Abstract—Planning and optimization of WDM networks hasraised much interest among the research community in the lastyears. Integer Linear Programming (ILP) is the most used exactmethod to perform this task and many studies have been publishedconcerning this issue. Unfortunately, many works have shownthat, even for small networks, the ILP formulations can easilyoverwhelm the capabilities of today state-of-the-art computingfacilities. So in this paper we focus our attention on ILP modelcomputational efficiency in order to provide a more effective toolin view of direct planning or other benchmarking applications.Our formulation exploits flow aggregation and consists in a newILP formulation that allows us to reach optimal solutions withless computational effort compared to other ILP approaches.This formulation applies to multifiber mesh networks with orwithout wavelength conversion. After presenting the formulationwe discuss the results obtained in the optimization of case-studynetworks.
I N RECENT optical networks, the introduction of the wave-length division multiplexing (WDM) technique has opened
the road to a new paradigm of transport infrastructure evolu-tion characterized by high capacity and high reliability. On theswitching equipment side, optical cross connects (OXC) sys-tems have become available, beside the more mature opticaladd-drop multiplexers. This opened up the road to the possibilityof deploying complex WDM networks based on mesh topolo-gies, while in the past single ring or overlaid multi-ring were themost used architectures for WDM. In order to transfer data be-tween two nodes, an optical connection needs to be set up androuted at the optical layer as in a circuit-switched network.
The increase in WDM complexity brought the need for suit-able network planning strategies into the foreground. Problemssuch as optimal dimensioning, routing and resource allocationfor optical connections must be continuously solved by new andold operators, to plan new installations or to update and expandthe existing ones. These problems can no longer be manuallysolved in complex network architectures, as it usually happenedin the earlier experimental WDM installations. Computer-aided
Manuscript received March 28, 2003; revised April 30, 2004, and November17, 2005. This work was supported in part by the EU Network of Excellence"E-Photon/ONe+. A preliminary version of this paper was presented at the IEEEINFOCOM 2002, New York, NY.
Digital Object Identifier 10.1109/TNET.2007.893158
planning tools are needed for the future which can determinehow to utilize efficiently the network resources in a reasonablecomputational time.
Since some years ago, research on optical networks has beeninvestigating design and optimization techniques. The variousproposed solutions can be classified into two main groups:heuristic methods and exact methods. The former returnsub-optimal solutions that in many cases are acceptable andhave the advantage of requiring a limited computational effort.The latter are much more computationally intensive and donot scale well with the network size, being even not applicablein some cases. However, since the exact methods are able toidentify the absolute optimal solution, they play a fundamentalrole either as direct planning tools or as benchmarks to validateand test heuristic methods.
The work we are presenting concerns exact methods to planand optimize multifiber WDM networks. In particular we focuson Integer Linear Programming (ILP), a widespread techniqueto solve exact optimization: we propose a new formulation ofthe optimization problem that we call source formulation, inthat it exploits the aggregation of all the flows generated in asingle source node [1]. Our source formulation is equivalent tothe well known flow formulation, but it allows a relevant reduc-tion of the number of variables and of constraints, thus sensiblydiminishing computation time and memory occupancy duringoptimization runs.
The paper summary is as follows. In Section II, we introduceour solution by presenting a short review of the literature re-garding ILP applications to WDM optimization. In Section III,the source formulation is presented and explained into details inthe two versions for network with or without wavelength con-version. Finally, in Section IV, results obtained by applying thesource formulation to case-study networks are shown and thenew formulation is compared to the traditional flow and routeformulations to point out the advantages of the method we areproposing. An Appendix is included to show the equivalence offlow and source formulation.
II. WDM NETWORK OPTIMIZATION BY INTEGER
LINEAR PROGRAMMING
Network design and planning is carried out with differenttechniques according to the type of traffic the network has tosupport. We investigate the static traffic case in which a knownset of permanent connection requests is assigned a priori to thenetwork. The connections requested by the nodes at a given timeto a WDM network all together form the offered traffic matrixvirtual topology (alias virtual topology). Each request is for oneor more point-to-point optical circuits (lightpaths) able to carry
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a given capacity from the source termination to the destinationtermination. We assume that all the WDM channels carry thesame capacity. Lightpaths are routed and switched by the OXCsof the network and the two lightpath terminations are located inthe source and destination OXCs.
We assume that the channels composing the lightpath (one foreach fiber it crosses) may have different wavelengths or may beall at the same wavelength, according to the availability of thewavelength conversion function in the transit OXCs. To sim-plify, we have considered two extreme cases referring to def-initions introduced in [2]: the virtual wavelength path (VWP)network case, in which all the OXCs are able to perform fullwavelength conversion, and the wavelength path (WP) networkcase, in which no wavelength conversion is allowed in the wholenetwork and lightpaths are subject to the “wavelength conti-nuity” constraint, that is absent in the VWP case. It is impor-tant to note that wavelength assignment to lightpaths in WP caseis an NP-complete problem (it is equivalent to the well-knowngraph-coloring problem) [3].
Today, WDM networks are often designed in order to be re-silient to failures that may occur to switching or transmissionequipment. Though automatic lightpath protection is very im-portant today (given the high bit-rates that a WDM channel usu-ally carries, e.g. 2.5 to 40 Gb/s), this feature will not be coveredin this work, for the reasons that will be explained later on.
