reduced. However, there are several hurdles yetto be crossed before there can be a truly all-optical network.

An all-optical network is capable of providing large datarates at the cost of omitting the fine granularity of packetswitching. This is acceptable for WANs that mostly performthe task of routing traffic meant for edge nodes inside someaccess network, and hence do not require individual packetanalysis at any stage. When we consider an access (MAN)network, the inability of the local traffic to maximally utilizethe full potential of the large bandwidth of an optical fibermakes it desirable to groom several traffic streams into a highcapacity wavelength channel.

The need for traffic grooming [2–5] is realized even morewhen a large user population can use only a limited poolof wavelengths. On the other hand, this particular aspect oftraffic management makes it difficult to have a fully opti-cal end-to-end lightpath. The access nodes then become thebottleneck point in the networks since they have to performslow opto-electrical conversions to separate the different traf-fic streams embedded in a single wavelength, but meant fordifferent destinations across the WDM network and pos-sibly other nodes inside the access network. Reduction ofsuch delay on the access nodes would need an elimination or

324 Photon Netw Commun (2006) 11:323–330

minimization of the requirement of electronic conversions.With the current level of technology it is not possible toanalyze data in the optical domain, so total elimination ofelectronic conversion would be impossible if traffic analysisis necessary. In such a case, clearly, reducing the number ofwavelengths that require packet level of switching is the otheroption. We will consider in this paper optimizing packet levelswitching to reduce the problem into grooming the traffic ina way that utilizes least number of wavelengths possible tocarry such mixed traffic.

Though several research efforts have addressed the prob-lem of traffic grooming for ring networks [5–7], and optimaltraffic grooming in WDM networks [8], none of the formu-lations have focused on metropolitan area networks exclu-sively, or exploited the network architecture to minimize thelatency occurred due to traffic grooming. In this paper, weconsider the traffic grooming problem for a single-hop net-work, typically a star network, an architecture suitable for ametropolitan and local area network. In such an architecture,the access node forms the nucleus of the network, with all theedge nodes around it. The access node is responsible for ana-lyzing the traffic that has to go out to the WAN, or redirectedto other nodes in the access network. And the only opto-elec-tronic conversion is done at the hub (access) node. The traffichas to be thus groomed in order to minimize the mixed wave-lengths requiring conversions at the hub node. We will termthis problem as the Minimal Cost Traffic Grooming Problem(MCTG). The problem turns out to be NP-hard, a complex-ity we will show in the paper, so we devise a simple greedyheuristics that performs within twice the optimal solution.

In Section Network model and problem formulation, wewill formally define the MCTG problem and reduce it to agraph-theoretical problem that is easier to analyze. We thenprovide proof of its complexity in Section Problem com-plexity and discuss a polynomial time heuristics and definea bound on its performance in Section A greedy heuristic.The numerical results for the heuristics in comparison to theoptimal solutions are also discussed. Finally, the conclusionfollows in the last section.

Network model and problem formulation

We consider the following WDM network for our problem.Consider G to be a star network with node set {v0, v1, . . . , vn},where v0 denotes the hub node. Each node vi , for 1 ≤ i ≤ n,is connected to v0 by two directed links (one in each direc-tion), and each link supports W wavelengths, and furthereach wavelength can support up to g low-rate circuits. Thehub node is equipped with 2nW LTEs (one for each channelin each direction) and a DCS. Lightpath Terminating Equip-ment (LTE) is an add-drop multiplexer, capable of pullingan individual wavelength signal from a multiplexed signal

stream, and inserting a different signal down the line, and aDigital Cross Connect Switch (DCS) is the actual equipmentthat provides groomed traffic. Thus, the hub node is capableof analyzing each traffic stream from any groomed channeland switching it back to a different groomed channel.

A traffic demand for the network is denoted by a ma-trix T [(t (i, j)], 1 ≤ i, j ≤ n, where t (i, j) denotes thetraffic (the number of low-rate circuits) from i to j , suchthat �(�{t (i, j)|1 ≤ j ≤ n})/g� ≤ W, (1 ≤ i ≤ n), and�(�{t (i, j)|1 ≤ i ≤ n})/g� ≤ W, (1 ≤ j ≤ n). The WANconnected to the star network can be assumed to be one of thenodes connected to the hub node. For the rest of the paper, weare concerned with costs in time related to network traffic.

