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Computer Networks xxx (2005) xxx–xxx

www.elsevier.com/locate/comnet

Uniform versus non-uniform band switchingin WDM networks q

Li-Wei Chen *, Poompat Saengudomlert, Eytan Modiano

Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139, United States

Abstract

We compare the effectiveness of uniform versus non-uniform waveband switching under the dual cost metrics ofswitching requirements and fiber capacity. We consider a star topology and begin by characterizing the optimal perfor-mance frontier achievable under no restrictions on waveband sizing, and provide algorithms employing non-uniformwaveband sizing that approach or achieve this optimum. We then consider the special case of uniform waveband sizing,and show that the performance compares very favorably. We also extend our results to general topologies.� 2005 Published by Elsevier B.V.

Keywords: WDM; Optical networks; Waveband switching

1. Introduction

The majority of the routing and switching liter-ature for WDM networks has focused on how tominimize the total number of wavelengths [1–8],since the number of wavelengths used specifiesthe amount of capacity required on each fiber.However, this single-resource approach does nottake into account the switching necessary at eachnode which routes each call to its destination.

1389-1286/$ - see front matter � 2005 Published by Elsevier B.V.doi:10.1016/j.comnet.2005.05.017

q This work was supported by the National Science Founda-tion (NSF) under Grant ANI-0073730.

* Corresponding author.E-mail addresses: lwchen@mit.edu (L.-W. Chen), tengo@

mit.edu (P. Saengudomlert), modiano@mit.edu (E. Modiano).

Switching costs can easily dominate bandwidthcosts in systems with a large number of calls. Ina conventional WDM system, each input fiber istypically demultiplexed into wavelengths, and eachwavelength relies on a N · N switch to route it tothe appropriate output fiber, where N is the nodaldegree. The number of switches required is equalto the number of wavelengths. For large networkswith many wavelengths, this approach can requiremany, often expensive switches.

Waveband switching, also known as bandswitching, attempts to address this problem. Theapproach is based on the observation that if thenumber of input wavelengths per fiber is large rel-ative to the number of output fibers, many of thewavelengths will need to be switched between the

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same fiber pairs. Waveband switching tries togroup the wavelengths into wavebands such thatall wavelengths in the same waveband can beswitched together, allowing the processing to beperformed at this coarser waveband level andreducing the number of switches required. Withwaveband switching each fiber would only bedemultiplexed into wavebands, and the numberof switches required would equal the number ofwavebands. Since the number of wavebands re-quired is typically much smaller than the numberof wavelengths, this can greatly reduce the process-ing and switching costs.

In this paper, we consider the resources of inter-est to be the number of wavelengths and wave-bands required by a given banding algorithm.Reducing the requirement for either quantity re-duces the costs in the network. Conceptually,every banding algorithm can be represented by apoint in a two-dimensional performance space, asillustrated in Fig. 1, indicating the number ofwavelengths and wavebands required by the algo-rithm. The shaded area in the figure represents theachievable region of performance over all possiblealgorithms. The goal is to characterize the optimalfrontier of achievable performance. This frontierwould give the optimal tradeoff between wave-lengths and wavebands achievable.

There has been some work in the literatureaddressing the waveband switching problem.Many papers consider the problem of wavebandallocation for static traffic. In [9–11], integer linear

wavebands

wavelengths

min-waveband algorithms

min-wavelength algorithms

achievable performance

regiontradeoffs

optimal min-wavelength algorithm

optimalmin-waveband

algorithm

Fig. 1. The region of achievable wavelength–wavebandtradeoffs.

programming formulations are given for a varietyof topologies, and the problem of optimal wave-band allocation is shown to be NP-complete. In[12], efficient algorithms for dynamic traffic areconsidered under a simplified traffic model thatlimits traffic to a single source node and does notallow wavelength overprovisioning, even if itwould result in fewer wavebands.

In this paper, we consider dynamic traffic in themore general problem of determining the optimaltradeoff between wavelengths and wavebands inband switching. Furthermore, we allow a moregeneral traffic model where every node is permittedto send traffic into the network. We first character-ize the optimal tradeoff with no restrictions onwaveband sizing, and derive efficient algorithmsfor achieving it. We then consider imposing therestriction that all wavebands must be uniformlysized, and show that the performance of uniformwaveband algorithms compare very favorably withthe optimum.

1.1. System model

In this paper, we adopt the P-port traffic modelfrom [4], which assumes that P transmitters andreceivers are available at each network node. Thisallows each node to send and receive a total of atmost P calls at any given time. If the instantaneoustraffic is represented by a matrix where each entry(i, j) consists of the number of calls sent from nodei to node j, the P-port model constrains each rowand column sum to be at most P. Any traffic setwith a matrix obeying this constraint is termedadmissible, and no calls in an admissible set maybe blocked. Under this model, sufficient resourcesmust be provisioned to support any admissible set.Call arrivals and departures may occur in arbitraryfashion, as long as the resultant traffic set remainsadmissible; these dynamic arrivals and departuresare represented by transitions between differentadmissible sets. This model is attractive becauseit limits traffic in a realistic fashion based on hard-ware constraints, and also allows dynamic aspectsof the traffic to be captured without makingassumptions about the call statistics.

We primarily consider the star topology in thispaper, with Section 5 describing extensions of our

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results to other topologies. This topology is repre-sentative of a hub or switch node in a network. Allnodes are connected via bidirectional fibers to acentral hub, which performs the switching. We as-sume no wavelength conversion, so calls must usethe same wavelength on all hops. To avoid colli-sion, no two calls may use the same wavelengthin the same direction on the same fiber. We con-sider both the single-source case, where only asingle node transmits, and the more generalmulti-source case, where every node may transmit.These situations are illustrated in Fig. 2.

The problem of band switching under this mod-el may be formulated as a matrix decomposition.Under the banding problem, we are given a trafficmatrix C where each entry [C]i,j represents thenumber of calls transmitted from source node i

to destination node j. In the single-source case,the traffic matrix is a vector of size 1 · N; for mul-ti-source traffic, the traffic matrix is a square N · N

matrix. Note that C may change over time due tocall arrivals and departures. For a fixed C, the goalis to group the calls into bands such that callswithin the same band that have the same sourcenode go to the same destination. This can beexpressed mathematically by

C 6 b1T 1 þ b2T 2 þ � � � þ bBT B; ð1Þ

where each bi is an integer representing the size ofwaveband i, and each Ti represents the corre-sponding switch configuration. Ti is thereforeeither a unit vector (single-source case) or a per-mutation matrix (multi-source).

a b

S

1

2

3

. . .

N

41

2 3

. . .

N

Fig. 2. (a) Single-source case, where a single source node sendsa total of at most P calls to up to N destinations. (b) Multi-source case, where each of N nodes sends and receives a total ofP calls.

Any set of wavebands specifying a valid decom-position is sufficient to support the particular traf-fic matrix C to which it applies. We impose twoadditional constraints: we require that B and {bi}must be fixed over all admissible traffic sets. FixingB is essential since B corresponds to the number ofswitches required, a hardware requirement thatshould not depend on random changes in the traf-fic. Fixing the band sizes {bi} removes the need fordynamically tunable filters, reducing costs. Underthese two constraints, we require that bandingalgorithms be characterized by fixed values of B

and {bi} such that for each admissible traffic setC, the algorithm is able to specify a decompositionaccording to (1) with specific switch configurationsTi for each waveband i. Recall that the perfor-mance of each banding algorithm can be judgedby the number of wavelengths and wavebands itrequires. The number of wavebands is given di-rectly by B, while the total number of wavelengthscan be calculated as

PBi¼1bi.

As illustrated in Fig. 1, our goal is to find theoptimal achievable frontier. The most general for-mulation of this problem is to allow the wavebandsizes {bi} to be non-uniform. Any uniform-wave-band algorithm will then be a valid special case.Under the general formulation, we can divide thisproblem into three parts. Note that Fig. 1 showstwo asymptotes to the achievable region: one cor-responding to the minimum possible number ofwavelengths required (the minimum-wavelength

asymptote, shown on the bottom of the achievableregion), and the other corresponding to the mini-mum number of wavebands (the minimum-

waveband asymptote, shown to the left of theachievable region). The first two parts of the prob-lem focus on these asymptotes, and attempt todetermine the optimal points on these lines. Specif-ically, we denote any algorithm with performanceachieving the minimum possible number of wave-lengths to be a minimum-wavelength algorithm.The problem of finding the best minimum-wave-length algorithm is known as the minimum-wave-

length problem. Similarly, any algorithm usingthe minimum possible number of wavebands is aminimum-waveband algorithm; finding the bestminimum-waveband algorithm is the minimum-

waveband problem.

