Lightpath Scheduling and Routing for Trafﬁc Adaptation in ...jingwu/publications/journal...slide...

Zhang et al. VOL. 2, NO. 10 /OCTOBER 2010/J. OPT. COMMUN. NETW. 803

Lightpath Scheduling and Routing forTraffic Adaptation in WDM Networks

James Yiming Zhang, Hussein Mouftah, Jing Wu, and Michel Savoie

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Abstract—We study the benefits and trade-offs of usingscheduled lightpaths for traffic adaptation. We propose anetwork planning model that allows lightpaths to slidewithin their desired timing windows with no penalty on theoptimization objective and to slide beyond their desired tim-ing windows with a decreasing tolerance level. Our modelquantitatively measures the timing satisfactions or viola-tions. We apply the Lagrangian relaxation and subgradientmethods to the formulated optimization problem, withwhich great computational efficiency is demonstrated whencompared with other existing algorithms. Our simulation re-sults show how timing flexibility improves network resourceutilization and reduces rejections.

Index Terms—Networks; Assignment and routingalgorithms; Combinatorial network design; Network optimi-zation.

I. INTRODUCTION

N etwork traffic at the optical layer periodically fluctu-ates. For core optical networks such as wavelength di-

vision multiplexing (WDM) networks, network traffic is ob-served following daily patterns [1] or weekly patterns [2].Such traffic patterns can be predicated from historical sta-tistics, which repeat every day (or week) with minor varia-tions in timing and volume. The knowledge about trafficpatterns provides an opportunity to schedule lightpaths toadapt to the changing traffic. Traffic adaptation is a networkreconfiguration process to adapt to traffic fluctuations.

Using scheduled lightpaths for traffic adaptation has dif-ferent timing requirements from the other lightpath sched-uling problems. Existing methods for the scheduled routingand wavelength assignment (RWA) problems assume that alightpath should be set up either at a given time or within agiven time window, which makes the lightpath schedulinginflexible for traffic adaptation. For example, in [3–15], net-work planning was conducted for a set of lightpath requests,each having prespecified starting and ending times. In[16–24], static network planning was conducted for fixedholding-time lightpath requests, each one being allowed toslide within its given time window. But, for traffic adapta-

Manuscript received April 13, 2010; revised August 2, 2010; accepted Au-gust 11, 2010; published September 29, 2010 �Doc. ID 126908�.

J. Y. Zhang and H. Mouftah are with the School of Information Technologyand Engineering, University of Ottawa, 800 King Edward Avenue, Ottawa,Ontario, Canada, K1N 6N5.

J. Wu (corresponding author e-mail: [email protected]) and M. Savoie arewith the Communications Research Centre (CRC) Canada, 3701 CarlingAvenue, Ottawa, Ontario, Canada, K2H 8S2.

Digital Object Identifier 10.1364/JOCN.2.000803

1943-0620/10/100803-17/$15.00 ©

ion, timing flexibility in lightpath scheduling is very impor-ant. Network operators are concerned about not only tim-ng violations but also resource utilization and lightpathejections and need a tool to make wise trade-offs betweenhese goals. Network operators would rather adjust light-ath scheduling timing than reject lightpaths that cannot beccommodated due to their strict timing requirements ormpractical timing windows.

For traffic adaptation, scheduled lightpaths are allowedo slightly slide in timing, without deteriorating the perfor-ance. Sliding-timing scheduling potentially provides bet-

er network resource utilization than fixed-timing schedul-ng. Since lightpaths are scheduled based on the statisticalraffic characteristics, minor timing slides should not impactuch on the performance of the traffic adaptation, while

ramatic timing slides should be avoided.

For traffic adaptation, the extent of timing satisfaction oriolation needs to be quantitatively measured. A timing win-ow should not be used in a “binary” way, i.e., it can eithere satisfied (thus the corresponding lightpath is accepted) orot (thus the corresponding lightpath is rejected). Networkperators often would prefer scheduled lightpaths being cen-ered on their desired timing, with a decreasing toleranceevel as scheduled lightpaths move away from their desirediming windows. For example, a network operator wants tochedule offline large database backup operations at nighthen the network traffic is light. The ideal timing windowould be after midnight and before 6:00 am. However, suchperations may start earlier than midnight but cause in-reasing penalty as they move toward evening traffic peakrom 7:30 pm to 9:30 pm, when the cost is higher. Similarly,he scheduled lightpaths could continue after 6:00 am butause increasing penalty as they move toward morning traf-c peak from 8:00 am to 10:00 am. This requires properodeling of the extent of timing satisfaction or violation,hich has not been done by the existing methods for the

tatic lightpath scheduling, to our knowledge, and motivateshis study. Furthermore, with the usage-based pricing [20],hich is a time-wise variable, a lightpath should be penal-

zed differently for the different time span being scheduledpon. However, this variation is not taken into consider-tion in [20].

Our study aims at planning scheduled lightpaths to adapto relatively stable traffic patterns. Static scheduled light-ath demands are known as inputs to our network planningroblem. Our problem is different from dynamic lightpathcheduling problems, which generally do not assume any

priori statistical information about traffic patterns

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804 J. OPT. COMMUN. NETW./VOL. 2, NO. 10 /OCTOBER 2010 Zhang et al.

[25–32]. In [33,34], static lightpath scheduling problemswere called “advance lightpath reservation problems,” whiledynamic lightpath scheduling problems were called “imme-diate lightpath reservation problems.” In our approach, oncea lightpath is preplanned, it becomes available to carry traf-fic at its scheduled time. In contrast, dynamic lightpathscheduling cannot guarantee the availability of a lightpath.Only when a request arrives, the network operator makesreal-time decisions depending on the network resourceavailability at the moment of the request. Our approachachieves a better coordination of lightpaths than dynamiclightpath scheduling by taking advantage of the extra infor-mation about traffic patterns.

Our method of using scheduled lightpaths for traffic ad-aptation has two major advantages over other virtual topol-ogy reconfiguration methods. First, our method provides aflexible framework to make trade-offs between the trafficadaptation and the network utilization. Second, we proac-tively coordinate lightpaths as the traffic changes, so thatinterruptions to the traffic are minimized. Essentially, wereconfigure the virtual topology by using scheduled light-paths, which is the same method used in all related work onlightpath scheduling. Other virtual topology reconfigurationmethods can be classified as reactive methods, prediction-based methods, etc. In [1,35,36], based on real-time trafficmeasurement, the online virtual topology is reconfigured toadapt to traffic fluctuations, where lightpaths are dynami-cally added or deleted. In [37], based on the predicted traffic,the virtual topology is reconfigured in time to adapt to traf-fic changes. In [38,39], the virtual topology is designed to fita wide range of traffic patterns so that the need for recon-figuring the virtual topology is reduced.

This paper is organized as follows: in Section II, we sum-marize the assumptions used in our model; in Section III,we present our model, followed by a solution based on theLagrangian relaxation and subgradient methods in SectionIV; we evaluate the performance of our solution methods inexperimental examples in Section V; and we conclude thispaper in Section VI. Derivation details are included in Ap-pendix A.

