Stable Logical Topologies for Survivable Traffic Grooming of Scheduled Demands

Jaekel et al. VOL. 2, NO. 10 /OCTOBER 2010/J. OPT. COMMUN. NETW. 793

Stable Logical Topologies forSurvivable Traffic Grooming of

Scheduled DemandsArunita Jaekel, Ying Chen, and Ataul Bari

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Abstract—There has been considerable research interestin the area of traffic grooming for WDM mesh networks. Thevast majority of the current work can be classified into oneof two categories, either static grooming or dynamic groom-ing. In many situations, the individual traffic demands re-quire bandwidth at certain predefined intervals, and re-sources allocated to nonoverlapping demands can be reusedin time. In this paper, we propose a new traffic groomingtechnique that exploits knowledge of the connection hold-ing times of traffic demands to lead to more efficient re-source utilization. We consider wavelength-convertible net-works as well as networks without any wavelength conver-sion capability and implement survivability using dedicatedand shared path protection. Although individual demandsmay be short lived, it is desirable to have a logical topologythat is relatively stable and not subject to frequent changes.Therefore, our objective is to design a stable logical topologythat can accommodate a collection of low-speed traffic de-mands with specified setup and teardown times. Our ap-proach results in lower equipment cost and significantly re-duced overhead for connection setupÕteardown. We presentefficient integer linear program (ILP) formulations that ad-dress the complete traffic grooming problem, including logi-cal topology design, routing and wavelength assignment,and routing of traffic demands over the selected topology.The primary focus of our ILP formulations is to minimizethe resource requirements. However, it is possible to modifyour formulations to maximize the throughput, if necessary.

Index Terms—Scheduled traffic model; Traffic grooming;WDM networks; RWA.

I. INTRODUCTION

W avelength division multiplexing (WDM) optical net-works are widely used for high-capacity backbone

networks due to their ability to carry large volumes of datawith a high degree of reliability and at a relatively low cost[1]. In such networks, the end-to-end optical communicationchannels called lightpaths [2] can be overlaid on top of thephysical fiber network. A lightpath may traverse one ormore fibers and must be assigned a single WDM channel oneach fiber link. Compared with the huge bandwidth of alightpath (2.5 Gbps to over 10 Gbps), individual requests forconnections are typically for data streams at a much lowerdata communication rate, of the order of megabits per sec-

Manuscript received July 15, 2009; revised July 10, 2010; accepted August16, 2010; published September 22, 2010 �Doc. ID 114259�.

The authors are with the School of Computer Science, University ofWindsor, Windsor, Canada (e-mail: [email protected]).

Digital Object Identifier 10.1364/JOCN.2.000793

1943-0620/10/100793-10/$15.00 ©

nd (Mbps). This tremendous mismatch between the capac-ty of individual lightpaths and the bandwidth requirementsf individual traffic demands has led to the emergence ofraffic grooming techniques. Traffic grooming techniques in

DM networks can be defined as a family of techniques forombining a number of low-speed data streams from users,o that the high capacity of each lightpath may be used asfficiently as possible [3]. These are normally classified asither static or dynamic traffic grooming, and there has beenonsiderable research interest in both of these areas in re-ent years. Static traffic grooming techniques [4–6] assumehat the set of low-speed traffic demands is known before-and and is persistent throughout the lifetime of the net-ork. Thus, there is no opportunity of sharing resourcesmong different demands. In dynamic traffic groomingDTG) [7,8], on the other hand, the arrival time of requestsre not known ahead of time. Hence, it is difficult to effi-iently design a stable topology that will be able to handlell traffic requests. Therefore, lightpaths are typically set upnd torn down based on the set of currently active demands.

Recently, a new traffic model, called the scheduled trafficodel, has been proposed in the literature [9]. This model is

ppropriate for applications that require periodic use ofightpaths (e.g., once per day) at predefined times. In this

odel, the setup and teardown times of demands are knownn advance, so resource allocation can be optimized in bothpace and time. Although routing and wavelength assign-ent [10] of scheduled lightpaths under the scheduled traf-c model have been considered in a number of recent papers11–15], these papers typically consider traffic demandsaving a coarse granularity (corresponding to the number of

ightpaths to be routed over the network). They do not ad-ress the problem of combining low-speed scheduled de-ands onto lightpaths in an efficient manner.

In this paper we propose a new approach for survivableraffic grooming and topology design that addresses theroblem of combining low-speed scheduled traffic demandsnto high-capacity lightpaths. Although individual (sub-avelength) demands may be short lived, our objective is to

reate a stable logical topology that is not subject to frequenthanges. The resulting logical topology should be capable ofccommodating all scheduled traffic demands, and these de-ands should be routed over the logical topology in a way

hat allows sharing of resources among nonoverlapping de-ands. The primary advantage of using a stable logical to-

ology is that it can be implemented using nonreconfig-

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794 J. OPT. COMMUN. NETW./VOL. 2, NO. 10 /OCTOBER 2010 Jaekel et al.

urable components, which is significantly less costly thanthe corresponding reconfigurable equipment. In addition, astable topology avoids short (but significant) disruptions toongoing traffic during topology changes and incurs no over-head, in terms of dynamic teardown and setup of lightpaths.