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Once solutions to the minimum-wavelength andminimum-waveband problems are obtained, twopoints on the optimal frontier are known. It thenremains only to find the best tradeoff betweenwavelengths and wavebands achievable betweenthese two points. Ideally, such a tradeoff shouldpresent a curve which adheres as closely as possi-ble to the asymptotes, presenting the best possibletradeoff. Obtaining the best possible such tradeoffis the subject of the final component of the band-ing problem.

In Section 2, the banding problem is investi-gated for the single-source traffic case. In Sections3 and 4, banding for general multi-source traffic isconsidered. We will compare the special case ofuniformly-sized wavebands to the more generalnon-uniform case, and show that uniform wave-band sizing compares very favorably. Finally, Sec-tion 5 describes extending the results of the paperto general network topologies.

S

1

2

S

1

2

S

1

2

4 calls 4

4 calls 3

1

4 calls 2

2

Legend:

2-wavelength band

1-wavelength band

ca

b

Fig. 3. The switching configurations for the three uniquemaximal traffic sets of Example 1. The traffic sets shown are:(a) [4,0], (b) [3,1] and (c) [2,2].

2. Waveband switching for single-source traffic

In this section, we consider the banding prob-lem for the case of single-source traffic. Under thisscenario, a single source node sends up to P unitsof traffic to be switched to N possible destinationnodes. The primary purpose of investigating thesingle-source model is to derive intuition for thedesign of good banding algorithms that will bebeneficial in addressing the multi-source trafficcase in the next section. The single-source modelalso has some merit in cases of a one-to-manytraffic scenario.

Example 1. Consider the case of P = 4, N = 2. Inthis example, the source sends four calls, distrib-uted among two destinations. There are only fivepossible maximal traffic sets, which (expressed invector form) are [4,0], [3,1], [2, 2], [1, 3], and [0,4].Clearly at least 4 wavelengths are required tosupport the traffic, since there are four calls. Wecan show that if we restrict ourselves to using onlyfour wavelengths (i.e. we consider the minimum-wavelength problem), the minimum number ofwavebands required is 3: one band of size 2, andtwo bands of size 1. By exhaustive verification we

can prove that this waveband sizing is sufficient forall possible traffic sets:

½4; 0� ¼ 2 � ½1; 0� þ 1 � ½1; 0� þ 1 � ½1; 0�¼ 2e1 þ e1 þ e1

½3; 1� ¼ 2e1 þ e1 þ e2

½2; 2� ¼ 2e1 þ e2 þ e2;

where ei is a unit vector with the ith entry equal to 1.Note that, as required, the sizes of each band and

total number of bands are fixed, and only theaccompanying unit vectors (which correspond tothe switch configurations for each band) changebetween traffic sets. The switching of each wavebandfor each scenario is illustrated in Fig. 3. In thisexample, the savings in switching is not largebecause the number of calls is not very large relativeto the number of destinations. As the number ofcalls increases, the savings will increase as well.

The number of wavebands can be furtherreduced if the use of additional wavelengths ispermitted. One possibility is to have one band ofsize 3, and one band of size 2. This reduces thenumber of wavebands to two, and these twowavebands can still support all five possible trafficsets. However, the total number of wavelengthsused has increased to 5. Efficient methods ofmaking these sorts of tradeoffs will be discussed.

We consider the two special cases of the mini-mum-wavelength and minimum-waveband prob-lems. Recall that the solutions to these twoproblems will provide two points on the optimalachievable performance frontier. We will deferthe discussion of obtaining good tradeoffs betweenthese two points until the multi-source case.

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2.1. The minimum-wavelength problem

Recall that for the minimum-wavelength prob-lem, we consider only banding algorithms thatuse the minimum possible number of wavelengths.Under the P-port model, a minimum of P wave-lengths are clearly necessary and sufficient: up toP calls can be sent, and if each wavelength is indi-vidually switched, P wavelengths can support allthe calls. The goal is to find the optimal algorithmthat uses only P wavelengths.

Since the number of wavelengths equals themaximum possible number of calls, given a maxi-mal admissible traffic set, each wavelength mustbe used to support a call. Therefore, the mini-mum-waveband problem is equivalent to findinga method of partitioning the P wavelengths intowavebands such that, for any admissible trafficset, there exists a method for assigning calls towavebands such that every wavelength is assigneda call. Furthermore, the optimal minimum-wave-length algorithm should accomplish this while atthe same time minimizing the total number ofwavebands. We show in this section that the opti-mal minimum-wavelength banding algorithm is agreedy algorithm. Specifically, the greedy algo-rithm chooses waveband sizes recursively, whereat each step a waveband is chosen to be as largeas possible subject to the constraint that everywavelength in that band can always be assigneda call under any maximal admissible traffic set.(We say that in this case every wavelength can befully utilized.)

Define bmax(N,P) to be the maximum wave-band size that we can guarantee will be fully uti-lized by any traffic set sending P calls to N

destinations. Since all calls in the waveband mustgo to the same destination, this is equivalent toproviding a guarantee that a destination nodecan always be found (under any admissible P-porttraffic set) which receives at least bmax(N,P) calls.To illustrate, when considering the example inFig. 3, we note that over all admissible traffic sets,a destination can always be found which receivesat least 2 calls, leading to the conclusion thatbmax(N,P) = 2 in that case. In general, we canguarantee that at least one of the destinations re-ceives dP/Ne calls. Furthermore, this is the largest

number for which we can make this guarantee; thisfollows from the fact that one admissible traffic setis where the traffic is divided evenly (up to a differ-ence of one wavelength due to integer constraints)among all destinations, and no destination receivesmore than dP/Ne calls under this traffic set. There-fore bmax(N,P) = dP/Ne.

Single-source greedy algorithm:

1. Let P1 = P be the number of calls remainingand N be the number of nodes. Let i = 1.

2. Let waveband i be of size bi = bmax(N,Pi) = dPi/Ne.

3. Locate a destination receiving at leastbmax(N,Pi) calls. Route waveband i to this desti-nation, and assign bmax(N,Pi) calls to it. Thenumber of calls remaining becomes Pi+1 =Pi bmax.

4. If Pi+1 > 0, let i i + 1 and go to Step 2.

Example 2. We revisit Example 1 and show howthe greedy algorithm is used to obtain the mini-mum-wavelength waveband sizes used in Fig. 3. Inthat example, P = P1 = 4 and N = 2. In the firstiteration, the greedy algorithm chooses the firstwaveband to be of size dP1/Ne = d4/2e = 2. As acorollary, we are guaranteed that 2 calls can beassigned to this waveband, leaving P2 = 2 callsunassigned. The next two iterations partition theremaining wavelengths into bands of a singlewavelength each, for a final partition of {2,1,1}.

We must now show that choosing the wavebandsizes using the greedy algorithm is optimal for theminimum-wavelength problem. The full proof,omitted here for brevity, is based on establishingthat the minimum number of wavebands requiredto support a given number of calls is non-decreas-ing in the number of calls. It therefore follows thatthe optimal approach for choosing each wavebandsize is to choose the band as large as possible,thereby minimizing the amount of traffic which re-mains (and therefore the subsequent number ofwavebands).

The greedy algorithm provides a method foroptimally determining waveband sizes for the min-imum-wavelength problem. This also implicitlyprovides a way of determining the minimum num-

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ber of wavebands required (i.e. by running thealgorithm and counting the number of wavebandsproduced). We can also derive an explicit upperbound on the minimum number of wavebands re-quired in the minimum-wavelength scenario. Weproceed by relaxing the integer constraints onbmax(N,P). Let Pk be the number of calls remain-ing after running the kth iteration of the greedyalgorithm. The series progresses as follows:

P 1 ¼ P PN¼ 1 1

N

� �� P ;

P 2 ¼ 1 1

N

� �� P 1 ¼ 1 1

N

� �2

� P ;

..

.