II. NETWORK OPERATIONS, MODELING, AND

ASSUMPTIONS

We consider wavelength-routed WDM networks withmesh topologies and capable of wavelength conversion. Wemodel a general topology WDM mesh network of N nodes in-terconnected by E links. Each link consists of a pair of fi-bers, each fiber for one direction and having W noninterfer-ing wavelength channels (WCs). Two nodes can be connectedthrough a lightpath defined as a concatenated sequence ofWCs [40]. The contribution of wavelength conversion hasbeen highlighted in reconfigurable WDM networks [41]. Weinclude wavelength conversion in our network model, sothat its contributions can be evaluated in the context ofscheduled lightpaths. Two lightpath segments that use dif-ferent wavelengths are allowed to be chained together by us-ing a wavelength converter installed at their junction node.In this way, a semilightpath is formed [42]. For simplicity,we use “lightpath” in this paper regardless of whether it

ses the same wavelength all the way or not. We assumehat all wavelength converters are capable of full-range con-ersion, which means any wavelength of an incoming light-ath segment can be converted into any wavelength of anutgoing lightpath segment. Each node has a predefinedumber of wavelength converters (could be zero if none). Aode shares its wavelength converters among all its incom-

ng and outgoing lightpath segments, i.e., if a wavelengthonverter is available at a given node, any incoming light-ath segment may choose to use it before entering any out-oing lightpath segment (see [43] for example node architec-ure).

Lightpaths are scheduled to be set up at the beginning ofheir starting time slots and be torn down at the end of theirnding time slots. Network-wide synchronous time slots aresed for resource allocations and lightpath scheduling. Allime slots have the same fixed duration, which should bemin or longer. Many network management systems moni-

or network status every 5 min; thus a time slot shorterhan this does not improve the performance but adds tre-endous planning and operation overhead. As we will show

hortly, the complexity of our scheduling problem directlyelates to the number of time slots in the planning time ho-izon. We assume the time to set up or tear down a lightpathi.e., signaling time) is negligible compared with the dura-ion of a time slot. The holding time of a lightpath is fixednd known in advance, measured by the number of timelots. Without losing generality, we number the time slots inur planning time horizon sequentially from 0 to Z−1 �0t�Z�.

We aim at scheduling and allocating network resources toightpaths, i.e., planning network operations for scheduledliding lightpath demands (SSLDs). The network resourcesrimarily include WCs and wavelength converters. We pro-ide an accepted SSLD with a RWA scheme, which is de-cribed as a list of allocated WCs and possibly together withavelength converters. The same RWA scheme is used for

he entire holding time of an SSLD, i.e., once an SSLD is ac-epted, it stays connected from its starting time slot to itsnding time slot. If there are insufficient resources for anSLD during its holding time, the SSLD is rejected. We al-

ow multiple SSLDs being set up between a given node pair.

III. PROBLEM FORMULATION

. Notations

For the remainder of this paper, the following notationsnd variables are used:

INPUT PARAMETERS:the set of all nodes in the network

ij the fiber between node i �i�V� and node j �j�V�the set of all fibers in the network, i.e., �eij�, (i�V, j�V)

ij the cost of using a WC on link eij �eij�E� for onetime slot

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oi the cost of using a wavelength converter on nodei �i�V� for one time slot

W the number of wavelengths used in the networklsdn the nth SSLD from node s �s�V� to node d �d

�V�. If it is accepted, we also use lsdn to denotethe RWA scheme for the SSLD.

L the set of all SSLDs, i.e., �lsdn�L the total number of SSLDstsdn the holding time of lsdn�bsdn ,bsdn� � the desired window of starting time for lsdn, 0

�bsdn�bsdn� �ZCsdn the routing cost of lsdn, i.e., the cost of resources

used by lsdnFi the number of wavelength converters installed

on node i �i�V�N the number of nodes in the networkP the penalty coefficient for rejecting an SSLDZ the total number of time slots of our scheduling

problemysdn the weight for the earliness penalty of lsdnrsdn the weight for the tardiness penalty of lsdn

DECISION VARIABLES:�sdn a binary integer variable indicating the admis-

sion status of lsdn. It is one, if lsdn is accepted.Otherwise, it is zero.

�sdn the starting time slot of lsdn, 0��sdn�Z.�ijct

sdn a binary integer variable representing the use ofthe cth WC �0�c�W� on fiber eij �eij�E� at timeslot t by lsdn �lsdn�L�. It is one, if lsdn uses suchthe WC at such a time slot. Otherwise, it is zero.

�itsdn a binary integer variable representing the use of

a wavelength converter on node i �i�V� at timeslot t �0� t�Z� by lsdn �lsdn�L�. It is one, if lsdnuses such a wavelength converter at such a timeslot. Otherwise, it is zero.

A the set of admission status of all lsdn’s, i.e., ��sdn�B the starting time slots of all lsdn’s, i.e., ��sdn��sdn the RWA scheme of lsdn �lsdn�L�, i.e., ��ijct

sdn�sdn�sdn the allocation of wavelength converters to lsdn

�lsdn�L�, i.e., ��jtsdn�sdn

� the RWA schemes of all lsdn’s, i.e., ��sdn�� the usage of wavelength converters by all lsdn’s,

i.e., ��sdn�

INTERMEDIATE VARIABLES FOR COMPUTATION ANDRESULT EVALUATIONS:

Dsdn the dual routing cost of lsdnEsdn the overall timing violation penalty of lsdnijct Lagrange multipliers corresponding to using a

WC c �0�c�W� on fiber eij �eij�E� at a time slott �0� t�Z�

it Lagrange multipliers corresponding to using awavelength converter on node i �i�V� at a timeslot t �0� t�Z�

ijc�,T the accumulated dual cost of using a WC c �0

�c�W� on fiber eij �eij�E�i

�,T the accumulated dual cost of using a wavelengthconverter on node i �i�V�

ˆijc�,T the continuous availability of a WC c �0�c

�W� on fiber eij �eij�E�ˆ

i�,T the continuous availability of using a wave-

length converter on node i �i�V�ˆ

ij�,T the accumulated cost of using a WC on fiber eij

�eij�E�ˆ i

�,T the accumulated cost of using a wavelength con-verter on node i �i�V�

Lsdn a sorted list of all the possible �sdn’s for lsdnbased on its dual profit in the dual solution.

. Objective Function

Our objective is to minimize the summation of the rejec-ion of requests, the resource usage, and the timing violationf lightpaths. We want to accept as many profitable requestss possible, and for the accepted requests, we want to findightpath schedules that respect their timing preference as

uch as possible, while at the same time provide them withWA schemes that use as few resources as possible. In order

o achieve this, we introduce a term in the objective functiono penalize the timing violation, if lsdn is not scheduled totart at time slot bsdn.

Our objective function is to minimize the function J, i.e.,inA,B,�,��J�, where

J � �lsdn�L

��1 − �sdn�P + �sdn�Csdn + Esdn��. �1�

The overall penalty consists of the rejection penalty (i.e.,; in practice, it generally equals the revenue of a given

ightpath), the resource usage cost (i.e., Csdn), and the tim-ng violation penalty (i.e., Esdn). The readers are referred to44] for examples for the penalty assignment. The timingiolation penalty could be either an earliness penalty or aardiness penalty. When lsdn is scheduled sooner than its de-ired starting time, the lightpath will be removed soonerhan the desired ending time, causing a shortage of the ef-ective service time after the lightpath is removed. This isecause we strictly respect the holding time of lsdn, i.e., if

sdn is accepted, its holding time will be exactly as requested.n the other hand, when lsdn is scheduled later than its de-

ired starting time of lsdn, there will be a shortage of effec-

ig. 1. (Color online) Shortage of effective service time caused bychedule earliness and tardiness.