Static optimization of a WDM network can be summarized asfollows: given a static traffic matrix, find the optimum values ofa set of network variables that minimizes a given cost (or objec-tive) function, under a set of constraints. The choice of variables,cost function and constraints greatly varies from case to case. Inthe past most of studies regarding WDM network planning wereaimed at virtual topology optimization with single-fiber WDMlinks [4], [5]. The cost function to be optimized was either thenumber of wavelengths necessary to route the static traffic or thenetwork load (the number of channels routed on the most loadedlink of the network) [6]. In [6] the authors introduce an ILPmodel based on aggregated flows applied to virtual topology op-timization. In the work we are proposing the virtual topology op-timization is accompanied by cost minimization of a multi-fiberphysical network: the number of fibers per link needed to sup-port a pre-assigned traffic is a variable of the problem to be min-imized, while the amount of wavelengths per fiber is preset [7].
WDM network optimization by ILP has been widely studiedin literature. We can subdivide research contributions into twogroups according to the type of networks to which they areapplied:
• WDM networks with single-fiber links;• multifiber WDM networks.
In the first group the problem consists in optimal routing andwavelength assignment (RWA) of the lightpaths. This is aNP-complete problem, as it was demonstrated in [3], [8]. Twobasic methods have been defined to model the RWA problem:flow formulation(FF) and route formulation (RF) [9]. In theformer the basic variables are the flows on each link relativeto each source-destination OXC pair; in the latter the basicvariables are the paths connecting each source-destinationtermination pair. Both these formulations have been employedto solve various sorts of problems and to investigate different
aspects of WDM networks. For example, in [9] the optimizationis carried out in order to emphasize the difference between WPand VWP scenarios. Ref. [6] studies the effects of imposing aconstraint on the average delay seen by a source-destinationpair and the amount of processing required at the nodes, whilein [10] possible utilization of bounds derived from the twoformulations by relaxation of the integer constraints are studiedand compared. In other works, the authors have selected ascost functions the number of wavelengths [9], [11] or the totalnumber of WDM channels in the network [12], [13]. In [14]authors propose new ILP formulations, which tend to have in-teger optimal solutions even when the integrality constraints arerelaxed, thereby allowing the problem to be solved optimallyby fast and highly efficient linear (not integer) programmingmethods. In [15] an exact linear formulation was presentedfor the logical topology design problem with no wavelengthconverters. In [16] the authors have investigated the so calledRWA-P, i.e., the RWA problem while allowing for degradationof routed signals by optical components.
In optimization of multifiber WDM networks optimal allo-cation of fibers has also to be solved, thus complicating theproblem of lightpath set up into routing, fiber and wavelengthassignment (RFWA). Solving RFWA becomes really chal-lenging even with relatively small networks, especially becauserouting and wavelength assignment is coupled to dimensioning.In this case, a new set of variables representing the numberof fibers of each physical link must be considered in additionto the flow or the route variables defined above for the twoformulations. This implies that RFWA has also to include thehighly complex localization problem. The choice of complexcost functions such as those comprising node or duct cost makesthe achievement of ILP optimal solution very challenging evenfor very small networks [17] (this is even worse in the caseof nonlinear objective function that require integer nonlinearprogramming [18]).
When the problem becomes computationally impractical,route formulation becomes more useful than flow formulation.If it is acceptable that RFWA is performed in a constrained way,then the solution complexity of the route formulation can becontrolled. For example, all the lightpaths can be constrained tobe routed along the first shortest paths connecting the sourceto the destination. Differently from the flow formulation, thecomplexity of which is strictly dependent on physical andvirtual topologies, the complexity of the route formulationdecreases with the number of paths that can be employed toroute the lightpaths. Multifiber network optimization with routeformulation and constrained routing has been studied in [9] and[19]–[23].
Beside route formulation with constrained routing, othermethods to control complexity have been proposed. A possi-bility is to stop the branch-and-bound algorithm (typically usedto solve ILP problems) after finding the first or a pre-definitenumber of integer solutions. Ref. [17] shows that acceptableresults (though quite far from the optimal solution) can beobtained when the branch-and-bound duration is fixed to 10minutes. Ref. [24] proposed that the whole RFWA problem canbe solved as a sequence of simpler problems (e.g. first routing,then fiber assignment, and so on). Other possible approaches
TORNATORE et al.: WDM NETWORK DESIGN BY ILP MODELS BASED ON FLOW AGGREGATION 711
Fig. 1. Example of (a) three distinct source-destination commodities and(b) the corresponding single source commodity, which will be exploited insource formulation.
are: exploitation of lagrangean relaxation [23], [25], relaxationof integer constraints [19] and randomized routing [12].
Undoubtedly, the massive need for computational resources(i.e., processing time and memory occupation) represents themain obstacle to an efficient application of ILP in optical net-work design. Constrained routing and the other simplificationtechniques are able to overcome this limitation, but the solutionthey produce is only an approximation of the actual optimal net-work design. The great advantage of ILP over heuristic methodsis the ability to guarantee that the obtained solution is the abso-lute optimum value. Any of the above techniques aimed at re-ducing the computational burden implies that the ILP approachloses its added value, even if the approximated solutions maybe close to the exact one. Our work develops and applies a newformulation of RFWA problem which is able to prune variablemultiplicity without introducing any approximation, thus pre-serving the added value of mathematical programming.
III. SOURCE FORMULATION OF THE RFWA PROBLEM
Let us consider a multifiber WDM network environmentunder static traffic, in which the number of wavelengths perfiber is given a priori, while the fiber numbers of eachphysical link are variables of the problem. Traditional ILPformulations based on flow or route paradigm1 solve the RFWAproblem managing source-destination commodity, that is tosay that these formulations route static connection requestsidentified by a source and a destination node on the graphrepresenting the WDM network [see Fig. 1(a)].
In our proposal, the ILP formulation will consider all the con-nections originating from a single source OXC as a single com-modity [see Fig. 1(b)]. Let us observe that single source com-modity on link assumes value equal to 2, because onthat link there are two source-destination commodities havingorigin in node . Thanks to this new model (that from now onwe will call source formulation or SF), we are able to prune thenumber of variables associated to traffic flows, thus reducingcomputational time and memory occupation compared to theflow formulation.