Pk ¼ 1 1

N

� �k

� P . ð2Þ

If P 6 N, then the number of bands B is simplyequal to P since each band is composed of onlya single wavelength. Therefore consider P > N

and determine the number of bands k requiredto reduce the number of unassigned wavelengthsto N.

Pk ¼ N ;

1 1

N

� �k

� P ¼ N ;

k ¼log N

P

� �log 1 1

N

� � .

Then the total number of wavebands is simplyk + N. Since relaxing the ceiling constraints under-estimates the size of each waveband, this gives anupper bound on the number of wavebands B,namely:

B 6N þ log N

Pð Þlog 11

Nð Þ ; P > N

P ; P 6 N

8<: . ð3Þ

From (3), we can also make the additionalobservation that if P 6 N, the number of bandsB equals the number of wavelengths P, and thereis no savings from banding in the minimum-wave-length case as each wavelength continues to beswitched individually.

2.2. The minimum-waveband problem

The optimal minimum-waveband algorithm isthe one that requires the fewest wavelengths sub-ject to using only the minimum number of wave-bands. In addition to providing a second pointon the optimal frontier, this will establish the min-imum cost in wavelengths required to obtain themaximum possible reduction in switching.

If P < N, then there exists an admissible trafficset where each call is sent to a different destination,and P wavebands are necessary. It follows from alittle further thought that P wavebands of a singlewavelength each are actually also sufficient to sup-port all admissible traffic sets, since each call canbe assigned a dedicated waveband under thisprovisioning.

The more interesting case arises when P P N.Since all wavelengths in the same waveband mustbe switched to the same destination, and thereare N possible destination, a minimum of N wave-bands are necessary. One (inefficient) approachthat requires only N wavebands is to staticallyswitch one waveband to each destination, and pro-vision P wavelengths per waveband; since there area total of only P calls, this is sufficient to supportany admissible traffic set. Our goal is to find a bet-ter, optimal algorithm using only N wavebandsthat minimizes the number of wavelengths used.We first obtain a lower bound on the number ofwavelengths required using the following lemma.

Lemma 1. Consider a banding algorithm that uses

N wavebands, and order the wavebands from small-est to largest. Let bi be the size of the ith waveband.

If the source sends up to P calls to the N destination

nodes, bi is bounded by

bi PP N þ i

i

; i ¼ 1; . . . ;N . ð4Þ

Corollary. The total number of wavelengths W

required is bounded by the sum of the bounds on

the individual waveband sizes, namely

W PXNi¼1

P N þ ii

. ð5Þ

This summation can be shown to increase asOðP logNÞ.

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Proof. The proof proceeds by constructing anadmissible traffic set which requires bi to have atleast PNþi

i

� wavelengths. Consider the traffic set

where the source S sends PNþii

� calls each to the

first i nodes, and a single call to each remainingnode. The total traffic in this construction is

i � P N þ ii

þ ðN iÞ 6 P

and therefore it is admissible. Since each destina-tion receives at least one call, each of the N wave-bands goes to a different destination.

Without loss of generality, we assign the largesti wavebands to nodes 1 through i. Each of thesewavebands must support PNþi

i

� calls. Therefore

bi P PNþii

� . h

Since Lemma 1 is a lower bound, any mini-mum-waveband algorithm which achieves thebound is optimal. We next present an algorithmthat can support any admissible traffic set usingwavebands of the minimal sizes specified by (4).Since this minimum-waveband algorithm woulduse no more wavelengths than the lower bound,it is therefore optimal.

Min-band algorithm:

1. Index the N waveband in order of decreasingsize, so that waveband i has size bi ¼ PNþi

i

� ,

where i = 1, . . . ,N. Note that b1 is the largestwaveband, and bN is the smallest.

2. Let i = 1.3. Locate the destination node with the greatest

number of remaining calls. Switch waveband i

to that node. Assign up to bi calls to wavebandi, and remove these calls from the traffic set.

4. If no calls remain, the algorithm terminates.Otherwise, increment i and return to Step 2.

By design, the algorithm uses only wavebandsof the minimum size, and therefore meets the lowerbound. It remains only to show that it is able tosupport any admissible set. First, suppose thateach destination receives at least a single call. Inthis case, we can rank each destination in decreas-ing order of number of calls received, so that thefirst destination receives the most calls. Then themin-band algorithm allocates the ith wavebandto the ith destination. Since each destination re-

ceives at least one call, the first i destinationsreceive at most P (N i) = P N + i calls,and the ith destination receives at mostb(P N + i)/ic calls. Since the ith waveband hassize b(P N + i)/ic, it suffices to accommodatethe calls. The proof in the case where some desti-nations do not receive calls is more cumbersomebut follows the same approach.

Example 3. Consider the case where P = 22 callsare distributed among N = 4 destinations. The opti-mum minimum-waveband algorithm requires 4wavebands. According to Lemma 1, the first wave-band is of size b1 = bP N + 1c = b224 + 1c =19. Similarly, b2 = 10, b3 = 7, and b4 = 5.

These wavebands can support any P-portadmissible traffic set, P = 22. For example, con-sider the traffic set C = [5,8,7,2]. The first wave-band is assigned to node 2, the destination with themost traffic, and carries all 8 calls. Similarly, b2 isassigned to node 3, b3 is assigned to node 1, and b4

is assigned to node 4. Using the matrix decompo-sition notation of (1), this can be written as

½5; 8; 7; 2� 6 19e2 þ 10e3 þ 7e1 þ 5e4

¼ ½7; 19; 10; 5�.

Note that here, the total number of wavelengthsavailable to each destination (represented by thevector on the right-hand side of the equation) isgreater than the number of calls: more wave-lengths were provisioned than absolutely requiredfor this particular traffic set. This over-provision-ing in wavelengths is necessary in order to guar-antee that all admissible traffic sets can beaccommodated.

It is instructive to compare this to the decom-position obtained by the greedy algorithm ofSection 2.1. The first waveband for the greedyalgorithm consists of dP/Ne = d22/4e = 6 wave-lengths. The remaining waveband sizes can beshown to be {4,3,3,2,1,1,1,1}. One possibleswitching configuration for these waveband sizes is

½5; 8; 7; 2� ¼ 6e2 þ 4e3 þ 3e1 þ 3e3 þ 2e1 þ e2

þ e2 þ e4 þ e4 ¼ ½5; 8; 7; 2�.

The greedy algorithm, since it is a minimum-wavelength algorithm, did not over-provision anywavelengths; however, more wavebands were

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required. This example also illustrates an impor-tant point. In the decomposition given by (1),equality is guaranteed to hold for all C if andonly if the wavebands were allocated using a min-imum-wavelength algorithm (such as the greedyalgorithm). Algorithms such as the min-band algo-rithm allow some overprovisioning of wavelengthsin order to further decrease the number of wave-bands required.

Eq. (5) gives total the number of wavelengthsrequired by the optimal minimum-waveband algo-rithm. We can approximate the total number ofwavelengths in closed form by relaxing the inte-grality constraint on the terms of the summation.This gives:

W min �XNi¼1

P N þ ii

¼ N þ ðP NÞXNi¼1

1

i

� N þ ðP NÞ logN � P logN ;

where the last approximation holds for P� N.Recall that any banding algorithm always re-

quires a minimum of P wavelengths. Our resultin this section shows that the best minimum-wave-band algorithm requires OðP logðNÞÞ wavelengths.One way of interpreting this result is that logðNÞrepresents the minimum wavelength inefficiency

necessary to achieve the minimum-wavebandbound. In other words, if we desire to use no morethan the minimum number of wavebands N, thenwe must pay a penalty of a factor of logðNÞincrease in the number of wavelengths used.

3. Waveband switching for multi-source traffic

We now consider the more general case of mul-ti-source traffic. In this scenario, N nodes are con-nected to a central hub. Each node is assumed tohave a hardware limitation of P transmitters andreceivers, and can therefore send and receive upto P calls. The hub must switch the calls, at a bandlevel, from the appropriate source to destinationnodes. In our discussion, we will assume thatself-traffic is allowed; the case without self-trafficis similar and leads to comparable results. We willshow that many of the concepts in this scenarioparallel those in the single-source case. (The pri-

mary differences are that the traffic set now con-sists of a traffic matrix rather than a vector, andthe switching configuration for each waveband willnow consist of a permutation matrix rather than aunit vector.) We will again begin by consideringtwo special cases, the minimum-wavelength andminimum-waveband problems, followed by inves-tigating algorithms that provide a tradeoff betweenthese two cases.