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tive service time before the lightpath starts. The shortage ofeffective service time caused by schedule earliness and tar-diness is shown in Fig. 1. The penalty margin should reflectthe network operator’s decreased tolerance level as the tim-ing of the lightpaths moves away from their desired timingwindows. In this paper, we adopt earliness and tardinesspenalties defined in [45]:

Esdn

=ysdn � �bsdn − �sdn�2 if �sdn � bsdn

�i.e., earliness penalty�

0 if bsdn � �sdn � bsdn�

�i.e., no penalty, since the lightpath starts

within its desired starting time window�

rsdn � ��sdn − bsdn� �2 if �sdn � bsdn�

�i.e., tardiness penalty�

, lsdn � L,

�2�

where ysdn and rsdn are the weights for earliness and tardi-ness penalties of lsdn. Our earliness and tardiness penaltiesreflect a decreasing tolerance level as a scheduled lightpathmoves away from its desired timing window (shown in Fig.2). When ysdn and rsdn are set to infinitely large positive val-ues and the resource costs are set to zero, this formulationthen becomes the same as the fixed time-window schedulingproblem in [14], which does not allow any timing violation.Note that other forms of the earliness and tardiness penaltyfunctions can also be used within our framework.

The cost of routing lsdn is denoted as Csdn and defined asthe total cost of using WCs and wavelength converters. Weuse this definition to illustrate that the cost of routing alightpath is a weighted summation of certain parameters re-lated to links and nodes. Such a definition may be extendedto incorporate other parameters without changing the solu-tion method proposed later in this paper.

Csdn = ��sdn�t��sdn+tsdn�

�dij � �eij�E

�0�c�W

�ijctsdn

+ oi � �i�V

�itsdn�, lsdn � L. �3�

Our design variables are the admission status of allSSLDs �A�, the starting time slots of all lightpaths �B�, the

Fig. 2. (Color online) Example of earliness and tardiness penaltiesreflecting a decreasing tolerance level as a scheduled lightpathmoves away from its desired timing window.

WA schemes for all lightpaths ��, and the allocations ofhe wavelength converter to all lightpaths ��. Our designariables are not completely independent. They representhree interrelated subproblems, i.e., lightpath request ad-issions, lightpath scheduling, and RWAs. Based on our as-

umption of the wavelength converter’s installation struc-ure, the allocations of wavelength converters to allightpaths may be derived from the RWA schemes for allightpaths.

. Constraints

) Flow Conservation Constraints: If lsdn is admitted, itsWA must be continuous along its path and be terminatedt its two end nodes:

�j�V

�0�c�W

�ijctsdn − �

j�V�

0�c�W�jict

sdn

= �sdn if �sdn � t � �sdn + tsdn, i = s

− �sdn if �sdn � t � �sdn + tsdn, i = d

0 otherwise ,

∀lsdn � L, i � V. �4�

If lsdn is accepted (i.e., �sdn=1), at the source node (i.e., is), there is one lightpath going out; at the destination node

i.e., i=d), there is one lightpath coming in; at any interme-iate node, this lightpath does not contribute to the numberf lightpaths that terminate at this node. For any node thats not related to this lightpath, or when lsdn is rejected (i.e.,sdn=0), this lightpath does not contribute to the number of

ightpaths that terminate at the node. These constraintsonfine that, if and only if �sdn=1, during the lifespan of lsdni.e., �sdn� t� ��sdn+ tsdn�), there must be a lightpath fromode s to node d.

) Wavelength Conversion Constraints:

�jtsdn = 1 if ∃ �m � V, k � V, b � c�, �mjbt

sdn = �jkctsdn = 1

0 otherwise � ,

∀lsdn � L, j � V, 0 � t � Z. �5�

One wavelength converter on an intermediate node j issed only when different wavelengths are assigned to lsdnor the incoming and outgoing wavelengths at this node.

) Exclusive WC Usage Constraints:

�lsdn�L

�ijctsdn � 1, ∀ eij � E, 0 � c � W, 0 � t � Z. �6�

Every WC at any time slot t cannot be used by more thanne lightpath.

) Converter Quantity Constraints:

�lsdn�L

�jtsdn � Fj, ∀ j � V, 0 � t � Z. �7�

The number of occupied converters on node j at any timelot t must not be more than the number of converters in-talled on the node.

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5) Lightpath Persistency Constraints:

�ijcxsdn = �ijcy

sdn, ∀ lsdn � L, eij � E, �sdn � x � �sdn

+ tsdn, �sdn � y � �sdn + tsdn �8�

During the lifespan of lsdn, its RWA scheme must remainthe same for all time slots. Note that when combined withwavelength conversion constraints, the allocation of thewavelength converters �jt

sdn stays the same, too.

IV. A SOLUTION BASED ON THE LAGRANGIAN

RELAXATION AND SUBGRADIENT METHODS

A. Overview of Existing Solution Methods for StaticLightpath Scheduling Problems

Existing static lightpath scheduling problems are solvedby different methods such as heuristics. As explained in ourintroduction, the existing problems assume simple caseswhere a lightpath starts either at its given time or within itsgiven time window. Our problem is different from theseproblems by allowing a lightpath to slide beyond its desired

TABSOLUTION METHODS FOR THE STATIC LIGHTPATH SCHEDULING

START

Solution Methods References

Heuristic algorithm(for the joint problem)

[3] Separately soFor the routioptimal solutthe wavelengused. A heurdemands arefirst fit wave

Heuristic algorithm [4] A global randnumber of re

Simulated-annealing-basedmethod

[5] A simulated-stopping con

Branch-and-bound method [6] Use the branproblems, pr

Heuristic algorithm [7,8] Sort demandlinks before searliest-setupdemand first

Heuristic algorithmassisted by CPLEX

[9] Precomputebest path sel

Heuristic algorithm [10] Partition demsame group a

Heuristic algorithm [11] Sort demandto allocate re

Heuristic algorithm [12] Partition demin the other

Heuristic algorithm [13] Partition demlinks of prevthe number o

Lagrangian-relaxation-basedmethod

[14,15] The constraicreate a dualof the originaLagrangian mnear-optimal

ime window but with an increasing penalty as it moves fur-her away. Therefore, solving our problem is more challeng-ng due to the additional flexibility in allowing extra timingiolation. In Table I, we summarize the solution methods forhe static lightpath scheduling problems where an SSLDust start at its specified starting time; otherwise the SSLD

s rejected. In Table II, we summarize the solution methodsor the static lightpath scheduling problems where an SSLDs allowed to slide within its specified time window; other-ise it is rejected.

The Lagrangian relaxation and subgradient methodsLRSM) have demonstrated their potential in solving large-cale optimization problems such as the static RWA problem46–49], traffic grooming and multicast routing [50], net-ork reconfiguration [41], the grade-of-service differenti-ted static RWA problem [44], the virtual path planning andacket routing problems in a layered network [51], and thewo-layer routing problem [52,53]. For example, the puretatic RWA problem in a mesh network with limited wave-ength conversion capability was studied in [46]. A link-ased formulation was proposed, considering the fairness ofemand acceptance among different node pairs. The LRSMramework was used to solve the proposed network recon-

IOBLEMS WHERE A LIGHTPATH MUST START AT ITS SPECIFIED

TIME

Description of Solution Methods

ng two subproblems: (1) routing and (2) wavelength assignment.subproblem, the branch-and-bound method is used to obtain the, while Tabu search is used to obtain approximate solutions. Forassignment subproblem, a greedy vertex coloring algorithm is

algorithm is used to jointly solve the two subproblems, whereted first and then apply the fixed-alternate routing, followed by ath assignment to each demand.search algorithm to compute RWA schemes that minimize the

ed SSLDsealing algorithm iteratively explores the solution space until a

on is satisfiedand-bound method for solving integer linear programminged by the CPLEX software packageone of the four policies and then assign costs to wavelength

ching for working and protection paths. Demand sorting policies:mand first, earliest-teardown demand first, most-conflictingst-conflicting demand first

didate paths for each demand and then use CPLEX to obtain theon from the candidate paths for all demandsds into disjoint (in time or in edge) groups. The demands in therouted at the same time and are assigned the same wavelength.ascending order of their starting time, then use greedy heuristics

rces to each demandds into groups (time-disjoint in one algorithm, time-overlappingrithm) and then route groups of demandsds into time-disjoint groups and then search for paths sharingly selected paths for the demands in the same group to increasenk-disjoint paths available to the demands in other groupsof no overbooking on any WC at any time slot are relaxed tooblem. The solution to the dual problem is a performance boundroblem. Then, the solution to the dual problem and the generatedtipliers are used to develop heuristic algorithms for achieving aution to the original problem.