In order to minimize the number of fibers needed to supporta certain amount of traffic, source and flow formulations areequivalent. In the Appendix, we show the equivalence of these
1From now on, the flow formulation case will be considered the main term ofcomparison.
two formulations by describing how to obtain an equivalent FF(SF) solution, given a SF (FF) solution. In other words, if theobjective is to evaluate the number and the distribution of thefibers in the network, we can simply apply SF in order to achievethe solution. Then, if we are interested also in the details of theRWA (i.e., the routing and wavelength assignment of each con-nection request), we have to transform the SF solution in a FF(or equivalent solution). This second step is absolutely negli-gible from a complexity point of view: if the first SF step is alocalization problem, the second step (needed to transform theSF solution in detailed RFWA description) has the same com-plexity of a mere max-flow algorithm (for further details referto the Appendix). So all the computational times reported in thefollowing are related to the SF step, disregarding the possiblefollowing transformation.
We explain now the details of the source formulation, forwhich two different versions are reported related to networkswith or without wavelength conversion capability.
A. Source Formulation for VWP Networks
First we consider a VWP network, provided with fullwavelength conversion as defined in Section II. The physicaltopology is modeled by the graph . Physical linksare represented by the undirected edges with ,while the nodes , with ,represent the OXCs. Each link is equipped with a certainamount of unidirectional fibers in each of the two directions;fiber direction is identified by the binary variable . Finally, thevirtual topology is represented by the set of known terms ,each one expressing the number of connections that must beestablished from the source node to the destination node .Unidirectional point-to-point connections are considered (thus,in the general case, ).
The variables in the source formulation are the following:• is the number of WDM channels on link on fibers
having direction which have been allocated to lightpathsgenerated at node ;
• is the number of fibers on link with direction .It should be noted that the flow variables are defined in
such a way that all the traffic originating from the same node andtraveling on the same link in the same direction is representedin an aggregated form, regardless of the destination. This is themain aspect that differentiates source from flow formulation.
The following additional symbols are defined:• identifies the set of fibers of link that are directed
as indicated by ; for sake of clarity, in the following wename a “unidirectional link”;
• is the set of “unidirectional links” having the nodeas one extreme and leaving the node; analogously, isthe set of “unidirectional links” having the node as a oneextreme and pointing towards the node;
• is the total number of requested connectionshaving node as source.
Now we can detail the source formulation. The cost functionto be minimized is the total fiber number
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Actually, the source formulation can be very easily extendedto solve optimization problems based on the length metric. Theonly change that must be made regards the cost function, whichbecomes
where is the geographical length of link .The set of constraints is the following:
(1)
(2)
(3)
integer (4)
integer (5)
Constraint (1) is a solenoidality constraint which imposes thatthe total flow (number of lightpaths) generated by node and ex-iting from it must be equal to the total number of connection re-quests having node as source. Note that the solenoidality con-straint is not applied on each node-pair (by which a connectionis requested) but on the aggregated traffic relative to a sourcenode: therefore it is not dependent on destinations.
Constraint (2) is again a solenoidality constraint. It corre-sponds to the following sequence. Let us take a node . Weexpress the flow conservation condition for each other node ofthe network , considering only traffic having as sourcenode. This condition states that the total flow generated by andleaving is given by the total flow generated by and incident on
minus the number of requested connections having as sourceand as destination .
In Fig. 2, we show the different application of thesolenoidality constraint in the flow and source formulationcases using two simple examples. The first example shownin Fig. 2(a) refers to solenoidality constraint in the classicalflow formulation. A single connection request has been routedbetween source node and destination node through nodesand (dotted line). The flows associated to this connection arerepresented by a solid arrow in the round windows that magnifythe situation in nodes , , and : in the source (destination) thenode leaving (entering) flow is equal to the offered traffic (i.e.,a traffic unit), while in the intermediate nodes the leaving flowsequal the entering flows (e.g., in node the leaving arrow has acorrespondent entering arrow).
Fig. 2(b) refers to solenoidality constraint working in sourceformulation case: a simple network case with two connectionrequests (between and and and ) is shown. At the sourcenode , the sum of the leaving flows is enforced to be equal to thesum of the traffics to be routed towards all the destinations (inthis example two traffic units, one destined to node , the otherdestined to node ). In the other nodes the sum of entering flowsequals the sum of leaving flows plus the traffic that is droppedat that node (e.g. in node in Fig. 2(b) we have two entering
Fig. 2. The solenoidality constraint in (a) flow formulation and (b) source for-mulation. In (c), two admissible solutions derivable from the previous sourceformulation outcome.
flows and just one leaving due to the flow which is dropped atthat node).
It is worth noting that the source formulation does not re-turn a detailed mapping of routing (i.e., a path for each singleconnection request), even if it optimally assigns the number offibers needed to support the traffic; a second step must be usedto identify the routing of the connections (see the Appendix). Inother words, the source formulation loses the information of therouting of each single connection due to the aggregation of flowson the basic variable. Let us refer to Fig. 2(c): over the sourceformulation outcome shown in Fig. 2(b), we can map two dis-tinct (yet admissible) routing assignments (RA): in a first RAthe two connections are routed on the two pathsand , while a second admissible RA could be
and .The capacity constraint (3) allows us to dimension the phys-
ical network capacity. In order to ensure a feasible resource al-location it imposes that on each link the sum of flows generatedby all the nodes is smaller than the product of the number offibers by the number of wavelengths per fiber. The remainingconstraints [(4) and (5)] enforce variable integrity.