3.1. The minimum-wavelength problem

Recall that for the minimum-wavelength prob-lem, we constrain ourselves to the domain of band-ing algorithms which use only the minimumpossible number of wavelengths. Since each nodemay send up to P calls, it is clear that at least P

wavelengths are necessary. In [13] it is shown thatthis is also sufficient. The challenge is therefore tofirst partition the P wavelengths into wavebands,and second, to develop an algorithm that will pro-vide a valid wavelength assignment for any admis-sible traffic set using these wavebands.

We first address the partitioning of the wave-bands. We consider the cases of interest to be max-imal traffic sets, since we can add fictitious calls toany non-maximal set to construct a maximal one.We define a wavelength to be fully utilized if it isused to carry a call on every link. Mathematically,this is equivalent to stating that the matrix of callssupported by that wavelength forms a permuta-tion matrix. We say that a waveband is fullyutilized if every wavelength in that waveband isfully utilized.

Every minimum-wavelength algorithm must beable to fully utilize every waveband under anyadmissible maximal traffic set. We have alreadyseen in Section 2.1 that a greedy approach is opti-mal for this type of problem. Recall that the gree-dy algorithm worked recursively by partitioning,at each step, the largest waveband size possiblesubject to the constraint that it could be fully uti-lized by any admissible traffic set. Intuitively, thiswas because the number of wavebands requiredturned out to be non-increasing in the amount oftraffic; therefore at each step the optimal solutionwas to minimize the residual amount of traffic.The only remaining problem is that largest fully

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utilizable waveband size, bmax(N,P), must bederived for the multi-source case.

We can view bmax(N,P) as the largest number ofidentical permutation matrices that we are guaran-teed to be able to find within any N · N matrixwith row and column sums equal to P. This isequivalent to stating that sufficiently many callsmust exist within any admissible traffic set withparameters N and P to fully utilize the waveband.

Example 4. Consider the case where P = 9 andN = 3. One admissible maximal traffic set satisfy-ing these constraints is given by

C1 ¼1 5 3

3 2 4

5 2 2

264

375.

The largest waveband that could be fully utilizedby this particular traffic set is 4 wavelengths, asshown below:

C1 ¼ 4 �0 1 0

0 0 1

1 0 0

264

375þ

1 1 3

3 2 0

4 2 2

264

375.

The first term consists of 4 identical permutationmatrices representing calls that can be used to fullyutilize a waveband of size 4 or less; the second termforms the remaining calls. Note that this showsonly that bmax must be at most 4; it may be possiblethat some other traffic set may require an evensmaller waveband for full utilization. For example,consider an alternate admissible set, formed by

C2 ¼2 4 3

4 2 3

3 3 3

264

375.

In this example, since no entry in the third rowcontains more than 3 calls, at best 3 identical per-mutations can be found, with one possible assign-ment shown below:

C2 ¼ 3 �0 1 0

1 0 0

0 0 1

264

375þ

2 1 3

1 2 3

3 3 0

264

375.

Therefore at best the largest waveband bmax canconsist of 3 wavelengths; any larger wavebandcould not be fully utilized by C2. Note that this

is still not enough to show that bmax might notneed to be lowered further; it may be possible thatsome other traffic set may require an even smallerwaveband for full utilization. The challenge will beto obtain in closed form an equation for bmax with-out performing this sort of exhaustive examinationof all possible maximal admissible traffic sets.

We will use results from graph theory to helpderive a solution to this problem. We can representany admissible traffic set as a bipartite graph. Thebipartite graph consists of two sets of N nodeseach. Denote these sets by V1 = {s1, . . . , sN} andV2 = {d1, . . . ,dN}. Nodes in V1 represent sources,and nodes in V2 represent destinations. We alsocreate a set of edges E that is dependent on thetraffic set under consideration: if there is at leastone call from a source node i to a destination nodej, create an edge connecting node si to node dj. Theedge is given a weight equal to the number of callsbetween that source–destination pair.

Define a maximal matching to be a subset ofedges from E such that exactly one edge is incidenton each node in V1 and V2. In our original matrixnotation, this corresponds to a matrix T where en-try [T]i,j is nonzero if and only if an edge between siand dj is contained in the matching. Note that T isa permutation matrix, since there is only one non-zero entry per row (since only one edge is adjacentto each si) and column (due to one edge per dj).The number of such permutation matrices thatcan be found in the particular traffic set is equalto the smallest edge weight in the maximalmatching.

Example 5. We consider again the set C1 given inExample 4. Fig. 4(a) gives the bipartite graphcorresponding to C1, and Fig. 4(b) locates abipartite matching within this graph. The bipar-tite matching corresponds to the permutationmatrix

T 1 ¼0 1 0

0 0 1

1 0 0

264

375.

Note that as the smallest edge weight on thematching is 4, we are able to locate 4 copies of Tin C1, as expected.

s1

s2

s3

d1

d2

d3

1

53

32

4

52

2

s1

s2

s3

d1

d2

d3

5

4

5

ba

Fig. 4. (a) The bipartite graph corresponding to C1 in Example5. (b) One possible bipartite matching from the graph.

10 L.-W. Chen et al. / Computer Networks xxx (2005) xxx–xxx

ARTICLE IN PRESS

Therefore, in this graph context, we wish tochoose bmax such that in any bipartite graph corre-sponding to an admissible traffic set, a maximalmatching can be found where the smallest-weightedge is at least bmax. Conveniently, a theorem ex-ists which provides necessary and sufficient condi-tions for the existence of maximal matchings onbipartite graphs.

Hall’s Theorem [14]: In a bipartite graphðV 1; V 2;EÞ, define the neighborhood of a subsetv � V1 to be those nodes in V2 which are con-nected via some edge in E to some node in v. Thenthere exists a maximal matching if and only if, forevery subset v � V1, its neighborhood N(v) has sizejN(v)jP jvj.

Hall�s Theorem provides the basis for determin-ing the existence of maximal matchings. The fol-lowing test is applied: a subset v of the sourcenodes V1 is chosen. If the neighborhood of thesubset N(v) is of size greater than or equal to thesize of the subset itself, the test is passed. This isshown in Fig. 5. The test is then repeated for allpossible subsets v of V1. If the test is passed for

1

2

3

4

1

2

3

4

v

1

2

3

4

1

2

3

4

N(v)

Fig. 5. To apply the test given by Hall�s Theorem, a subset v ofthe source nodes V1 is first chosen. In this case, v consists of 2nodes, and the neighborhood N(v) contains 4 nodes. Thereforethe test is passed for this choice of v. This test must be repeatedfor all possible choices of v.

all subsets, then a maximal matching exists. If atleast one test is failed, then no maximal matchingexists.

We can determine if a waveband of a given sizeb can be fully utilized by a given traffic set as fol-lows. Determine the bipartite graph correspondingto the traffic set, and delete any edges with weightless than b. This removes from consideration anymaximal matchings with edge weight less than b,since such matchings cannot fully utilize the wave-band. The tests given by Hall�s Theorem can thenbe applied to this graph to determine if a maximalmatching exists that is sufficiently large to fully uti-lize the waveband. If the test fails, then a wave-band of size b is too large to be sufficientlyutilized. This test should be applied to all maximaltraffic sets to guarantee that any maximal set canfully utilize the waveband.

In principle, the preceding approach could beused to determine bmax(N,P) numerically by bruteforce. However, we will see that a closed-formsolution can be obtained. The method for obtain-ing the closed-form expression for bmax(N,P) relieson attempting to construct a bipartite graph whichcauses the test given by Hall�s Theorem to befailed. (In a slight abuse of notation, we call sucha bipartite matching a ‘‘counterexample’’.)bmax(N,P) is then the largest waveband size forwhich no counterexample exists.

In order for the test to be failed, a maximal traf-fic set must be found for which we can choose a v

such that the size of the neighborhood N(v) issmaller than the size of v. We therefore wish toconstruct a counterexample where, if jvj = n, thenjN(v)j = m, where m < n.