Minimal cost traffic grooming (MCTG) problem

Given such a network G, we define the MCTG problem to bethe problem of minimizing the total number of circuits thatneed to be groomed at the hub node, i.e., of maximizing thetotal number of circuits that can be carried without optical-electronic-optical(O/E/O) conversion in between source anddestination. More precisely, the problem is to define an n ×ndecision matrix A[a(i, j)], where a(i, j) = 1 if and only ifthe traffic stream t (i, j) is chosen to be groomed such that

(1) a(i, j) ∈ {0, 1} for 1 ≤ i, j ≤ n,(2) �(�{t (i, j)a(i, j)|1 ≤ j ≤ n})/g� ≤ W − �{1 −

a(i, j)|1 ≤ j ≤ n}, (1 ≤ i ≤ n),(3) �(�{t (i, j)a(i, j)|1 ≤ i ≤ n})/g� ≤ W − �{1 −

a(i, j)|1 ≤ i ≤ n}, (1 ≤ j ≤ n),(4) satisfying (1)–(3), �1≤i, j≤nt (i, j)a(i, j) is minimized.

Constraints (2) and (3) ensure the availability of at leastone wavelength per non-groomed traffic stream, on each out-going link and incoming link of a node respectively. Con-straint (4) is equivalent to maximizing the total number ofcircuits that can be carried without O/E/O conversion in be-tween source and destination. Constraint (4) can be changeddepending on the viewpoint of optimization as follows: first,if the delay is proportional to the number of wavelengths togo through O/E/O conversion, maximizing the number ofwavelengths that carry a single traffic stream is the appropri-ate goal of optimization. Then constraint (4) can be rewrit-ten as �1≤i, j≤n(1 − a(i, j)) while satisfying constraints (1)through (3). Second, if the delay is proportional to both thenumber of wavelengths to go through O/E/O conversion andthe number of low level circuits to be groomed, one wantto maximize the number of wavelength that carry a singletraffic stream and yet minimize the number of traffic to begroomed at the hub. We, thus, have two optimization criteria.Depending on which criteria is important, we can change theconstraint (4) to either (a) or (b) as follow: (a) maximizing�1≤i, j≤n(1−a(i, j)) while satisfying constraints (1) through

Photon Netw Commun (2006) 11:323–330 325

DCS

LTE

LTE

LTE

LTE

LTE

LTE

LTE

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LTE

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LTE

LTE

LTE

LTEV1

V2

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V4 V4

V1

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OXC

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16 16 16 16 16 16 16 16

Fig. 1 Traffic grooming in star network

(3) and minimizing �1≤i, j≤nt (i, j)a(i, j). (b) minimizing�1≤i, j≤nt (i, j)a(i, j)while satisfying constraints (1) through(3) and maximizing �1≤i, j≤n(1 − a(i, j)).

Note that in the above formulation, for any t (i, j) = 0, anyoptimal solution should assign a(i, j) = 1. The followingexample illustrates the formulation of the MCTG problem.

Consider an example of the star network shown in Fig. 1.The four nodes are attached to the hub node 0. It is assumedthat W = 2 and g = 16. The traffic demand T [t (i, j)]is given as: t (1, 2) = 7, t (1, 3) = 6, t (1, 4) = 9, t (2, 1)

= 2, t (2, 4) = 8, t (3, 1) = 8, t (3, 2) = 5, t (3, 4) = 8,and t (4, 3) = 6. One feasible solution A[a(i, j)] is thata(1, 4) = 0, a(2, 1) = 0, a(3, 2) = 0, a(4, 3) = 0, andother a(i, j)s are all 1. With this arrangement, the four wave-lengths (channels) carrying traffic between (1, 4), (2, 1), (3,2), and (4, 3) do not require any traffic analysis at the hubnode. All the traffic on these wavelengths will be switched totheir corresponding destination nodes as soon as it is receivedat the hub node. On the other hand, an optimal solution wouldbe a(1, 4) = 0, a(2, 1) = 0, a(3, 1) = 0, a(4, 3) = 0, andthe rest of them being equal to 1. This reduces the numberof grooming circuits by three from the non-optimal solution.This can be interpreted that the optimal solution carries threemore circuits with transparent wavelengths than the non-opti-mal trivial solution.

Next, we introduce a graph-theoretic formulation of theMCTG problem and show its equivalence to the MCTG prob-lem. For a given graph G(V, E), E(v) denotes the set ofedges in E incident on v in the rest of the paper.

