1. IntroductionOptical transport networks based on wavelength-division multiplexing (WDM) areconsidered to be the most appropriate choice for the future Internet backbone. Since afailure in a WDM network such as a cable cut may result in a tremendous amount ofdata loss, efficient protection of data transport in WDM networks is thus essential. Ina WDM optical mesh network with path-based protection, each connection has twolink-disjoint paths: one working path and one protection path. A connection request isconsidered blocked if no sufficient resources are available to route either path. Effi-cient resource utilization can be achieved through backup resource sharing, a tech-nique that allows multiple protection paths to share some common wavelength linksas long as their corresponding working paths are link-disjoint while 100% restorabil-ity [1] is maintained. Shared path protection outperforms, in terms of resources used,other protection techniques based on the dedicated reservation of backup capacities.
A great deal of research has been conducted on survivability provisioning in WDMoptical networks. Previous work has considered several types of traffic model, e.g.,static traffic, dynamic random traffic, admissible set, and incremental traffic, wherethe connection-holding time of demands is not known in advance. While these differ-ent traffic models are valid and useful in many circumstances, they are not able tocapture the traffic characteristics of applications that require bandwidth during spe-cific time intervals or circuit leasing on a short-term basis. For instance, a client com-pany may request scheduled demands for bandwidth from a service provider to satisfyits communication requirements at a specific time, e.g., between headquarters andproduction centers during office hours or between data centers during the night whenbackup of databases is performed, and so on. These applications require provisioningof scheduled dedicated channels or bandwidth pipes at a specific time with certainduration. These types of traffic are termed scheduled traffic.
In this work, we study the optimal survivability design problems under the sched-uled traffic model in wavelength-convertible WDM optical mesh networks with single-
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failure scenarios. We consider scheduled traffic only to characterize the extent of sur-vivability performance gain under the scheduled traffic model. We assume thatnonbifurcated routing is employed, the cross-connects are capable of full wavelengthconversion, and the network resources are unlimited. In the model, a set of demandsis given, and the setup and tear-down time of a demand are known in advance. Weuse SRWA-S to denote the survivable routing and wavelength assignment (RWA)problem under the scheduled traffic model. We first formulate the SRWA-S problem astwo jointly optimized RWA integer linear programs (ILPs). The objective is to mini-mize the total network resources (e.g., number of wavelength links) used by bothworking paths and protection paths of all demands while 100% restorability is guar-anteed against any single-failure scenarios. The additional information on connection-holding time gives a service provider a better opportunity to optimize the networkresources jointly in space (i.e., backup resource sharing) and in time, since a demandis considered accommodated as long as it is provisioned during its holding time andtime-disjoint demands (working path and protection path alike) can therefore sharenetwork resources. To solve large problems, we propose efficient heuristic algorithmswith different demand-ordering policies. The heuristic algorithms are also applicablein dealing with dynamic traffic demands that arrive sequentially. Our simulationresults indicate that the optimization of resource sharing in space and time enabledby our connection-holding-time-aware protection schemes can achieve significantlybetter resource utilization than schemes that are holding-time unaware. In addition,the proposed heuristic algorithms are shown to be indeed very effective.
The rest of the paper is organized as follows. Section 2 introduces the scheduledtraffic model and summarizes some related work. In Section 3, we present our jointRWA ILP problem formulations. Heuristic algorithms using different demand-ordering policies that efficiently solve the problems are described in Section 4.Numerical results and performance evaluation are reported in Section 5. Conclusionsare presented in Section 6.
2. Scheduled Traffic Model and Related Work2.A. Traffic ModelIn this paper, we consider a wavelength-convertible WDM optical mesh network. Thenetwork has an arbitrary topology represented by a directed symmetric graph G= �N ,L�, where N, L are the set of nodes and the set of directed links, respectively.Each link, represented by a pair of ordered nodes, has a set of wavelengths K. We con-sider networks with directed links so that our problem formulations in Section 3 aregeneral. A set of scheduled traffic demands, D, is given, each demand of which is rep-resented by a tuple �sr ,dr ,nr ,�r ,�r�, where sr and dr are the source and destinationnodes of the demand r, respectively; nr is the number of requested light paths; and �rand �r are the setup and tear-down times of the demand, respectively.