In this paper, we address the complete problem of surviv-able traffic grooming and topology design under the sched-uled traffic model in WDM mesh networks and proposethree integer linear program (ILP) formulations to solve thisproblem optimally. The objective of all three formulations isto minimize the amount of resources used to accommodateall the demands (assuming all demands can be accommo-dated). ILP1 and ILP2 do not assume wavelength conver-sion capabilities, so each lightpath is allocated the samechannel on each fiber it traverses. ILP1 considers fault-freenetworks only. For the design of survivable grooming ca-pable networks, ILP2 provides protection at the lightpathlevel, such that the network can survive single link failures,using either dedicated path protection (DPP) or shared pathprotection (SPP) techniques. Wavelength-convertible net-works can be handled by ILP3.

Our formulations solve the following subproblems:i) design of a stable logical topology that does not change

over time,ii) routing and wavelength assignment (RWA) for each

primary and backup lightpath included in the logicaltopology,

iii) combining of low-speed traffic demands onto high-capacity lightpaths, and

iv) sharing of resources among time-disjoint demands.

The resources being minimized may be at the optical level(e.g., number of transceivers) or at the electronic level (e.g.,the amount of electronic switching needed for each demand).The actual resource being minimized depends on the par-ticular objective function used by the ILP (as discussed inSubsection III.F). Our formulations also take into accountresource limitations, such as the number of transmittersand receivers at each node. It is well-known that the com-plexity of an ILP is, in general, exponential and heavily de-pendent on the number of integer variables. As a result,most of the existing ILP formulations for traffic groomingand topology design can only handle very small networkswith a small number of traffic demands. However, one of theimportant features of our proposed ILP formulations is thatwe employ novel techniques to develop efficient formula-tions that can generate optimal solutions to practical sizedproblems. For example, we have used special constraintsthat allow continuous variables to replace integer variablesin certain cases. This reduces the number of integer vari-ables and hence the overall complexity of the formulation.Furthermore, to the best of our knowledge, this is the firstapproach that jointly considers survivable network design,traffic grooming, and the RWA problem for scheduled sub-wavelength demands.

The main contributions of this paper are as follows:1. We propose a new approach for survivable traffic

grooming and topology design that exploits the knowl-edge of connection holding times of low-speed sched-uled traffic demands to design a stable logical topologyand maximize resource sharing.

2. We present efficient ILP formulations that optimallysolve the complete traffic grooming problem (includinglogical topology design and RWA) and provide protec-tion against single link faults at the lightpath level.This allows us to avoid using suboptimal heuristics,which often have no specified performance bounds.

3. We extend our proposed ILP to handle networks withand without wavelength converters.

4. We demonstrate through simulations that, unlikemost existing ILP formulations for traffic grooming,our formulation can be used for practical networkswith hundreds of individual traffic demands. Theproblem sizes considered in this paper are comparablewith those used for existing heuristic approachesavailable in the literature [4,16,17].

The remainder of the paper is organized as follows. Sec-ion II reviews the scheduled traffic model and traffic groom-ng techniques for fault-free networks as well as survivableraffic grooming. Section III presents our ILP formulationsor optimal topology design with traffic grooming under thecheduled traffic model. We discuss and analyze our resultsn Section IV and present our conclusions in Section V.

II. RELATED WORK

. Scheduled Traffic Model

The models of traffic demand, usually considered in theiterature for the design of WDM networks include static,ynamic, incremental, and scheduled traffic models. Intatic traffic demand models, the set of lightpaths to be es-ablished is known in advance. The dynamic traffic demandodels consider random arrival time and random duration

f the demand, whereas demands are incrementally addedo the network in the incremental traffic demand model.

In the fixed-window scheduled traffic model, each sched-led lightpath demand (SLD) [9] is represented as (s, d, n,

s, te), where s and d are the source and destination, n rep-esents the number of requested lightpaths for the demand,nd ts and te are the setup and teardown times of the de-and, respectively. Resource allocation for lightpaths under

he scheduled traffic model has received considerable re-earch attention in recent years [11–15], and it has beenhown that connection-holding-time-aware approaches con-istently outperform traditional RWA algorithms for sched-led lightpath demands. Kuri et al. [9] present a branch-nd-bound algorithm and a tabu-search-based algorithm toolve the routing problem. A generalized graph coloring ap-roach is used to solve the wavelength assignment problemeparately. Skorin-Kapov [11] improves the tabu-search-ased routing algorithm proposed in [9]. Instead of relyingn a randomized neighborhood search, the author develops

neighborhood reduction technique to reduce the searchpace significantly. In [12] the authors propose optimal ILPormulations for the design of survivable wavelength-onvertible networks, under the fixed-window scheduledraffic model. Heuristic solutions for the same problem haveeen presented in [13,14]. In [15], an ILP formulation andeuristic are presented for prioritized demands under thexed scheduled traffic model, with wavelength conversion.

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The fixed-window traffic model can be augmented so thatthe setup and teardown times are no longer fixed but canslide within a larger window [18,19]. In [19], the authors in-vestigate the relationship between wavelength efficiencyand time flexibility of the scheduled demands. In [20], theauthors provide ILP formulations for jointly optimizing de-mand scheduling and resource allocation in survivableWDM networks.