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figuration problem. Great computational efficiency has beendemonstrated when compared with other existing algo-rithms. Despite the advances made by the LRSM in thestatic RWA problems, lightpath scheduling cannot directlyadopt the same approach because the static RWA problemdoes not consider the timing attribute.

A simple formulation employing the LRSM for lightpathscheduling was presented in [14,15] to maximize the net-work revenue. However, the formulation provided only al-lows simple network decisions, i.e., to reject/accept a light-path demand if the time constraint is not satisfied withoutany tolerance of the timing violation, which makes networkscheduling inflexible. Moreover, no wavelength conversionor the penalty of network resource usage was considered,and thus only limited network parameters can be controlledin the scheduling.

B. Application of the Lagrangian Relaxation andSubgradient Methods to the Static LightpathScheduling and RWA Problem

Our formulated problem faces a big challenge in comput-ing its optimal solution because of the large number of de-sign variables and constraints. For a network with N nodes,E links, W wavelengths, Z time slots, and L SSLDs, our for-mulation uses �A�+ �B�+ ��+ ��=L+L+L�Z�E�W+L�Z�N binary integer variables, where |"| denotes the numberof elements in the set. The number of constraints in our for-mulation is very large too (shown in Table III). Conse-quently, finding the exact optimal solution for a medium-size problem is infeasible even for the static RWA problem

TABSOLUTION METHODS FOR THE STATIC LIGHTPATH SCHEDULING

ITS SPECIFIED

Solution Methods References

Heuristic algorithmassisted by CPLEX

[16,17] Schedule demandsmethods in Table I)starting time

Heuristic algorithm [18,19] Three steps: (1) divRWA algorithm onstep (2)

Heuristic algorithm [20] Heuristic algorithmsingle-fiber

Heuristic algorithm [21] Partitioning SSLDsHeuristic algorithm [22–24] Heuristic algorithm

TABLE IIINUMBER OF CONSTRAINTS IN OUR FORMULATION

Type of ConstraintsNumber ofConstraints

Flow conservation constraints in Eq. (4) L�N�ZWavelength conversion constraints in Eq. (5) L�N�ZExclusive WC usage constraints in Eq. (6) E�W�ZConverter quantity constraints in Eq. (7) N�ZLightpath persistency constraints in Eq. (8) L�E�Z� �Z−1�

46], which is only a simplified version of our current prob-em by setting Z (the number of time slots) to 1. Thus, weim at obtaining near-optimal solutions to our problem,hile providing a tight performance bound that can be used

o evaluate the optimality of our solutions.

We propose a method based on the LRSM and derive aual problem (DP) by using the Lagrangian relaxation (LR)ethod. We solve the DP using the subgradient method.hen, we project the DP’s solution to a feasible solution to

he original problem by using a proposed heuristic algo-ithm. The overall flow chart is shown in Fig. 3. The advan-age of solving the problem in the dual space first is that theP is a convex integer problem, which can be efficiently

olved by the subgradient method, while there is no conve-ient way to solve the original problem directly [54]. The ap-lication of the subgradient method is illustrated in Fig. 4.ote that the solution space and dual space use differentariables and might have different dimensions of the solu-ion space as well.

The computational time complexity of our proposedethod is polynomial with respect to all design variables, in

ontrast to the exponential time complexity of the branch-nd-bound method used in CPLEX, e.g., O�KL�, where K is aredefined integer constant �K 2�. Unlike heuristic algo-ithms, our LRSM provides a theoretical performance boundnd a feasible near-optimal solution at the same time. Theuality gap, which indicates the optimality of a near-ptimal solution, is well controlled.

The LR method is used to derive the DP of the originalroblem. The derivation details are given in Appendix A. Weefine the dual routing cost of lsdn as Dsdn:

Dsdn = min�sdn,�sdn,�sdn

�Esdn + �eij�E

�0�c�W

��sdn�t��sdn+tsdn

�ijctsdn�ijct

+ dij� + �i�V


�itsdn�it + oi�� ,

∀lsdn � L. �9�

Then, the dual problem in Eq. (A7) (see Appendix A) maye simplified as

IIOBLEMS WHERE A LIGHTPATH IS ALLOWED TO SLIDE WITHIN

ME WINDOW

Description of Solution Methods

imally in time by CPLEX and then use existing methods (e.g.,the subproblem where a lightpath must start at its specified

g SSLDs into disjoint windows, (2) using a greedy-window-basedSLDs, and (3) rearranging the SSLDs that are not satisfied in

schedule SSLDs and to assign wavelength to lightpaths in a

o time-disjoint groupsjointly schedule lightpaths and to allocate resources

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�min�sdn

��1 − �sdn�Psdn + �sdnDsdn��, �10�

which we refer to as the relaxed problem.

We are able to decompose the overall relaxed problem intoindependent subproblems. The total number of the subprob-lems is L, each corresponding to lsdn. The decision on �sdn isstraightforward based on the penalty Psdn for rejecting lsdnand dual routing cost Dsdn of lsdn (may be thought of as thepenalty for accepting lsdn). A lower penalty should be chosen:if Psdn�Dsdn, we set �sdn to 1 (i.e., lsdn is accepted); other-wise, we set �sdn to 0. We break a tie arbitrarily.

C. Algorithm for Obtaining Dual Routing Cost

The dual routing cost Dsdn in Eq. (9) may be simplified as

Dsdn = min�sdn

Esdn + min�sdn,�sdn

� �eij�E

�0�c�W


�ijctsdn�ijct

+ dij� + �i�V


�itsdn�it + oi�� ∀ lsdn � L,

�11�

Fig. 3. (Color online) Illustration of our solution fr

Fig. 4. (Color online) Application of the subgradient method in thedual space.

ubject to the constraints (4) and (5).