Given a set of scheduled demands, some demands may not overlap in time. Forexample, Table 1 shows an example of a scheduled demand set that includes sevendemands. The time correlation of these demands is shown in Fig. 1. It is easy to findthat demands r1 and r4 are time-disjoint. Since the network resources used bydemand r1 have been released when demand r4 is scheduled, the resources can becompletely reused by demand r4. This motivates us to take into account the time-disjointness (if any) among demands along both working and protection paths in addi-tion to optimizing the spatial network resource sharing based on backup resourcesharing, to achieve a higher degree of overall network resource shareability.
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2.B. Related WorkThe literature on survivability provisioning in optical networks is abundant (see Refs.[2–4] and references therein). Previous work has considered several types of trafficmodel, e.g., static traffic, dynamic random traffic, admissible set, and incrementaltraffic, where the connection-holding time of demands is not known in advance. Forexample, in Refs. [4–10] several joint working and protection paths planningapproaches in survivable WDM networks were proposed. The corresponding optimiza-tion problems have been proved to be NP-complete. Some recent work [10–14] consid-ered dynamic routing and wavelength assignment of light paths with protectionrequirements.
Previous work that considered scheduled a traffic model is very limited. In the workof Kuri et al. [15,16], a scheduled light-path demand model was proposed. The routingand wavelength assignment problem is solved using a branch and bound algorithmand a tabu search algorithm. The issue of diverse routing of scheduled light-pathdemands was addressed in another work of Kuri’s [17]. The problem was formulatedas an optimization model, which is basically a two-step optimization approach, and asimulated-annealing-based algorithm was proposed to find approximate solutions tothe optimization problem. The work of Tornatore et al. [18] exploited the connection-holding-time information to dynamically provision shared-path-protected connectionsby use of heuristic algorithms. Saradhi [19] considered the provisioning of fault-tolerant scheduled light-path demands based on a two-step optimization that uses aset of precomputed routes for working and protection paths. In Ref. [20], we proposeda general sliding scheduled traffic model. In this model, the setup time �r of a demandr whose holding time is � time units is not known in advance. Rather, �r is allowed tobegin in a prespecified time window �tste� subject to the constraint that ts��r� te−�.We then consider two problems: how to (1) properly place a demand within its associ-ated time window to reduce overlap in time among a set of demands and (2) route andassign wavelengths (RWA) to a set of demands under the proposed sliding scheduledtraffic model in mesh reconfigurable WDM optical networks without wavelength con-version. In Ref. [21], we designed a two-step optimization approach to solve theSRWA-S problem. The first step precomputes a set of candidate routes for workingand protection paths. The ensuing ILPs then explore the large solution space to obtainthe best working and protection paths. In Ref. [22], we formulated the SRWA-S prob-lem into joint RWA ILPs that maximally exploit network resource reuse in both spaceand time.
3. Problem FormulationIn this section, we consider dedicated and shared protection schemes in survivableWDM optical mesh networks under the scheduled traffic model. We call these twoarchitectures DP-S and SP-S, respectively. To obtain optimal solutions for the SRWA-Sproblem, we formulate two ILPs for these two architectures and conduct joint routingand wavelength assignment for all the demands in a given traffic demand set D. Theobjective of both ILPs is to minimize the total number of wavelength links used. Theresults of ILPs will be compared with those of more computationally efficient proposedheuristic algorithms.
3.A. Input Parameters and Variables(1) The following are given as program inputs:
Fig. 1. Time correlation of the example demand set in Table 1.