B. Traffc Grooming

Traffic grooming techniques are used to combine low-speed data streams onto high-speed lightpaths with the ob-jective of minimizing the network cost or maximizing thenetwork throughput. The optimal design of WDM mesh net-works with traffic grooming capabilities deals with jointlysolving the topology design, RWA, and traffic routing (of low-speed demands) subproblems. Traffic grooming can use ei-ther the bifurcated model or the nonbifurcated model. In thenonbifurcated (bifurcated) model, each user data stream iscommunicated using a single (one or more) logical path(s)from the source of the data stream to its destination. The bi-furcated model allows more efficient use of network re-sources, but the nonbifurcated model has a number of tech-nological advantages [4]. In this paper, we adopt thenonbifurcated model.

For static traffic grooming, it is assumed that the set ofdemands are known beforehand and the logical topology re-mains stable after deployment. A number of ILP formula-tions for solving the complete traffic grooming problem havebeen proposed in the literature [4–6]. Such formulationsquickly become computationally intractable, even formoderate-sized networks, and the problem is usually solvedby applying heuristics for practical networks.

For dynamic traffic grooming [7], traffic demands are pre-sented to the network sequentially or based on a specifiedarrival rate. Each demand can then be accommodated by i)routing over the logical topology using available bandwidthon the existing lightpaths or ii) establishing new light-path(s) if the required resources are available. If an estab-lished lightpath is not carrying any traffic, it may be torndown. Since lightpaths are created and/or destroyed in re-sponse to the current demand set, the logical topology typi-cally varies with time. In [8], the authors propose setting upa static topology a priori and then routing the dynamic re-quests to minimize blocking probability. This paper consid-ers single-hop routing only.

Although there has been much interest in traffic groom-ing techniques for both the static and dynamic traffic mod-els, relatively little work has been done in terms of trafficgrooming of scheduled demands. In [21], the authors pro-pose a number of heuristic algorithms, including a custom-ized tabu search scheme to schedule demands in time andallocate resources to lightpaths under the sliding scheduledtraffic model. However, this approach does not build a stablelogical topology but requires a topology that changes basedon the traffic demands active at any given time. In[16,22,23], the authors consider the cost savings achieved byimplementing a static logical topology for scheduled low-speed demands. These works typically focus on the topologydesign and grooming aspects (without RWA) and have not

ddressed survivability of the demands. In [22] lightpath to-ology configuration under a bifurcated traffic model is ad-ressed. The authors consider two variants of the problem,ith and without rerouting of IP/MPLS tunnels between the

ime periods, and propose ILP formulations that optimizeifferent objectives. In [16] the authors present MILP for-ulations to solve the topology design, RWA, and traffic

outing problem for scheduled traffic demands inavelength-convertible WDM networks. For large net-orks, space-reduction and decomposition heuristic algo-

ithms are used to reduce the problem size. In [23], the au-hors present ILP and heuristic algorithms to solve theopology design and grooming problem (without RWA) forcheduled demands. The traffic grooming schemes in16,22,23] all focus on fault-free networks and do not con-ider protection of lightpaths.

. Survivable Traffc Grooming

The survivable traffic grooming problem for WDM opticalesh networks has been addressed in [17,24–26]. Surviv-

bility can be achieved at the connection level or at theightpath level by using standard dedicated path protectionDPP) or shared path protection (SPP) techniques. Path pro-ection schemes establish two lightpaths, a primary (ororking) path and an edge-disjoint backup (or protection)ath for each logical edge. In SPP, resources allocated to arotection path can be shared with other protection paths ifhe corresponding primary paths are edge disjoint. In dedi-ated protection, such sharing is not allowed. This leads to aimpler implementation at the cost of using additional re-ources.

Heuristic grooming algorithms for solving the problem ofurvivable connections under various failures, such as fiberut and duct cut, have been studied in [17,24], under theeneral shared risk link group diverse routing constraints.he work in [24] considers static traffic and focuses on theroblem with protection at the subwavelength connectionevel, whereas path protection at the lightpath level haseen considered in [17]. The authors also outline ILP formu-ations to generate optimal solutions for very small net-orks with a limited number of requests, which wereainly used to validate the results of the proposed heuris-

ics. In [25], an efficient ILP formulation is proposed for theomplete survivable traffic grooming problem, including to-ology design, traffic routing, and routing and wavelengthssignment, using both dedicated and shared protection athe lightpath level. However, this approach is for staticrooming and does not consider scheduled demands. Theork in [26] focused on different frameworks for protecting

ow-speed connections against single link failures in WDMesh grooming networks. In this study, the authors have

roposed three approaches for the protection of connections,amely, the protection-at-lightpath (PAL), mixed protection-t-connection (MPAC), and separate protection-at-onnection (SPAC) levels. They have provided a qualitativeomparison among these methods and shown that, for bothhared and dedicated protection, SPAC performs better withsufficient number of grooming ports, and PAL performs

etter with a small to moderate number of such ports. Inhis paper, we have used PAL because the primary goal is touild a “stable” logical topology, i.e., we do not want the to-

