Design and Dimensioning of Logical Survivable Topologies Against Multiple Failures

Brigitte Jaumard and Hai Anh Hoang VOL. 5, NO. 1/JANUARY 2013/J. OPT. COMMUN. NETW. 23

Design and Dimensioning of LogicalSurvivable Topologies Against Multiple

FailuresBrigitte Jaumard and Hai Anh Hoang

Abstract—In IP-over-WDM networks, protection can beoffered at the optical layer or at the IP layer. Today, itis well acknowledged that synergies need to be developedbetween the IP and optical layers in order to optimize theresource utilization and to reduce the costs and the energyconsumption of future networks. In this paper, we study thedesign of logical survivable topologies for service recoveryagainst multiple failures, including SRLG—shared risk linkgroup—failures in IP-over-WDM networks. We propose a newoptimization model, called SURLOG_CGILP, based on a columngeneration path formulation. It is highly scalable and allowsthe exact solution of several benchmark instances, which haveonly been solved with the help of heuristics so far. In thenumerical experiments, we investigate the dimensioning of thephysical links assuming IP restoration against multiple-linkfailures. We observe that the redundancy ratios (recoveryover primary ratios for the bandwidth requirements) that areobtained are similar to the redundancy ratios reported foroptical protection.

Index Terms—IP-over-WDM networks; IP restoration; Linkdimensioning; Multiple-link failures; Optical protection; Sur-vivability.

I. INTRODUCTION

T he design and management of future networks mayrely on an all-IP design [1,2], where synergies will

need to be developed between the IP and the optical layersin order to optimize the resource utilization, to reducethe energy consumption and the cost of the networks, aswell as to guarantee the service level agreements (SLAs),while bandwidth intensive applications, like, e.g., video/IPTVservices and on-line games, will continue to emerge [3,4].

Designers of networks have to contend with frequentfailures at or below the IP layer: fiber cuts, router hard-ware/software failures, malfunctioning of optical equipment,protocol misconfigurations, etc. While most failures result fromthe digging up of cables arising in construction work andaccount for about 60% of failure events in optical networks[1], there are also multiple failures, which usually have acommon cause [5] (e.g., they share a common component which

Manuscript received April 30, 2012; revised October 17, 2012; acceptedOctober 19, 2012; published December 14, 2012 (Doc. ID 167727).

The authors are with the Computer Science and Software Engineering (CSE)Department of Concordia University, Montreal (QC) H3G 1M8, Canada (e-mail:[email protected]).

Digital Object Identifier 10.1364/JOCN.5.000023

fails and causes all links to go down together). It followsthat single/multiple link or node failures are unavoidableevents and therefore need to be addressed in any networkmanagement through a backup mechanism which must ensurenetwork connectivity in the case of any multiple failure.

When a failure occurs, the backup mechanism establishesan alternate route to carry the interrupted connections.Depending on whether this alternate route is on-line or off-linegenerated, the corresponding backup mechanism is referredto as restoration or protection. Both layers, the IP layer andthe optical layer, need to be resilient to failures. Restorationmechanisms are widely deployed at the IP layer, while theoptical layer most often uses protection mechanisms [6].

The IP layer is referred to as the logical/virtual layer, whereeach logical link (called LSP—label switch path—in the contextof MPLS—multiprotocol label switching) is mapped onto alightpath in the optical/physical layer. Therefore, a networkfailure, such as, e.g., a fiber cut, can result in several logicalbroken links because the physical resource (e.g., a duct hostingseveral fibers) can be shared by several optical lightpaths,and those logical broken links, in turn, can make the logicaltopology disconnected. However, a necessary condition for theexistence of a restoration scheme in the IP layer is that thelogical topology remains connected (survivable) when somefailures occur.

In the present study, we investigate further the logicalsurvivable topology design problem with a new mathematicalmodel, called SURLOG_CGILP, which is much more scalablethan previously published ones. We also consider multiple-linkfailures (including the so-called shared risk link group (SRLG),see, e.g., [7]). As mentioned in [5], where the authorsstudied the characterization of failures in an operationalSprint IP backbone network, multiple failures do occur,and need to be addressed in the design of a survivablelogical topology. The third and last motivation of this paperis to investigate the proper link dimensioning in order toensure successful IP restoration, and evaluate the resultingredundancy ratio, i.e., restoration over primary ratio for thebandwidth requirements in the optical layer.

The paper is organized as follows. A literature review of themost recent relevant papers is given in Section II. The newlyproposed mathematical SURLOG_CGILP model is describedin Section III and its solution in Section IV. Numericalexperiments are discussed in Section V. Conclusions are drawnin the last section.

1943-0620/13/010023-14/$15.00 © 2013 Optical Society of America

24 J. OPT. COMMUN. NETW./VOL. 5, NO. 1/JANUARY 2013 Brigitte Jaumard and Hai Anh Hoang

II. LITERATURE REVIEW

There exist quite a number of studies on the design ofsurvivable logical topologies. They can be classified into thoseconsidering only single-link failures, and those addressingmultiple failures, including single node failures. Some studieshave looked at mathematical models and then very oftenturned to heuristics as their models do not usually scale wellenough in order to solve meaningful data instances, while otherstudies consider heuristics directly. We review them below.

In addition, most studies only focus on IP restoration,assuming no optical protection, in order to address single ormultiple failures. This is the case as well for the present study.Consequently, we mainly review the IP restoration proposalsfor survivable IP-over-WDM networks (Subsection II.A), andonly briefly mention some papers discussing both opticalprotection and IP restoration in Subsection II.C.

A. IP Restoration

Modiano and Narula-Tam [8,9] proposed the first ILP(integer linear programming) models for designing logicalsurvivable topologies, in the context of bidirectional physicaland logical topologies (with no wavelength continuity assump-tion). Their models are based on some cutset requirementsto guarantee the survivability of logical topologies againstsingle-link failures. Numerical experiments have been con-ducted on ring logical topologies where the number of cutsetsis manageable, under the objective of minimizing the numberof wavelengths per logical link (rather than, e.g., the number ofwavelengths per physical link as an estimate of the bandwidthrequirements).

Kan et al. [10] studied jointly the capacity assignment(i.e., how much capacity on each logical/physical link shouldbe reserved for spare capacity) and the logical survivabilityin IP-over-WDM networks. After introducing two metrics, theload factor (fraction of the capacity of a logical link usedfor working routing) and the share factor (for given workingcapacities, minimize the overall amount of required sparecapacity in order to ensure survivability), they studied theimpact of the logical link routing on network survivabilityand spare capacity requirements. The authors showed that thelightpath routing has a significant impact on the spare capacityrequirements against single-link failures and suggested thatlightpath routing should be carefully planned in view ofdesigning logical survivable topologies.

Todimala and Ramamurthy [11] proposed an improvementof the ILP model of [9], under the wavelength continuityassumption, subject to SRLG constraints. Although morescalable than the ILP model of [8,9], the improved ILP modelwas only scalable on particular topologies as its number ofconstraints still included the exponential overall number ofcutsets in the graph underlying the logical topology. Theauthors showed that the number of cutsets is, however,polynomial when the logical topology corresponds to a planarcyclic graph (i.e., if it has a drawing of a simple cyclic graphconnecting all the vertices and having chords that do not cross).Numerical results were only provided for the latter graphs.