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• N: the set of nodes in the network.• L: the set of links in the network.• K: the set of wavelengths on each link.• D, the set of scheduled traffic demands. Each demand r�D is represented as
�sr ,dr ,nr ,�r ,�r�, where sr, dr, nr, �r, and �r are the source node, the destination node,the number of requested light paths, and the setup and tear-down times of demand r,respectively.
• Trp,rq� �0,1�: indicates whether demands rp and rq overlap in time �=1� or not�=0� �p�q�. Since Trp,rq and Trq,rp are the same, we consider only Trp,rq �p�q� toreduce the number of constraints in the ILP. Hereafter, we assume that rp and rq areordered in demand set D without loss of generality.
(2) The problem solves the following variables given a set of traffic demands D:• �i,j
r � �0,1� indicates whether the working path of demand r traverses link �i , j��=1� or not �=0�.
• �i,jr � �0,1� indicates whether the protection path of demand r traverses link �i , j�
�=1� or not �=0�.• Ai,j
r,�� �0,1� indicates whether the working path of demand r traverses link �i , j�using wavelength � �=1� or not �=0�.
• Bi,jr,�� �0,1� indicates whether the protection path of demand r traverses link �i , j�
using wavelength � �=1� or not �=0�.• Si,j
rp,rq� �0,1� indicates whether the working paths of demands rp and rq are link-joint with respect to link �i , j� �=1� or not �=0�. These variables are only used in ILP2,presented in Subsection 3.C.
• TSrp,rq� �0,1� indicates whether demands rp and rq overlap in time and theirworking paths are link-joint �=1� or not �=0� �p�q�. These variables are used only inILP2, presented in Subsection 3.C.
• Xi,j� � �0,1� indicates whether some working paths or protection paths traverse
link �i , j� using wavelength � �=1� or not �=0�.
3.B. ILP1: Dedicated Path Protection under the Scheduled Traffic Model (DP-S)This ILP is developed for joint RWA optimization using dedicated path protectionunder the scheduled traffic model (i.e., holding-time aware). In this optimizationmodel, demands last only during specified time intervals (i.e., holding-time aware),and hence network resources can be reused among demands that are time-disjoint.However, all protection paths of the demands that overlap in time are not allowed toshare resources even if their corresponding working paths are link-disjoint.
Objective
minimize � �"�i,j��L
�"��K
Xi,j� �1�
Subject to (r�D , �i , j��L ,��K, if not specified otherwise) Flow conservation con-straints on working paths:
�"o:�sr,o��L
�sr,or = 1, " sr:r � D; �i,sr
r = 0, " r � D, " i:�i,sr� � L, �2�
�"i:�i,dr��L
�i,dr
r = 1, " dr:r � D; �dr,or = 0, " r � D, " o:�dr,o� � L, �3�
�"i:�i,j��L
�i,jr − �
"o:�j,o��L�j,o
r = 0, " j � N�j � sr,dr�. �4�
Flow conservation constraints on protection paths:
�"o:�sr,o��L
�sr,or = 1, " sr:r � D; �i,sr
r = 0, " r � D, " i:�i,sr� � L, �5�
�"i:�i,d ��L
�i,dr
r = 1, " dr:r � D; �dr,or = 0, " r � D, " o:�dr,o� � L, �6�
r
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�"i:�i,j��L
�i,jr − �
"o:�j,o��L�j,o
r = 0, " j � N�j � sr,dr�. �7�
The working path and protection path of demand r should be link-disjoint:
�i,jr + �i,j
r � 1. �8�
Requested capacity of demand r should be satisfied:
�"��K
Ai,jr,� = �i,j
r nr, �"��K
Bi,jr,� = �i,j
r nr. �9�
Wavelength � on link �i , j� should not be shared by two demands if they overlap intime:
Ai,jrp,� + Ai,j
rq,�� 1, Ai,j
rp,� + Bi,jrq,�
� 1, Bi,jrp,� + Ai,j
rq,�� 1,
Bi,jrp,� + Bi,j
rq,�� 1, " rp,rq � D�p � q and Trp,rq = 1�. �10�
Constraints indicating whether wavelength � on link �i , j� is used by some workingpaths or protection paths are
Xi,j� � �
"r�D�Ai,j
r,� + Bi,jr,��, D Xi,j
� �"r�D
�Ai,jr,� + Bi,j
r,��. �11�
Note that in this formulation wavelength links of the working path of demand r canbe reused by the working path or the protection path of other demands that do notoverlap in time with r. The same is also true for wavelength links of the protectionpath of a demand.