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III. TRAFFIC GROOMING FOR SCHEDULED DEMANDS

In this section, we present our ILP formulations for trafficgrooming of scheduled demands with fixed setup/teardowntimes. These approaches can also be used in conjunctionwith standard dynamic traffic grooming schemes. A stablelogical topology capable of supporting the scheduled de-mands is set up first. Subsequently, if unscheduled demandsare presented to the network, they can be accommodated us-ing existing dynamic traffic grooming techniques. The statictraffic grooming problem can be treated as a special case ofour formulations, where the duration of each demand spansthe entire network lifetime. This means that all the de-mands overlap in time so that resource sharing among indi-vidual demands is not possible. A preliminary version of ourapproach is presented in [27]; however, it can only handlenetworks with full wavelength conversion capabilities anddoes not consider shared path protection.

We assume that a set of scheduled, low-speed, periodictraffic demands with fixed setup and teardown times arespecified as input. The demands are arranged in increasingorder of their start times and used to partition the entiretime period into disjoint intervals. In our scheme, it is quitepossible to have multiple demands with the same setupand/or teardown times. Figure 1 shows a simple exampleconsisting of five demands with start times ts1, ts2, ts3, ts4,and ts5 and end times te1, te2, te3, te4, and te5, respectively.The demands partition the entire time period into six inter-vals �l1 , l2 , . . . , l6�, as shown in Fig. 1. In this example, thestart times ts2 and ts3 coincide, the end times te2 and te4 co-incide, and finally end time te1 coincides with start time ts4.

Our formulation designs a logical topology that can ac-commodate all the demands active during each time inter-val �lj�, without exceeding the capacity of the lightpaths.Unlike static traffic grooming algorithms, our ILP also al-lows sharing of resources among time-disjoint demands. Fi-nally, the formulation also finds a feasible RWA for the set oflightpaths (primary and backup) in the logical topology.

We note that in Fig. 1, there are no demands scheduledduring interval l5. However, it does not make sense to sim-ply tear down all lightpaths during this interval. Thus, eventhough the bandwidth requirements vary from one intervalto another, our logical topology remains stable for the entireduration. Any available spare capacity (for example, during

Fig. 1. Scheduling of partially overlapping demands.

nterval l5) can be used to route unscheduled demands ar-iving at the network, using existing dynamic traffic groom-ng techniques. This reduces the overhead required for light-ath setup and teardown. Since our topology does nothange with time, it is important to design the topology asfficiently as possible. Therefore, the objective for all threef our formulations is to minimize the amount of resourcessed. This can be done either at the logical topology level ort the physical topology level by choosing an appropriate ob-ective function. The objective function can also be modifiedo maximize the network throughput, as explained in Sub-ection III.F.

. Network Parameters

The following network parameters are given as inputs.• A physical fiber network G�N ,E�, where N is the set of

nodes, and E is the set of links.• n=number of nodes in the network.• A set of channels K that each fiber can accommodate.• A set of potential logical edges P that may be included

in the logical topology.• TXi

�RXi�=number of transmitters (receivers) available

at node i.• g=capacity of a lightpath in OC-n notation. In this pa-

per, we have used g=OC-160.• A set of demands Q= ��sq ,dq ,nq , tsq , teq��, where the

bandwidth requirement nq of a demand q is specified inOC-n notation. We assume that the data rate of the in-dividual demands varies between OC-3 to OC-24 and isalways less than the capacity of a lightpath.

• A set of R precomputed routes, over the physical topol-ogy, between each source–destination pair.

• fesd,r=1, if and only if the rth route between source s and

destination d uses fiber link e.• lmax: The total number of intervals in the network.• L: The set �lj ,1� j� lmax� of time intervals.• A sequence of disjoint time intervals lj, j=1,2,3, . . .

used to partition the entire time period (24 hours in ourexperiments).

• lj,q=1, if and only if demand q is active during intervallj ,1� j� lmax.

. ILP Variables

The variables required for the ILP are defined in this sub-ection. We define the following integer variables:

• bp=1, if and only if logical edge p is included in the logi-cal topology.

• fpq=1, if and only if demand q is routed over logicaledge p.

• xr,p�yr,p�=1, if and only if logical edge p uses the rthroute to establish the primary (backup) lightpath, fromits source s to destination d.

• wk,p�zk,p�=1, if and only if channel k is assigned to theprimary (backup) lightpath corresponding to logicaledge p.

• mq=1, if and only if demand q is accommodated.

We also define the following continuous variables:• sp,e�tp,e�=1, if and only if logical edge p uses link e on its

primary (backup) route.

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• �k,pe ��k,p

e �=1, if and only if channel k on physical link eis assigned to the primary (backup) lightpath corre-sponding to logical edge p.

• �ke =1, if and only if channel k on link e is used by one or

more backup lightpaths.

Although sp,e�tp,e�, �k,pe ��k,p

e �, and �ke are defined as con-

tinuous variables, they are restricted to take on integer val-ues only by the corresponding constraints.