To deal with the complexity of designing logical survivabletopologies in IP-over-WDM networks, Kurant and Thiran [12]introduced the concept of piecewise survivable mapping, i.e., asurvivable mapping of the logical topology onto the physicaltopology exists if and only if there exists a survivable mappingfor a contracted logical topology, that is, a logical topologywhere a specified subset of edges is contracted. The resultingalgorithm, called SMART (survivable mapping algorithm byring trimming) was not scalable as the reduction correspondedto an NP-complete problem. Their first study dealt withsingle-link failures only, while subsequent papers extended theresults to single node failures [13] and multiple failures [14].The exact SMART algorithm was modified and the resultingheuristic SMART-H was tested on various data instances,against other heuristics.

Enhancements of [12] were provided in [15,16] in theparticular case of cycles. The authors proposed an enhancedversion of the SMART-H heuristic while using the concept ofrandomized rounding to find disjoint paths. In this way, theirheuristic achieved a higher success rate than SMART-H.

Still based on the results of [12], Thulasiraman et al. [17,18]investigated the well studied duality between circuits and cutsin a graph in order to improve the SMART algorithm (basedon circuits). Subsequently, they proposed a new algorithm andvariants of it based on cuts. In addition, they presented twoother heuristics, one based on maximum matching theory andthe other based on both the primal and dual algorithms. Intheir computational experiments, the different heuristics werecompared in terms of computational time, protection capacityand survivability success rate. The question then remained onhow many survivable logical topologies remained undetected,as heuristics may fail to detect survivability.

Liu and Ruan [19] considered the survivable mappingproblem of an IP-over-WDM network from a new angle. Theystudied the question of adding logical links when no survivablelogical topology exists. They also showed that, even if thegiven logical topology is survivable, it may be of interest toadd logical links in order to reduce the minimal survivablemapping cost. They designed two ILP models, but, again, dueto their requirements of including all cutset constraints, with avery limited scalability. Thulasiraman et al. [20] extended theirmodel given in [18] in order to take into account the addition oflogical links that ensure the existence of a survivable lightpathrouting. Their study mainly investigated the particular case ofchordal graphs and no numerical results were reported.

Lin et al. [21] specified two levels of survivability byintroducing the concepts of weakly and strongly survivablenetworks. A network was defined to be weakly survivableif there existed a mapping such that the logical layerremained connected after a single-link failure, while a stronglysurvivable network was a weak one with an additionalcondition stating that the mapping provides enough capacityto support all disrupted traffic. The authors proposed tosolve two design problems: the first problem was to find amapping such that the weak survivability was guaranteedwhile maximizing the logical link demand satisfaction after aphysical link failure, and the second problem was to determinea mapping that ensured strong survivability with minimumspare capacity requirements. The method that was used intheir study in order to tackle those problems consisted of two


steps where the routing with weak survivability requirementwas established in the first step and the protection aftera single-link failure was determined in the second step.Despite the fact that ILP formulations were introduced, theauthors had to develop heuristics to get around the scalabilityissues. Moreover, multiple-link failures as well as wavelengthcontinuity were not considered in this research.

B. Our Contribution

To date, most proposed ILP models have been based oncutset constraints, and consequently, have scalability issues.Indeed, most ILP models become intractable when the sizeof the logical networks gets significant. A great effort hasbeen used in order to reduce the number of generatedcutset constraints by exploiting some special graph structures.However, so far, there are no clear tools to identify the usefulcutsets without jeopardizing the optimality of the solutions,although the design of efficient heuristics has been allowed.Also, nearly all studies (indeed all except Kan et al. [10] in theabove cited studies) consider unit demands only, i.e., considerthe input of a set of logical links (or lightpaths when they havebeen mapped to a physical path) with a unit demand. Whilethis can be justified in the context of unlimited physical linkcapacities, it is restrictive in the context of a limited number ofwavelengths (i.e., transport capacities).

In this study, we use decomposition techniques to solvethe logical survivable topology problem, under the wavelengthcontinuity assumption, subject to multiple failures includingsingle node failures and SRLGs, on arbitrary networks inthe context of directional physical and logical topologies. Inaddition, instead of using a cutset formulation, we propose apath formulation with the on-line generation of augmentingpaths1 in order to guarantee scalability. The recourse todecomposition techniques allows the design of exact scalablemodels. As we will see in Section V, this allows us to solvelarger instances than in the previous studies.

In addition to a scalable ILP model, we also consider mul-tiple failures under the wavelength continuity assumption forthe primary routing, and investigate the proper dimensioningof the physical links in order to guarantee a successful IPrestoration, if no connectivity issues prevent it.

C. IP Restoration Versus Optical Protection

While most authors have studied IP restoration for thedesign of IP-over-WDM networks, several authors have raisedand discussed the synergies to be developed between the IPand optical layers in order to optimize the use of network re-sources, while still meeting the quality of recovery expectationsor standards [1]. For early papers, we can cite [6,22,23]. Morerecent investigations can be found in, e.g., [2,24,25].

1 That is, paths whose addition in the constraint matrix ensures theimprovement of the incumbent solution.

III. THE SURLOG_CGILP MODEL

A. Generalities

We propose to develop a mathematical model for thedesign of a survivable IP-over-WDM network, with thefollowing design statement. For given logical and physicaldirected topologies with physical transport capacities, checkingwhether the logical topology is survivable requires mappinglogical links to physical paths, under the wavelength continuityassumption, and verifying whether, under any failure of asingle link, or a single node, or a set of links, there alwaysexists a path linking the source to the destination of everylogical link. If such a mapping exists, the objective is to findthe mapping with the minimum cost, i.e., minimum bandwidthrequirements as estimated by the sum, over the set of physicallinks, of the number of required wavelengths per physical link.However, a network may not be able to implement a certainmapping either due to the shortage of routing bandwidth ordue to the shortage of spare capacity in either the logical orthe physical network. As opposed to many studies, our modelallows the estimation of the additional required capacity notonly in the logical network, but in the physical network, inorder to guarantee survivability assuming connectivity exists.The only study we are aware of where the authors deal withthe spare capacity in the physical network is [21].

The mapping of logical links onto physical ones is managedin a hierarchical process of multi-criteria decisions, in whichthe following objectives are considered in descending order ofpriority:

1) maximize the number of provisioned lightpaths;

2) minimize the additional bandwidth requirements in orderto support the traffic provisioning;

3) minimize the number of unprotected lightpaths after asingle/multiple link/node failure.

The hierarchical multi-criteria optimization is handled withrelative weights to each of the multiple criteria on the basisof their relative priority and range values.

B. An Illustrating Example

An illustration of the network design we propose toinvestigate is depicted in Fig. 1. The physical network and itsinitial transport capacity are represented in Fig. 1(a), while thelogical network and its demand are represented in Fig. 1(b). Asshown in Fig. 1(a), there are three transport capacity units oneach physical link at the outset. A mapping of the logical linksonto the physical ones is represented in Fig. 1(c), where eachphysical lightpath is represented by a solid line. For instance,logical links v6–v3 and v5–v4 are mapped on v6 → v7 → v3 andv5 → v6 → v4, respectively.