To obtain the dual routing cost Dsdn for a given lsdn, weropose an algorithm to optimize the scheduling and RWAor the lightpath, i.e., to decide its starting time slot �sdn, asell as its RWA scheme �sdn and �sdn. Our proposed algo-

ithm calculates an augmented property attribute (calledaccumulated dual cost” or ADC) of each resource, and thenses the shortest path algorithm on wavelength graphSPAWG) to solve the second part of Dsdn in Eq. (11), i.e., theWA scheme optimization:

min�sdn,�sdn

� �eij�E

�0�c�W


�ijctsdn�ijct + dij�

+ �i�V


�itsdn�it + oi��. �12�

For a given resource (e.g., a WC or a wavelength con-erter), its accumulated dual cost relates to its two usagearameters: the starting time and the duration of using it.hen a resource’s corresponding Lagrange multiplier is

iven, its accumulated dual cost is fixed and is representedy a trimmed two-dimensional array. No matter which light-ath uses the resource, the accumulated dual cost does nothange. Thus for a given set of all resources’ Lagrange mul-ipliers, the accumulated dual costs are calculated once andre presented to all lightpaths for the lightpaths. Whenolving the DP, we choose resources for lightpaths based onhe accumulated dual costs of the resources. The competi-ion among lightpaths for the same resource is reflected byhe Lagrange multiplier of the resource, which will be dis-ussed shortly in Subsection IV.D.

ework and its relation to the mathematical model.
am

Twtsc

wtiotu

E

loaiiv

tmirttm

mr�l

Fd


The accumulated dual cost ijc�,T of using a WC c �0�c

�W� on fiber eij �eij�E� is

ijc�,T = �

��t��+T�ijc,t + dij�, ∀ eij � E, 0 � c � W, �13�

where � denotes the starting time slot of using the WC, andT denotes the duration of using it.

Similarly, the accumulated dual cost i�,T of using a wave-

length converter on node i �i�V� is

i�,T = �

��t��+T�i,t + oi�, ∀ i � V. �14�

We incrementally calculate ijc�,T and i

�,T as the startingtime slot � increases by reusing the calculated results fromthe previous time slot, except for the initial time slot (i.e.,�=0).

To obtain the dual routing cost in Eq. (11), we compare thedual routing cost for all possible starting time slots. Sincethe accumulated dual cost of all resources are calculated, weuse the SPAWG to solve the second part of Dsdn in Eq. (11),which is now simplified as min�sdn,�sdn

��eij�E�0�c�Wijc�,T

+�i�Vi�,T� by using the definition of the accumulated dual

cost of WCs in Eq. (13) and wavelength converters in Eq.(14). Essentially, the SPAWG is a shortest path algorithmcustomized for WDM networks [46]. In this way, the optimalscheduling and RWA scheme for a given lightpath is ob-tained by solving the dual routing cost in Eq. (11).

For each lsdn, we define its dual profit to be �Psdn−Dsdn�.We sort all the possible �sdn’s for lsdn based on its dual profitin the dual solution and create a list SLsdn. This sorted listwill be used in our heuristic algorithm later in SubsectionIV.E.

D. Updating Lagrange Multipliers

We employ the subgradient method to solve the DP maxi-mization. The subgradient method [54] is only useful for in-teger programming problems. Since all our variables are in-tegers, the subgradient method is applicable to our problem.

The variables of the DP are Lagrange multipliers, whichwe represent as a multiplier vector z= � ,�. The vector isupdated by the following formula:

z�h+1� = z�h� + ��h�g�z�h��, �15�

where z�h� denotes the value of the vector z obtained at thehth iteration, and ��h� denotes the step size of the hth itera-tion. The vector g�z� is the subgradient of the dual functionq with respect to z, i.e., g�z�= �g�� ,g��. The vectors g��and g�� are composed of g�ijct� and g�it�, respectively,where

g�ijct� = �lsdn�L

�ijctsdn − 1, ∀ eij � E, 0 � c � W, 0 � t � Z,

�16�

g�it� = �lsdn�L

�itsdn − Fi, ∀ j � V, 0 � t � Z. �17�

he updating process of the vector is illustrated in Fig. 5,here in each iteration, the variables move along the direc-

ion of their subgradient for a distance determined by itstep size. Z* denotes the optimal solution. The step size isalculated by

��h� = � �qU − q�h�

gT�z�h��g�z�h��, �18�

here qU is an estimate of the optimal solution (which weake the best value of the objective function J obtained), q�h�

s the value of q at the hth iteration, and gT is the transposef g. The parameters � and qU are changed adaptively ashe algorithm converges. Convergence can also be speededp by using the method in [46].

. Constructing a Feasible RWA Scheme

The DP’s solution may be infeasible for the original prob-em because some of the original constraints are relaxed. Inther words, the exclusive WC usage constraints in Eq. (6)nd the converter quantity constraints (7) might be violatedn the DP’s solution. Note that all other constraints are sat-sfied, which is ensured by the DP’s solution procedure pro-ided in Subsections IV.B and IV.C.

To facilitate our heuristic algorithm, we define two at-ributes for each resource: continuous availability and accu-ulated cost. Similar to the definition of the ACD described

n Subsection IV.C, the two new attributes relate to a givenesource’s two usage parameters, i.e., the starting time andhe duration of using this resource, Moreover, both at-ributes are independent of the routing of lightpaths anday be represented by a trimmed two-dimensional array.

The continuous availability of a resource is defined as itsinimal capacity from a given starting time through the du-

ation of using it. The continuous availability of a WC c �0c�W� on fiber eij �eij�E� is denoted as �ijc

�,T and is calcu-ated as

ig. 5. (Color online) Updating the variables by using the subgra-ient method.

phadttFdag(nfafoitif[st

F

ttOlBlceftf


�ijc�,T = min

��t��+T�1 − �

lsdn�L�ijc,t

sdn�, ∀ eij � E, 0 � c � W,

�19�

where � denotes the starting time slot of using the WC, andT denotes the duration of using it.

Similarly, the continuous availability of a wavelength con-verter on node i �i�V� is denoted as �i

�,T and is calculated as

�i�,T = min

��t��+T�Fi − �

lsdn�L�i,t

sdn�, ∀ i � V. �20�

As the counterpart of the accumulated dual cost definedin the dual problem space, we define the accumulated cost inthe primal problem space. The accumulated cost of using aWC on fiber eij �eij�E� is denoted as dij

�,T and is calculated as

dij�,T = �

��t��+Tdij, ∀ eij � E. �21�

Similarly, the accumulated cost of using a wavelengthconverter on node i �i�V� is denoted as oi

�,T and is calcu-lated as

oi�,T = �

��t��+Toi, ∀ i � V. �22�

We propose a heuristic algorithm to construct a feasiblescheduling and RWA scheme based on the obtained DP’s so-lution. Using our heuristic algorithm, we determine

• which SSLDs should be rejected, rescheduled, or re-routed;

• which starting time slots the rescheduled lightpathsshould take;

• which paths and WCs the rerouted lightpaths shouldtake.

We sort SSLDs according to our proposed SSLD priorityrules and then sequentially search for feasible schedulingand RWA schemes for the sorted SSLDs. We set two SSLD

Fig. 6. (Color online) Flow chart of our heuristic algo

riority rules: (1) an accepted SSLD in the dual solution hasigher priority over a rejected SSLD in the dual solution,nd (2) the priority of accepted SSLDs in the dual solution isetermined based on their dual profit (defined in the solu-ion to the relaxed problem in Subsection IV.B). We break aie arbitrarily. Our algorithm is shown in the flow chart inig. 6. For lsdn, we decide its best scheduling based on itsual profit, which is stored in the sorted list SLsdn. Then, forgiven starting time slot of lsdn, we adopt the heuristic al-

orithm called the feasible route searching algorithmFRSA) proposed in [46] to obtain a feasible RWA scheme. Ifo feasible RWA scheme is found, the next-best schedulingor the same lsdn is tried until all possible starting time slotsre searched or a feasible scheduling and RWA scheme isound. The FRSA is implemented on the wavelength graphf the network, which is formed by separating every nodento W vertices representing the wavelengths available onhe node and connecting these vertices with arcs represent-ng the WCs and the wavelength converters. The details toorm the wavelength graph can be found in Appendix B of46], while the availability and the weight �WEijc� of a re-ource should be replaced by the continuous availability andhe accumulated cost, respectively.