C. ILP for Scheduled Taffic Grooming WithoutWavelength Conversion Capability (ILP1)

Minimize �p�P

�q�Q

fpq · nq �1�

Subject to:

a) Flow and capacity constraints:

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− 1, if i = dq,

0, otherwise ,

∀q � Q, ∀ i � N, �2�

�q�Q

fpq · nq · lj,q � g · bp, ∀ j � L, ∀ p � P. �3�

b) Transceiver constraints:

�p:from�p�=i

bp � TXi, ∀ i � N, �4�

�p:to�p�=i

bp � RXi, ∀ i � N. �5�

c) RWA constraints:

�r

xr,p = bp, ∀ p � P, �6�

sp,e = �r

xr,p · fespdp,r, ∀ p � P, ∀ e � E, �7�

�k�K

wk,p = bp, ∀ p � P, �8�

wk,p + sp,e − �k,pe � 1, ∀ k � K, ∀ e � E, ∀ p � P,

�9a�

wk,p � �k,pe , ∀ k � K, ∀ e � E, ∀ p � P, �9b�

sp,e � �kpe , ∀ k � K, ∀ e � E, ∀ p � P, �9c�

�p�P

�k,pe � 1, ∀ k � K, ∀ e � E. �10�

The objective function in Eq. (1) minimizes the totalamount of electronic switching required for all the demandsby minimizing the weighted hop count. This approach at-

empts to maximize the total amount of spare capacityvailable on all the lightpaths.

Constraint (2) is the standard flow equation [28] and issed to route each demand over the logical topology, using aingle multihop logical path, in accordance with the nonbi-urcation model used in this paper.

Capacity constraint (3) ensures that there is no flow on aogical edge p if it is not selected for the logical topology (i.e.,p=0). If a logical edge is selected (i.e., bp=1), then Eq. (3)nsures that the total flow on p during any given time inter-al lj does not exceed the capacity g of the lightpath. Weote that the selected logical edges are obtained from P,hich may contain multiple potential edges, with the same

ource and destination nodes, but each with a distinctd_number. Thus, the ILP can handle multiple lightpathsetween some (or all) node pairs.

Constraints (4) and (5) ensure that the total number ofightpaths originating and terminating at node i does notxceed the number of transmitters and receivers availablet node i.

Constraint (6) is the routing constraint for the primaryightpath. It ensures that if the pth logical edge is includedn the logical topology, then the corresponding primaryightpath is allocated exactly one route over the physical to-ology, from the R precomputed routes for each node pair.

Constraint (7) is used to define variables sp,e. The variablep,e will have a value of 1 if the primary lightpath, fromource sp to destination dp, uses the rth route, i.e., xr,p=1,nd edge e is on the rth route, i.e., fe

spdp,r=1. In other words,p,e=1 if and only if the primary lightpath corresponding toogical edge p uses physical link e.

Constraint (8) assigns a channel for each lightpath in-luded in the logical topology. It also enforces the wave-ength continuity constraint by allocating exactly one chan-el for each selected lightpath.

Constraints (9a)–(9c) are used to define the continuousariable �k,p

e , which is set to 1, if both wk,p and sp,e are 1, andotherwise.

Finally, constraint (10) states that a channel k on a link ean be assigned to at most one lightpath.

. ILP for Survivable Scheduled Traffic GroomingILP2)

In this paper we implement protection at the lightpathevel, rather than at the connection level. Thus, for each pri-

ary lightpath, we also set up a backup lightpath using ei-her DPP or SPP. The initial formulation (ILP1) can be aug-ented with additional constraints that handle resource

llocation for the backup paths. In this subsection, weresent the ILPs for survivable logical topology design andraffic grooming of scheduled demands.

Minimize �p�P

�q�Q

fpq · nq �11�

Subject to:

Constraints (2)–(8) and (9a)–(9c)

d) RWA constraints for backup lightpaths:

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�r

yr,p = bp, ∀ p � P, �12�

tp,e = �r

yr,p · fespdp,r, ∀ p � P, ∀ e � E, �13�

�k�K

zk,p = bp, ∀ p � P, �14�

zk,p + tp,e − �k,pe � 1, ∀ k � K, ∀ e � E, ∀ p � P,

�15a�

zk,p � �k,pe , ∀ k � K, ∀ e � E, ∀ p � P, �15b�

tp,e � �kpe , ∀ k � K, ∀ e � E, ∀ p � P, �15c�

xr,p + yr,p � 1, ∀ p � P, r = 0,1,2, . . . ,R − 1, �16�

�ke � �k,p

e , ∀ k � K, ∀ e � E, ∀ p � P, �17a�

�ke � �

p�P�k,p

e , ∀ k � K, ∀ e � E, �17b�

�p�P

�k,pe + �k

e � 1, ∀ k � K, ∀ e � E. �18�

e) Dedicated protection constraint:

�p�P

�k,pe � 1, ∀ k � K, ∀ e � E. �19�

f) Shared protection constraint:

�k,p1e1 + �k,p2

e1 + sp1,e2 + sp2,e2 � 3,

∀ k � K, ∀ e1 � e2 � E, ∀ p1 � p2 � P. �20�

The same as Eq. (1), the objective function given in Eq.(11) also minimizes the amount of electronic switching re-quired by all demands by minimizing the weighted hopcount over the logical topology. We also use flow and capac-ity constraints (2) and (3) and transceiver constraints (4)and (5) as well as the RWA constraints for primary light-paths (6)–(8) and (9a)–(9c) given in ILP1.