3.C. ILP2: Shared Path Protection under the Scheduled Traffic Model (SP-S)This ILP is developed for joint RWA optimization using shared path protection underthe scheduled traffic model (i.e., holding-time aware). This optimization model enablesresource optimization in both space and time, i.e., network resources can be reusedamong demands that are time-disjoint, and resources can also be shared by protectionpaths whose corresponding working paths are link-disjoint, even if their demandsoverlap in time.
Objective
minimize � �"�i,j��L
�"��K
Xi,j� �12�
Subject to (r�D , �i , j��L ,��K, if not specified otherwise) are flow conservation con-straints on working paths and protection paths (2)–(7), link-disjointness constraints(8), as well as capacity constraints (9) that are used as in ILP1.
Constraints indicating whether the working paths of demands rp and rq are link-joint with respect to link �i , j� (Si,j
rp,rq is set to 1 only when both �i,jrp and �i,j
rq take on 1)are:
�i,jrp + �i,j
rq � Si,jrp,rq + 1, �i,j
rp + �i,jrq 2 Si,j
rp,rq, " rp,rq � D�p � q�. �13�
Constraints indicating whether demands rp and rq overlap in time and their workingpaths are link-joint are:
TSrp,rq � Trp,rq �"�i,j��L
Si,jrp,rq,
L TSrp,rq Trp,rq �"�i,j��L
Si,jrp,rq, " rp,rq � D�p � q�. �14�
Wavelength � on link �i , j� should not be used by two demands if they overlap in time:
Ai,jrp,� + Ai,j
rq,�� 1, Ai,j
rp,� + Bi,jrq,�
� 1, Bi,jrp,� + Ai,j
rq,�� 1,
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"rp,rq � D�p � q and Trp,rq = 1�. �15�
Wavelength � on link �i , j� should not be shared by two protection paths if their corre-sponding demands overlap in time and their corresponding working paths are link-joint:
Bi,jrp,� + Bi,j
rq,� + TSrp,rq � 2, " rp,rq � D�p � q�. �16�
Finally, constraints (11) indicate whether wavelength � on link �i , j� is used by someworking paths or protection paths.
Note that this formulation maximally exploits the network resource reuse in bothspace and time, resulting in the minimization of total network resources used.
4. Heuristic AlgorithmsDue to the high computational complexity [23], the ILP formulations of Section 3 canbe solved in a reasonable amount of time only when the network size is small, eachlink has a small number of wavelengths, and the set of traffic demands is not large.Therefore, we propose efficient heuristic algorithms to solve large survivable routingand wavelength assignment problems under the scheduled traffic model by processingdemands sequentially. Hence our proposed heuristic algorithms are also applicable indealing with dynamic traffic demands. In particular, our approach strives to use theminimum total network resources by exploiting network resource reuse in both thespace and time domains simultaneously, i.e., the algorithms approximate the protec-tion scheme SP-S presented in Section 3.
