IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 9, NOVEMBER 2003 1441

Scheduling Bursts in Time-Domain WavelengthInterleaved Networks

Kevin Ross, Nicholas Bambos, Member, IEEE, Krishnan Kumaran, Iraj Saniee, Member, IEEE, andIndra Widjaja, Senior Member, IEEE

Abstract—We consider the problem of scheduling bursts of datain an optical network with an ultrafast tunable laser and a fixedreceiver at each node. Due to the high data rates employed on theoptical links, the burst transmissions typically last for very shorttimes compared with the round trip propagation times betweensource-destination pairs. A good schedule should ensure that1) there are no transmit/receive conflicts; 2) propagation delaysare observed; and 3) throughput is maximized (schedule length isminimized). We formulate the scheduling problem with periodicdemand as a generalization of the well-known crossbar switchscheduling. We prove that even in the presence of propagationdelays, there exist a class of computationally viable scheduling al-gorithms which asymptotically achieve the maximum throughputobtainable without propagation delays. We also show that anyschedule can be rearranged to achieve a factor-two approximationof the maximum throughput even without asymptotic limits.However, the delay/throughput performance of these schedulesis limited in practice. We consequently propose a schedulingalgorithm that exhibits near optimal (on average within 7%of optimum) delay/throughput performance in realistic networkexamples.

Index Terms—Birkhoff-von Neumann theorem, burst sched-uling, constrained queueing systems, dynamic throughputmaximization, optical networking, protection, stability.

I. INTRODUCTION

PROGRESS in optical transmission technology has enabledhigh-bandwidth and cost-effective transfer of information

based on wavelength-division multiplexed (WDM) systems. Ina transparent optical network, end-to-end wavelength servicesare provided without costly electronic regenerators at interme-diate nodes. Currently, services based on end-to-end wavelengthconnections are generally not economical since the total trafficdemand between node pairs has not scaled to the wavelength ca-pacity. Optical packet switching (OPS) [1], [2] and optical burstswitching (OBS) [3], [4] have been proposed to improve the uti-lization of wavelength connections in transparent networking.In OPS, packets (on the order of 100 s of nanoseconds) fromthe input fibers are forwarded to the output fibers in the opticaldomain. Thus, OPS requires ultrafast optical cross-connects or

Manuscript received January 2, 2003; revised August 5, 2003.K. Ross and N. Bambos are with the Departments of Electrical Engineering

and Mathematical Sciences and Engineering, Stanford University, Stanford, CA94305 USA.

K. Kumaran is with Exxon Mobil Research and Engineering, Annandale, NJ08801-0998 USA.

I. Saniee and I. Widjaja are with Mathematical Sciences, Bell Laboratories,Lucent Technologies, Murray Hill, NJ 07974 USA (e-mail: iis@research.bell-labs.com.

Digital Object Identifier 10.1109/JSAC.2003.818230

switches for forwarding packets and optical buffering for re-solving packet contention. In OBS, the switch forward bursts(consisting of multiple packets on the order of microseconds)optically from input fibers to output fibers. Depending on theprotocols used for resolving burst contention, OBS may or maynot need optical buffering. However, OBS still needs fast opticalswitching capable of reconfiguration at microsecond timescaleand the associated contention algorithm at each switch. Thesetechnological barriers do not seem to be easily overcome in theforeseeable future.

In [5], we proposed a new optic-based transport network,called time-domain wavelength interleaved network(TWIN),that bridges the gap between end-to-end demands and wave-length capacity without the need for fast (microsecond or evenmillisecond) switch reconfiguration or optical buffering. TWINalso performs efficient traffic grooming without resorting toelectronic cross-connects, as is done today in SONET/SDH(e.g., see [6] and [7]). The TWIN architecture consists ofa simple core that is based on passive wavelength-selectiveswitches (WSS) capable of routing incoming wavelengths tothe appropriate outgoing ports (or fibers) [8] and an intelligentedge utilizing fast tunable lasers to emulate fast switching ofdata in the core.1

Fig. 1 shows a simple TWIN architecture. In its simplestform, each destination nodeis assigned a unique wavelength

. When a source node has data to send to destination,the source tunes its laser to for the duration of the datatransmission.2 To ensure that the optical signal of a givenwavelength launched by a source will be routed to the intendeddestination, the WSS at each node is preprovisioned so thatan incoming optical signal of a given wavelength is routedto the appropriate output port. The preprovisioning essentiallydetermines the route for each source-destination pair. Oncepreprovisioned, each WSS performs self-routing of variousoptical signals on-the-fly based on their wavelengths. Note thatfast active optical switching is physically eliminated in thisarchitecture as the WSS reconfiguration timescale is equivalentto the timescale routes are modified, which is relatively longunder normal operating conditions (e.g., no failures). Tunablelasers at the sources emulate fast switching and enable optical

1A transmitter with a multifrequency laser can switch wavelengths in sub-nanoseconds [9].

2Without loss of generality and for simplicity of exposition, we assumethroughout this paper that the total demand from/to a given source/destinationis no more than the capacity of a single�. In general, a uniqueset of�s may be assigned to each node when transmitted/received load exceedscapacity of a single�. The scheduling algorithms in this paper apply tothis more general case with minor modification.