Constraints (12)–(14) are used for routing and wave-length assignment of backup lightpaths and are analogousto constraints (6)–(8). Constraints (15a)–(15c) are used todefine the variable �k,p

e for backup lightpaths and are simi-lar to (9a)–(9c). Constraint (16) states that the same physi-cal route cannot be selected for both the primary and backuplightpaths corresponding to a given logical edge. Since allthe R routes between each source–destination pair are pre-computed to be edge disjoint, this ensures that the primaryand backup lightpaths are edge disjoint.

Constraints (17a) and (17b) are used to define the vari-able �k

e , which is set to 1 if channel k on link e is assigned toat least one (possibly more) backup lightpath; otherwise, �k

e

is set to 0. This is needed since a channel may be assigned tomore than one backup lightpath (for shared protection only).

ven if multiple backup lightpaths are allocated to channelon link e, the value of �k

e does not exceed 1.

Constraint (18) replaces constraint (10) and ensures thatf a channel k on link e is assigned to a primary lightpath, itannot be assigned to any other primary or backup light-ath.

Constraint (19) is used for dedicated protection and en-ures that two backup lightpaths are not allowed to shareny resources, i.e., cannot be assigned the same channel oncommon link. For shared protection, constraint (20) is

sed instead of constraint (19). This constraint states that ifackup lightpaths for two logical edges, p1 and p2, share ahannel k on a common link e1, then for any other link e2 its not possible for both primary paths to share that link.

. Handling Wavelength-Convertible Networks (ILP3)

The formulation given in ILP2 can be easily modified toandle wavelength-convertible networks as well. In theodified formulation (ILP3), the variables �k,p

e and �k,pe are

efined as binary variables, rather than continuous vari-bles. In wavelength-convertible networks a lightpath maye assigned a different wavelength on each link it traverses.herefore constraints (8) and (14), which assign a singleavelength for each primary and backup lightpath, are notppropriate for this case. Instead, these two constraints areeplaced by constraints (21) and (22), respectively, as shownelow:

�k�K

�k,pe = sp,e, ∀ p � P, ∀ e � E, �21�

�k�K

�k,pe = tp,e, ∀ p � P, ∀ e � E. �22�

Constraint (21) states that a channel is assigned on link eo the primary lightpath for the pth logical edge if the light-ath traverses link e (i.e., sp,e=1); otherwise no channel isssigned. In a similar manner constraint (22) assigns ahannel on each traversed link for a backup lightpath. How-ver, there is no restriction stating that a lightpath must bessigned the same channel on successive links. Finally, con-traints (9a)–(9c) and (15a)–(15c) are no longer needed forLP3. All other constraints remain unchanged.

. Possible Modifications to the ILPs

All of our proposed formulations minimize the weightedum of the number of logical edges or lightpaths traversedy each traffic demand. A number of different objective func-ions can also be easily implemented by slightly modifyingur ILPs. For example, one objective may be to minimize theumber of lightpaths (i.e., minimize �p�Pbp), which reduceshe total transceiver cost. Another possible objective is to re-uce the optical resources at the physical level in terms ofhe number wavelength links, i.e., the number of hops re-uired to route the lightpaths over the physical topologyminimize �e�E�p�Psp,e+ tp,e).

The objectives mentioned so far all minimize the amountf resources required to accommodate a set of demands.owever, if sufficient resources are not available in the net-

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work, it makes sense to maximize the amount of traffic thatcan be supported, i.e., maximize the throughput. This can bedone by changing the objective function and one constraint.The objective function (1) or (11) should be changed to thefollowing:

Maximize �q�Q

nq · mq. �23�

Also, flow constraint (2) should be replaced by the follow-ing constraint:

�p:from�p�=i

fpq − �p:to�p�=i

fpq = �mq, if i = sq

− mq, if i = dq

0, otherwise. �24�

Constraint (24) must satisfy ∀q�Q and ∀i�N. The otherconstraints remain the same. This modified formulationuses an integer variable mq, which specifies whether trafficdemand q can be accommodated in the network. If demandq is handled by the above formulation, mq=1. Thus nq ·mqgives the contribution of demand q to the network through-put. Equation (23) is the objective function that maximizesthe weighted sum �q�Qnq ·mq of traffic demands that can behandled by the network. This value depends on available re-sources such as the number of transceivers per node, the ca-pacity of a lightpath, and the number of available channelsper fiber.

Constraint (24) gives the standard flow conservation con-straint as in constraint (2) except that it enforces that flowconstraint only for the commodities that are accommodatedin the network (i.e., mq=1).