. Analysis of Computation Complexity

The total computation complexity of our algorithm is de-ermined by two factors: the total number of iterations andhe complexity of computing solutions within each iteration.ur algorithm uses iterations to improve the quality of so-

utions. The convergence of the iterations is very efficient.ecause in the worst case the total number of iterations is

imited to a constant, we are able to simplify the analysis ofomputation complexity to a single iteration. Within each it-ration, the computation complexity is determined by twoactors: the complexity of computing the DP’s solution andhe complexity of the heuristic algorithm in constructing aeasible solution.

m to obtain a feasible scheduling and RWA solution.
rith

GS

lndtamTlatosteL

wp(salocuwsjibtduSStt

pWtp

Fw


The computation of the DP’s solution has the computationcomplexity of O��N+W�N2WZ+ �N+WE�Z2�. The optimaldual solution is achieved when all the subproblems areminimized. The computation complexity of the accumulateddual cost for a given starting time and a given holding timeis O�N+WE�. There are altogether Z�Z−1� combinations ofstarting time and holding time, which result in a computa-tion complexity of O��N+WE�Z2� in calculating all combina-tions of the accumulated dual cost. The complexity of solvingeach dual routing cost Dsdn for a given starting time �sdn isthe same as the complexity of the modified minimum-costsemilightpath [46], i.e., O��N+W�NW�. By grouping theDsdn’s with the same source node, the complexity of solvingall Dsdn’s is O��N+W�N2W�. We need to solve it for everypossible starting time �sdn, and thus the overall complexityto solve all dual routing costs is O��N+W�N2WZ�. The totalcomplexity of solving the DP is O��N+W�N2WZ+ �N+WE�Z2�.

The heuristic algorithm to construct a feasible solutionhas the computation complexity of O�L3 log L+L2�NW�2

+L�N+WE�Z3+LZ log Z�. The computation complexity ofthe continuous availability for a given starting time and agiven holding time is O��N+WE�Z�. There are altogetherZ�Z−1� combinations of starting time and holding time,which result in a computation complexity of O��N+WE�Z3�in calculating all combinations of the continuous availabil-ity. The complexity of the FRSA is O�L2 log L+L�NW�2�,which was analyzed in [46]. The time complexity for thesorting operation is O�Z log Z�. Therefore the overall com-plexity of the heuristic algorithm is O�L3 log L+L2�NW�2

+L�N+WE�Z3+LZ log Z�.

Our algorithm has a polynomial computation complexitywith respect to all design variables. The total complexity ofour algorithm is dominated by the total number of SSLDsand the total number of time slots. In Table IV, we show acomparison with some solution methods for the static light-path scheduling problems where a lightpath must start atits specified starting time. Since our problem allows slidingin the time window and scheduling beyond the desired timewindow, our computation complexity highly depends on thetotal number of time slots.

TABLE IVCOMPUTATION COMPLEXITY FOR THE STATIC LIGHTPATH

SCHEDULING PROBLEMS WHERE A LIGHTPATH MUST START AT

ITS SPECIFIED STARTING TIME

SolutionMethods References

Branch andbound

In [3], O��KMAX�L� for the lightpath routingsubproblem, where KMAX denotes the maximumnumber of alternative paths between a given nodepair. Note that its complexity is exponential withrespect to the number of SSLDs.

Heuristicalgorithms

In [10], O�N3+L log�L�+L2N2� for its constructiveheuristics (which end deterministically); in [18],O�L2�EW+N log N��.

. Evaluation of a Feasible Scheduling and RWAcheme

We use the duality gap to evaluate the quality of our so-utions. The duality gap is defined as �J*−q*�, where J* de-otes the optimal value of the optimization objective, and q*

enotes the optimal value of the dual function. The value ofhe objective function J of any feasible solution obtained isn upper bound on the optimal objective J*. The DP’s opti-al solution q*, on the other hand, is a lower bound of J*.he value of the dual function q of any DP’s solution is a

ower bound of q*. In conclusion, the value �J−q� providesn upper bound of the duality gap. Even without obtaininghe exact optimum, we are able to estimate the distance ofur suboptimal solution to the optimal solution. We are en-ured, by using the duality gap, that our near-optimal solu-ion is within a certain range from the optimum. The read-rs are referred to [46] for the performance comparison ofR with other methodologies.

V. PERFORMANCE EVALUATION

We evaluate the performance of our algorithm in a net-ork example operating under randomly generated trafficatterns. Our example network is a mesh topology networki.e., NSFNET) with 14 nodes and 21 links. Its topology ishown in Fig. 7, which marks the sequence number of nodesnd links. In this section, we present results for one particu-ar traffic pattern. Please note that the performance of ourptimization framework is not contingent upon the statisti-al property of the traffic and the same trends are observednder several other traffic matrices, as well as various net-ork topologies. The number of SSLDs for all node pairs is

hown in Table V, where the number on the ith row and theth column represents the total number of SSLDs from nodeto j over all the time slots. We randomly assign their valuesetween 0 and 3. The total number of SSLDs is 286. Theiriming requirement is shown in Table VI, which is also ran-omly generated. The array at the ith row and the jth col-mn represents the timing and duration requirements ofSLDs from node i to j. In an array, each row represents oneSLD, where the first number is its earliest desired startingime slot, the second number is its latest desired startingime slot, and the third number is its duration.

In our first example, we study how timing flexibility im-roves network resource utilization and reduces rejections.e first fix the tardiness penalty and study the impact of

he earliness penalty. We vary the weight of the earlinessenalty for SSLDs, so that when an SSLD’s starting time is

ig. 7. Example network for performance evaluation (NSFNETith 14 nodes and 21 links).

nprttivn(sit(5w

tptozscoti

WoWh


earlier than its desired starting time, an earliness penalty isincurred. We introduce two measurements of timing viola-tion: the sum of earliness violations (SEV) and the sum oftardiness violations (STV), defined as

Sum of Earliness Violations �SEV�

= �lsdn�L

min�0,�bsdn − �sdn��, �23�

Sum of Tardiness Violations �STV�

= �lsdn�L

min�0,��sdn − bsdn� ��. �24�

TABLE VNUMBER OF SSLDS FOR ALL NODE PAIRS

TABTIMING AND DURATION REQUIREME

In Fig. 8, we present the trade-off between SEV and theumber of rejected SSLDs, as the weight of the earlinessenalty ysdn increases. The SEV decreases as the number ofejections increases, which indicates that fewer time viola-ions may be achieved at the cost of more rejections. For aotal of 286 SSLDs, an increase of SEV from 0 to 55 resultsn a reduction of rejected SSLDs from 9 to 3. If no timingiolation is allowed, the weight of earliness penalty ysdneeds to be above 50 with the current parameter settingshown on the top of the figure). In such a situation, the re-ult is the same as the traditional sliding window schedul-ng. As the earliness penalty varies, the achieved optimiza-ion objective and its bound remain almost unchangedshown in Fig. 9). The duality gap is constantly lower than%, and the values have some minor changes only when theeight of the earliness penalty ysdn approaches 0.