0733-8716/03$17.00 © 2003 IEEE

1442 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 9, NOVEMBER 2003

Fig. 1. Logical multipoint-to-point trees, one for each of the two destinations, overlayed on top of a physical topology.

grooming through dynamic tuning of appropriate wavelengthsat designated times. To prevent collisions at a node, TWINrelies on scheduling to coordinate the tunable lasers. In thispaper, we consider the problem of scheduling bursts of periodicdata among a multiplicity of source and destination nodes thatare geographically separated.

Fig. 1 shows an example where multipoint-to-point wave-length connections rooted at two destinations (nodes 6 and7) have been configured. Note that the multipoint-to-pointconnections require the wavelength-selective switches to becapable of merging incoming wavelengths intended for thesame destination.3 For example, node 5 must be capable ofmerging incoming wavelength from two inputs connectedto nodes 2 and 4 to the same output connected to node 7. Bydesigning multipoint-to-point connections, TWIN only needsa total of wavelengths in a network of nodes. Further,by employing only one fast tunable laser at each source, TWINmaximizes the utilization of the transmitter; that is, a singletunable laser at the source shares its transmissions to dif-ferent destinations by using different wavelengths interleavedin time domain. The brief data transmission on a particularwavelength is called a burst. For example, in Fig. 1, node 1interleaves its transmissions to nodes 6 and 7 by tuning itslaser to when it is transmitting a burst to node 6 and to

when it is transmitting a burst to node 7. Each node in thenetwork simply performs self-routing of bursts based on theircolors (i.e., wavelengths). For example, node 2 will route anincoming burst of to node 4 and an incoming burst ofto node 5, while node 5 will route an incoming burst ofto node 4 and an incoming burst of to node 7, which isthe intended destination. Finally, note that an incoming burstof arriving at node 4 will be routed to node 6, which isthe intended destination.

Burst scheduling is an important component of the overallTWIN architecture. Because a source laser can only transmitbursts one at a time, the scheduler has to avoid conflicts bothat the source and at the destination. In the following, we as-sume that time is slotted at each receiver and each fixed-size slot

3See [5] for a description of wavelength-selective switches with merging .

can contain exactly one burst. At each time slot, the schedulerneeds to assign the appropriate wavelength to each tunable laser.The scheduler must not only avoid source/destination conflicts,but also make efficient use of the available time slots. Unlikethe well-known problem of scheduling packets in a single node(e.g., see [10] and [11]), scheduling in TWIN needs to deal witha network of arbitrary topology and link propagation delays. Tothe best of our knowledge, such a scheduling problem has notyet been investigated in the past. We note, however, that the needfor scheduling in all optical networks has been recognized ear-lier in [12].

We represent the scheduling problem for periodic demandsas a (periodic) sequence of matchings in a bipartite graph whereeach node in the original network is represented by a pair ofsource-destination nodes in the bipartite graph. The propagationdelay between a source nodeand a destination nodeis due tothe length of the fiber on the transmission tree withas its root(see Section II-B).

We consider a class ofbatching policies that asymptoti-cally achieve maximum throughput for admissible load. Thebatching procedure involves creating a sequence of (carefullyconstructed) basic static schedules. Such basic static schedulesare derived from permutation matrices (and their derivatives)which give 100% throughput when there is no propagationdelay. Observing that such an algorithm does not take advantageof the interleaving opportunity in the problem, and may lead tounnecessary delays, we propose an alternative algorithm whichperforms well under manageable demands.

This paper proceeds as follows. Section II describes the keyassumptions and the model in detail. In Section III, we formulatethe TWIN scheduling problem with nonzero propagation delay.In Section IV, we construct simple batch policies that achieveoptimal throughput asymptotically with and without protectionrequirements. In Section V, we give a constructive factor-twoapproximation that is shown to be achievable foranyschedulingscheme. In Section VI, we propose a computationally moresophisticated algorithm TWIN iterative independent set (TIIS),which performs well in realistic network settings. Finally,Section VII shows simulation results for the proposed algorithmon a set of real network settings with real traffic demands.

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Fig. 2. Example of a two-node scheduling problem in which propagation delay increases the scheduling interval.

II. PRELIMINARIES

A. Model Attributes

Consider a TWIN network whereby a periodic traffic demandin units of bursts per cycle is to be scheduled for each node

pair . In what follows, we consider bursts of fixed durationonly. For example, with the burst size of 50s and the trans-mission rate of 10 Gb/s, a DS3 ( Mb/s) demand between anode pair translates to bursts/s.

In the case where there is no propagation delay between thetime a burst is sent and the time the same burst is received, wecould model the scheduling problem as a bipartite maximumweight matching problem [13]. A matching is a set of concur-rent nonconflicting transmissions between node pairs. Weightededges between source nodes on the left and destination nodes onthe right in the bipartite graph represent the demands. A feasibleschedule is a sequence of matchings such that each source-desti-nation pair is included in exactly matchings. The lengthof time needed to complete all matchings for a givenis re-ferred to as theperiodof the schedule.