The initial physical transport capacities may not besufficient for a feasible mapping with respect to the demandvalues, so that extra capacity needs to be sought. Extracapacity consists of additional routing capacity, either toprovision working lightpaths, or to properly ensure sufficient


Fig. 1. (Color online) Example of the proposed network design.

protection capacity. For instance, as logical link v5–v4 isassociated with a six-unit demand, a six-unit transportcapacity is required on physical link v5 → v6 as well as on v6 →v4. Yet, physical link v6 → v4 has only three units; therefore,an additional three units are required in order to ensure aproper primary lightpath provisioning. In the case of physicallink v5 → v6, as it is traversed by lightpaths v5 → v6 → v7and v5 → v6 → v4 (corresponding to logical links v5 → v7 andv5 → v4, respectively), ten units (six units for the demand oflogical link v5 → v7 plus four units for the demand of logicallink v5 → v4) are required, i.e., seven additional units in orderto fulfill the demand.

Except for [21], previous studies did not investigate therequirements on the spare optical transport capacities inorder to ensure successful IP restoration. Here, we not onlycheck whether the spare optical transport capacities are largeenough, but also compute the additional required bandwidthrequirements if the current values are not large enough inorder to guarantee successful IP restoration. Coming back tothe example, let us consider a physical span failure on v2–v7,meaning that we need to reroute the working traffic of thelogical links mapped onto v2 to v7. In the example, there is onlyone such logical link, from v2 to v7. A possible IP restorationpath is v2 → v1 → v5 → v7 (corresponding to physical pathv2 → v1 → v3 → v5 → v6 → v7). Then, each physical link thatis involved in that restoration path needs three units of sparebandwidth in order to take care of the disrupted traffic. Herephysical link v2 → v1 has a three-unit transport capacity, ofwhich one unit is used for the provisioning of the supportingphysical path of demand (logical link) v2 → v1. Consequently,we need one more unit of spare transport capacity in orderto guarantee a successful IP restoration, if needed, of logicallink (v2,v7). For each physical link, we need to compute thepossibly additional spare transport capacity that is requiredfor a given single failure, whether a single link, or a singlenode, or a multiple-link failure. Then, the resulting additionalrequired spare capacity is the maximum required one, over allsingle failures. We next very formally explain how to computethe possibly additional spare transport capacity requirements.

Let CAP` be the original overall bandwidth (number of wave-lengths) available on each physical link `, CAPR

`be the

required bandwidth for a successful IP restoration and CAPW`

be the bandwidth requirement for primary routing. LetADDR

`be the possibly additional protection capacity needed

on physical link ` in order to guarantee a successful IP

TABLE IADDITIONAL WORKING AND RESTORATION BANDWIDTH

REQUIREMENTS

Physical link ` ADDW`

ADDR`

v1 → v2 2 0v2 → v1 1 0v1 → v3 3 0v3 → v1 2 0v4 → v2 6 3v2 → v4 2 0v7 → v2 0 0v2 → v7 3 0v4 → v6 5 2v6 → v4 6 3v7 → v6 5 2v6 → v7 4 1v7 → v3 5 2v3 → v7 0 0v3 → v5 3 0v5 → v3 0 0v5 → v6 10 7v6 → v5 0 0

restoration. We have

ADDW` =max

{0, CAPW

` −CAP`

}, (1)

ADDR` =max

{0, CAPR

` −max{

CAP`+ADDW` −CAPW

` ,0}}

, (2)

where ADDW is the possible additional bandwidth requirementin order that the mapping allows the provisioning of the overalldemand, and CAPR

`is computed as described in detail in

Subsection IV.E. Finally, the sum over the additional capacityof protection as well as of routing is what we need to addto a network in order to make a corresponding mappingfeasible. Table I shows the additional required spare capacityper physical link, whether for the mapping to satisfy the wholedemand, or for a successful IP restoration, with respect to themapping of Fig. 1(c).

C. Definitions and Notation

Before we set our new scalable mathematical model, calledSURLOG_CGILP, we need to introduce some definitions andnotation.


Let the physical topology represented by a directed graphGP = (Vp,Ep), where Vp is the set of nodes and Ep is the setof links indexed by ` (where each link is associated with adirectional fiber link), and let the logical topology representedby a directed multi-graph GL = (VL,EL), where VL is the set ofnodes and EL is the set of logical links, indexed by `′. Eachlogical link is associated with a demand. The GP graph isassumed to be strongly connected, i.e., there is a path fromeach node in GP to every other node, but this is not necessarilythe case for GL. Each demand unit is associated with a singlelogical link, and we allow multiple logical links between apair of source and destination nodes in the case of multi-unitdemands.

For a given logical link `′, let SRC(`′) be its source nodeand DST(`′) be its destination node. We denote by ω+

G (v)(respectively ω−

G (v)) the set of outgoing (respectively incoming)links of node v in any given graph G.

We denote by F the set of potential failure sets, indexed byF, where each set F is a set of edges (undirected links) whichmight fail at the same time (as in an SRLG). In a study on 100%protection against single-link failures, each set F contains asingle physical edge (or bidirectional link), and

⋃F∈F F = EL.

Indeed, if link ` ∈ EP fails, then the physical directional link inthe opposite direction of ` will also fail. In other words, for eachlink failure in the physical network, in order to avoid confusion,we should write bidirectional link failure or edge failure, but itis common understanding. Consequently, we will go on withthe commonly accepted terminology.

The SURLOG_CGILP model relies on the concept of configu-rations, where a configuration is a one-unit mapping on a givenwavelength. In this way, not only is wavelength continuitytaken care of, but a decomposition scheme is also allowed thatentitles the use of decomposition techniques such as columngeneration ones for solving the resulting model (see Section IVfor a detailed explanation on the use of column generationtechniques).

Let C be the overall set of configurations, indexed by c. Eachconfiguration c is associated with a wavelength, say λc, andis defined by the list of logical links (a subset of EL) routedon physical lightpaths associated with wavelength λc. Moreformally, a configuration is characterized by coefficients f c

``′such that f c

``′ = 1 if virtual link `′ is routed over a physicallightpath containing link ` in configuration c, 0 otherwise.For each logical link `′ taken care of by c, there exists asequence of physical links defining a path from the source tothe destination of `′, with λc assigned to each of those links,therefore defining a physical lightpath on which `′ is routed.

Let ac`′ = 1 if one lightpath has been found in GP in order to

route logical link `′, 0 otherwise. Observe that

ac`′ = max

`∈EP

f c``′ . (3)

We propose to write the SURLOG_CGILP model in sucha way that it always has a solution, whether or not thereexists a survivable logical topology. To do so, we introducedecision variables with penalties for identifying whether itis possible to protect a logical link. In this way, we alwaysoutput a logical topology, but it can be incomplete regardingthe protection of the mapping of a logical link to a physical

path, with respect to some failure sets F ∈ F . Moreover, weallow the addition of more bandwidth to not only provision theworking traffic, but also guarantee a successful restoration forthe disrupted traffic. We minimize the additional bandwidthwhich is required to fully provision all working lightpathsand, thanks to a heuristic, compute the additional requiredbandwidth to ensure enough bandwidth to all IP restorationpaths.