Let us now discuss the source formulation complexity for aVWP network. Table I shows the relations expressing the totalnumber of variables and constraints as functions of the physical
TORNATORE et al.: WDM NETWORK DESIGN BY ILP MODELS BASED ON FLOW AGGREGATION 713
TABLE ICOMPARISON ON CONSTRAINT AND VARIABLE NUMBERS BETWEEN SOURCE AND FLOW FORMULATIONS
topology size and the number of node pairs requiring connec-tions. The corresponding relations for the flow and route formu-lation are reported for comparison (symbols reported in Table Ihave been previously described, except for that represents themean number of possible alternative routes between two nodesin the network). In the table, is the number of source-destina-tion node-pairs requiring connections, that is upper-bounded bythe number of node pairs of the virtual topology .
The number of variables of the source formulation grows withthe product of the number of links by the number of nodes. In theflow formulation it grows instead with the product of the numberof links by the number of node pairs requesting connections.So, from the variable number point of view, source formulationshould be more efficient than flow formulation under the condi-tion , that is presumably a common situation in real net-works. If there is at least one lightpath requested by each nodepair, then we could set (the maximal value thatcould be achieved by ). Source formulation in this case allowsa reduction of the number of variables by a factor comparedto flow formulation. The same order of reduction is obtainedon the number of constraints, whose complexity decreases from
to . The previous comparison is focused onthe difference between flow and source formulations. As far theroute formulation without constrained routing is concerned, wecan easily notice that the variables number is dependent on theterm , i.e., the total number of possible alternative paths foreach node pair requiring connections in a network. This meansthat the number of variables tends to grow very quickly withnetwork connectivity and dimension, so that, for example in ourcase study-networks, flow formulation is modelized by a lowernumber of variables than route formulation.
B. Source Formulation for WP Networks
The source formulation can be extended to networks withoutwavelength conversion. ILP complexity in the WP case growswith the number of wavelengths per fiber and constraintsbecome more complicated because wavelength continuity hasto be imposed on the lightpaths. Nevertheless, the advantagesof the source over flow formulation are still relevant.
The cost function is the same as in the VWP caseSection III-A. A new index must beadded to identify the wavelength of the WDM channels, inorder to impose the wavelength continuity constraint along alightpath. Flow variables defined in the VWP case are trans-formed: now indicates the number of WDM channelshaving wavelength which on the “unidirectional link”carry lightpaths generated at node . The known terms and
have to be split, originating the new variables and. The set of constraints is modified as follows:
(6)
(7)
(8)
(9)
(10)
integer
integer
integer
integer
The solenoidality constraints are split into the sets (6) and(8) in order to impose flow conservation independently for eachwavelength. Also, the capacity constraint (10) is modified. Thenew constraints (7) and (9) express the distribution of the totalnumber of connections among the different wavelengths for asource node and for a source-destination pair respectively.
Table I compares the complexity of source and flow formu-lations also in the WP case. In both formulations the number ofconstraints and variables increases linearly with . It is impor-tant to notice that the increase of in the WP scenario is ac-companied not only by a growth of variable and constraint num-bers, but also by the extension of range of possible values thatthe variable can take. Although not directly arguable from thetable, this has a great impact on computational time and memoryrequirement. The advantage of the source formulation can beevaluated in a simple way by considering a fully-connected vir-tual topology in which . Under such assumptionthe dominant term of the number of variables is and
for source and flow formulation, respectively. As forthe number of constraints, the two dominant terms areand , respectively.
Finally, we shall mention a limitation of the source formula-tion. Unfortunately, this formulation can not be extended to op-timize path-protected WDM networks. In fact path protectionrequires to route lightpaths under the link-disjoint constraint, sothat a working lightpath can not share any physical link with itsprotection lightpath. The basic variables contains informa-tion concerning all the connections having the same source nodeaggregated together. No explicit reference can be inferred
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Fig. 3. Physical topologies of two case-study networks: (a) NSFNET and(b) EON.
TABLE IIILP VARIABLES AND CONSTRAINTS FOR NSFNET AND EON
IN THE VWP CASE
regarding lightpaths having the same source and the same des-tination, so that the link-disjoint constraint can not be enforced.Anyway, other protection techniques, such as link protection,could be planned using source formulation. In fact an approachto link protection consists in providing for each link (i.e., for allits fibers) an alternative route in order to face link failure: sucha feature does not need information related to traffic destinationnode. A source formulation based model for link protection inboth dedicated and shared cases is currently under study.
IV. CASE STUDIES AND RESULT COMPARISON
In this section we present and discuss the results obtainedby ILP optimization exploiting source formulation on two case-study network types in comparison with results obtained usingtraditional flow or route formulations. Well-known mesh net-works are considered first, that is the National Science Foun-dation Network (NSFNET) and the European Optical Network(EON). Then a class of networks called “wheel networks” areconsidered, in which the variation of the connectivity index [2],[11] defines a set of topologies ranging from the ring to thefull-mesh network.
A. NSFNET and EON
Data regarding the physical topology of NSFNET and EON,represented in Fig. 3(a) and (b), have been taken from [21] and[26], respectively. NSFNET has 14 nodes and 22 links, whileEON has 19 nodes and 39 links. The virtual topologies arebased on the static (symmetric) traffic matrices derived fromreal traffic measurements which are reported in the same refer-ences. The two traffic matrices comprise 360 and 1380 unidirec-tional connection requests for NSFNET and EON, respectively,while the distinct node pairs requiring connections are respec-tively 108 and 342 (each node pair can require more than oneconnection). Both VWP and WP cases have been analyzed.
Fig. 4. ILP variables and constraints for NSFNET in the WP case.