We fix the earliness penalty and study the impact of theardiness penalty by varying the weight of the tardinessenalty rsdn for SSLDs. In Fig. 10, we show the trade-off be-ween STV and the number of rejected SSLDs as the weightf the tardiness penalty rsdn increases. The achieved optimi-ation objective and its bound are shown in Fig. 11. We ob-erve a similar trend as shown in Figs. 8 and 9, which indi-ates that we may adjust either one of the two parametersr both to control timing violations, with minimal impact onhe achieved optimization objective. Note that the optimal-ty of the results is ensured by their tight lower bounds.

In our second example, we study the impact of the cost ofCs (denoted by dij). We set the parameters (Fi=4, W=14,

i=0, Psdn=100, ysdn=49, rsdn=20) and vary the cost of WCs.e grouped the SSLDs into three groups based on their

olding time. We can see in Fig. 12 that as dij increases from

VIS OF SSLDS FOR ALL NODE PAIRS

LENT

tsasbsiivciAvwn

tp

Fd

Fb


0 to 40, the average hop counts of each group drop at differ-ent rates. We use hop count zero to denote the status atwhich all the associated SSLDs are rejected. As dij in-creases, the number of rejected SSLDs increases as shownin Fig. 13. In Fig. 14, we demonstrate that the performanceof our scheduling results is mostly optimal for this study.The network operator can thus easily control the hop num-ber of the routings by adjusting the dij value. Please notethat we only set all Psdn’s to the same value for the simplic-ity of our numerical experiments. Our model allows the easycontrol of fairness of the rejection/acceptance by adjustingthe Psdn values. The readers are referred to [44] for a com-plete study on the assignments of the Psdn’s.

In our third example, we study wavelength conversion’simpacts on the design objective and the reduction of rejectedSSLDs. We set the cost of all wavelength converters to zero,i.e., oi=0, so that if a wavelength converter is available,

Fig. 8. (Color online) Number of rejected SSLDs versus the earli-ness violations as ysdn varies.

Fig. 9. (Color online) Achieved optimization objective and itsbound as y varies.
sdn
here is no cost for a lightpath to use it. In each parameteretting, we compare two cases: no wavelength conversionnd four wavelength converters available at each node. Wehow the achieved value of the design objective and itsound, the number of rejected SSLDs in Table VII. Our re-ults show that wavelength conversion does not significantlymprove the LR bound, but it helps our heuristic algorithmn finding a better feasible solution; thus a slightly betteralue of the design objective is achieved. In summary, theontribution of wavelength conversion for lightpath schedul-ng is minimal. Note that the total number of SSLDs is 286.lthough fewer SSLDs are rejected with wavelength con-erters than without wavelength converters, the impact ofavelength converters on the optimization objective is insig-ificant considering the total number of SSLDs.

Our algorithm demonstrates consistent near-optimum op-imized solutions under various network settings and trafficatterns. Duality gaps on other network topologies and traf-

ig. 10. (Color online) Number of rejected SSLDs versus the tar-iness violations as rsdn varies.

ig. 11. (Color online) Achieved optimization objective and itsound as r varies.
sdn

Wpdowm

mopnto

pWos

sTtdblmm�awl

Fb


fic matrices are shown in Table VIII. Our simulation resultsfor different network and traffic scenarios show the sametrend of improvement of network resource utilization andrejection reductions by timing flexibility.

Most of our results for the NSFNET example are obtainedwithin 3 hours of computation on a personal computer withWindows XP, Centrino 1.99 GHz CPU, and 1 GB RAM. Thecomputation time complexity is much higher (although stillpolynomial) than the RWA algorithm in [46] due to the extratime dimension.

VI. CONCLUSIONS

In this paper, we have studied the benefits and trade-offsof using scheduled lightpaths for traffic adaptation. Know-ing traffic patterns allows us to schedule lightpaths adap-tively to the changing traffic at the network planning stage.

Fig. 12. (Color online) Impact of dij on the average hop count.

Fig. 13. (Color online) Impact of d on the rejection of SSLDs.
ij
e proposed a network planning model as an optimizationroblem. Our model allows lightpaths to slide within theiresired timing windows with no penalty on the optimizationbjective and to slide beyond their desired timing windowsith a decreasing tolerance level. Our model quantitativelyeasures the timing satisfaction or violation.

We applied the Lagrangian relaxation and subgradientethods to the formulated optimization problem. Our meth-

ds demonstrated great computational efficiency when com-ared with other existing algorithms. We aimed at obtainingear-optimal solutions to our problem, while providing aight performance bound that can be used to evaluate theptimality of our solution.

Our simulation results show how timing flexibility im-roves network resource utilization and reduces rejections.e also studied the impact of the cost of network resources

n the optimization objective. Our future work includestudying the variable holding time.

APPENDIX A

We choose to relax the exclusive WC usage con-traints (6) and converter quantity constraints (7).he reason for choosing these constraints to relax is

hat the derived DP can thus be decomposed into in-ependent subproblems, whose optimal solutions cane easily obtained. Associated with the Lagrangian re-axation, additional elements are added into the pri-

al function J by using the corresponding Lagrangeultipliers ijct (eij�E, 0�c�W, 0� t�Z) and it (jV, 0� t�Z). In this way, the “hard” constraints (6)

nd (7) are transformed into a “soft” price in the DP,hich leads to the following Lagrangian dual prob-

em:

ig. 14. (Color online) Achieved optimization objective and itsound as dij varies.


max, 0

�q� = minA,B,�,�

�lsdn�L

��1 − �sdn�Psdn + �sdn�Csdn + Esdn��

+ �eij�E

�0�c�W

�0�t�Z

ijct� �lsdn�L

�ijctsdn − 1�

+ �i�V

�0�t�Z

it� �lsdn�L

�itsdn − Fi�� , �A1�

subject to constraints (4), (5), and (8). We use to de-note �ijct� and to denote �it�.

After regrouping the relevant terms, the dual func-tion leads to the following problem:

minA,B,�,�

�lsdn�L

��1 − �sdn�Psdn + �sdn�Csdn + Esdn�

+ �eij�E

�0�c�W

�0�t�Z

�ijctsdnijct + �

i�V�

0�t�Zit�it

sdn�− �

eij�E�

0�c�W�

0�t�Zijct − �

i�V�

0�t�ZitFi� . �A2�

TABUSING WAVELENGTH CONVERSION TO IMPROVE THE ACHIEVED

SS

Parameters

W ysdn rsdn dij

12 49 24 4

8 49 24 4

10 49 24 4

10 49 24 8

12 20 20 4

12 20 20 8

TABLCOMPUTATION RESULTS USING VARIO

Network Settings

Network NameNumber ofNodes �N�

Number ofLinks �E�

NumberWavelength Ch

NSFNET 14 21 1214 21 414 21 7

Random Network 22 35 1022 35 622 35 8

European Network 28 61 828 61 1028 61 428 61 12

Average Duality

By using the fact that

�ijctsdn = �sdn�ijct

sdn, ∀ lsdn � L, eij � E,

0 � c � W, 0 � t � Z, �A3�

�itsdn = �sdn�it

sdn, ∀ lsdn � L, j � V, 0 � t � Z.