Such a formulation has the advantage of providinginformation when no survivable logical topology exists, i.e., ittells us how many logical links have not been routed, orhave been routed without any possible restoration feature.Let PENALAW be the penalty associated with one additionalbandwidth unit for primary/working routing, PENALNR be thepenalty for no possible restoration (due to connectivity issues)for a logical link with respect to a given failure (single linkor single node or multiple link) set and WEIGHTNR be theweight of the bandwidth requirements for the restoration inthe objective. As we a priori favor restoration with additionalbandwidth over no restoration, PENALAW À PENALNR.

D. Variables

We introduce the four sets of variables of the SURLOG_CGILP

model. The first set of variables comprises the decision ones:(zc)c∈C such that zc = 1 if configuration c is selected (i.e., aproposal for the mapping of a subset of logical links), 0otherwise. The second set of variables (ϕF

`′1`′2)F∈F ;`′1,`′2∈EL

is used in order to take care of identifying a restorationlogical path for each lightpath which has been selected forthe mapping of a unit logical link. The constraints satisfiedby those variables are described in the next paragraph. Thevariable ϕF

`′1`′2∈ {0,1} is equal to 1 if the restoration logical path

for protecting logical link `′1 goes through `′2, and 0 otherwise.

The last set of variables, (y`′ )`′∈EL, takes care of the possibly

unprotected mappings, with respect to some particular failuresets. Variables are defined as follows: variable xF

`′ = 1 if logicallink `′ cannot be protected when failure set F occurs, and 0otherwise.

E. The SURLOG_CGILP Model

We are now ready for the description of the optimizationSURLOG_CGILP model. We aim at minimizing the cost(i.e., bandwidth requirements) of the logical survivabletopology throughout the sum, over the set of physical links,of the number of required wavelengths per physical link. Ifthe transport capacities are not large enough, we aim atminimizing the additional required bandwidth in order to mapthe overall demand, and we minimize the number of pairs(`′,F), where `′ ∈ EL,F ∈ F , without any protection. The lastterm in the objective corresponds to the minimization of thebandwidth requirements (for an approximation of them, seethe discussion at the end of this section) in order to guarantee


a successful restoration to the logical links that may fail.

min∑c∈C

( ∑(`,`′)∈EP×EL

f c``′

)zc +PENALAW × ∑

`∈EP

ADDW`

+PENALNR × ∑(`′,F)∈EL×F

xF`′

+WEIGHTBR × ∑(`′1,`′2,F)∈EL×EL×F

ϕF`′1,`′2

. (4)

The constraints can be written as follows:

∑c∈C

ac`′ zc ≥ 1, `′ ∈ EL, (5)

∑c∈C

∑`′∈EL

f c``′ zc ≤ CAP`+ADDW

` , ` ∈ EP, (6)

ϕF`′1,`′2

≤ ∑c∈C

f c``′1

zc, `′1,`′2 ∈ EL,

` ∈ F,F ∈F , (7)

ϕF`′1,`′2

≤ 1− ∑c∈C

f c``′2

zc, `′1,`′2 ∈ EL,

` ∈ F,F ∈F , (8)∑`′2∈ω+

GL(SRC(`′1))

ϕF`′1,`′2

= ∑`′2∈ω−

GL(DST(`′1))

ϕF`′1,`′2

= 1− xF`′1

,

`′1 ∈ EL,F ∈F , (9)∑`′2∈ω+

GL(v)ϕF`′1,`′2

= ∑`′2∈ω−

GL(v)ϕF`′1,`′2

≤ 1,

`′1 ∈ EL,F ∈F ,v 6∈ {SRC(`′1), DST(`′1)}, (10)∑`′2∈ω−

GL(vs)

ϕF`′1,`′2

= ∑`′2∈ω+

GL(vd )

ϕF`′1,`′2

= 0,

`′1 ∈ EL,F ∈F , (11)

zc ∈ {0,1} c ∈ C, (12)

ϕF`′1`

′2∈ {0,1}, F ∈F ,`′1,`′2 ∈ EL. (13)

There are two blocks of constraints. The first block is madeof constraints (5) and (6) and deals with the mapping of thelogical links onto lightpaths. Constraints (5) ensure that eachlogical link is routed on the physical topology in at least oneconfiguration, i.e., on at least one physical lightpath. Con-straints (6) are transport capacity constraints, i.e., ensure that,for a given physical link ` ∈ EP, no more than CAP` + ADDW

`lightpaths are routed on it. Indeed, CAP` corresponds to thetransport capacity, while ADDW

`is equal to the minimum

number of additional units in order to be able to find a feasiblesolution, expressed in number of wavelengths.

The second block of constraints (7)–(13) take care of therestoration paths. First, we need to establish restoration pathsonly for the logical links which are impaired by a failure: thisis the purpose of constraints (7). We next discuss the design ofthe required restoration paths. If ` ∈F belongs to the physicalrouting path of logical link `′2 in the selected configurations(i.e.,

∑c∈C f c

``′2zc = 1), then logical link `′2 cannot be used by an

alternate route for routing any logical link, and in particular`′1, i.e., ϕF

`′1,`′2= 0, in case links of F fail.

If ` ∈ F does not belong to the physical routing pathof logical link `′2 in the selected configurations, then∑

c∈C f c``′2

zc = 0 and, consequently, `′2 can be considered in an

alternate route for routing a logical link in case links of F fail.

If∑

c∈C f c``′1

zc = 1 and ` ∈ F, logical link `′1 needs an

alternate path if links of F fail. Consequently, there is a needfor one flow unit from the source to the destination of `′1 incase the links of F fail: this is the purpose of constraints (10) to(13), which compute a path in GL from SRC(`′1) to DST(`′1) forlogical link `′1 if it is impacted by failure F. However, if due to alack of network connectivity such a path cannot be found, thenxF`′1

= 1. Note that constraints (13) forbid consideration of both

incoming links for the source nodes and outgoing links for thedestination nodes. If a mapping has been found for logical link`′1, but no protection is possible, it is taken care of by variable

xF`′ in constraints (10).

When dealing with mathematical modeling for restorationpaths, one has to worry about unnecessary loops in therestoration paths. The first type of loop occurs at a nodebelonging to the restoration paths, and can be alleviatedby forcing the incoming/outgoing flows not to exceed 1(remember that each logical link is associated with a one-unitdemand). This is guaranteed thanks to constraints (8) forthe source and destination nodes, and thanks to constraints(11) for the intermediate nodes. The second type of loophas to do with isolated loops, which are not connected torestoration paths. Those are taken care of with minimizingbandwidth requirements that override these loops that wouldotherwise artificially increase the bandwidth requirements.As the bandwidth sharing induces nonlinear constraints, thecurrent objective only takes care of an approximation of thebandwidth consumption with the term weighted by PENALBR;see, however, Subsection III.F for an accurate estimation ofthe bandwidth requirements for the restoration paths, whichintegrates the functionality of bandwidth sharing.