Under the P-port model, the nodes in v cansend at most mP calls to nodes in N(v). Theremaining residual traffic is therefore at least(n m)P. These calls are sent to nodes outsidethe neighborhood, and hence must belong to edgesadjacent to a non-neighborhood node. Call theseedges non-neighborhood edges. Non-neighborhoodedges are those edges with weight less thanbmax(N,P) which are not eligible to belong to amaximal matching, since they do not contain en-ough calls to fully utilize the matching. This isillustrated in Fig. 6. There are at most n Æ (N m)non-neighborhood edges.

n

m

mPmP

n N-m

(n-m)P

(n-m)P

ba

Fig. 6. At most mP calls can be sent to nodes in N(v), and hence(n m)P calls must go to non-neighborhood nodes.

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ARTICLE IN PRESS

We can therefore construct a counterexample ifand only if the residual traffic can be dividedamong the non-neighborhood edges such that nonon-neighborhood edge has weight greater thanor equal to bmax(N,P). Since there are (n m)Presidual calls and n(N m) non-neighborhoodedges, there is at least one non-neighborhood edgewith weight at least

ðn mÞPnðN mÞ

� �.

If bmax(N,P) is chosen to be at most this num-ber, then no counterexample exists for the givenvalues of n and m. We can choose bmax(N,P) toguarantee that no counterexample exists byminimizing over n and m and choosing bmax(N,P)smaller than this minimum:

bmax 6 minn;m

ðn mÞPnðN mÞ

� �¼ min

n;m

1 mn

� �P

N m

� �. ð6Þ

We fix for the moment m and consider the min-imization over n. Since n > m, the minimization issubject to the constraint 0 < m < n < N. Eq. (6) isminimized by choosing n as small as possible.Since n and m are both integer quantities, weshould choose n = m + 1; conversely, m = n 1.

Making this substitution, the minimizationbecomes:

bmax 6 minn

Pn½N ðn 1Þ�

� �. ð7Þ

Since the ceiling function is monotonic,

minn

Pn½N ðn 1Þ�

� �¼ min

n

Pn½N ðn 1Þ�

� �.

Ignoring integrality constraints, the right-handside is easily shown to be minimized at n� ¼ Nþ1

2.

If N is odd, then Nþ12

is an integer and hence is avalid choice for n*. We subsequently obtain a valuefor bmax of

bmax ¼P

Nþ12

� �Nþ1

2

� �& ’

¼ 4P

ðN þ 1Þ2

& ’; if N odd.

If N is even, then since P=nNðn1Þ is convex, the

minimizing value of n must be one of the integersadjacent to Nþ1

2, namely either bNþ1

2c ¼ N

2or

dNþ12e ¼ N

2þ 1. It is easy to verify that either case

results in the same value of

bmax ¼P

N2

� �N2þ 1

� �& ’

¼ 4PNðN þ 2Þ

� �; if N even.

In summary,

bmax ¼4P

NðNþ2Þ

l m; N even

4PðNþ1Þ2

l m; N odd

8><>: . ð8Þ

Now that bmax(N,P) is known for general P-port traffic, (8) can be used in conjunction withthe greedy algorithm to optimally partition thewavebands. Again, note that the waveband sizingis independent of the particular traffic set and de-pends only on the network parameters N and P.Therefore wavebands of constant size can be usedfor any admissible traffic set. The assignment ofcalls to the wavebands can be obtained by a bipar-tite matching algorithm; by using the expressionbmax(N,P) to size each waveband, we have guaran-teed that maximal matchings can be found whichfully utilize each waveband for any admissibletraffic set.

Example 6. We continue with the 3-node networkfrom Example 4 where P = 9. Using the greedyalgorithm, we would determine that the largest

waveband should be 4PðNþ1Þ2

l m¼ ð4Þð9Þ

ð4Þ2

l m¼ 3.

After this step, 9 3 = 6 wavelengths remain tobe partitioned. We repeat this process until allwavelengths have been assigned to bands. Thefinal waveband partition is {3, 2, 1, 1, 1, 1}. Bychoice of bmax, this partitioning can be fully

12 L.-W. Chen et al. / Computer Networks xxx (2005) xxx–xxx

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utilized by any admissible traffic set. For example,consider the traffic set C1 from Example 4. Onepossible decomposition is

1 5 3

3 2 4

5 2 2

264

375 ¼ 3 �

0 1 0

0 0 1

1 0 0

264

375þ 2 �

0 0 1

1 0 0

0 1 0

264

375

þ 1 �0 0 1

0 1 0

1 0 0

264

375þ 1 �

0 1 0

1 0 0

0 0 1

264

375

þ 1 �1 0 0

0 1 0

0 0 1

264

375þ 1 �

0 1 0

0 0 1

1 0 0

264

375.

Note that the equality in this decomposition indi-cates that all wavelengths are fully utilized. Fur-thermore, by the optimality of the greedyalgorithm, we are guaranteed that this is the min-imum possible number of wavebands subject tothe minimum-wavelength constraint.

Notice that there are ðNþ1Þ24 1 ¼ 3 wavebands

consisting of a single wavelength. This results fromthe fact that once P becomes less than thedenominator, the greedy algorithm will chooseall remaining wavebands to be of size 1. Thisindicates that banding is only useful when P > N2/4 if no wavelength inefficiency is allowed. Essen-tially, this fact is based on the intuition that if thenumber of nodes is large relative to the number ofcalls, then we cannot guarantee that sufficientlymany calls will belong to the same source–desti-nation pairs for banding to take place.

In principle, with the greedy algorithm, we canobtain the exact minimum number of wavebandsrequired under the minimum-wavelength con-straint simply by iterating through the algorithmand counting the number of wavebands produced.We can also obtain in closed form an upperbound on the number of wavebands by relaxingthe integer constraints and using argumentsanalogous to the single-source case. Let Pk bethe value of P after running the kth iteration ofthe greedy algorithm. The series progresses asfollows:

P 1 ¼ P 4

ðN þ 1Þ2� P ¼ 1 4

ðN þ 1Þ2

" #� P ;

P 2 ¼ 1 4

ðN þ 1Þ2

" #� P 1 ¼ 1 4

ðN þ 1Þ2

" #2

� P ;

..

.

Pk ¼ 1 4

ðN þ 1Þ2

" #k� P . ð9Þ

If P 6ðNþ1Þ2

4, then the number of bands B is sim-

ply equal to ðNþ1Þ24

since each band is composed ofonly a single wavelength. Therefore considerP > ðNþ1Þ2

4and determine the number of bands k

required to reduce the number of unassignedwavelengths to ðNþ1Þ2

4. Then the total number of

wavebands would be k þ ðNþ1Þ24

.

Pk ¼ðN þ 1Þ2

4;

1 4

ðN þ 1Þ2

" #k� P ¼ ðN þ 1Þ2

4;

k ¼log ðNþ1Þ2

4P

h ilog 1 4

ðNþ1Þ2

h i .

Again, since removing the ceiling constraintunderestimates the size of each waveband, thisgives an upper bound on the number of wave-bands B, namely:

B 6

ðNþ1Þ24þ log

ðNþ1Þ24P

� log 1 4

ðNþ1Þ2

h i ; P > ðNþ1Þ24

P ; P 6ðNþ1Þ2

4

8>><>>: . ð10Þ

It can be show from (10) that the number ofwavebands required grows as OðN 2 logðP=N 2ÞÞ.Since the number of wavebands required by thegreedy algorithm is the minimum possible forany minimum-wavelength algorithm, this allowsus to quantify the maximum switching reductionpossible without wavelength inefficiency.

Example 7. Recall the case of Example 6, whereN = 3 and P = 9. From (10), we obtain an upperbound on the number of wavebands of 6.82. Sincethe number of wavebands must be an integer, we

L.-W. Chen et al. / Computer Networks xxx (2005) xxx–xxx 13

ARTICLE IN PRESS

can safely round the bound down to 6, which isexactly equal to the true value of 6 wavebands. Wecan also use (10) to approximate the number ofwavebands by 5.4, which is still close to the truevalue of 6 wavebands.