Suppose that the transmission time of each burst takes exactlyone time slot of a fixed duration. The minimum number of timeslots to meet all demands in such a graph (with zero propagationdelays) is determined by the node with maximum total weightas follows:

(1)

In TWIN, the nodes in the network are geographically sepa-rated. Therefore, a physical distance between nodes implies anonnegligible propagation delay ( s/km) from sending toreceiving a burst. In this paper, we study how propagation delaysinfluence the burst scheduling problem. Observe that the effectof propagation delays complicates the problem in the sense thatsimultaneous burst transmissions at sources do not necessarilyimply simultaneous burst arrivals at destinations. More impor-tantly, time-separated burst transmissions at different sourcesmay arrive at the same destination simultaneously. Thus, it isclear that various propagation delays must be accounted forcarefully to avoid conflicts. With nonzero propagation delays,we say that a schedule is feasible if a source sends at most one

burst and a destination receives at most one burst in any timeslot.

To see how the propagation delays can affect the schedule,consider a simple network of four nodes in which two nodes 1and 2 transmit one burst to each of the two destination nodes 3and 4 as shown in Fig. 2 (left). Denote a burst fromto as

. If there are no propagation delays in this network,then all bursts can be sent in two time slots (i.e., ) asfollows: Send and in the first time slotfollowed by and in the second time slot, seeFig. 2 (middle). Now, consider the case where a propagationdelay of one time slot is introduced to a single node pair, say(1, 3), with other node pairs still having zero propagation delays.Then, the above schedule (in fact, any schedule) fails to sendall bursts within two time slots. The arrival of sentin the first time slot will coincide with the arrival ofsent in the second time slot. Fig. 2 (right) shows a minimumlength schedule with three time slots. From this example, wesee that propagation delay may increase the minimum time slotsrequired to meet a given demand.

We note that for a schedule of length no larger thanslots,the maximum queueing delay that any burst can experience isbounded by . For example, for tunable lasers with transmis-sion rates of 10 Gb/s at each node, with slot size of 50s [5],the storage requirement at each node is in the order of a singlescheduling period. For a scheduling periods of 200–400 slots(see Section VI), this translates to – 1010 – 10 bits or 12–25 Mbytes of storage.

B. Destination Trees

One key driver to reduce the network cost in TWIN is thateach source is equipped with a single tunable laser that can tuneto any wavelength. In general, a scheduler must avoid con-flict throughout every link of a network. Specifically, multiplebursts of the same wavelength transmitted by different sourcesmust not traverse each link simultaneously. Therefore, a sched-uler generally needs to keep track of the entire network state—apotentially prohibitive computational task. However, we use asimple procedure to ensure that such complicated mechanism isnot required.

One simplification in TWIN comes from the fact that eachdestination is assigned aunique receiving wavelength.This

1444 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 9, NOVEMBER 2003

Fig. 3. Destination tree with a unique wavelength: a schedule with no conflictsat the destination has no conflicts in any node or link of the tree.

condition ensures that all bursts in the network at a particularwavelength will be destined for a single destination node. Thus,conflicts can only occur among bursts with a common destina-tion, and we can consider each destination separately.

The use of destination trees in TWIN eliminates conflicts fur-ther. As shown in Fig. 3, since the union of paths traversed bythe sources to reach a given destination lie on a tree, a burst’sposition at a given time defines a unique arrival time at thedestination. Burst conflicts in the network, thus, correspond toa common arrival time at a destination. Correspondingly, if aschedule is such that arrival times have no conflicts, then thereare no conflicts anywhere in the network.

Destination trees can be constructed in a variety of ways. Forexample, one can use some spanning tree for the whole network(e.g., the minimum spanning tree). For a given destination, thebursts are sent to the destination along the path between theleaves of the source-destination pair. In the second example, onecan consider shortest paths in the network. It is well known thatthe shortest paths from all nodes to a single destination form atree rooted at the destination. Finally, one can consider a globalring structure. In this case each burst must be sent either all inone direction around the ring, or all along the shortest directionaround the ring to the destination. Either of these options leadsto a destination tree. Fig. 4 illustrates these three examples.

III. T RANSMISSION SCHEDULING IN TWIN:PROBLEM FORMULATION

Consider a network of nodes and denote their set by. For each communicating source-destina-

tion pair , we assign4 a logical queue, where databursts originating in nodeand destined to nodeare queuedup while waiting to be transmitted. The system operates inslotted time and one data burst can be transmitted in each timeslot .

The scheduler chooses a service configuration vectorat each time slot , where is 1 when a

burst is sent (if any is available) fromto at time slot , and

4In what follows, we take the indicesi; j to represent nodes inC = f1; 2; 3; . . . ; Cg. For reasons of notational simplicity, we do notinclude the index sets in the formulas where the aforementioned indices appear,but implicitly assume them, unless explicitly stated otherwise.

, otherwise. Hence, each service configuration—in the set ofall feasible ones —describes the set of queues from which aburst is transmitted (if any available) if is chosen. The systemqueue length or residual transmission demand vector is denotedby , where is the number of bursts inqueue at the endof time slot . The initial demand matrixis .