IV. RESULTS

We have simulated our formulations with different de-mand sets on a number of networks ranging from a small6-node network to practical sized networks such as the 14-node NSFNET and 20-node ARPANET [29]. For each net-work, we ran our experiments with at least 5 demand sets,where the size of each demand set ranged from about 60 de-mands (for the 6-node network) to over 800 demands (for the20-node network). Table I shows the simulation parametersused for different network sizes and the average number oflightpaths established for each network size. In this table, nindicates the network size, i.e., the number of nodes in thenetwork. The column Tx /Rx indicates the number of trans-

TABLE INUMBER OF DEMANDS AND NUMBER OF LIGHTPATHS ESTAB-

LISHED FOR EACH NETWORK SIZE

No. ofmodes (n) Tx/Rx

Number of Demands

Number ofLightpathsOC-3 OC-6 OC-12 OC-24 Total

6 3 14 14 16 19 63 1810 5 43 42 50 57 192 5014 5 92 92 91 110 385 7020 10 188 190 186 242 806 200

itters and the number of receivers in each node. For ourimulations, we assumed that each node has the same num-er of transmitters and receivers. However, such uniformitys not required in our formulations, and the proposed ILPsan easily handle different numbers of transceivers at eachode. The source and the destination of each demand as wells its data rate, in OC-n notation, were randomly generated.he results were obtained using ILOG CPLEX [30]. For ourxperiments, we considered demands scheduled over a4-hour period. However, any suitable scheduling periodsuch as hourly, daily, weekly, or even monthly scheduling)an be used with our formulation.

. Degree of Overlap of Traffic Demands

We have compared the performance of our approach tohe conventional approach that does not consider the con-ection holding time of individual demands. We refer touch a model as a holding-time-unaware (HTU) model. TheTU model was simulated in our experiments by setting theuration of every demand to the full 24-hour period. Theelative performance of the proposed ILP over the HTUodel depends directly on the amount of time overlap

mong the individual subwavelength demands in the de-and set Q. Therefore, we considered three different de-and sets as follows:i) Low demand overlap (LDO): duration of each demand

is set randomly between 1 and 10 hours.ii) Medium demand overlap (MDO): duration of each de-

mand is set randomly between 1 and 24 hours.iii) High demand overlap (HDO): duration of each de-

mand is set randomly between 10 and 24 hours.he start time and duration for each individual demand inach set was randomly generated. The main idea in gener-ting the demand sets was that given the same overall timeeriod and same number of demands, the amount of overlapetween demands would in general increase as the length ofhe individual demands are increased. We have used the de-and overlap factor ��, as a metric to characterize the de-

ree of overlap of a set of demands. The demand overlap fac-or is defined as

� =�

i

lmax

�j=1

n−1

�k=j+1

n

Tj,ki

0.5 · lmax · n�n − 1�. �25�

ere, Tj,ki =1 if demands j and k overlap during interval i.

he value of � varies between 0 (there are no overlappingemands in any interval) and 1 (all demands overlap in allntervals). If the value of � for a given set of demands is closeo 0, it indicates low demand overlap, and its value in-reases as the amount of overlap increases. Obviously, a de-and set with �=1 corresponds to the HTU model. The val-

es of � for the LDO, MDO, and HDO sets used in ourimulations are 0.05, 0.28, and 0.5, respectively.

. Results of Fault-Free Networks

In general, our simulations indicated that, if a demandet can be accommodated with fewer channels (e.g., K=16),o additional advantage is gained by setting it with a largeralue (e.g., K=32). However, if a demand set cannot be

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handled with K=16, it is possible to accommodate the de-mand set by increasing the number of available channels to32. The time required to obtain an optimal solution rangedfrom less than 1 s for the smaller networks to several thou-sand seconds for the 20-node network. These solution timesare reasonable, since this type of topology design is expectedto be done offline.

We first present the results for networks without wave-length conversion using ILP1. Figures 2 and 3 show themaximum amount of resources used during any time inter-val (in terms of the weighted hop count at the logical level)by the scheduled demand set, with 16 and 32 channels perfiber (i.e., K=16 and K=32), respectively. It is clear fromFigs. 2 and 3 that consideration of connection holding timesallows more efficient use of available resources. The amountof improvement increases as the demand overlap decreases.This means that considerable savings can be achieved by al-lowing resource sharing among time-disjoint demands. Theaverage improvement varies from about 20% for a demandset with a high degree of overlap to over 65% for low demandoverlap.

We note that since our proposed ILP generates an optimalsolution for a given set of demands, it is guaranteed to per-form as well as or better than a dynamic traffic-grooming-based approach for the same demand set. Figure 4 showsthe percentage reduction in resource requirements with re-spect to the HTU case for different network sizes and differ-ent amounts of demand overlap.

We see that the amount of improvement is significant inall cases, even when demand overlap is high. Considerationof connection holding times seems to have a greater impactfor the larger networks (14-node and 20-node networks).

Fig. 2. (Color online) Resource requirements versus network sizewith K=16 in fault-free networks without wavelength conversion.

Fig. 3. (Color online) Resource requirements versus network sizewith K=32 in fault-free networks without wavelength conversion.

We next consider results for wavelength-convertible net-orks. Figure 5 indicates network resources used (in termsf the weighted hop count at the logical level) on differentetwork sizes under different demand overlap models.