Maximal weighted local constraint subgraph (MWLCS)problem

Given an integer C and a bipartite graph G = (X, Y ; E),where each edge e ∈ E is associated with an integer w(e)and each vertex v ∈ X ∪ Y is associated with an integerα(v) such that �(�e∈E(v)w(e))/C� ≤ α(v) for each v ∈X ∪ Y , the problem is to find a subset E0 ⊆ E of edges suchthat �e∈E0w(e) is maximized, and �(�{w(e)|e ∈ E(v) \E0})/C� ≤ α(v) − |E(v) ∩ E0| for each v ∈ X ∪ Y .

Theorem 1 The MCTG problem is equivalent to theMWLCS problem when α(v)s are equal for all v ∈ X ∪ Y .

Proof The equivalence can be verified by transforming anyinstance of the MCTG problem to a corresponding instanceof the MWLCS problem. Recall that MCTG problem is pre-sented in terms of G, A = {a(i, j)}, T = {t (i, j)}, g, n,and W , and MWLCS problem is presented in terms of G =(X, Y ; E), w(e), α(v), C , and E0. For any instance G(V, E)

of MCTG, the transformation to the MWLCS instance G ′(X, Y, E ′) can be done by keeping X = V −{v0}, Y = V −{v0}, E ′ = {(i, j)|t (i, j) = 0} and w(i, j) = t (i, j), wherei ∈ X, j ∈ Y ; C = g; and α(v) = W for each v ∈ X ∪ Y .By defining E0 = {(i, j)|a(i, j) = 0}, it can be verified eas-ily that the feasibility constraint of MWLCS reduces to thesecond and third constraint of the MCTG problem. Similarly,the optimization objective also becomes equivalent. The in-verse transformation from MWLCS to MCTG is also thesame. �


Fig. 2 Transformation of3-Partition problem intoMWLCS problem 1 i 3m

1 m 1 i 3m

(m-1)B + 2g - (m-1)a

(m-1)B + 2C - (m-1)a

(m-1)B + 2g - (m-1)a

ia

ia1a 1a

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Problem complexity

The problems we described in the last section are not easyto solve and turn out to be NP-complete. We prove herethat the decision version of the MWLCS problem (calledD-WLCS problem) is NP-complete, consequently provingNP-hardness of the MWLCS problem. The D-WLCS prob-lem is stated as: given an integer C , a bipartite graph G withw(e) and α(v) defined similarly as in the MWLCS problem,and an integer γ , the problem is to decide whether there existsa subset E0 ⊆ E such that

1. �(�{w(e)|e ∈ E(v) \ E0})/C� ≤ α(v) − |E(v) ∩ E0|for each v ∈ X ∪ Y ,

2. �e∈E0w(e) ≥ γ .

The proof of NP-completeness is based on the followingknown NP-complete problem.

3-Partition problem

Given a set A = {a1, . . . , a3m} of 3m integers such thatB/4 < ai < B/2 for each ai (1 ≤ i ≤ 3m) where B =�1≤i≤3mai/m, the problem is to decide whether A can be par-titioned into disjoint m sets A1, . . . , Am such that�ai ∈A j ai =B for each 1 ≤ j ≤ m.

Theorem 2 The D-WLCS problem is NP-complete.

Proof It is easy to see that the D-WLCS problem can besolved non-deterministically in polynomial time, hence weonly discuss a polynomial time transformation of the 3-Par-tition problem to the D-WLCS problem.

Let A be an instance to the 3-Partition problem, then aninstance to the D-WLCS problem is constructed as follows,

by first assuming B < C < (m − 1)B. The vertex set X ∪ Yis defined as X = {xi |1 ≤ i ≤ 3m} and Y = {yi |1 ≤i ≤ m} ∪ {zi |1 ≤ i ≤ 3m}. The edge set is defined asE = {(xi , y j )|1 ≤ i ≤ 3m, 1 ≤ j ≤ m} ∪ {(xi , zi )|1 ≤i ≤ 3m}. The weight w(e) of each edge e is defined as:(i) w(xi , y j ) = ai for each edge of type (xi , y j ) and (ii)for each edge of type (xi , zi ), 1 ≤ i ≤ 3m, w(xi , zi ) =(m −1)B +2C − (m −1)ai . Let α(v) = �(m −1)B/C�+3for all v ∈ X ∪ Y . Finally, γ is defined to be m B. It can beverified that �(�e∈E(v)w(e))/C� ≤ α(v) for each v ∈ X ∪Yas follows: Since it is clear for v ∈ Y , let us consider vi ∈ Xonly. �(�e∈E(vi )w(e))/C� = �mai +(m −1)B +2C −(m −1)ai/C� = �(m − 1)B + 2C + ai/C� = �(m − 1)B/C� +2 + �ai/C� ≤ �(m − 1)B/C� + 3.