Table II shows the number of variables and constraints thatare involved in the ILP problem applied to the two networks inthe VWP case. They clearly show the advantage achieved usingthe aggregation of flows. Data are computed using the relationsreported in Table I except the number of variables in route for-mulation, which needs as input variable all admissible paths inthe network between nodes requiring connections. So, in orderto run an optimization based on non-constrained route formu-lation we have precomputed all the possible alternative pathsusing a greedy routine: our algorithm takes about five hours to-compute the 14 604 paths in the NSFNET network, while in theEON, due to its greater dimension, our algorithm takes about tenhours to compute about 1.57 10 paths between a single nodeand all the other nodes taken as destinations. The huge numberof variables in this last case induced us not to proceed on EONoptimization based on unconstrained route formulation.
For the WP case, a comparison can be done taking into ac-count route formulation complexity. In this latter case, as we canargue from Fig. 4, the number of constraints is smaller than insource formulation, but the number of variables grows rapidly.In fact it is associated to the number of all the possible routesconnecting each node couple, that increases exponentially withnetwork dimension and in particular it is influenced by the con-nectivity index of the network. In the following we will see howthe increase in network dimension and connectivity will affectILP model performance. To solve the ILP problems we used
TORNATORE et al.: WDM NETWORK DESIGN BY ILP MODELS BASED ON FLOW AGGREGATION 715
TABLE IIIVWP NSFNET OPTIMIZATION: COMPUTATIONAL TIME
the software tool CPLEX 6.5 based on the branch-and-boundmethod [27]. As hardware platform a workstation with a 1 GHzprocessor was used. The available memory (physical RAM +swap) amounted to 900 MByte.
Before the presentation of numerical results, it is crucial to re-member that the the source and the flow formulation are equiva-lent (see the Appendix). This equivalence is confirmed in all thenetwork cases in which both formulations succeed in finding theoptimum value: this value in fact results to be the same in thetwo formulations.
We have already shown the advantage of source formula-tion versus flow and route formulation in terms of variable andconstraint numbers. It is important to see how much this ad-vantage affects the actual computational performance of ILP.Tables III and IV display computational time and memory oc-cupation measurements of NSFNET optimization in the VWPcase (s, m, h, and d stand for seconds, minutes, hours and days,respectively, while MB stands for megabyte). Computationaltimes in bold are associated to runs succeeding in finding op-timal values.
To clearly understand the reported data, a particular aspectof ILP must be clarified. The branch-and-bound algorithmprogressively occupies memory with its data structure whileit is running. When the optimal solution is found, the algo-rithm stops and the computational time and the final memoryoccupation can be measured. In some cases, however, all theavailable memory is filled up before the optimal solution canbe found. In these cases, CPLEX returns the best but non-op-timal solution that branch-and-bound has been able to findand forces the execution to quit. These cases are identifiedby the out-of-memory tag (O.O.M.) and the computationaltime measures how long it has taken to fill up memory. Thisinteger solution, forced to be returned because of the limitedamount of memory, is associated to the so-called gap parameterthat expresses the percentage difference between the integersolution found and the minimal possible value the solutioncould reach (i.e., a lower bound returned by branch and boundalgorithm). This parameter returns an estimation of the qualityof the non-optimal integer solution found in terms of maximalpossible distance from the optimum. From Tables III and IVwe can see that the out-of-memory event is less frequent with
TABLE VVWP EON OPTIMIZATION: COMPUTATIONAL TIME AND gapBETWEEN INTEGER SOLUTION FOUND AND LOWER BOUND
RETURNED BY BRANCH AND BOUND ALGORITHM
TABLE VIWP NSFNET OPTIMIZATION: TIME REQUIRED TO FILL UP
100 MB OF MEMORY
the source formulation than with flow formulation. Moreoversource formulation always requires a smaller memory amountand a shorter run duration than the other two formulations. Thegap in run duration between source and the other two tendsto increase with the parameter. This is probably due to theextension of the range of the possible values that the variablecan take.
Table V shows resource occupation comparison betweensource and flow formulation in EON network (the route formu-lation is not feasible due to the huge number of variables); againcomputational times in bold are associated to runs succeedingin finding optimal values. We will show here the gap parameterto compare the quality of integer solution found: neither sourceformulation nor flow formulation succeeds in demonstrating theoptimality of the returned integer solution except for ,but quality of source formulation solutions is evidently betterthan quality of flow solutions. So O.O.M event happens in allthe optimizations, while for the amount of occupiedmemory is about 10 MB in both cases.
NSFNET optimization in the WP scenario takes a very longtime with the hardware we employed. In some cases it was toolong to wait either for the optimal result or for an out-of-memoryevent. Thus, in Table VI we have reported the time necessary tofill up of the first 100 MB of memory. The case withhas proved to be too complex to be solved in a reasonable timeand therefore it has been omitted (except for in thesource formulation). The speed of the branch-and-bound algo-rithm applied to the flow formulation decreases dramatically forhigh values of . Although in the source formulation the speeddoes not decrease so much, the model simplification allows asignificant computational time decrement compared to flow for-mulation. The route formulation, despite the great difference onthe number of variables, has comparable results regarding com-putational performance; this is probably due to the structure ofroute variables that make simpler to set the wavelength conti-nuity constraints.
Now we are going to compare the source and the flow formu-lations on the basis of the final value of the cost function. In allthe cases in which, for both formulations, the branch-and-boundends up before an out-of-memory event, the final values ob-tained are coincident, thus proving the equivalence of source
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Fig. 5. Source-flow comparison on the final number of fibers in the VWP case, as (a) absolute and (b) percent relative difference.
and flow formulation. In all the other cases, the best integer so-lution returned by the SF is smaller than or equal to the FF bestinteger solution. Furthermore, the gap parameter associated in-teger solutions is always smaller for SF than for FF, showingthat SF optimization runs are able to get closer to lower boundof the problem.
We focus our analysis on the performance comparison be-tween flow and source formulation. The following parametersare introduced:
• : total fiber number returned by ILP basedon the source (flow) formulation;
• : difference ;• : percent relative difference
.We will show results concerning the two case-study networks.