Given a schedule of service vectorsapplied in consecutive time slots, we can easily see that the evo-lution of each queue is governed by

(2)

The operator ensures that the demand in a queue remains0 after the queue empties for the first time. We denote bythe timespan of the schedule, that is, the time untilall queuesare cleared, given the initial demand.

For each source-destination pair , we assume that thereis a fixed, known propagation delay for sending a data burstfrom node to node . Hence, a burst sent at timefrom towill arrive at its destination at time . We assume that thepropagation delay is an integer multiple of time slots. (This as-sumption is made for simplicity in the exposition, but the anal-ysis can easily be extended to the case of nonintegral delays.)We denote the propagation delay matrix by

.Note that in the case of zero propagation delays, the ser-

vice policy must satisfy the followingconstraints:

(3)

for each time slot . These constraints are due to the fact thatat most one burst is sent by each source and one is receivedby each destination in each time slot. We note that the serviceconfiguration set in this case is the set of permutationmatrices and their derivatives

(4)

With a nonzero propagation delay matrix ,the service configuration constraints must involve time delays.For example, two different sourcesand may send to desti-nation at different time slots, but their propagation delaysand , respectively, may cause a conflict upon arrival at thedestination. Hence, the service schedulemust satisfy

(5)

(6)

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Fig. 4. Destination trees from the original network (a) can be constructed by using: (b) spanning tree, (c) shortest-path trees, and (d) ring.

Fig. 5. Scheduling as a matching with zero (middle) and nonzero (right) propagation delays.

for all times slots . We denote the service configuration set inthis case by

(7)

We aim to construct efficient service schedules which havehigh average throughput, and clear the initial demand overacceptable timespans.

Note that given an initial demand in the queues, the min-imum number of slots to clear the demand and deliverthe data bursts to their destinations can be obtained by solvingthe following optimization problem:

(8)

In the case of zero propagation delays, the problem simplyreduces to one of bipartite matching and is well known that

. Whennonzero propagation delays are introduced, the timespan for anyfeasible schedule can exceed . Therefore, it is of interest tocompare the throughput performance of schedules under

nonzero propagation delay against the lower bound andconsider the ratio

(9)

in various regimes, especially in the asymptotic regime of largedemands.

In the case of zero propagation delay, the set of availableconfigurations corresponds to the set of source-destination(input-output) matchings. This property is lost when propaga-tion delays are introduced. Nevertheless, many of the matchingproperties can be retained. Consider the 22 scheduling casewith propagation delays with one unit of demand between eachnode pair, as shown in Fig. 5. The reason the scheduling problemcannot be thought of as a simple matching problem is due to therelationship between sequential matches within a schedule. Thescheduled bursts at one time slot have a direct effect on the fea-sibility of sending bursts in the next time slot. Instead of usinga single graph to describe the network, we can draw the sched-uling problem according to a layered matching. A burst sent atone time slot arrives at a later time slot, with the correspondingdelay taken into account. We look for a matching which se-lects at least edges corresponding to input–output pair .Fig. 5 shows that the two-input, two-output problem over threetime slots can be represented by an abridged matching. Weightsor demands are over the set of edges corresponding to a pair overmultiple time slots. The figure illustrates the scheduling graph

in the matching formulation. As discussed earlier, with zeropropagation delays the problem has a schedule of two time slotswhile with propagation delay, three time slots would be required(Fig. 2). We define the bipartitescheduling graph as fol-lows. If there are initial nodes and time slots, sendingnodes (on the left) correspond to bursts being sent at each time

1446 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 9, NOVEMBER 2003

slot, and a further receiving nodes (on the right) correspondto those bursts arriving. These nodes are indexed according to

for sending at time slot and for receiving at timeslot , respectively. An edge joins node to nodefor each demand pair and each time slot .

IV. SIMPLE BATCHING POLICY WITH

ASYMPTOTIC OPTIMALITY

In this section, we show how schedules with zero propaga-tion delays (0-delay schedules) can be exploited in a simpleway to constructbatching policiesthat are valid schedules forthe system with nonzero propagation delays (-delay sched-ules). Further, we show that a-delay schedule constructed inthis manner can provide maximum possible (100%) asymptoticsystem throughput for any given fixed and finite propagationdelays between source-destination pairs. We note that dynamicschedules with similar properties have been considered in [14].Surprisingly, we find that this asymptotic result continues tohold even when the schedule is to accommodate failures of anarbitrary subset of connections that need to be rerouted on pathswith different propagation delays.

A. Constructing Batched Schedules

We note that the Birkhoff-von Neumann theorem [16], [17]permits us to construct a 0-delay schedule using a fixed numberof permutation matrices. More precisely, given the demand ma-trix , where is the number of nodes, thefollowing lemma holds as a direct consequence of this theorem:

Lemma 1: As before, letand denote a permutation matrix or

its derivative. It is then true that , wherefor some choice of permutation matrices

and their derivatives and .The lemma expresses the fact that, among the set of all pos-

sible permutation matrices and their derivatives , it ispossible to construct a schedule forconsisting of less than

permutation matrices, with matrix being repeated ex-actly times. Each matrix has unit entries iff thesource-destination pair is picked for transmission, and 0otherwise. The lemma also guarantees that, if theare all in-tegral, so are the . A valid 0-delay schedule can then be ob-tained by successively applying the in any chosen sequenceso that each appears times. Decomposing the demand ma-trix into permutation matrices efficiently is discussed in [11].