Figure 2 shows an example to construct a bipartite graphfor a 3-partition problem instance.

We now show that there exists a desired partition of A intoA1, . . . , Am if and only if there exists a solution E0 ⊆ E tothe D-WLCS problem such that �e∈E0w(e) = m B.

Let A1, . . . , Am be a solution to the 3-Partition prob-lem. Without loss of generality, assume that a1, a2, a3 ∈A1, a4, a5, a6 ∈ A2, . . . , a3m−2, a3m−1, a3m ∈ Am . We canthen construct an edge set E0 such that E0 = {(x3i−2, yi ),

(x3i−1, yi ), (x3i , yi )|1 ≤ i ≤ m}. Clearly, �e∈E0w(e) =m B, and E0 is a feasible solution.

Suppose the D-WLCS problem has a feasible solution,say E0. From the construction of G, we note that in any fea-sible solution, the number of edges incident on each xi , 1 ≤i ≤ 3m, is at most one. To obtain �e∈E0w(e) ≥ m B,the only possible arrangement is to have exactly one edgeof weight ai in E0 for each 1 ≤ i ≤ 3m. Furthermore,�e∈E(yi )∩E0w(e) = B for each 1 ≤ i ≤ m. This providesa solution to the 3-Partition problem, which completes theproof of the theorem. �


The result in Theorem 2 implies that the MWLCS prob-lem is NP-hard. This together with the equivalence result inTheorem 1 establishes the following main theorem in thissection.

Theorem 3 The MCTG problem is NP-hard.

A greedy heuristic

The MWLCS problem (and consequently, the MCTG prob-lem) was shown to be NP-hard in the previous section. Inthis section, we present a greedy heuristic algorithm for theMCTG problem, and analyze the performance bound for thealgorithm.

Algorithm

Since the MCTG problem has already been shown to beequivalent to the MWLCS problem, we present a heuris-tic algorithm for the MWLCS problem. Our greedy heuristicalgorithm works in time linear to the number of edges in thegiven bipartite graph G(X, Y ; E). Edges are initially sortedsuch that w(e1) ≥ · · · ≥ w(em), where m is the numberof edges in G. Initially, we assume that α(v) = W (whereW is the number of wavelengths supported by each link inthe formulation of the MCTG problem) for each v ∈ X ∪ Y .

Algorithm 4.1MWLCS–HEURISTIC

Input: A bipartite graph G(X, Y ; E), w(e) for each e,α(v) = W for all v, and C

Output: E0 ⊆ Ebegin

Set E0 = ∅.Let w(e1) ≥ · · · ≥ w(em).for i = 1 to m do

if E0 ∪ {ei } is a feasible solution, add ei to E0.endfor

end Algorithm.

The time complexity of the algorithm is O(n2 log n2),which comes from O(n2 log n2) for sorting, and O(n2) tocheck the feasibility condition for each edge.

2-Optimality analysis of MCTG

In this section, we show that the MWLCS heuristic algorithmguarantees an 1

2 -optimal solution to the MWLCS problem,which in turn guarantees a 2-optimal solution to the MCTGproblem. In the rest of the paper, Egreedy

0 and Eoptimal0 are

used to define solution sets to the MWLCS problem fromgreedy and optimal algorithms, respectively.

Lemma 1 2 ∗ |Egreedy0 | ≥ |Eoptimal

0 |Proof Let E = {e1, e2, . . . , en} be the edge set for theMWLCS problem, where w(ei ) > w(e j ), 1 ≤ i < j ≤ nfor simplicity. Assume that any solution for the problem isrepresented by a 0/1 vector of size n.

Let X = (x1, x2, . . . , xn) be an optimal solution as op-posed to Y = (y1, y2, . . . , yn) which is a greedy solution.Let i denote the first index such that xi = yi . We build solu-tion X from solution Y by removing edges from solution Yand possibly adding more edges to it.