The difference between the two formulations obtained in theVWP scenario are represented in Fig. 5(a) (absolute values) andin Fig. 5(b) (percent) as functions of the parameter . The ab-solute difference is on average greater for the EON which hasa larger number of nodes and links. Convergence between thetwo formulations occurs, for example, in the NSFNET case for
and , in accordance with Table IV.2 It should benoted that sub-optimal solutions with the flow formulation canbe up to 18% worse than the corresponding solutions producedby the source formulation.
In Fig. 6, and are displayed for NSFNET in the WPcase. In this case, the strong increase of variable and constraintnumber with causes a relevant increase of the differencesbetween the two formulations. It is worth noting that both SFor FF are unable to reach the end of optimization runs (exceptfor the case). We evaluate the quality of the integersolution not only comparing its numerical value (as shown inFig. 6), but also comparing the gap parameter associated to theinteger solution: for 2, 4, 8, 16 the value of the gap isequal to 0%, 1.9%, 5.5% and 12% for SF and 0.3%, 2.4%, 8%,and 30% for FF. In conclusion, integer solutions provided bySF outperform solutions provided by FF as far as the numericalvalues, computational times and proximity to the lower boundare concerned. Up to this point of the paper we have described
2In the case W = 4, SF and FF return the same integer solution, which isvery likely to be the optimal, but the both of them fail in proving the optimalityof integer solution found.
Fig. 6. Source-flow comparison on the final number of fibers in the WP casefor NSFNET.
Fig. 7. Source-flow comparison on the final total fiber length in the VWP case,as percent relative difference.
multifiber WDM network optimization having the total numberof fibers as cost function. This interpretation of network costis called hop metric and it models a situation in which all thefibers of the network have the same cost. However, in real net-works the cost of a link also depends on its geographical length,which for example determines the number of optical line ampli-fiers that must be installed. Measuring the cost of a fiber in thissituation becomes much more complicated and the hop metric
TORNATORE et al.: WDM NETWORK DESIGN BY ILP MODELS BASED ON FLOW AGGREGATION 717
Fig. 8. NSFNET total fiber number optimized by ILP source formulation and by a deterministic heuristic, in the (a) VWP and (b) WP cases.
is not appropriate anymore. Another simple alternative is thelength metric, which assigns a cost to each fiber proportionalto the geographical length of the link it belongs to. Althoughstill not completely realistic (e.g., it does not take into accountthat the cost of the duct should be shared by all the fibers of alink), it could be useful in many situations (e.g., when the costof optical line amplifiers is an important issue). Clearly, the hopmetric can be regarded as a particular case of length metric inwhich all the links have unity length.3 We have tested sourceformulation based on length metric on NSFNET and EON inthe VWP case. In a fashion similar to hop metric, the followingparameters have been defined:
• : total fiber length returned by ILP based onthe source (flow) formulation;
• : difference ;• : percent relative difference
.Link lengths were assigned for the two networks accordingto [21] and [26].
Fig. 7 displays the percent relative difference betweenthe total fiber lengths obtained applying the source and theflow formulation. The same conclusions drawn for the hopmetric can be extended to these new optimization experiments.Source formulation performs better in all the cases in whichan out-of-memory event occurs; otherwise, the results arecoincident, but source formulation converges more rapidly(computational times are omitted for brevity). It is worth notingthat the length metric results in more solutions found for theconsidered set of with respect to hop metric case: this isbecause there are less tie-breaks in B&B algorithm than in thehop metric case, where the weight assigned to each link is thesame. In confirmation of this observation, the values of the gapparameter for the length metric are always smaller than thosereported for hop metric in Table V.
Finally, we show a comparison between the ILP optimizationcarried out by source formulation and the optimization by theheuristic approach described in [28]. Let us consider NSFNETand the hop metric. In Fig. 8 ILP and heuristic final results aredisplayed in the VWP [Fig. 8(a)] and WP case [Fig. 8(b)]. The
3Neither hop nor length metric take the node cost into account. Node-costoptimization issues are not covered by this paper.
TABLE VIIWHEEL NETWORKS OPTIMIZATION: RESULTS ON FIBER NUMBER
comparison shows that the results of the two techniques are quiteclose: the heuristic approach is able to provide good sub-optimalresults, but only the exact approach allows to reach the absoluteoptimum (or to come closer to it when limitations on memoryor computational time prevents branch-and-bound to converge).As far as the computational time is concerned, we have noticedthat heuristic and source-formulation ILP behave similarly forVWP networks. In the WP case, however, heuristic methods aremuch faster than ILP, even when source formulation is adopted.It can be noticed in Fig. 8(b) that heuristic has been the onlypossible approach to obtain a result with , given thehardware limitations of our workstation. Concluding, SF hasprovided a useful benchmark to evaluate the performance of aheuristic strategy on NSFNET and EON, that can be consideredtwo significant test-case networks.
B. Wheel Networks
In order to show the effectiveness of our source formulation,we have performed optimization experiments also on the set of8-node “wheel networks” shown in Fig. 9. This network classdefines topologies with increasing connectivity degree, startingfrom the ring network and ending with a full mesh network. Thisis obtained by increasing the number of edges with respect toinitial ring topology, so that the connectivity index (i.e., theratio between the number of links in the considered network andthe number of link in the full-mesh network case) assumes thevalues 0.29 (ring), 0.43, 0.57, 0.71, 1 (full-mesh). Again, wehave assumed different values of , that is {2,4,8,16,32}.This new class of network topologies will allow us to better ap-preciate the behavior of SF with respect to FF and RF, whilevarying one crucial network parameter, the connectivity index.
718 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 3, JUNE 2007
Fig. 9. Wheel networks with different connectivity degrees.