Given a set of scheduled traffic demands D, the proposed heuristic algorithms firstsort the demands based on a chosen ordering policy, which results in a particularorder of demands. Different ordering policies are proposed below, and their impact onperformance will be studied. The ordered demands are saved in a new demand set D�and are processed sequentially. To process demand r�D� �1�r� D��, a two-stepapproach called least-cost time-path (LCTP) routing and wavelength assignment algo-rithm is devised that is based on a modified Dijkstra algorithm to first search for theleast-cost path in both the space and time domains as the working path WPr and thencompute as the protection path PPr a link-disjoint path with the least additional cost.The crucial step of this algorithm is to assign costs to all wavelength links in the net-work before searching for WPr or PPr. To approximate the protection scheme SP-Swith shared path protection under the scheduled traffic model, paths WPr and PPralso have to satisfy constraints with respect to demands already accommodated in thenetwork. Therefore the working path and the protection path of demand r, WPr andPPr, respectively, have to satisfy the following constraints [24]:
• C1: WPr and PPr are link-disjoint.• C2: For demand r�1, WPr cannot use wavelength-link �� if it has been used by
WPr� or PPr� �1�r��r� and demand r� overlaps with demand r in time. In otherwords, if demand r� is time-disjoint with demand r, all wavelength links used by WPr�and PPr� can be reused by WPr.
• C3: For demand r�1, PPr cannot use wavelength link �� if it has been used byWPr� �1�r��r� and demand r� overlaps with demand r in time.
• C4: For demand r�1, PPr cannot use wavelength link �� if it has been used byPPr� �1�r��r� whose corresponding working path WPr� shares a link with WPr anddemand r� overlaps with demand r in time.
The proposed LCTP algorithm will intelligently update the wavelength-link costsbased on the above constraints to maximally exploit the network resource reuse inboth space and time domains, the details of which are given in Subsection 4.B.
4.A. Demand-Ordering Policies
4.A.1. Earliest-Setup Demand First (ESDF)This policy schedules demands in increasing order of their setup times, i.e., the earliera demand starts, the earlier it will be scheduled. For demands with the same setuptime, the demand with an earlier tear-down time will be scheduled first.
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4.A.2. Earliest-Tear-Down Demand First (ETDF)This policy schedules demands in increasing order of their tear-down times, i.e., theearlier a demand ends, the earlier it will be scheduled. For demands with the sametear-down time, the demand with an earlier setup time will be scheduled first.
4.A.3. Most Conflicting Demand First (MCDF)Given a traffic demand r�D, we define T�r� as the set of demands in D that overlapwith r in time, i.e.,
T�r� = �r�r and r� overlap in time, r� � D,r� � r�. �17�
T�r� represents the number of demands that overlap with demand r in time and istermed the time conflict index of demand r.
This policy schedules demands in decreasing order of their time conflict indices.That is, the more a demand overlaps in time with other demands in D, the earlier itwill be processed. For demands with the same time conflict index, policies ESDF andETDF will be applied in order.
4.A.4. Least-Conflicting Demand First (LCDF)In contrast with MCDF, this policy schedules demands in increasing order of theirtime conflict indices. For demands with the same time conflict index, policies ESDFand ETDF will be applied in order.
4.B. Wavelength-Link Cost UpdateAfter a chosen ordering policy is applied, a new ordered demand set D� is obtained.The objective of the heuristic algorithm is then to accommodate all traffic demands inthe set. Given a traffic demand r�D� to be processed next, the algorithm dynamicallyupdates the costs of all wavelength links in the network before searching for a work-ing path WPr and a protection path PPr.
4.B.1. Finding Working Path WPrWhen processing the first demand r of the ordered demand set D�, the costs of allwavelength links are initialized to the default costs of their corresponding links, i.e.,Cw����=C���, "��E, "��K, where Cw���� denotes the assigned cost of wavelength-link �� and C��� denotes the default cost of link �.
When processing a demand r after the first one is processed, the algorithm divideswavelength links into three categories, unusable, sharable, and unused wavelengthlinks:
• Unusable wavelength links: All wavelength links that satisfy constraint C2 can-not be used by WPr.