Assume now that one has performed such a decompositionand obtained and for a 0-delay schedule. How doesone construct a valid-delay schedule by utilizing the 0-delayschedule? The main idea is to transmit a batch of bursts with

copies of permutation , wait until all transmissions havereached their destinations and repeats the process with thenext batch of copies of . More formally, see thefollowing.

Theorem 1: Let and

, where is the delay fromsource to destination . Then, the following procedure gives avalid -delay schedule: Order the in any fixed manner,

say as and repeat the following steps for.

• Apply the schedule exactly times in successive timeslots; i.e., as abatch.

• Wait time slots.Proof: It is easy to see that this procedure produces a valid

-delay schedule. Repeated successive applications of the samedoes not produce any conflicts within batches, since succes-

sively transmitted bursts only arrive successively for any givenpair. The waiting time between distinct permutations is

chosen in such a way that the last arrival from the previous batchoccurs before the first arrival of the next batch, which avoids aconflict across successive batches.

This simple batched schedule with appropriately chosenwaiting times between batches then has a total length given

B. Asymptotic Optimality

We now show that thebatch scheduleof Theorem 1 is asymp-totically optimal in terms of achievable throughput. To do so,we first make the following simple observations regarding thescaling behavior of the schedule.

Lemma 2: With the 0-delay schedule, when the demands arescaled using a global parameteras , the de-composition is only changed by the scaling

.Lemma 3: With fixed nonzero propagation delays, scaling

the demands only preserves the structure of the schedule exceptfor the simple scaling of the batch sizes as . In partic-ular, the waiting times are independent of, whichleads to the scaling

where is the cumulative waiting time andis the maximum propagation delay.

We can now state the theorem of asymptotic throughputoptimality.

Theorem 2: As the scaling parameteris increased, the ratioof throughputs of the-delay schedule and the 0-delay scheduleapproaches unity arbitrarily closely. In other words, these twoschedules have identical throughput as .

Proof: The 0-delay schedule has length for any fixed, while the -delay schedule has length , as observed

in Lemmas 2 and 3. The theorem then simply follows from thefact that

From this result, we see that it is possible to improve thethroughput of the batch scheduled system with nonzero prop-agation delays by increasing the batch sizes while keeping thewaiting time overhead fixed between batches. The tradeoff oc-curs in the form of increased latency, which grows with, ascan be seen in Fig. 6.

The throughput ratio is given bywhile the latency ratio is given by .

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Fig. 6. (a) Behavior of throughput and latency ratios with scaling. (b) Latency-throughput tradeoff.

Notice the absence of in the denominator of the latter, sincethe 0-delay schedule does not require increased batch sizes togain efficiency and, hence, can provide low latency.

C. Protection

We now investigate batched schedule with protec-tion. Consider a scenario where the traffic between eachsource-destination pair is protected with two paths:pri-mary andbackup . In general, the primary and backup pathshave different propagation delays and , respectively.Typically, the primary paths are chosen to be shorter thanthe backups, so that . To improve survivability ofconnections, each such pair of paths is chosen so that only oneof them can fail at a time in most cases (i.e., in “single-failure”scenarios). Upon such a failure, all affected pairs woulddetect the failure and automatically switch to the backup path.To minimize service interruption, the pairs unaffected bythe failure do not alter their existing schedule.

It may appear that batched schedule with protection would in-crease , and cause a reduction in throughput due to the longerbackup delays. However, observe that, asymptotically, the over-head due to backup delays is not significant since the overheadonly translates to replacing the waiting times between batchesby longer times computed with and taking bothunfailed operation and all possible failure modes into account,that is

where . It is clear that making the waiting timelonger between batches does not affect the validity of theschedule. Thus, the asymptotic throughput in a batch scheduledsystem that takes failures into account is the same as a 0-delaysystem. Thus, accounting for failures introduces no overheadasymptotically.

Lemma 4: Batch schedules with (failure) protection have thesame asymptotic throughput as the 0-delay schedule withoutprotection.

While the above results establish asymptotic optimality of thebatch schedules, we observe that for finite scaling of demands

(latency), these schedules could be very inefficient. The waitingtimes between batches can be themselves comparable to,thus, making the actual efficiency factor of the order of.Hence, it may not be desirable to use the batched schedule inpractice. In the following sections, we show that it is possible toprove bounds even for simple greedy algorithms for finite de-mands and to construct a practical algorithm that performs sub-stantially better than the bounds offered by greedy algorithms.

V. ALTERNATIVE STATIC POLICIES

Consider a scheduling problem with a single batch. If weadopt the algorithm described previously which only uses theservice configurations , then the bound of maximumtime required to service the batch is , where

and . This bound is usually not acceptablewhen is small compared with .