Fig. 10. Computational time fo SF, FF, RF for different � and W values.
We assume that offered traffic is uniform and equal to one con-nection request for each node couple.
The numerical results obtained by SF for these networks aresummarized in Table VII. Let us now analyze in detail these re-sults. SF, FF and RF all lead to the optimum solution for0.29, 0.43, 0.57 and all values of , for 0.71 with2, 16, 32 and for with 16, 32 (the correspondingnumerical values are in bold in Table VII. When all the three for-mulations succeed in finding optimal values, SF takes sensiblylower computational times than the other two formulations (seethe corresponding values in Fig. 10, where we have drawn thecomputational times for each value of connectivity index).
Moreover SF allows us to obtain optimal values also in somenetwork cases in which FF and RF fail; in fact for 0.71 with
4, 8 and with 16 SF succeeds in finding op-timal values that are not reached by FF and RF (these values arereported in italic style). For these three cases, Fig. 10 comparesthe computational times SF takes to reach the optimal value withthe computational times required by FF and RF to reach theirbest integer values under O.O.M. condition.
Finally, in the cases with 2, 4, 8, the three formu-lations fail in proving the optimality of the best integer solutionreturned, so the computational times refer all to O.O.M cases(time needed to occupy the whole memory). So, to better appre-
TORNATORE et al.: WDM NETWORK DESIGN BY ILP MODELS BASED ON FLOW AGGREGATION 719
TABLE VIII“WHEEL” NETWORK WITH � = 1: RESULTS ON FIBER NUMBER
ciate the performance of SF in full-mesh network, in Table VIIIwe have reported the best integer solutions reached by the threeformulations before O.O.M event.4 The results obtained by SFare strong candidates to be the effective optimal values, becausethey are equal to the optimal values for 0.71. FF showsworse results than SF on fiber number, but in particular RF re-turns results far from SF, due to the exponential increase ofadmissible paths in the full-meshed case. Concluding, Fig. 10clearly shows the difference in computational time among thethree approaches: SF outperforms FF and RF for all the values of
.5 In particular, for increasing values of , a larger number ofvariables and constraints results in larger computational times,but SF keeps returning the better results.
V. CONCLUSION
We have presented and discussed a novel formulation,called source formulation, to solve static-traffic WDM net-work optimization by ILP. This formulation has been definedfor multifiber networks with or without wavelength conver-sion capability supporting unidirectional unprotected opticalconnections. Thanks to the source formulation, we are ableto substantially prune the multiplicity of both variables andconstraints compared to the well-known flow formulation.Exploiting source formulation we thus obtain a competitiveoptimization tool capable to solve ILP problems with rela-tively low computational time and memory occupation. Thecase-studies discussed in the paper prove the advantages ofsource formulation in several network-planning experiments.
APPENDIX
As we have seen in Section III SF manages single sourcescommodities, while FF manages source-destination commodi-ties. From the ILP solutions point of view (i.e., if we considerthe integrality constraint on capacity and flow variables), the twoformulation are equivalent. To prove this last statement, in thefollowing we will show how to obtain a SF solution from a FFsolution and vice versa. It is worth noting that the value of theobjective function does not change passing from a model to theother. So, if we can always obtain a FF (SF) solution from aSF (FF) solution, then the solution that minimizes SF (FF) willminimize also a corresponding solution in FF and the optimalRFWA can be obtained equivalently by means of SF or FF.
4Except for W = 32 for which all the three approaches find the optimumsolution.
5We have omitted to report the ring network case, which is rapidly solvedby each of the three approaches. Anyway a deeper analysis of this simple caseshows that B&B algorithm explores a lower number of nodes in the case of SFcompared to FF and RF.
Let us consider the VWP case. The variable aggregation thathas been exploited to transform FF in SF formulation is
(11)
The solenoidality and capacity constraint in flow formulationcan be written as follows:
ififotherwise
(12)
(13)
A. From FF to SF
Let us consider a generic admissible solution for FF. To obtain a SF solution, just sum the flows with
fixed source on all the destination to obtain a SF routing. Thecapacity constraint (3) is automatically enforced combining(13) and (11).
B. From SF to FF
Let us consider a generic admissible solution for SF. Let us isolate a single source commodity with fixed
source .6 Let us try to route the FF flows (having assource node and any possible destination ) on the commodityidentified by , considering its value as a term of capacityfor each link of the graph. To route the traffic we can use theusual techniques of max-flow theory loading flows along pathswith positive remaining capacity, until all FF commodities aredealt with. The capacity of the single source commodity willbe enough to support all the traffic, because from (11):
(14)
Alternatively, for each single source commodity from SF wecould solve this simple ILP problem (as stated before, is fixed):
(15)
(16)
if
ifotherwise
(17)
and, as capacity constraint, we can use (14).
REFERENCES
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6Then the following FF on SF mapping can be extended to all the single sourcecommodity returned by SF.
720 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 3, JUNE 2007
[2] K.-I. Sato, Advances in Transport Network Technologies—PhotonicNetworks, ATM, and SDH, 1st ed. Norwood, MA: Artech House,1996.
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[4] B. Mukherjee, Optical Communication Networks. New York: Mc-Graw-Hill, 1997.
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[6] R. Ramaswami and K. N. Sivarajan, “Design of logical topologies forwavelength-routed optical networks,” IEEE J. Sel. Areas Commun.,vol. 14, pp. 840–851, Jun. 1996.
[7] S. Baroni, P. Bayvel, R. J. Gibbens, and S. K. Korotky, “Analysis anddesign of resilient multifiber wavelength-routed optical transport net-works,” J. Lightw. Technol., vol. 17, pp. 743–758, May 1999.