• Sharable wavelength links: If a wavelength link �� has been used by the workingpath or the protection path of some demand r� that has been accommodated, and thedemand r� does not overlap demand r in time, then �� can be shared by WPr. Weencourage WPr to use �� by assigning a small cost �C�l� to it, where � represents avery small number. For example, � takes 0.01 in our simulation.
• Unused wavelength links: For wavelength links that have not been used by anyaccommodated demands, the default costs of their corresponding links are assigned.
The cost of a wavelength link is then assigned according to the following scheme:
Once the working path WPr of demand r has been determined, the demands whoseworking paths share a link with WPr can then be identified. We define S�r� as the setof such demands, i.e.,
S�r� = �r WP and WP share a link, r � D , 1 � r � r�. �19�
� r� r � � �
.
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4.B.2. Finding the Protection Path PPrWhen processing the first demand r of the ordered demand set D�, the links used byWPr cannot be used by the protection path PPr. The costs of other wavelength linksare initialized to the default costs of their corresponding links as follows:
Cp���� = �+ if � � K ٠�� � WPr
C��� otherwise . �20�
For demands that follow, similar to searching for WPr, the algorithm divides wave-length links into categories of unusable, sharable, and unused wavelength links:
• Unusable wavelength links: All wavelength links that satisfy constraints C1, C3,and C4 cannot be used by PPr
• Sharable wavelength links: If a wavelength-link �� has been used by the work-ing path of some demand r� that has been accommodated but does not overlap withdemand r in time, or �� has been used by the protection path of some demand r� thathas been accommodated but does not overlap in time or share a link with demand r,then �� can be shared by PPr. Therefore, the algorithm assigns a small cost to �� toencourage resource sharing.
• Unused wavelength links: For the wavelength links that have not been used byany accommodated demands, the default costs of their corresponding links areassigned.
The algorithm then assigns different costs to unusable, sharable, and unused wave-length links according to the following scheme:
Cp����
=�+ if � � K ٠�� � WPr or if $ r�,1 � r� � r ٠�� � WPr� ٠r� � T�r�
or if $ r�,1 � r� � r ٠�� � PPr� ٠r� � T�r� ٠r� � S�r�
� C��� if $ r�,1 � r� � r ٠�� � WPr� ٠r� � T�r�
or if $ r�,1 � r� � r Ù �� � PPr� Ù �r� � T�r� Ú r� � S�r��
C��� otherwise�
�21�
4.C. Least-Cost Time-Path Routing and Wavelength Assignment Algorithm(LCTP)We assume nonbifurcated routing in this study. Given a traffic demand r that requiresnr light paths, the objective is therefore to find a pair of link-disjoint paths (i.e., work-ing path and protection path) such that the working path (protection path) will useonly one physical route and nr wavelengths will be used on all the links along theroute. The proposed algorithm (LCTP) finds the pair of paths as follows. It updatesthe costs of all wavelength links before searching for a working path using the schemeof Subsection 4.B.1, and before searching for a protection path, the costs of all wave-length links are updated using the scheme of Subsection 4.B.2. For each link �, thefirst nr wavelength links with the minimum costs are then chosen. The cost summa-tion of these nr wavelength links is then assigned as the cost of link �, i.e.,
Cw��� = ��i=�1
�nr Cw��i� C���1� � Cw���2
� � . . . � Cw���K� ٠Cw���nr
� � +
+ C���1� � Cw���2
� � . . . � Cw���K� ٠Cw���nr
� = + � , �22�
Cp��� = ��i=�1
�nr Cp��i� C���1� � Cp���2
� � . . . � Cp���K� ٠Cp���nr
� � +
+ C���1� � Cp���2
� � . . . � Cp���K� ٠Cp���nr
� = + �. �23�
The least-cost path in both space and time is first computed as the working pathWPr and then a link-disjoint path of least additional cost is computed as the protec-tion path PPr. Since full wavelength conversion is assumed and the greedywavelength-link selection described above assigns the least cost to each link (if atleast n wavelengths are available on it), the Dijkstra algorithm based on the com-
r
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puted link costs is able to find the least-cost path while the demand’s light-pathrequirement is also satisfied if such a path does exist. The step of the heuristic algo-rithm is shown in Algorithm 1.