If is small, we can consider other alternative algorithms tofind a schedule; for example, schedulingburst by burstinsteadof using a single linear programming solution, see (8). Specif-ically, consider any such algorithm which schedules bursts oneat a time with a condition that each new burst being schedulednot to conflict with a previously scheduled burst. Here, we es-tablish an upper bound on the time required for the schedule.

Lemma 5: Any sequential burst-scheduling algorithm willcreate a feasible schedule in time slots.

Proof: Consider a schedule of lengthwhere bursts are added sequentially. If a burst has not beenscheduled, there are at most time slots when is sched-uled to send bursts, at most time slots when is scheduledto receive bursts, and at most time slots which are not fea-sible for receiving bursts. Hence, there is at least one feasibletime slot when the burst can be sent at time and re-ceived at time .

Note that these bounds are for very general burst-based sched-uling algorithms. Any such algorithm will schedule an unsched-uled burst if it is possible. We conclude this section with a simpleconstructive upper bound for any feasible schedule.

Lemma 6: An optimal schedule for a TWIN network witharbitrary propagation delays requires no more than timeslots. Further, given any schedule, it can be made to require nomore than time slots. In general, for any reasonableschedule, .

1448 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 9, NOVEMBER 2003

Fig. 7. Scheduling a matching problem as an independent set problem: the maximal matching of the left hand graph is equivalent to maximal independent set onthe right.

Proof: Take any (possibly infeasible) schedule of lengthand reduce it to a partial (feasible) schedule of

length by removing all 1) conflicts and 2) transmissionsscheduled after . Take any such removed transmission,between, say, . We argue that this transmission could be ac-commodated within existing vacant time slots atand . Indeedthe selected transmission can be scheduled because atthereare at most arrivals, by definition 1). Since there are

slots at in total, by our construction, there must be atleast vacant slots. The inverse imageof these vacant slots at(by subtracting the fixed propagationdelay from these slots at) are also exactly slots. Again,by definition 1) only at most of these slots may havebeen assigned. Thus, there is at least one va-cant slot amongst these at. The selected transmission can nowbe scheduled at for . Proof follows by induction.

VI. TWIN I TERATIVE SCHEDULER

In the previous sections, we demonstrated that schedulingwith nonzero propagation delays asymptotically has the samelength as scheduling with zero propagation delays, andprovided a (rough) upper bound for any reasonableschedule. In this section, we construct schedules that attempt toachieve the lower bound even without asymptotic limits.

We propose a burst-based scheduling algorithm which pri-oritizes the order of edges selected from the scheduling graph

in the abridged matching formulation. To motivate the op-eration of the algorithm, we first note that maximal matchingproblems can be translated to problems of finding an indepen-dent set. Fig. 7 illustrates this translation on a simple network.

In [15], Kahale and Wright considered an independent setgraph model of wireless data networks. In their treatment, theyproposed a simple heuristic to find a max-weight independentset, where each node in a graph is assigned a weight. Theiralgorithm for a graph can be described as follows:

1. Let the max independent set .2. Let .3. and .4. Set and

and .5. Repeat steps 2–4 until .

We propose an extension of this algorithm, the TIIS sched-uling algorithm, to find an abridged matching to our schedulinggraph. This extension uses a weight for asetof edges of the same

source-destination pair. To simplify the exposition, we first as-sume that all s are to be scheduled for one period only. Theset of edges between the same source-destination pairatdifferent time slots is assigned a weight equal to the number ofbursts to be scheduled. When an edge is selected, the weightassigned to that edge set is decreased by one.

Further, to guarantee that link pairs are not oversubscribed,we propose a more frequent updating scheme. We update theweight every time an edge is selected. This prevents the schemefrom selecting a node which has already met its demand, andalso gives more up-to-date prioritizing of nodes.

TIIS begins with a candidate number of time slots. Itfirst determines if is feasibleand if so provides an explicitschedule with time slots. A schedule with near-minimallength is then obtained by binary search in .

Given a partial schedule , wesay that a burst is feasible at time if and

. Hence, we define

(10)

for all pairs at time slot . This indicator function is zeroif there is either a transmission scheduled from nodeat timeor an arrival at node at time .

We also defineneighborsof each edge. Two edges are saidto be neighbors if they could not be scheduled simultaneously,that is

(11)

The TIIS algorithm is as follows:

1. Set2. Set3. Let

4. Set .Set .

5. Repeat (3) until either for all or .

Note that helps determine if a neighbor of hasalready been scheduled, in which case that neighbor does notcontribute to the denominator in step 3). Also, note that there are

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two termination conditions, corresponding to success or failureof schedule creation. If , then the constructed schedulehas met all demands.

Remark 1) Time Complexity of TIIS:Let bethe total demand and be the number of nodes in the orig-inal graph. The scheduling graph created has nodes and

edges. At each iteration, the total weight of the graph isreduced by one, and the algorithm searches all remaining edgesand their neighbors. Since each edge has neighbors, therequired computation time is .