�A4�

We further rewrite Eq. (A2) as

minA,B,�,�

�lsdn�L

��1 − �sdn�Psdn + �sdn�Csdn + Esdn

+ �eij�E

�0�c�W

�0�t�Z


i�V�

0�t�Zit�it

sdn��− �

eij�E�

0�c�W�

0�t�Zijct − �

i�V�

0�t�ZitFi� . �A5�

VIISIGN OBJECTIVE AND TO REDUCE THE NUMBER OF REJECTED

S

Results

q J

Number ofRejectedSSLDs

9122 9256 29146 9414 510165 11178 3110174 11762 379263 10005 149287 10330 19

17820 17862 3017829 17922 379278 9781 99288 10145 15

17723 17882 3517726 17942 41

VIIITOPOLOGIES AND TRAFFIC MATRICES

Total Number ofSSLDs in Traffic Matrix J q

DualityGap (%)els �W�

231 11653 11344 2.72105 5379 5081 5.86165 8198 8035 2.03352 18278 17641 3.61211 11015 10715 2.80263 13708 13346 2.72403 19421 19162 1.35605 28180 27222 3.52201 9673 9217 4.95807 37837 35434 5.43

p (%) 3.50

LEDE

LD

F

404040404040

EUS

ofann

Ga


Since the last two terms are independent of the de-cision variables, the problem can be further simplifiedas

minA,B,�,�

�lsdn�L

��1 − �sdn�Psdn + �sdn�Csdn + Esdn

+ �eij�E

�0�c�W

�0�t�Z


i�V�

0�t�Zit�it

sdn�� , �A6�

subject to constraints (4), (5), and (8).Due to constraints (8), for every lsdn, if �sdn� t

� ��sdn+ tsdn�, we may set variables �ijctsdn and �it

sdn toconstants (denoted by �ijc

sdn and �isdn, respectively); oth-

erwise, set them all to zero. Considering the cost ofrouting a lightpath in Eq. (3), we thus rewrite Eq. (A6)as

minA �

lsdn�L��1 − �sdn�Psdn + �sdn min

�sdn,�sdn,�sdn�Esdn

+ �eij�E

�0�c�W


�ijctsdn�ijct + dij�

+ �i�V


�itsdn�it + oi��. �A7�

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[3] J. Kuri, N. Puech, M. Gagnaire, E. Dotaro, and R. Douville,“Routing and wavelength assignment of scheduled lightpathdemands,” IEEE J. Sel. Areas Commun., vol. 21, no. 8, pp.1231–1240, Oct. 2003.

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[5] J. Kuri, N. Puech, and M. Gagnaire, “Diverse routing of sched-uled lightpath demands in an optical transport network,” in4th Int. Workshop on Design of Reliable Communication Net-works (DCRN 2003), Banff, Alberta, Canada, 2003, pp. 69–76.

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James Y. Zhang obtained his Ph.D. degreein electrical engineering from the Univer-sity of Ottawa, Ontario, Canada. He re-ceived both his B.S. and M.S. degrees inelectronic science from Zhejiang University,China. He has done extensive researchwork in various fields including optimiza-tion for communication networks andmanufacturing processes, object detectionin images and pattern recognition, machinelearning, high-speed digital hardware de-

ign, and biometrics. He is currently working on natural languagerocessing at Idilia Inc., Montreal, Canada.

Hussein T. Mouftah (F’90) joined theSchool of Information Technology and Engi-neering (SITE) of the University of Ottawain 2002 as a Tier 1 Canada Research ChairProfessor, where he became a UniversityDistinguished Professor in 2006. He waswith the ECE Department at Queen’s Uni-versity (1979–2002), where he was prior tohis departure a Full Professor and the De-partment Associate Head. He has 6 years ofindustrial experience mainly at Bell North-

rn Research of Ottawa (became Nortel Networks). He served theEEE Communications Society as Editor-in-Chief of the IEEE Com-unications Magazine (1995–1997), Director of Magazines (1998–

999), Chair of the Awards Committee (2002–2003), Director ofducation (2006–2007), and Member of the Board of Governors

1997–1999 and 2006–2007). He has been a Distinguished Speakerf the IEEE Communications Society (2000–2007). Currently heerves IEEE Canada (Region 7) as Chair of the Awards and Recog-ition Committee. He is the author or coauthor of 6 books, 32 bookhapters, and more than 950 technical papers, 10 patents, and 140ndustrial reports. He is the joint holder of 12 Best Paper and/or

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Outstanding Paper Awards. He has received numerous prestigiousawards, such as the 2008 ORION Leadership Award of Merit, the2007 Royal Society of Canada Thomas W. Eadie Medal, the 2007–2008 University of Ottawa Award for Excellence in Research, the2006 IEEE Canada McNaughton Gold Medal, the 2006 EIC JulianSmith Medal, the 2004 IEEE ComSoc Edwin Howard ArmstrongAchievement Award, the 2004 George S. Glinski Award for Excel-lence in Research of the University of Ottawa Faculty of Engineer-ing, the 1989 Engineering Medal for Research and Development ofthe Association of Professional Engineers of Ontario (PEO), and theOntario Distinguished Researcher Award of the Ontario InnovationTrust. Dr. Mouftah is a Fellow of the Canadian Academy of Engi-neering (2003), the Engineering Institute of Canada (2005), and theRoyal Society of Canada RSC: The Academy of Science (2008).

Jing Wu (M’97-SM’08) obtained a B.Sc. de-gree in information science and technologyin 1992 and a Ph.D. degree in systems en-gineering in 1997, both from Xi’an JiaoTong University, China. He is now a Re-search Scientist at the Communications Re-search Centre Canada (Ottawa, Canada),an Agency of Industry Canada. In the past,he worked at Beijing University of Postsand Telecommunications (Beijing, China) asa Faculty Member, at Queen’s University

(Kingston, Canada) as a Postdoctoral Fellow, and at Nortel Net-works Corporate (Ottawa, Canada) as a System Design Engineer.Currently, he is also appointed as an Adjunct Professor at the Uni-versity of Ottawa, School of Information Technology and Engineer-ing. He has contributed over 70 conference and journal papers. Heholds three patents on Internet congestion control and one patenton control plane failure recovery. His research interests include con-trol and management of optical networks, protocols and algorithmsin networking, and optical network performance evaluation and op-timization. Dr. Wu is a co-chair of the subcommittee on “network ar-

hitecture, management and applications” of the Asia Communica-ions and Photonics Exhibit and Conference (ACP), 2009 and 2010.r. Wu is a member of the technical program committees for many

nternational conferences such as IEEE ICC and IEEE GLOBE-OM.

Michel Savoie is the research programmanager for the Broadband Applicationsand Optical Networks group of the Broad-band Network Technologies ResearchBranch at the Communications ResearchCentre Canada (CRC), an Agency of Indus-try Canada. He maintains expertise inbroadband systems and related technolo-gies such as user-controlled lightpaths(UCLP), infrastructure as a service (IaaS),green ICT, application-oriented networking

AON), advanced IP, and ATM- and WDM-based optical networks inrder to provide advice on important national initiatives and toemonstrate the application of CRC technologies in a real opera-ional environment. He has managed two UCLP projects funded un-er CANARIE’s directed research program and was involved withUCALYPTUS: a Service-oriented Participatory Design Studioroject funded under the CANARIE Intelligent Infrastructure Pro-ram (CIIP). He was involved with PHOSPHORUS: a Lambda Userontrolled Infrastructure for European Research integrated project

unded by the European Commission under the IST 6th Frame-ork. He is also involved with the Health Services Virtual Organi-

ation (HSVO) project under CANARIE’s Network-Enabled Plat-orms program, the High Performance Digital Media NetworkHPDMnet) research initiative, and the GreenStar Network projectnder CANARIE’s Green IT Pilot program. CRC and its partnersre currently engaged in the further development of the Argia/CLP and the IaaS Framework software. Mr. Savoie holds a B.Sc.nd M.Sc. in electrical engineering from the University of Newrunswick.

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