Algorithm 1. Steps of the LCTP algorithm.
LCTP �V ,E ,K ,D��1. Process demand r�D� in order;2. For each wavelength link ��, assign cost Cw����;3. For each link �, compute Cw���;4. Use the Dijkstra algorithm to find the least-cost path as the workingpath WPr for demand r; exit if a path cannot be found;5. For each wavelength link ��, assign cost Cp����;6. For each link �, compute Cp���;7. Use the Dijkstra algorithm to find the least-cost path as the protectionpath PPr for demand r; exit if a path cannot be found;8. If r� D�, go to step 1; exit otherwise.
5. Performance EvaluationIn this section, we evaluate the performance of the ILP formulations and the heuris-tic algorithms described in the previous sections. The objective of all the optimizationmodels and heuristic algorithms is to determine a pair of working path and protectionpath and assign wavelengths to them for each traffic demand in a demand set, so thatthe total number of wavelength links used is minimized given that network resourcesare sufficient to accommodate all demands. Therefore, the default cost of each link,i.e., C���, takes on 1 in the simulation of heuristic algorithms.
In the simulation, we also developed and implemented ILP formulations for dedi-cated and shared protection schemes (not shown here), respectively, under the conven-tional static traffic model in which all demands are known in advance and do notchange over time. They are referred to as DP and SP, respectively. These two architec-tures are connection-holding-time unaware and their performance are used for com-parison purpose only.
We first evaluate the joint RWA ILPs (i.e., ILP1 and ILP2). We then evaluate theperformance of the heuristic algorithms and compare it against that of ILP2 sinceshared path protection under the scheduled traffic model is employed in the heuristicalgorithms and ILP2. Finally, we apply the heuristic algorithms with different order-ing policies to large networks with large demand sets to further investigate and com-pare their performance.
We use the three networks used in Ref. [6] [Figs. 5(a)–5(c) of Ref. [6]], which areredrawn in Fig. 2, and the NSFNET topology shown in Fig. 3 for performance evalua-tion and comparison. We assume that links in the example networks are directed byreplacing a link with a pair of directed links. The source and destination of a demandare generated randomly, and the bandwidth requirements in terms of number of lightpaths are drawn from a uniform distribution in Refs. [1,3]. In addition, the setup andtear-down times of a demand are also generated randomly between 0 and 24 h andmeet the demand time correlation requirements. We use the demand time correlationdefined in Ref. [15] to characterize the time-overlapping behavior among a set ofdemands. We consider three classes of demand set with a demand time correlationfactor being weak (0.01), medium (0.1), and strong (0.5) to measure the extent of timeoverlapping among demands in a set. The ILP optimization problems are solved usingCPLEX 8.1 running on a 2.5 GHz Pentium IV processor with 2 Gbyte RAM. Feasiblesuboptimal solutions are recorded after 4 h of execution if optimal solutions are notobtained before the time limit. In Tables 2 and 3, numbers with asterisks indicate theoptimal solutions found, and numbers without asterisks indicate the current bestsolutions reported by CPLEX within the time limit.
5.A. Performance of ILPsTable 2 shows the total number of wavelength links needed to satisfy different trafficdemand sets in different optimization models with weak, medium, and strong demandtime correlation, respectively. From the table, we observe that in the first two cases
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the size of example networks is small, the set of traffic demands is small, and thenumber of wavelengths on each link is not large. Under such scenarios, all the optimi-zation models investigated are able to find optimal solutions. In larger networks withlarger demand sets, however, SP, DP-S and SP-S cannot find optimal solutions withinthe time limit, as shown in the last four cases of Table 2.