Figure 6 shows the percent reduction of the weighted hopounts using the LDO, MDO, and HDO models comparedith the HTU model for networks with different sizes. Theumber of channels (|K|) for the networks with 6 nodes to4 nodes was considered as 16, whereas for a 20-node net-ork, |K| was 32. It is evident from Figs. 5 and 6 that sig-ificant savings can be achieved using our approach com-ared with holding-time-unaware models. As noted in Figs.and 3, the amount of improvement increases as the de-and overlap decreases.

Figure 7 shows that the resource requirements inavelength-continuous and wavelength-convertible net-

ig. 5. (Color online) Resource requirements versus network sizeith K=16 in fault-free networks with wavelength conversion.

ig. 4. (Color online) Percentage improvement in resource require-ents compared with the HTU case using ILP1 with K=32.

ig. 6. (Color online) Percentage improvement in resource require-ents compared with HTU case in fault-free networks with wave-

ength conversion.

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works under LDO are very close, regardless of the networksize. The results for the MDO, HDO, and HTU models followa similar pattern. This seems to indicate that in most caseslittle or no improvements are realized in terms of resourceutilization at the logical level by allowing wavelength con-version capabilities at the network nodes. However, wave-length conversion may still lead to some improvements interms of the number of wavelength links used by allowinglightpaths to use shorter routes.

In addition to evaluating the performance of our proposedformulations, we also studied the performance (after modi-fying ILP1 appropriately) for each of the alternative objec-tive functions on 6-node and 10-node networks. The resultsfor the 6-node network are shown in Fig. 8, and the resultsfor the 10-node network are similar.

The above results are consistent with the results of ourproposed formulations and indicate that noticeable improve-ments can be expected, even for these alternative objectives.In order to include the results for different objectives on thesame graph, we have expressed the values in terms of therelative improvement achieved (over the HTU model) foreach case. The first (second) set of bars shows the reductionin the number of lightpaths (wavelength links) obtained byusing min �p�Pbp (min �e�E�p�Psp,e+ tp,e) as the objective.The third set shows the relative increase in the number ofdemands that can be accommodated using the objectivemax �q�Qnq ·mq.

C. Results of Survivable Networks

In this subsection, we consider results for survivable net-works without wavelength conversion. Results for

Fig. 7. (Color online) Comparison of resource requirements inwavelength-continuous and wavelength-convertible networks underthe low-demand-overlap model.

Fig. 8. (Color online) Relative improvement with alternative objec-tive functions.

avelength-convertible networks followed a similar patternnd are not reported separately. We implement network sur-ivability using both dedicated and shared path protections.s in the previous cases, our objective is to minimize theeighted hop count. However, the weighted hop count (at

he logical level) remains the same whether or not backupightpaths are implemented. Therefore, for comparison pur-oses, we have compared the number of wavelength linksequired for fault-free (i.e., no backup paths), dedicated, andhared protection schemes. Figure 9 compares the averageumber of wavelength links required for wavelength-ontinuous networks for the case with no protection, sharedrotection, and dedicated protection under the low-demand-verlap model. As expected, shared path protection requiresewer resources compared with dedicated path protection.esults for the other cases follow a similar pattern.

In summary, we have experimented with a number of dif-erent networks; for each network we considered several de-and sets and different amounts of overlap between de-ands. The significance of the results can be noted as

ollows:i) The results not only show that knowledge of connection

holding times can decrease the amount of resources re-quired to accommodate a given set of demands (whichwas expected), but they also indicate how the relativeimprovements are affected by the amount of demandoverlap.

ii) The results clearly indicate (Table I) that it is possibleto accommodate a large number of low-speed demandsusing a stable logical topology with relatively few logi-cal edges when the data rates of the individual de-mands are relatively small compared with the capac-ity of a lightpath.

iii) The results indicate that wavelength conversion doesnot lead to significant benefits in terms of the objec-tive used in our simulations.

V. CONCLUSIONS

In this paper, we have introduced a new method for sur-ivable topology design and traffic grooming of low-speed,cheduled traffic demands whose setup and teardown timesre known in advance. We proposed the design of a stableogical topology capable of supporting the specified demandet and sharing resources allocated to nonoverlapping de-

ig. 9. (Color online) Resource requirements versus network sizeith K=16 in survivable wavelength-continuous networks under

he LDO model.


mands. We addressed the joint topology design (includingRWA) and traffic grooming problem and presented a numberof efficient ILP formulations to solve the problem optimallyfor different design scenarios. We considered networks bothwith and without wavelength converters and implementedsurvivability using shared and dedicated path protection.Static traffic grooming can be considered a special case ofour model where all demands overlap in time. Experimentalresults demonstrated that our approach is feasible for prac-tical sized networks with hundreds of demands. Using astable topology reduces network costs due to expensive re-configurable components and eliminates the overhead fordynamic connection setup and teardown. Based on the in-sights gained from this work, we are currently developingheuristic approaches that can be used for more complex traf-fic models, such as the sliding window model.

ACKNOWLEDGMENT

The work of A. Jaekel is supported by a research grant fromthe Natural Sciences and Engineering Research Council ofCanada (NSERC).

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Documents

Stable Logical Topologies for Survivable Traffic Grooming of Scheduled Demands