Remark 2) Nonintegrality of Propagation Delays:So far, wehave assumed that propagation delays are all integer multiplesof the burst/slot size. For 50-s slot size, this requirement trans-lates to distances that are integer multiples of 10 km. Althoughthis assumption simplifies the exposition of the scheduling al-gorithm, it isnota requirement for TWIN. When the integralitycondition does not hold, the destination conflict conditions ofSection III need to be modified to disallow overlap of two ormore arriving bursts at the same destination. In half of the sim-ulations reported in Section VII, the underlying networks do notsatisfy the integrality condition.

Remark 3) Periodic Schedules:If we schedule the burstsaccording to an abridged matching as described above, theschedule will be feasible in time slots. Since the schedulehas a lag time at the end for arrivals to happen, this can beinefficient.

An alternative is to take advantage of the periodicity of theschedule. If we are dealing with a system which has periodicdemands (e.g., from TDM applications) and delays each periodon a repeated sequence, then it is reasonable to schedule burstssent in one batch to arrive in the following. This approach mayallow the same number of bursts to be scheduled in less timeslots, thus, increasing efficiency. Periodic schedules are also lessdemanding in terms of computation time since the demand re-mains fixed for each period. Thus, the schedules are quasi staticin normal operation.

To solve the periodic scheduling problem, we introduce edgescorresponding to arrivals in future periods. Node is joinedby an edge to node .

Remark 4) Protection in the Network:A practical issue isconcerned with network protection to improve reliability. Wehave assumed a single tree structure for each receiver for thecase when the network is unprotected. This means that if eithera link or a node fails, a burst traversing this link or node willnever reach its destination.

Protection requires two disjoint paths (one for primary andthe other for backup) from each source to each destination. Anumber of mechanisms can be put in place, and we discuss theirimplications particularly in the context of the implementation ofthe TIIS algorithm. In general, the mechanism can be dividedinto online and offline backup. With online backup, a scheduleis first computed for the primary paths. When a failure occurs,a new schedule is recomputed using the back path(s) that areaffected by the failure. With offline backup, a joint schedule iscomputed for both primary and backup paths, and no recompu-tation is needed when failures occur. Because offline backup isgenerally much faster in terms of restoration time than offlinebackup, we focus on offline backup in the following.

TABLE INETWORKSUSED IN THE SIMULATION STUDY

The protection schedule for offline backup can be dealtwith in several ways. The traditional , 1:1 and sharedprotection schemes all have natural interpretations in TWIN.Here, we consider a 1:1 slot-shared backup mechanism, whereeach burst is scheduled for a primary and a backup transmis-sion. A key property of this protection mechanism is that asource-destination pair will switch to backup when its primarypath is unavailable and the switching mustnot create a conflictat the destination withanyother burst scheduled. The primaryand backup transmission for the same source-destination paircan share the same departure time at the source or the samearrival time at the destination since primary and backup routesare never used simultaneously. However, these transmissionscannot occur at the same time for any source-destination pair.(This distinguishes the proposed scheme from its less efficient

counterpart where primary and backup paths are usedsimultaneously and must, therefore, avoid conflict at bothsource and destination).

Incorporating (shared) protection into the TIIS algorithmturns out to require a relatively small change in the definition ofneighbors between the matchings to include a duplicate copy ofeach demand routed on a separate path. For the schedule withprotection, each demand is duplicated. While the primarypath has propagation delay , the backup path may have adistinct delay . Note that since the primary and backup pathsare not used simultaneously, their schedules can conflict. Theneighborhood structure ensures that the protection time slot foreach node pair is not in conflict with the rest of the scheduleexcluding the assignment of its primary path; that is,

(12)

With this alteration in the definition of neighbors, theTIIS algorithm with protection can be implemented as givenpreviously.

VII. SIMULATION RESULTS

In this section, we compare the TIIS algorithm against thelower bound corresponding to zero propagation delays. We alsoinclude a greedy algorithm that blindly selects node pair withthe highest demand at each iteration for comparison.

We consider various realistic transport networks (the realnames have been replaced with network numbers) based onactual traffic demand, as depicted in Table I. In our results, we

1450 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 21, NO. 9, NOVEMBER 2003

TABLE IISCHEDULE SIZES FORNETWORK SAMPLES WITH THEIR THROUGHPUTRATIOS (TR)

assume that the burst duration is 50s. The propagation delayon the fiber is approximately 0.2 km/s. Thus, the delay insending a message from hundred to a thousand kms is of theorder of tens to hundreds of such time slots.

The longest delay could happen if there is a single burst sentfrom node to node in period . In this case, the round-triptime from sending a burst fromto to receiving a confirmationwould be the length of a period plus the propagation delay. Themaximum delay is, therefore, .

Table II shows our simulation results. We first note that theTIIS schedules are on average within 7%

of the zero-delay lower bound while the greedyalgorithm requires 30% more time slots. For protection, we ob-serve a roughly 13% average increase in time slots for (shared)protection with TIIS while the greedy algorithm needs morethan twice the time slots. Based on these tests, we conclude that:1) creating efficient schedules in case of nonzero propagationdelays is nontrivial; 2) TIIS scheme is typically within a smallpercentage of a lower bound (the zero propagation case) on theoptimal schedule; and 3) shared protection with TIIS is usuallyeffective.