The restoration paths that are obtained as a result of thesolution of the SURLOG_CGILP model are not necessarily opti-mized. The addition of an exact computation in the expression(4) of the objective that would take care of the minimization ofthe bandwidth requirements when identifying the restorationentails some nonlinearities in the constraint set. For thatreason, we have used the term

∑(`′1,`′2,F)∈EL×EL×FϕF

`′1,`′2,

which is only an approximation of the required restorationbandwidth (as it does not take the bandwidth sharing intoaccount). In the next section, we propose a second ILP model,once the mappings of the logical links onto the physical pathsare done, where we attempt to optimize the choice of therestoration paths, and then minimize exactly the bandwidthrequirements for a successful restoration.

F. Optimization of the Selection of the Restoration Paths

Assuming that we are given the mapping of the logical linksonto the physical links, the objective is to optimize the selectionof the restoration paths in order to minimize the bandwidthrequirements. The mappings are assumed to be described byparameters f``′ such that f``′ = 1 if logical link `′ is mappedon a physical path containing `.


The variable ϕF`′1`

′2∈ {0,1} is equal to 1 if the restoration

logical path for protecting logical link `′1 goes through `′2, and0 otherwise.

Let EL(F) be the set of all logical links, which are impairedby a failure of one of the links of F, and EL( F) be the set of alllogical links, which are not impaired by a failure of one of thelinks of F.

The objective can be written as follows:

min∑`∈EP

ADDR` . (14)

Constraints are expressed as follows:∑`′1∈EL(F)

∑`′2∈EL

f``′2ϕF`′1,`′2

≤ CAP`+ADDW` −CAPW

` +ADDR` ,

` ∈ EP,F ∈F , (15)

ϕF`′1,`′2

= 0, `′1 ∈ EL( F),`′2 ∈ EL,` ∈ F,F ∈F , (7)

ϕF`′1,`′2

= 0, `′1 ∈ EL(F),`′2 ∈ EL(F),` ∈ F,F ∈F , (8)

∑`′2∈ω+

GL(SRC(`′1))

ϕF`′1,`′2

= ∑`′2∈ω−

GL(DST(`′1))

ϕF`′1,`′2

= 1,

`′1 ∈ EL(F),F ∈F , (9)∑`′2∈ω+

GL(v)ϕF`′1,`′2

= ∑`′2∈ω−

GL(v)ϕF`′1,`′2

≤ 1,

`′1 ∈ EL(F),F ∈F ,v 6∈ {SRC(`′1), DST(`′1)}, (10)∑`′2∈ω−

GL(vs)

ϕF`′1,`′2

= ∑`′2∈ω+

GL(vd )

ϕF`′1,`′2

= 0,

`′1 ∈ EL(F),F ∈F , (11)

CAPR` ≥ 0, ` ∈ EP, (16)

ϕF`′1`

′2∈ {0,1}, `′1 ∈ EL,`′2 ∈ EL,F ∈F . (17)

In constraints (17), we compute the bandwidth require-ments on physical link ` following a failure of the links ofF. We first need to identify all the logical links `′2 whichare impaired by such a failure: they correspond to the logicallinks belonging to EL(F) (first summation). Next, for anyimpaired logical link, we examine the lightpath mapping, andcompute the number of times a lightpath goes through link `

(second summation). Last, in order to obtain the bandwidthrequirements for restoration on link `, we look at the failureset with the largest restoration bandwidth requirements (thatis, where we take into account bandwidth sharing among thefailure sets). The remaining constraints are similar to those inthe previous models, with the only difference being that themappings are already known: those constraints compute therestoration paths.

Fig. 2. (Color online) Solution scheme.

IV. SOLUTION OF THE SURLOG_CGILP MODEL

A. Why Use a Column Generation Model?

The optimization SURLOG_CGILP model described in theprevious section has a priori an exponential number of vari-ables, as the number of possible configurations is exponential.Therefore, we need to look at a solution scheme which doesnot require explicit embedding of all possible configurations.Such a scheme corresponds to the use of column generationtechniques which allow optimal (or near optimal) solutions tobe reached with only an implicit enumeration of the set ofconfigurations. The outline of these techniques is as follows,and is summarized in the inner loop of the flowchart of Fig. 2.We start with an initial set of configurations (only very few ofthem), and set the so-called restricted master problem (RMP),in contrast to the master problem defined by the model ofSection III in which all configurations are considered. Then,we search for an augmenting configuration, i.e., whether thereexists a configuration which allows improvement of the currentvalue of the linear relaxation of the RMP. If we succeed infinding such a configuration, we add it to the current RMP, andwe iterate. If not, we can claim that we have found the optimalsolution of the linear relaxation of the master problem, and wenext look for an integer solution of it (see the outer loop of theflowchart of Fig. 2).

The search for an augmenting configuration is made possibleby the use of the so-called pricing problem, described in thenext paragraph, while the search for an integer solution isdescribed in Subsection IV.C. Readers who are not familiarwith decomposition techniques may refer to, e.g., [26,27].

B. The Pricing Problem

The pricing problem looks for augmenting configurations. Itsobjective is defined by the so-called reduced cost, fed with thevalues of the dual variables of the constraints in which variablezc appears (see [26] if not familiar with linear programmingtools). If there is a configuration with a negative reduced cost,


then it is an augmenting one, otherwise, the linear relaxationof the master problem has been optimally solved (no moreaugmenting configurations exist). We next provide the detaileddescription of the pricing problem.

1) Variables: There are two sets of variables. The first set ismade of decision variables (a`′ )`′∈EL

such that a`′ = 1 if logicallink `′ is routed over a physical lightpath associated with λc, 0otherwise. The second set of variables, ( f``′ )`∈EP ,`′∈EL

, is usedfor the establishment of lightpaths in the physical topology.They are defined as follows: f``′ = 1, ` ∈ EP, `′ ∈ EL, if a unitflow of logical link `′ goes through physical link ` with λcassigned to it, 0 otherwise.

Note that, a priori, both sets of variables should also beindexed with c, i.e., the index of the configuration underconstruction. However, we did not add it to alleviate thenotation, with the understanding that each pricing problembuilds a single configuration. In addition, observe that, whilea`′ and f``′ are parameter values in the master problem (seeSection III), they are variables in the pricing model. Althoughthere may be a slight abuse of notation, we refrain fromintroducing new notation, so as to facilitate the understandingof the column generation techniques, i.e., the sequence ofalternate solutions of the restricted problem and of the pricingproblem, which are feeding each other (values of the dualvariables for the pricing problem, configurations or columns forthe RMP), until an optimal solution of the linear relaxation ofthe master problem is found.

2) Objective: Reduced Cost: The reduced cost expression,for a given configuration (the c index is removed in order toalleviate the notation), is as follows:

COST = COST− ∑`′∈EL

uD`′a`′ +

∑`∈EP

∑`′∈EL

uC` f``′

+ ∑F∈F

∑`∈F

∑`′1∈EL

∑`′2∈EL

(−uF,1

``′1`′2

f``′1+uF,2

``′1`′2

f``′2

), (18)

where uD`≥ 0 (respectively uC

`≥ 0, uF,1

``′1`′2≥ 0, uF,2

``′1`′2≥ 0) are

the values of the dual variables associated with constraints (5)(respectively (6), (7) and (8)), and where

COST =∑(`,`′)∈EP×EL

f``′ .