3.2. The minimum-waveband problem

Recall that a minimum-waveband algorithm isdefined to be a banding algorithm that uses theminimum possible number of wavebands. Sinceall wavelengths in the same waveband must go tothe same destination, and there are N possible dif-ferent destinations, a minimum of N wavebandsare required. One way to achieve this is to stati-cally provision P wavelengths between eachsource–destination pair, using a total of PN wave-lengths. The minimum-waveband problem istherefore to find a better, dynamic algorithm thatuses fewer wavelengths. Since the single-sourcetraffic model is a special case of the multi-sourcemodel, we can also use (5) to provide a lowerbound on the number of wavelengths required:

W PXNi¼1

P N þ ii

.

However, in this case it is possible to show that thebound is not tight. We therefore do not know theachievable minimum number of wavelengths, onlythat it cannot be less than that specified by (5).

We next propose a wavelength-efficient mini-mum-banding algorithm which requires OðP

ffiffiffiffiNpÞ

total wavelengths, which improves on the O(PN)worst case. The algorithm operates by decompos-ing the traffic set into N sub-matrices, and at-tempts to group entries with heavy weights andlight weights into separate sub-matrices. This willallow some wavebands to use less than the worstcase of P. The algorithm relies on the followinglemma:

Lemma 2. Consider a P-port traffic set on an N-

node star. For any value of k such that 1 6 k 6 N,

there exists a decomposition satisfying (1) where

b1 ¼ � � � ¼ bk ¼ P ;

bkþ1 ¼ � � � ¼ bN ¼P

k þ 1

� �.

Proof. See Appendix A. h

Corollary. Any P-port N-node traffic set can be

routed using k bands of size P and N k bands of

size Pkþ1

l m.

The proof of Lemma 2 forms the basis of theSQRT(N) algorithm. Since it holds for any valueof k, it is logical to use the value of k which resultsin the fewest total number of wavelengths used. Todetermine this, we write down the expression forthe number of wavelengths required:

W k ¼ kP þ ðN kÞ Pk þ 1

� �. ð11Þ

It can be shown that this expression is minimizedat k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiN þ 1p

1. If we relax the integer con-straint on k and substitute this back into the equa-tion, we obtain:

W ¼ 2P ðffiffiffiffiffiffiffiffiffiffiffiffiN þ 1p

1Þleading to the observation that the SQRT(N) algo-rithm requires OðP

ffiffiffiffiNp

) wavelengths. The resultsof this section show that the maximum amountof switching reduction can be achieved by, atworst, a factor of

ffiffiffiffiNp

increase in the number ofwavelengths.

Example 8. We examine the case of Example 6under the minimum-waveband restriction. Usingthe SQRT(N) algorithm, we see that for N = 3 andP = 9, we should choose k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiN þ 1p

1 ¼ 1. Wetherefore require only 1 waveband of size 9 andN 1 = 2 bands of size dP/(k + 1)e = 5, producinga final waveband sizing of {9, 5, 5}, for a total of 19wavelengths and 3 bands. We can compare thiswith the optimal minimum-wavelength solution of{3, 2, 1, 1, 1, 1}, which uses the minimum of 9wavelengths but requires 6 wavebands.

Shown below is one possible call assignmentusing these waveband sizes for the specific trafficset C1.

1 5 3

3 2 4

5 2 2

24

35 6 9 �

0 1 0

0 0 1

1 0 0

24

35þ 5 �

0 0 1

1 0 0

0 1 0

24

35

þ 5 �1 0 0

0 1 0

0 0 1

24

35 ¼ 5 9 5

5 5 9

9 5 5

24

35.

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Note that wavelengths were overprovisioned inthis case for some source–destination pairs. How-ever, the additional wavelengths are required toguarantee that all admissible traffic sets can beaccommodated by the 3 wavebands.

3.3. Hybridization: wavelength–waveband tradeoffs

The greedy algorithm previously discussed pro-vides us with one point on the optimal frontier.The SQRT(N) algorithm provides us with a sec-ond point which, while not necessarily optimal,nonetheless provides good performance in theminimum-waveband case. As in the single-sourcetraffic case, we can now investigate the use of adual-algorithm approach to obtaining a tradeoffbetween these two performance points.

To review, in the dual algorithm approach, theinitial g wavebands are allocated using the greedyalgorithm. The motivation for this is that the gree-dy algorithm performs well when the number ofcalls is large relative to the number of nodes. Theremaining calls are allocated using the SQRT(N)algorithm, which always uses the minimum num-ber of wavebands N. By varying the number ofwavebands g given over to the greedy algorithm,more emphasis can be given to reducing eitherwavelengths or wavebands. At g = 0, the dual-algorithm approach is reduced to exactly theSQRT(N) algorithm; conversely for g sufficientlylarge, only the greedy algorithm will be used.

Example 9. Consider a 10-node star with P =1000. Suppose the greedy algorithm is used toassign the first g = 25 wavebands, and the remain-ing calls use the SQRT(N) algorithm. Using theequation for bmax(N,P) given by (8), the greedyalgorithm chooses waveband sizes of {34, 33, 32,31, 29, 29, 28, 27, 26, 25, 24, 23, 22, 22, 21, 20, 20,19, 18, 18, 17, 17, 16, 15, 15}. Summing the bandsizes shows that a total of 581 wavelengths areused.

After this assignment, 419 calls per noderemain. Using the SQRT(N) algorithm, 10 addi-tional wavebands are required to accommodatethem. Since

ffiffiffiffiffiffiffiffiffiffiffiffiN þ 1p

1 ¼ 2:317, the minimizingvalue of k must be either 2 or 3; substituting bothvalues into (11) shows that k = 2 is optimal.

Therefore we choose 2 wavebands of size 419,and 8 wavebands of size d419/(2 + 1)e = 140, for atotal of 1958 wavelengths.

Therefore in this example a total of 35 wave-bands, containing 1958 + 581 = 2539 wavelengths,are required.

We can obtain an approximate expression forthe tradeoff curve from the dual-algorithm ap-proach as follows. The greedy algorithm is initiallyemployed for g iterations. From (9), we know thatafter g iterations, the remaining traffic is given by

P 0 ¼ 1 4

ðN þ 1Þ2

" #g� P .

The greedy algorithm is therefore responsiblefor using g wavebands, of total size P P 0. Theremaining traffic is handled using the SQRT(N)algorithm, which uses N wavebands and¼ 2P 0ð

ffiffiffiffiffiffiffiffiffiffiffiffiN þ 1p

1Þ wavelengths. Therefore the to-tal number of wavelengths and wavebands usedcan be expressed by

W dual ¼ W greedy þ W min -band

¼ 1 1 4

ðN þ 1Þ2

" #g !� P

þ 1 4

ðN þ 1Þ2

" #g� 2P

ffiffiffiffiffiffiffiffiffiffiffiffiN þ 1p

1$ %

¼ 1þ 2ffiffiffiffiffiffiffiffiffiffiffiffiN þ 1p

3$ %

1 4

ðN þ 1Þ2

!g" #� P

� 1þ 2ffiffiffiffiNp

1 4

ðN þ 1Þ2

!g" #� P ;

Bdual ¼ Bgreedy þ Bmin-band ¼ g þ N .

Combining, we obtain

W dual ¼ 1þ 2ffiffiffiffiNp

1 4

ðN þ 1Þ2

!BdualN" #� P .

The dual-algorithm approach actually producesa family of algorithms. Choosing the number ofwavebands B specifies the number of wavelengthsW, and vice versa. Fig. 7 shows the performanceof this approach for a star with N = 10, P =1000. The two asymptotes represent the SQRT(N)

0 50 100 1500

1000

2000

3000

4000

5000

6000

7000

8000

wavebands B

wav

elen

gths

W

dual algorithm tradeoff

SQRT(N) algorithm

greedy algorithm

uniform waveband sizing approach

Fig. 7. Performance of the uniform waveband algorithm for astar with N = 10, P = 1000. The red (dashed line) and blue(dotted line) asymptotes represent the SQRT(N) and greedyalgorithms, respectively.

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ARTICLE IN PRESS

and greedy algorithms, which operate in the mini-mum-waveband and minimum-wavelength re-gimes, respectively. The dual-algorithm approachessentially interpolates between the minimum-waveband and minimum-wavelength cases to pro-duce algorithms with intermediate waveband andwavelength requirements.