VIII. C ONCLUSION

We considered scheduling of bursts in a TWIN networkwhere a tunable laser at each node is shared for transmission tomultiple destinations by changing the wavelength transmittedbased on the assigned wavelength to each destination. Weshowed that the scheduling problem with zero propagationdelay is equivalent to the well-known optimal matchingproblem in bipartite graphs. When propagation delays arenonzero, TWIN scheduling requires extensions to bipartitegraph matching and we showed that the optimal bipartitematching is, in general, inadequate.

We demonstrated that the stability region when there arepropagation delays is indeed the same as the stability regionfor the problem with no propagation delays. This followedfrom construction of a class of a batching policy that asymp-totically has the same (maximal) efficiency as the case withno propagation delays. In particular, the batching policy usedpermutation schedules derived from matching in the zero prop-agation network via the Birkhoff–von Neumann theorem. Weshowed, somewhat surprisingly, that batching policies retaintheir maximal asymptotic throughput even when additionalmatchings are needed for protection on redundant paths withpossibly distinct propagation delays.

We described the TIIS algorithm, which is our proposed ap-proximation technique to construct explicit schedules in TWIN

networks with arbitrary propagation delays. The efficiency ofthe TIIS algorithm for a set of real network problems was onaverage within 7% of the lower bound on the optimal schedulewhile other (natural) greedy schemes required twice as manytime slots. We further showed that with shared protection, TIISrequired only an additional 13% time slots compared with thecase of no protection while the greedy scheme required twice asmany time slots.

We note that it is possible to extend these results withminor modifications to the case, where some nodes havemultiple tunable lasers or some destinations have multipleburst receivers. Both the asymptotic results proved and thecomputational results presented support the conjecture thatscheduling with propagation delays should in general requireno more than time slots, where is the schedulelength corresponding to zero propagation delay. A completedemonstration of this result is yet to be furnished.

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Kevin Rossreceived the B.Sc. degree in mathematicsfrom the University of Canterbury, Christchurch,New Zealand, in 1998 and the M.Sc. degree in engi-neering, economic systems, and operations researchfrom Stanford University, Stanford, CA, in 2001,where he is currently working toward the Ph.D.degree in management science and engineering.

He has also worked as a Research Science Internin the Mathematical Sciences Department, Bell Lab-oratories, Lucent Technologies, Murray Hill, NJ. Hisresearch interests are in the development of scalable

scheduling algorithms for quality of service control, with a focus on their appli-cation to communication systems.

Nicholas Bambos(S’84–M’89), photograph and biography not available at thetime of publication.

Krishnan Kumaran received the B.Tech. degree inmechanical engineering from the Indian Institute ofTechnology, Madras, and the Ph.D. degree in physicsfrom Rutgers University, New Brunswick, NJ.

He is currently a Member of the Complex SystemsModeling Section, Corporate Strategic Research ofExxon Mobil Corporation, Clinton, NJ. Formerly, hewas a Member of Technical Staff in the Mathematicsof Networks and Systems Research Department, BellLabs, Lucent Technologies, Murray Hill, NJ, wherehis research interests were in modeling, analysis and

simulation of resource management, and scheduling issues in wireless networksand ATM.

Iraj Saniee (M’98) received the B.A. and M.A.(honors) degrees in mathematics and the Ph.D.in operations research and control theory fromCambridge University, Cambridge, U.K.

He is head of the Mathematics of Networks andSystems Research Department, Bell Laboratories,Lucent Technologies, Murray Hill, NJ. The emphasisof research in his department is on the modeling,analysis and optimization of emerging processesin data, wireless, and optical networks. His recentresearch has focused on control and optimization

of resource-sharing optical networks, performance evaluation of multiscalingand limiting models for data traffic, network design and development ofalgorithms for the underlying mathematical and computational models. Priorto Bell Labs, he headed the Network Design and Traffic Research Group inthe Information Sciences Laboratory, Bellcore (Telcordia), where he directedresearch in performance analysis of new models of data traffic and developmentof light-weight tools for network design and management. He has publishednumerous articles in IEEE, SIAM, and INFORMS journals and proceedings.

Dr. Saniee is a Member of International Federation of Information Processing(IFIP) Working Group 7.3, serves on the Editorial Board of theOperations Re-search Journal, and on the program committees of several conferences.

Indra Widjaja (S’83–M’92–SM’02) received thePh.D. degree in electrical engineering from theUniversity of Toronto, Toronto, ON, Canada.

From 1994 to 1997, he was an Assistant Professorof electrical and computer engineering at theUniversity of Arizona, Tucson. From 1997 to 2001,he was with Fujitsu Network Communications,where he worked on multiservice switching andSONET transport products. Since 2001, he has beena Research Member at Bell Laboratories, LucentTechnologies, Murray Hill, NJ, performing research

in communication networks. He is coauthor ofCommunication Networks:Fundamental Concepts and Key Architectures(with Leon-Garcia).