3) Constraints: There is only one set of constraints, whichis constituted of multi-flow constraints, in order to route asmany logical links as possible on routes to which wavelengthλc is assigned. Indeed, a one-unit logical link `′ is mapped toa route in the physical network if a route (flow) can be foundfrom SRC(`′) to DST(`′).

∑`∈ω−(v)

f``′ −∑

`∈ω+(v)f``′ =

a`′ if SRC(`′)= v

−a`′ if DST(`′)= v

0 otherwise

v ∈VL.

(19)

C. Search for an Integer Solution

In order to guarantee reaching an optimal integer solution,one must use a branch-and-price method (see, e.g., [27]) whena column generation model is used. However, it requires someeffort in order to identify an efficient branching scheme whichis scalable. Instead, let z̃ILP be the optimal integer solution ofthe ILP model such that its constraint matrix corresponds tothe one of the last solved RMP. It is well known that z̃ILP is notnecessarily the value z?ILP of an optimal integer solution of themaster problem. However, z?LP, the optimal value of the linearrelaxation of the master problem, provides a lower bound onz?ILP. It follows that the accuracy of z̃ILP can be measured bythe following optimality gap (in percentage):

|z̃ILP − z?LP|/z?LP ×100. (20)

In our experiments, we use the MILP solver of CPLEX [28]in order to get z̃ILP. If the optimality gap is not satisfactory,i.e., the precision is more than, e.g., 1%, then we select a smallset of configurations among the ones that have a zc value closeto 1 in the optimal solution of the linear relaxation of the RMP,and solve again the linear relaxation of the master problemwith some variables zc set to 1. This then allows the generationof more configurations, and consequently reinforcement of thelinear relaxation of the master problem (see the outer loop inFig. 2). Although such a process is not theoretically guaranteedto always allow the reduction of the optimality gap, it workswell in practice.

It is critical to obtain integer solutions with a smalloptimality gap in order to accurately conclude on the existenceof a fully survivable logical topology, i.e., a topology thatsurvives all failures of F , or, if appropriate, that survives thelargest number of failures. The smaller the optimality gap, themore likely the solution z̃ILP provides the proper conclusion.

D. Hierarchical Multi-criteria Objective

The values of the penalties in the objective function reflectthe priorities of its various components. PENALAW, being thecoefficient with the largest value, implies that the mappingsare defined requiring the least amount of additional workingbandwidth, if the transport capacities are not large enough.Next, with the second largest coefficients, PENALNR, we lookfor a mapping with the minimum number of pairs (`′,F) ofunprotected logical links with respect to some failure sets.Last, we worry about the minimization of the bandwidthrequirements in order to ensure a successful recovery, in caseone failure occurs.

It is theoretically possible to directly solve the SUR-LOG_CGILP model with the PENAL values, and guaranteethe priority hierarchy. However, the penalty values mayneed to be quite large in order to properly differentiate thepriorities. Consequently, the optimization process may facesome numerical difficulties owing to the large range for theobjective coefficients. In order to get round these numericalstability difficulties, we propose a sequential solution schemein order to solve the SURLOG_CGILP model. We next explainit using generic notation. Let us assume that we need to


TABLE IINETWORK TOPOLOGIES

Topology # nodes # edges = (# links)/2 Average nodal degree Reference

NJLATA 11 23 4.2 [29]NSF 14 21 3.0 [30]EURO 19 37 3.9 [31]24-NET 24 43 3.4 [11]

solve the following optimization problem, with a hierarchicalmulti-criteria objective, where PENAL2 À PENAL1 À 1:

min f0(z)+PENAL1 f1(y)+PENAL2 f2(x)

subject to: A0z+ A1 y+ A2x = b,

z ∈ Z, y ∈Y , x ∈ X .

The iterative solution process is as follows. First, we solve thefollowing ILP model, say ILP1:

min f2(x)


z ∈ Z, y ∈Y , x ∈ X .

Let f ?2 be the optimal solution of ILP1. We next solve ILP2,defined as follows:

min f1(y)


f2(x)= f ?2 ,

z ∈ Z, y ∈Y , x ∈ X .

Let f ?1 be the optimal solution of ILP2. We next solve ILP3,defined as follows:

min f0(z)


f1(y)= f ?1 ,

f2(x)= f ?2 ,

z ∈ Z, y ∈Y , x ∈ X .

Note that, in the objective under study, there are threepenalty coefficients. The above procedure can be easilyextended from two to three penalty coefficients.

E. Computing the Required Spare Capacity for aSuccessful IP Restoration

Assuming that we use the restoration paths which areprovided by the solution of the SURLOG_ILP model, we needto compute their bandwidth requirements. We next provide analgorithm to do this.

Let CAPR`

be the spare capacity that is required on link ` inorder to ensure a successful IP restoration.

For all F ∈F doLet LF be the list of logical links which fail,

following a failure of the links of FFor all ` ∈ EP, CAPF

`← 0 EndFor

For all `′ ∈ LF doCompute RP`′ , the IP recovery pathCompute PP`′ , the physical recovery path underlying

RP`′For all ` ∈ PP`′ , CAPF

`← CAPF

`+1 EndFor

EndForEndForCAPR

`=max

F∈FCAPF

`

The redundancy ratio is defined as the fraction of the sparecapacity over the working bandwidth; thus, it is given in thefollowing formulation:

∑`∈EP

CAPR`∑

`∈EP ,`′∈ELf``′

.

V. NUMERICAL RESULTS

A. Data Instances

We conducted experiments on the same four differentphysical topologies as in Todimala and Ramamurthy [11],i.e., NJLATA, NSF, EURO and 24-NET, which are describedin Table II. As in [11], we used randomly generated degreek regular undirected graphs and m-edge general undirectedgraphs for the virtual topologies, and assumed that eitherVL = VP or 1

2 |VL| = |VP| (indications are given in the tables).Undirected graphs were converted to directed graphs byreplacing each edge by two links between the same node pair,but of opposite directions.

We used four different sets of failure sets: the single (F1),dual (F2), triple (F3) and quadruple (F4) sets of failure sets.Set F1 corresponds to the set of single-link failure sets, and Fk(k = 2,3,4) designates a set of failure sets, where each failureset contains k links. Links belonging to the same failure set arelinks that are likely to fail at the same time. Moreover, as oftenobserved in networks [32] where multiple-link failures occur,we require that links belonging to the same failure set alwayshave a node in common. Based on the Fk, we define differentfailure scenarios that are described in Table III. The firstscenario corresponds to the classical failure scenario limited tosingle-link failures, while the subsequent scenarios contain anincreasing number of multiple failures, with a larger number of


TABLE IIIMULTIPLE FAILURE SET SCENARIOS

Scenario Failure sets

1 F1 only single-link failure sets2 F1 ∪F7%

23 F1 ∪F7%

2 ∪F7%3

4 F1 ∪F10%2 ∪F10%

35 F1 ∪F10%

2 ∪F10%3 ∪F5%

4

Fp%k consists of p% (randomly selected) of all possible

p-uples of links with a common node.

links per failure set. The number of multiple failures is definedas a percentage as the cardinality of the Fk set; details of thesenumbers can be found in Table III for each network instance.Note that for ease of comparison, sets of failure sets are definedincrementally, i.e., F7%

2 ⊆F10%2 , and so on.