4. The uniform waveband approach

Thus far in discussing multi-source traffic, wehave allowed wavebands to be non-uniformlysized. Here we consider fixing all wavebands to aconstant size b and derive the number of wave-lengths and wavebands required. By varying b,a family of banding algorithms with varyingnumbers of wavebands and wavelengths can beobtained. Somewhat surprisingly, we will see thatin the multi-source case, by using uniform wave-band sizes, most of the maximum banding gaincan be obtained at a very small cost in additionalwavelengths.

We begin by first deriving the minimum numberof wavebands required for a fixed waveband size b.

Theorem 1. Given a fixed band size b, the neces-

sary and sufficient minimum number of wavebands

required to support P-port traffic in an N-nodestar is

Buniform ¼ N þ P Nb

. ð12Þ

Corollary. The corresponding necessary and suffi-

cient minimum number of wavelengths required is

W uniform ¼ bN þ bP N

b

. ð13Þ

We first prove necessity of (12) by providing anexample which requires at least this number ofwavelengths. Sufficiency will be shown by con-struction. Eq. (13) then follows directly from thefact that each band is of size b.

Consider the traffic set where node 1 sends asingle call to nodes 1 to N 1, and P (N 1)calls to node N. In this case, N 1 bands are re-quired to support traffic to the first N 1 nodes,

while PðN1Þb

l mbands are required to support traf-

fic to node N. This gives a lower bound on thenumber of wavebands of

Buniform P ðN 1Þ þ P ðN 1Þb

� �

¼ N þ P Nb

;

where the last step follows from the observation

that PðN1Þb

l m¼ PN

b

� þ 1.

We will show using a bipartite matching ap-proach that these quantities are sufficient as well.We will first construct a bipartite multi-graph. Amulti-graph differs from a graph in that it allowsmultiple edges between the same two nodes. Weconsider two sets of nodes V1 = {s1, . . . , sN} andV2 = {d1, . . . ,dN}. For a given admissible trafficset, define the number of calls from node i to nodej to be ci,j. Then create dci;jb e edges connecting nodesi to dj. The complete set of edges E represents thetraffic to be carried, now in units of wavebands(each of size b) instead of wavelengths.

The number of edges adjacent to each sourcenode si can be obtained by summing

ci;jb

& 'over j.

Let the number of destinations receiving non-zerotraffic from node i be Ni. Without loss of general-ity, assume these are nodes 1 through Ni. Wedecompose the traffic to these destinations by

ci;j ¼ xi;j þ ri;j; ð14Þ

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where ri,j is chosen such that xi,j is a nonnegativeinteger multiple of b, and 1 6 ri,j 6 b. We can thenexpress the summation of interest as

XNi¼1

ci;jb

l m¼XNi¼1

xi;j þ ri;jb

l m

¼XNi

i¼1

ri;jb

l mþXNi

i¼1

xi;j

b

¼ Ni þ1

b

XNi

i¼1

xi;j; ð15Þ

where the second step relies on xib being integer,

and the third on the fact that ri 6 b.By summing (14) over i and noting thatPNii¼1ci;j ¼ P , we can obtain the following useful

relation:

P ¼XNi

i¼1

xi;j þXNi

i¼1

ri;j )XNi

i¼1

xi;j ¼ P XNi

i¼1

ri;j

6 P Ni;

where the last line results from observing thatri,j P 1. Using this result, (15) becomes

XNi¼1

ci;jb

l m6 Ni þ

P Ni

b¼ 1 1

b

� �Ni þ

Pb

6 1 1

b

� �N þ P

b¼ N þ P N

b.

Since both the summation on the left and N on theright are integers, by taking the floor of both sideswe can conclude

XNi¼1

ci;jb

l m6 N þ P N

b

.

Similar arguments can be used to show that thenumber of edges adjacent to each destination nodedj is given by

XNj¼1

ci;jb

l m6 N þ P N

b

.

Therefore each node has at most N þ PNb

� edges

adjacent to it. The following lemma will now proveuseful.

Lemma 3. In a bipartite multi-graph where each

node is adjacent to at most k edges, a partitioning

exists that divides the edges into at most k

matchings.

Proof. See [14]. h

By Lemma 3, at most an equal number ofmatchings are required. Since calls in each match-ing can share the same waveband, at mostN þ PN

b

� wavebands are required. The number

of wavelengths follows directly from the fact thateach waveband is of size b.

Example 10. Consider a star with N = 10 andP = 1000, and consider uniform band sizesof b = 40. By Theorem 1, N þ PN

b

� ¼ 10þ

10001050

� ¼ 29 wavebands are required. Since each

waveband consists of 50 wavelengths, a total of50 Æ 29 = 1711 wavelengths are used as well.

We now have a method for obtaining necessaryand sufficient conditions on the number of wave-bands and wavelengths are necessary and sufficient.By ignoring the integrality constraints, we cansolve for the number of wavelengths as a functionof the number of wavebands and obtain an approx-imate characterization of the waveband-wave-length tradeoff using uniformly-sized wavebands:

W uniform ¼ P þ P NBuniform N

1

� �N . ð16Þ

Fig. 7 illustrates the performance of the uniformwaveband algorithm for a star with N = 10,P = 1000. The two asymptotes represent theSQRT(N) and greedy algorithms, which operatein the minimum-waveband and minimum-wave-length regimes, respectively. Note that althoughthe uniform waveband algorithm performs poorlyin the minimum-waveband regime (where it re-quires many more wavelengths than the SQRT(N)algorithm), the performance improves dramati-cally once a few additional wavebands are intro-duced. By around 40 wavebands, it requires onlyslightly more wavelengths than the greedy algo-rithm, which uses 121 wavebands. We observe thatby allowing slightly more wavelengths than theminimum-wavelength case, the fixed-wavebandalgorithm can greatly reduce the number of wave-bands required, approaching the minimum-wave-band bound significantly.

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From this graph, two observations can be made:

1. As the number of wavebands increases, the per-formance of the uniform-waveband algorithmappears to approach the optimal performanceof the greedy algorithm. In particular, at theright endpoint, it appears to be almost wave-length-efficient.

2. Because of the slow increase in the number ofwavelengths required as the number of wave-bands decreases, it appears that the majorityof the reduction in the number of wavebandscan be achieved at very little cost in wavelengthinefficiency (as compared to the greedyalgorithm).

The first observation can be verified bycomparing the number of wavelengths used bythe uniform-waveband algorithm to the greedyalgorithm. For P large we can approximate Bgreedy

by N2

4log 4P

N2

$ %. For this number of bands, the num-

ber of wavelengths used by the uniform-wavebandalgorithm is approximately

W uniform � P þ PNB¼ 1þ 1

N2

4log 4P

N2

$ %24

35 � P

¼ ð1þ aÞ � P ;

where a is a term that goes to zero as P increases.Recall that the greedy algorithm, which was wave-length-efficient, uses the minimum ofPwavelengths.Therefore the performance of the uniform-wave-band algorithm approaches the optimum asymptot-ically in the minimum-wavelength regime.

It is also possible to show analytically by slopeanalysis that Wuniform approaches its final valuevery quickly; this gives rise to the second observa-tion, which is extremely significant from a practicalperspective. If we are interested in building an ac-tual implementation, it indicates that a majorityof the gain from using banding can be achievedwith very little wavelength inefficiency. For exam-ple, in the graph of Fig. 7, the processing granular-ity can be reduced from 1000 wavelengths (withoutbanding) to 30 wavebands, a reduction of 97%, at acost of only a 50% increase in the number ofwavelengths.

5. Banding on general topologies

Thus far all our results have been for the startopology. In this section we extend the precedingbanding results to general topologies for whichrouting algorithms for P-port traffic are known.

Recall that we have shown that banding can beconsidered as a matrix decomposition problem,where for a given admissible traffic set C, our goalis to decompose it into the sum of a fixed numberB of weighted permutation matrices:

C 6 b1T 1 þ b2T 2 þ � � � þ bBT B; ð17Þwhere the band sizes {bi} and the total number ofwavebands B are constant for all traffic sets. Thegoal was to minimize, over all possible admissibletraffic sets, the two cost parameters correspondingto the number of wavebands B and the number ofwavelengths

PBi¼1bi.