From the table, we observe that DP-S and SP-S schemes use many fewer wave-length links than DP and SP schemes in all scenarios, especially when the demandtime correlation is weak. For example, in case 3, the performance improvement ofDP-S over DP is 51%, 38%, and 22% when the demand time correlations are weak,medium and strong, respectively, while that of SP-S over SP is 47%, 33%, and 16%,respectively. This is because the protection schemes under the scheduled traffic modelare able to exploit the time-disjointness among demands and reuse wavelength linksas much as possible.
5.B. ILP Versus Heuristic AlgorithmsTable 3 shows the total number of wavelength links used to satisfy different trafficdemand sets in ILP2 and different heuristic algorithms with weak, medium, andstrong demand time correlation, respectively. As we pointed out earlier, ILP2 findsoptimal solutions in the first two cases and obtains suboptimal solutions within cer-
Fig. 2. Three example networks used in Ref. [6].
Fig. 3. NSFNET topology.
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tain time limit in the last four cases. In the first three cases (Table 3), we observe thatall the four heuristic algorithms achieve the same results in almost all scenarios andthe results are very close to that of ILP2. As the network size, demand set size orwavelength set size increases, the computational complexity of ILP2 prevents it fromfinding an optimal solution within the time limit. In contrast, the heuristic algorithmsresult in much better solutions in much less time compared with those of ILP2, whichare obtained within the time limit, as shown in cases 4, 5, and 6. In addition, weobserve that in the last three cases, the algorithms employing ordering policies ESDFand MCDF slightly outperform other two algorithms, and the algorithm using policyLCDF needs the most wavelength links to accommodate the same sets of demands.
Table 2. Total Number of Wavelength Links Used in ILPs ( *Indicates theOptimal Solution Found)
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5.C. Heuristic Algorithms in Large NetworksIn this subsection, we investigate the performance of the heuristic algorithms byusing traffic demand sets with various sizes in the NSFNET topology. Figures4(a)–4(c), show the total number of wavelength links used by the heuristic algorithmswith different ordering policies and weak, medium, and strong demand time correla-tion, respectively, versus the number of demands in a demand set. Although the per-formance of the four algorithms appears to be close, we observe that the algorithmemploying the MCDF policy consistently achieves the best performance, and the oneusing the LCDF policy needs more wavelength links than the other three algorithms.
Fig. 4. Total number of wavelength links versus number of demands.
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6. ConclusionsWe have studied the optimal survivability design problem under a scheduled trafficmodel in wavelength-convertible WDM optical mesh networks. In this model, a set ofdemands is given, and the setup time and tear-down time of a demand are known inadvance. We formulate the joint routing and wavelength assignment problems asILPs that maximally exploit network resource reuse in both space and time. The over-all objective is to minimize the total number of wavelength links used by workingpaths and protection paths of all traffic demands while 100% restorability is guaran-teed against any single-failure scenarios. Moreover, we propose efficient heuristicalgorithms with different demand-ordering policies to solve large survivable routingand wavelength assignment problems under the scheduled traffic model. Our simula-tion results indicate that the optimization of resource sharing in space and timeenabled by our connection-holding-time-aware protection schemes can achieve signifi-cantly better resource utilization than schemes that are holding-time unaware. Inaddition, the proposed heuristic algorithms are shown to be indeed very effective.
AcknowledgmentThe work reported in this paper was supported in part by the U.S. Department ofEnergy Early Career Award DE-FG02-03ER25580, and a Dayton Area Graduate Stud-ies Institute graduate scholarship. Any opinions, findings, and conclusions or recom-mendations expressed in this paper are those of the authors and do not necessarilyreflect the views of the funding agencies. We thank the anonymous reviewers and theeditor for their comments and suggestions, which significantly improved the paper.
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23. The problems can easily be shown to be NP-complete.24. Assume that demands in D� are indexed from 1 to D�.