Programs were developed using the OPL modeling languageand the (integer) linear programs were solved using Cplex12.2 [28]. We used a computer with a 4-core 2.2 GHz AMDOpteron 64-bit processor to run the programs.

Note that, for the penalty coefficients of the objectivefunction, we use PENALAW = 108, PENALNR = 104 andWEIGHTBR = 0.01.

B. Transport Capacity and Link Dimensioning

In order to set meaningful transport capacity values, somebasic (physical) link dimensioning is needed. We thereforecomputed the shortest path routing of all logical links(forgetting about wavelength continuity), and then computedthe transport capacities required for such a routing andmapping of the logical links onto the shortest physical paths.Let CAPE

`be the resulting required estimated transport

capacity for each physical link `. We then set

CAP` = ALEA{CAPE` −20%, CAPE

` +20%},

where ALEA{a,b} is a function which randomly generates avalue of set {a,b}.

Although most network designers use a shortest pathrouting for working lightpaths, we aim at simulating themapping of logical links onto physical links in a contextwhere the existing transport capacities do not necessarilyallow a shortest path routing, due, e.g., to the evolution or thedynamicity of the traffic.

As will be seen in the experiments in the forthcomingsections, we computed the minimum additional transportcapacity that is required in order to (i) route all demands(additional routing capacity) and (ii) ensure enough capacityfor a successful IP restoration at the logical layer (additionalrestoration capacity).

C. The Performance of the Proposed Model

We now discuss the performance of the column generationSURLOG_CGILP model proposed in Section III in the contextof 100% recovery against single-link failures, i.e., when F =F1 = {Fe = {e}, e ∈ E}, where E is the set of edges in the physicalnetwork (edges are undirected links between two nodes). Theresults that are reported in Table IV come from a set ofrandomly generated virtual topologies with some degree k reg-ular undirected graphs and some m-edge general undirectedgraphs.

The number of generated configurations as well as the num-ber of selected configurations in the integer solutions are givenin the third and fourth columns, respectively. The numbersclearly show that a very small number of configurations outof the overall number of potential configurations is needed inorder to reach the optimal solution of the linear relaxation of

TABLE IVPERFORMANCE OF THE SURLOG_CGILP MODEL

Configurations Optimality gap# wavelengths

per link

Instance Logical topology # generated # selected z̃ILP (%) µ(W) σ(W)

NJLATA

degree 3 128 34 46 0.1 1.0 0.920-edge 313 40 69 0.0 1.5 1.540-edge 472 80 131 0.1 2.8 2.470-edge 903 140 242 0.1 5.3 4.6

NSF

21-edge 246 42 91 0.1 2.2 1.225-edge 368 60 108 0.1 2.6 1.350-edge 532 100 218 0.0 5.2 2.280-edge 826 160 346 0.0 8.2 3.2

EURO

degree 3 916 58 132 0.0 1.8 1.230-edge 712 60 151 0.0 2.0 1.835-edge 640 70 167 0.0 2.3 1.870-edge 1540 140 326 0.0 4.4 3.190-edge 2117 180 414 0.0 5.6 3.9

24-NET40-edge 1398 80 241 0.0 2.8 1.870-edge 1933 140 400 0.0 4.7 3.290-edge 2409 180 518 0.0 6.0 3.9

24-NET |VL | = 1/2|VP |40-edge 888 80 226 0.0 2.6 2.590-edge 2166 180 548 0.0 6.4 6.0120-edge 2759 240 702 0.0 8.2 7.7


TABLE VEXISTENCE AND DIMENSIONING OF A SURVIVABLE LOGICAL TOPOLOGY (SINGLE-LINK FAILURES)

Bandwidth

Restoration

# non-survivablelogical links Primary routing Method 1 Method 2

InstanceLogicaltopology %

# failuresets To be added (%)

To be added(%)

Redundancyratio

To be added(%)

Redundancyratio

NJLATA degree 3 0.0 0.0 0.0 81.6 1.7 78.9 1.620-edge 25.0 2.5 1.0 83.3 1.3 75.0 1.2

VP =VL 40-edge 0.0 0.0 1.0 94.9 1.3 80.8 1.170-edge 0.0 0.0 7.0 107.8 1.3 82.5 1.0

NSF 21-edge 34.0 1.7 0.0 76.3 1.2 62.3 1.025-edge 16.0 1.0 0.0 104.0 1.4 84.8 1.1

VP =VL 50-edge 0.0 0.0 5.0 85.0 1.1 61.7 0.880-edge 0.0 0.0 7.0 71.7 0.9 50.8 0.7

EURO degree 3 35.7 1.1 1.0 110.7 1.5 93.7 1.330-edge 52.8 1.5 0.0 89.7 1.4 82.6 1.3

VP =VL 35-edge 34.7 1.5 0.0 90.5 1.3 68.1 1.070-edge 8.7 1.0 3.0 89.5 1.2 62.5 0.890-edge 8.0 1.0 8.0 95.8 1.2 65.7 0.8

24-NET 40-edge 12.5 2.8 2.0 117.6 1.5 84.6 1.1VP =VL 70-edge 0.0 0.0 6.0 95.3 1.2 69.5 0.9

90-edge 0.0 0.0 14.0 94.4 1.2 65.4 0.8

24-NET 40-edge 0.0 0.0 1.0 113.6 1.4 76.9 1.0|VL | = 1/2|VP | 90-edge 0.0 0.0 31.0 112.8 1.3 77.7 0.9

120-edge 0.0 0.0 47.0 115.8 1.3 79.9 0.9

the master problem on the one hand, and a nearly optimalinteger solution on the other hand. The next two columnscontain the characteristics of the ILP solutions: (i) the integervalue z̃ILP of the optimization SURLOG_CGILP model, and(ii) the optimality gap in order to measure the accuracy of z̃ILP.We can observe that all optimality gaps are less than 1%, andsome are even equal to 0, meaning that the integer solutionsare either optimal or very close to optimal ones.

The last two columns of Table IV provide the mean (µ)and variance (σ) of the number of required wavelengths (forworking routing) on a physical link. We can use these numbersin order to evaluate the impact of the transport capacity valuesselected by Todimala and Ramamurthy [11]: 6 wavelengths forNJLATA and 8 for NSF (values are not available for EURO and24-NET). Comparing these values with the average numbersof required wavelengths, together with the variance values, asindicated in the penultimate and last columns of Table IV, weconclude that most probably the transport capacities had noimpact on the output solutions in [11]. Indeed, the averagewavelength requirements (± indicates the variance) on thephysical links are 1.0 ± 0.9(degree-3 logical network) and1.5±1.5(20-edge logical network) for NJLATA, and 2.2±1.2 (21-edge logical network) and 2.6 ± 1.3 (25-edge logical network)for NSF.

D. Networking Performances

We now discuss the results in terms of the logicalnetwork connectivity, physical network dimensioning and thebandwidth requirements for recovery (IP restoration), whetherwe use the restoration paths as output by the solution ofthe SURLOG_CGILP model (referred to below as method 1),or optimize the selection of the restoration paths, once the

mapping of the logical links onto the physical paths iscompleted, referred to below as method 2.