In the star, each permutation Ti could beaccommodated using a single waveband consistingof bi wavelengths. This approach can be extendedto other topologies in a straightforward manner,with the main difference being that each permuta-tion Ti may now require multiple wavebands ofsize bi to support it. In general, if the RWA algo-rithm requires /(N) wavelengths for permutationtraffic on the topology, and a banding algorithmis considered which uses B wavebands and W

wavelengths on a star, then the extension of thatbanding algorithm to the new topology requires:

Btotal ¼ B � /ðNÞ;W total ¼ W � /ðNÞ.

Routing algorithms for permutation traffic existin the literature for rings with [8] and without [4]conversion, trees [13], and torus networks [7].

For example, in the case of a ring, [4] providesan optimal RWA algorithm for the bidirectionalring topology without conversion using the mini-mum number of wavelengths. Specifically, it showsthat dN/3e wavelengths are necessary and sufficientto support any single-port traffic set. Since Ti is apermutation matrix, it can be supported usingdN/3e wavelengths. Each set of calls biTi fromthe decomposition of (17), can therefore be sup-ported using dN/3e wavebands, each consistingof bi wavelengths.

18 L.-W. Chen et al. / Computer Networks xxx (2005) xxx–xxx

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Using this approach, the entire traffic set T canbe supported using B sets of wavebands, whereeach such set i consists of dN/3e wavebands of biwavelengths. The total numbers of wavebandsand wavelengths are

Btotal ¼ BN3

� �;

W total ¼N3

� �XBi¼1

bi ¼ WN3

� �.

6. Conclusion

In this paper, we considered waveband switch-ing as a method for reducing overall network costs.We provide wavelength-efficient algorithms thatuse the minimum possible number of wavebands,and show that the optimal approach is to use agreedy algorithm. We provided minimum-wave-band algorithms that allow for small wavelengthinefficiencies in return for reducing the number ofwavebands down to just the nodal degree. We usethese results to help characterize the optimal per-formance frontier. We also provided a uniformwaveband approach that compares very favorablyto the optimal performance frontier and achieveslarge reductions in switching requirements at verylittle cost in wavelength inefficiency. Finally, we ex-tend our results to general topologies where permu-tation traffic routing algorithms are known.

Appendix A

The proof of Lemma 2 will be by construction.We will first decompose the traffic matrix C intotwo sub-matrices: the ‘‘heavy’’ matrix CH, contain-

ing all entries with weight greater than Pkþ1

l m, and

the ‘‘light’’ matrix CL, containing no entries great-

er than Pkþ1

l m.

We first assign any entry in C greater than Pkþ1

l mto CH. Note that at this point each row and col-umn in CH contains at most k entries. (If anyrow or column exceeds k entries, then that rowor column in C must have had a sum greater thanP, meaning C is not an admissible traffic set.) We

next continue assigning entries in C to CH untileach row and column of CH has exactly k entries.

Suppose there exists a row in CH that containsfewer than k entries. Then there must also be a col-umn that has fewer than k entries. (This followsfrom the fact that CH is square; if all columns havek entries, and each row has no more than k entries,then all rows must also have k entries.) Locate theentry corresponding to that row and column in C,and assign it to CH. Repeat until each row has k en-tries. By the same reasoning as before, all columnsmust now have k entries also. It is well known thatany such matrix can be decomposed into at most kmatrices with only one non-zero entry per row andcolumn. Therefore, by performing this furtherdecomposition and noting that all entries in CH

are at most P, we have shown that CH can besupported by at most k wavebands of size P.

Assign all remaining entries in C to CL; thisgives CL therefore has exactly N k entries perrow and column. This can similarly be decom-posed into N k matrices with only one entryper row and column; since each entry is at most

Pkþ1

l m, we can support CL using at most N k

wavebands of size Pkþ1

l m.

References

[1] L. Li, A.K. Somani, Dynamic wavelength routing usingcongestion and neighborhood information, IEEE/ACMTrans. Network. 7 (1999) 779–786.

[2] D. Banerjee, B. Mukherjee, Wavelength-routed opticalnetworks: linear formulation, resource budgeting tradeoffs,and a reconfiguration study, IEEE/ACM Trans. Network.8 (2000) 598–607.

[3] R. Ramaswami, K.N. Sivarajan, Routing and wavelengthassignment in all-optical networks, IEEE/ACM Trans.Network. 3 (1995) 489–500.

[4] A. Narula-Tam, P.J. Lin, E. Modiano, Efficient routingand wavelength assignment for reconfigurable WDMnetworks, IEEE J. Select. Areas Commun. 20 (2002) 75–88.

[5] O. Gerstel, G. Sasaki, S. Kutten, R. Ramaswami, Worst-case analysis of dynamic wavelength allocation in opticalnetworks, IEEE/ACM Trans. Network. 7 (1999) 833–845.

[6] P. Saengudomlert, E. Modiano, R.G. Gallager, An on-linerouting and wavelength assignment algorithm for dynamictraffic in a WDM bidirectional ring, in: Proc. JCIS, 2002.

[7] P. Saengudomlert, E. Modiano, R.G. Gallager, On-linerouting and wavelength assignment for dynamic traffic inWDM ring and torus networks, in: Proc. IEEE INFO-COM, 2003.

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[8] L.-W. Chen, E. Modiano, Efficient routing and wavelengthassignment for reconfigurable WDM networks with wave-length converters, in: Proc. IEEE INFOCOM, 2003.

[9] A.A.M. Saleh, J.M. Simmons, Architectural principles ofoptical regional and metropolitan access networks, J.Lightwave Technol. 17 (1999) 2431–2448.

[10] M. Lee, J. Yu, Y. Kim, C. Kang, J. Park, Design ofhierarchical crossconnect WDM networks employing atwo-stage multiplexing scheme of waveband and wave-length, IEEE J. Select. Areas Commun. 20 (2002) 166–171.

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Li-Wei Chen (S�97) received the B.A.Sc.degree in Electrical and ComputerEngineering from Queen�s University,Kingston, ON, Canada, in 1998, and theM.A.Sc. degree in Electrical and Com-puter Engineering from the UniversityofToronto, Toronto, ON, Canada, in2000. He is currently working toward hisPh.D. degree in the Laboratory forInformation and Decision Systems(LIDS) at the Massachusetts Institute ofTechnology (MIT), Cambridge.

His research interests are in the area of network architecture,with an emphasis on optical and highspeed networks.

Poompat Saengudomlert received theKing Scholarship from Thailand in1991 to earn his B.S.E. degree in Elec-trical Engineering from PrincetonUniversity in 1996. He then obtainedhis M.S. and Ph.D. degrees, both inElectrical Engineering and ComputerScience, from MIT in 1998 and 2002respectively. From 2002 to 2004, he wasa Postdoctoral Associate at the Labo-ratory for Information and Decision

Systems at MIT. From 2005, he is an assistant professor at the

Asian Institute of Technology, Thailand. His research interest

includes communication theory, optical networks, arrayprocessing, and resource allocation problems.

Eytan Modiano, Senior Member ofIEEE, received his B.S. degree inElectrical Engineering and ComputerScience from the University of Con-necticut at Storrs in 1986 and his M.S.and Ph.D. degrees, both in ElectricalEngineering, from the University ofMaryland, College Park, MD, in 1989and 1992 respectively. He was a NavalResearch Laboratory Fellow between1987 and 1992 and a National

Research Council Post Doctoral Fellow during 1992–1993

while he was conducting research on security and performanceissues in distributed network protocols.

Between 1993 and 1999 he was with the CommunicationsDivision at MIT Lincoln Laboratory where he designed com-munication protocols for satellite, wireless, and optical net-works and was the project leader for MIT Lincoln Laboratory�sNext Generation Internet (NGI) project. He joined the MITfaculty in 1999, where he is presently an Associate Professor inthe Department of Aeronautics and Astronautics and theLaboratory for Information and Decision Systems (LIDS). Hisresearch is on communication networks and protocols withemphasis on satellite, wireless, and optical networks.

He is currently an Associate Editor for CommunicationNetworks for IEEE Transactions on Information Theory andfor The International Journal of Satellite Communications. Hehad served as a guest editor for IEEE JSAC special issue onWDM network architectures; the Computer Networks Journalspecial issue on Broadband Internet Access; the Journal ofCommunications and Networks special issue on Wireless Ad-Hoc Networks; and for IEEE Journal of Lightwave Technologyspecial issue on Optical Networks. He is the Technical Programco-chair for Wiopt 2006 and vice-chair for Infocom 2007.