• Case 1: xi = 1, yi = 0. Clearly, this is a contradiction.When the greedy algorithm considers ei , ei is an edgewith the largest weight among edges that can be addedto the solution without violating the feasibility constraint(as indicated by xi = 1). Therefore, it should have beenadded to the greedy solution. Yet, it has not been addedto the greedy solution.

• Case 2: xi = 0, yi = 1. If edge ei is removed from thegreedy solution, at most 2 edges, each incident on verti-ces at either end of ei and belonging to X , can be added tothe solution without violating the feasibility constraint.For example, one can add xk or x j after removing e j

as shown in Fig. 3. Recall the feasibility constraint is�(�{w(e)|e ∈ E(v)\E0})/C� ≤ α(v)−|E(v)∩ E0| foreach v ∈ X ∪ Y . For each edge added to the solution,E0, the right hand side value, α(v) − |E(v) ∩ E0|, ofthe feasibility constraint decreases by 1 for some vertex,for example u′ or v′. However, the left hand side value,�(�{w(e)|e ∈ E(v)\E0})/C�, decreases by less than 1.Edges can be added to E0 only if it decreases the lefthand value by exactly 1. Therefore, each edges in solu-tion Y, Egreedy

0 , can be replaced with at most two edges

in solution X , Eoptimal0 .

Thus, we can transform solution Y to solution X by add-ing to Y at most twice the number of edges that have beenremoved from Y . Since the total number of edges that canbe removed from Y cannot exceed the number of edges inY , and any edge that already exists in X also exists in Y ,2 ∗ |Egreedy

0 | ≥ |Eoptimal0 |. �

u

v u'

v'

v

u v'

u'

u

v

xx xxe

(a) (c)(b)

i ei eij k k j

Fig. 3 Example for possible replacement of edges in X and Y


Fig. 4 Average deviation ofgreedy heuristics from theoptimal solution

0

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0 5 10 15 20 25 30 35

Per

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n fr

om th

e O

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al

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Greedy and Optimal Traffic Grooming Performance

Wavelengths = 5Wavelengths = 10

For any edget set E ′ ⊆ E , let w(E ′) = �{w(e)|e ∈ E ′}.We then establish the following result.

Theorem 4 2 ∗ w(Egreedy0 ) ≥ w(Eoptimal

0 )

Proof From Lemma 1, while transforming a greedy solutionY to an optimal solution X , each edge removed was the larg-est weight edge that did not exist in X but existed in Y . Foreach removed edge ei , at most two edges were added, and anynew edges considered have to have weight at most w(ei ), Ifnot, either they already exist in both X and Y , or the assump-tion that ei is the largest weight edge not in X but in Y , is acontradiction. If the two added edges are el and em , then

w(ei ) ≥ w(el) (1)

w(ei ) ≥ w(em) (2)

2 ∗ w(ei ) ≥ w(el) + w(em). (3)

Hence from Lemma 1 and the above result, we show 2 ∗w(Egreedy

0 ) ≥ w(Eoptimal0 ). �

We will further strengthen our claim in Theorem 4 fromnumerical results discussed in Section Experimental results,and show that the actual performance of the greedy heuristicsis much better than the theoretical bound of at least half ofoptimal.

Experimental results

We implemented an Integer Programming formulation forthe MWLCS problem to obtain optimal solutions. We use

these optimal solutions to compare with the performance ofour greedy heuristics. Our integer program formulation wassolved using ILOGs CPLEX. To cover a wide range of solu-tions, the test network traffics were randomly generated fornetworks of different sizes. The average was taken over 500test cases for each network size. For all the test cases, g = 16.The traffic between each pair of nodes in the network hasbeen randomly chosen to occupy at most half the bandwidthavailable for each wavelength.

The performance has been measured in terms of percent-age deviation of the greedy solution (SG) from the optimalsolution(SO), given by 100 ∗ (1 − SG

SO). As is evident from

Fig. 4, the average performance degradation for the greedyheuristics remains well within 10% of the optimal solution,and improves as more wavelengths are available in the net-work. Figure 5 provides the maximum deviation observedbetween the greedy and optimal solutions among all the testtraffics for a particular size of the network. It follows thetrend of the average deviation from the optimal solution forthe previous figure, and remains close to the optimal solu-tion and well within the bound of 2 shown in the previoussection. The maximum deviation however increases if thebandwidth resources are tight with respect to the networksize. For a network with 5 nodes and 2 wavelengths, themaximum deviation was observed to go as high as 36%.