1) Recovery From Single-Link Failures: The first set ofexperiments is again with single-link failures only and theresults are reported in Table V. The results include

(i) the percentage of logical links which cannot survive thelink failure of at least one failure set;

(ii) for the logical links of (i), the average number of failuresets they cannot survive;

(iii) the additional bandwidth (% with respect to the initialtransport capacities) which is required in order to be ableto map and properly provision all logical links;

and for both methods for computing the required bandwidthfor restoration:

(iv) the additional bandwidth (% with respect to the adjustedtransport capacities after the routing/mapping of alllogical links onto the physical links) which needs to beadded in order to completely support IP restoration;

(v) the redundancy ratio assuming enough spare capacity isprovided in order to allow a successful IP restoration inthe case of single-link failure.

We observe that, when the number of logical links increases,the connectivity issues which prevent the definition of a properrestoration path disappear. Moreover, assuming a reasonablenumber of logical links, when some of them are not fullyrecoverable, they cannot be recovered only for a small numberof failure sets, less than 1 or 2 link failures. It is then the choiceof the network operator or of the Internet service provider to


TABLE VITHE EXISTENCE AND DIMENSIONING OF A SURVIVABLE LOGICAL TOPOLOGY (MULTIPLE-LINK FAILURES)

Bandwidth

Restoration

Primaryrouting Method 1 Method 2

Number ofmultiplefailures

#non-survivable

logical links

InstanceFailurescenario |F2| |F3| |F4| %

# failuresets

To beadded (%)

To beadded (%)

Redundancyratio

To beadded (%)

Redundancyratio

NJLATA2 7 25.0 2.8 1.0 84.4 1.3 77.1 1.23 7 10 43.8 3.0 1.0 201.0 2.8 176.0 2.5

|VL | = |VP | 4 10 13 56.2 3.1 1.0 207.3 2.8 184.4 2.520-edge 5 10 13 7 56.2 3.7 1.0 221.9 3.0 197.9 2.7

NJLATA2 7 0.0 0.0 7.0 134.2 1.6 97.8 1.23 7 10 0.0 0.0 7.0 208.6 2.4 139.8 1.6

|VL | = |VP | 4 10 13 0.0 0.0 7.0 217.1 2.5 133.8 1.670-edge 5 10 13 7 0.0 0.0 7.0 234.2 2.7 145.4 1.7

EURO2 10 8.0 1.5 8.0 100.2 1.2 68.4 0.93 10 11 8.0 2.9 8.0 135.6 1.6 83.5 1.0

|VL | = |VP | 4 14 16 8.0 4.9 8.0 139.6 1.6 89.0 1.190-edge 5 14 16 7 8.0 4.9 8.0 173.6 2.0 100.9 1.2

24-NET2 9 1.5 1.0 14.0 111.8 1.3 77.1 1.03 9 6 1.5 1.0 14.0 135.1 1.6 87.3 1.1

|VL | = |VP | 4 13 9 1.5 1.0 14.0 148.3 1.7 90.8 1.190-edge 5 13 9 2 1.5 1.0 14.0 154.9 1.8 92.9 1.1

24-NET2 9 0.0 0.0 47.0 146.7 1.6 100.4 1.13 9 6 0.0 0.0 47.0 176.0 1.9 115.2 1.3

|VL | = 1/2|VP | 4 13 9 0.0 0.0 47.0 187.0 2.0 121.6 1.3120-edge 5 13 9 2 0.0 0.0 47.0 191.2 2.0 131.7 1.4

offer services without recovery (or no service for the demandsassociated with those logical links).

Except for the 24-NET instance with |VL| = 1/2|VP|, thenetwork dimensioning that is proposed in Subsection V.Bseems reasonable as, on average, only a small percentageof additional bandwidth is required in order to map alllogical links successfully. For the last instance, logical linksare restricted to be between some pairs of links, thereforeincreasing on average the lengths of the physical pathson which the logical links are mapped. Therefore, for anidentical number of logical links, the additional bandwidthrequirements for a successful working routing are often largerwhen the number of logical links increases, from 6% to 31% forthe 90-edge instance (i.e., 180 logical links).

We observe that the redundancy ratio is similar to what hasbeen observed for, e.g., link protection in optical networks; see,e.g., [33]. The redundancy ratios with method 1 are larger thanthose with method 2, as expected, meaning than in practicethere are several choices for the restoration paths. Moreover,when the number of logical links increases, as the connectivityof the network is increasing as well, the redundancy ratios tendto decrease.

Note that there is no direct correlation between the requiredadditional bandwidth for a successful restoration and the re-dundancy ratios as, on the one hand, not all transport capacityis necessarily used for working routing, and therefore part ofit may be used for restoration, and, on the other hand, theredundancy ratio only refers to the used bandwidth, whetherfor working or restoration routing on the physical links.

2) Recovery From Multiple-Link Failures: In the next setof experiments, we look at multiple failure scenarios 1–5as described in Table III, and the results are reported inTable VI. For each set Fk,k = 2,3,4, we indicated the numberof failure sets. Again, we observed that the redundancy ratiosare smaller with method 2, and that the differences betweenthe two methods get larger when the number of failure sets,and their cardinality, increase. Indeed, method 2 allows thebandwidth sharing to be maximized, while optimizing thechoice of the restoration paths. Accordingly, method 2 requiresless bandwidth for successful IP restoration than method 1; thedifference can be up to 35%, which is significant.

E. Computing Times

Computing times (i.e., wall times) range from minutes totens of hours, as described in Table VII.

VI. CONCLUSION

We have proposed a first scalable ILP model for the designof survivable logical topologies, which allows the exact solutionof most of the data instances considered so far in the literature.In addition, the SURLOG_CGILP model not only allows checkingfor whether a survivable logical topology exists or not, but,when none exists, it still constructs the largest possiblesurvivable logical topology, with indication of an appropriatedimensioning of the physical links.


TABLE VIICOMPUTING WALL TIMES

Single-link failures hh:mm

NJLATA—70 edges 0:07NSF—80 edges 0:08EURO—90 edges 6:0024NET (VP =VL)—90 edges 6:3024 NET (VP = 1/2VL)—120 edges 2:40

Multiple-link failures hh:mm

Scenario

NJLATA edge 70

2 0:263 1:304 3:005 6:00

EURO—90 edges

2 23:003 39:004 36:005 39:00

24NET (VP =VL)—90 edges

2 35:003 20:004 11:005 10:00

24NET (VP = 1/2VL)—120 edges

2 8:003 13:004 17:005 25:00

In future work, we will investigate how to improve thesolution of the SURLOG_CGILP model with heuristics in orderto solve larger data instances (including data instances withmulti-unit logical links), closer to the ones encountered in thecurrent IP-over-WDM networks. We believe that there is a lotof room to improve the current solution scheme and to combineit with heuristics in order to speed it up.

ACKNOWLEDGMENTS

The first author was supported by a Concordia UniversityResearch Chair (Tier I) and by an NSERC (Natural Sciencesand Engineering Research Council of Canada) grant.

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Design and Dimensioning of Logical Survivable Topologies Against Multiple Failures

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