Conclusion

All-optical networks are capable of satisfying the needs ofthe current and future high-speed network traffic. However,


Fig. 5 Maximum deviation ofgreedy heuristics from theoptimal solution

0

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Maximum Deviation for Greedy and Optimal Traffic Grooming Performance

Wavelengths = 5 Wavelengths =10

it is unlikely for the local triaffic in an access network toexploit the full potential of all optical network. This make itdesirable to groom several traffic stream into a high capacitywavelength channel. The network architecture for the opticalnetwork with grooming capability inherently contains bottle-neck points that require slower opto-electronic conversionsand hence do considerable damage to the latency of the sys-tem. We studied the reduction of such damage by efficientlygrooming traffic that requires less analysis by the accessnodes that form the focal point of the star network archi-tecture most suitable for all-optical networks at the MAN orLAN level. We then presented in the paper the hardness of theproblem and proposed a simpler but highly effective greedymethod as a solution to it. The numerical results show thatthe greedy heuristics performs closer to the optimum solu-tion than the theoretical bound, making it suitable to adaptfor traffic grooming in a metropolitan area network with startopology.

References

1. Baldine, I., Jackson, L.E., Rouskas, G.N.: Helios: A broadcast opticalarchitecture, Networking 2002, LNCS 2345, pp. 887–898 (2002)

2. Zhu, K., Mukherjee, B.: A review of traffic grooming in WDM opti-cal networks: architecture and challenges. SPIE Optical NetworksMagazine 4(2), 55–64 (2003)

3. Modiano, E., Lin, P.: Traffic grooming in WDM Networks. IEEECommun. Magazine 39(7), 124–129 (2001)

4. Chiu, A.L., Modiano, E.H.: Traffic grooming in algorithms for reduc-ing electronic multiplexing cost in WDM ring networks. IEEE/OSAJ. Light-wave Technol. 18(1), 2–12 (2000)

5. Dutt, R., Rouskas, G.N.: An optimal traffic grooming in WDM rings.IEEE J. Selected Areas Commun. 20(1), 110–121 (2002)

6. Zhang, X., Qiao, C.: An effective and comprehensive approachfor traffic grooming and wavelength assignment in SONET/WDMRings. IEEE/ACM Trans. Networking 8(5), 608–617 (2000)

7. Wan, P.J., Calinescu, G., Liu, L., Frieder, O.: Grooming of arbitrarytraffic in SONET/WDM BLSRs. IEEE J. Selected Areas Commun.18(10), 1995–2003 (2000)

8. Dutta, R., Huang, S., Rouskas, G.N.: On optimal traffic groomingin elemental network topologies. Proc. OPTICOMM 2003, Dallas,Texas, October 2003 pp. 12–24

Hongsik Choi received the D.Sc. degree incomputer science from the George Washing-ton University, Washington, DC, in 1997. Heis an Assistant Professor with the Departmentof Computer Science, Virginia Common-wealth University, Richmond, VA. Beforejoining VCU, he was an Assistant Professorin the Department of Computer Engineering,Hallym University, Korea from 1997 to 2001and in the Department of Computer Science,George Washington University, Washington

DC from 2001 to 2003. His research interests include network designand analysis, combinatorial design and application of algorithms.


Nikhil Grag received his MS Degree in com-puter science from The George WashingtonUniversity, Washington DC, in 2003. He iscurrently working as a program manager atMicrosoft India.

Hyeong-Ah Choi is a Professor in theDepartment of Computer Science at theGeorge Washington University where shehas been since 1988. She was an AssistantProfessor in the Department of Computer Sci-ence at Michigan State University from 1986to 1988. Her research interests include opticalnetworks, wireless networks, and schedul-ing theory. She published over 100 research

articles in journals and conference proceedings. She is an AssociateEditor of the Journal of the High Speed Networks, and has servedas a Guest Editor of a special issue on Survivable Optical Networksand as a member of technical program committee for numerous con-ferences and workshops. Her research has been supported by NSF,NSA, NIST, DARPA, and DISA.

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Minimal Delay Traffic Grooming in WDM Optical Star Networks