[8] S. Even, A. Itai, and A. Shamir, “On the complexity of timetableand multicommodity flows problems,” SIAM J. Comput., vol. 5, pp.691–703, 1976.
[9] N. Wauters and P. M. Deemester, “Design of the optical path layerin multiwavelength cross-connected networks,” IEEE J. Sel. AreasCommun., vol. 14, pp. 881–891, Jun. 1996.
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[11] S. Baroni and P. Bayvel, “Wavelength requirements in arbitrarily con-nected wavelength-routed optical networks,” J. Lightw. Technol., vol.15, no. 2, pp. 242–251, Feb. 1997.
[12] D. Banerjee and B. Mukherjee, “A practical approach for routing andwavelength assignment in large wavelength-routed optical networks,”IEEE J. Sel. Areas Commun., pp. 903–908, Jun. 1996.
[13] M. Alanialy and E. Ayanoglu, “Provisioning algorithms for WDM op-tical networks,” IEEE/ACM Trans. Netw., vol. 3, no. 5, pp. 767–778,Oct. 1999.
[14] A. E. Ozdaglar and D. P. Bertsekas, “Routing and wavelength assign-ment in optical networks,” IEEE/ACM Trans. Netw., vol. 11, no. 2, pp.259–272, Apr. 2003.
[15] R. M. Krishnaswamy and K. N. Sivarajan, “Design of logical topolo-gies: A linear formulation for wavelength-routed optical networks withno wavelength changers,” IEEE/ACM Trans. Netw., vol. 9, no. 2, pp.186–198, Apr. 2001.
[16] M. Ali, B. Ramamurthy, and Deogun, “Routing and wavelengthassignment with power considerations in optical networks,” Comput.Commun., no. 32, pp. 539–555, 2000.
[17] B. V. Caenegem, W. V. Parys, F. D. Turck, and P. M. Deemester,“Dimensioning of survivable WDM networks,” IEEE J. Sel. AreasCommun., pp. 1146–1157, Sep. 1998.
[18] C. M. Lee, C. C. Hui, F. Tong, and P. Yum, “Network dimensioningin WDM based all optical networks,” in Proc. IEEE GLOBECOM’98,1998, vol. 1, pp. 328–333.
[19] D. Banerjee and B. Mukherjee, “Wavelength-routed optical networks:Linear formulation, resource budgeting tradeoffs and a reconfigurationstudy,” IEEE/ACM Trans. Netw., pp. 598–607, Oct. 2000.
[20] F. Glenstrup, “Full design of robust optical networks,” in Proc. 15thNordic Teletraffic Seminar, Lund, Sweden, Aug. 2000.
[21] Y. Miyao and H. Saito, “Optimal design and evaluation of survivableWDM transport networks,” IEEE J. Sel. Areas Commun., vol. 16, pp.1190–1198, Sep. 1999.
[22] M. S. Kumar and P. S. Kumar, “Static lightpath establishment in WDMnetworks—New ILP formulations and heuristic algorithms,” Comput.Commun., no. 25, pp. 109–114, 2002.
[23] M. Saad and Z. Luo, “A Lagrangian decomposition approach for therouting and wavelength assignment in multifiber WDM networks,” inProc. IEEE GLOBECOM’02, Taipei, Taiwan, Nov. 2002.
[24] F. Poppe and P. Demeester, “Wavelength requirement of mesh-restor-able multi-wavelength optical networks,” IEEE/ACM Trans. Netw., vol.3, no. 5, pp. 767–778, Oct. 1999.
[25] H. Yen and F. Y. Lin, “Near optimal design of lightpath routing andwavelength assignment in purely optical WDM networks,” in Proc.ONDM’01, 2001, pp. 89–100.
[26] A. Fumagalli, I. Cerutti, M. Tacca, F. Masetti, R. Jagannathan, and S.Alagar, “Survivable networks based on optimal routing and WDM self-healing rings,” in Proc. IEEE INFOCOM’99, 1999, vol. 2, pp. 726–733.
[27] “Ilog CPLEX 6.5, User’s Manual,” ILOG, Mar. 1999.[28] G. Maier, A. Pattavina, L. Roberti, and T. Chich, “Static-lightpath de-
sign by heuristic methods in multifiber WDM networks,” SPIE Opt.Netw. Mag., vol. 3, no. 5, pp. 52–66, Sep.–Oct. 2002.
Massimo Tornatore (S’03–M’07) received theLaurea degree in telecommunication engineering inOctober 2001 and the Ph.D. degree in informationengineering in May 2006 from the Politecnico diMilano, Milan, Italy.
He worked in collaboration with CoreCom, PirelliSubmarine Telecom Systems and Telecom ItaliaLabs, and he visited the Networks Laboratory,University of California at Davis and the CTTClaboratories in Barcelona. He is currently an As-sistant Researcher at the Politecnico di Milano. He
is a co-author of more than 20 papers. His research interests include design,protection strategies, traffic grooming in optical WDM networks, and groupcommunication security.
Guido Maier (S’97–M’99) received the Laurea de-gree in electronic engineering from the Politecnico diMilano, Milan, Italy, in 1995 and the Ph.D. degree intelecommunication engineering from the same uni-versity in 2000.
From 1996 to 2006, he was a Researcher atCoreCom, where he held the position of Head of theOptical Networking Laboratory. Since 2006, he hasbeen an Assistant Professor with the Department ofElectronics and Information, Politecnico di Milano.His main areas of interest are optical network
modeling, design and optimization, ASON/GMPLS architecture, and WDMswitching systems. He is author of more than 30 papers in the area of opticalnetworks published in international journals and conference proceedings. He iscurrently involved in industrial, national and European research projects.
Achille Pattavina (M’85–SM’93) received the de-gree in electronic engineering (Dr. Eng. degree) fromthe University “La Sapienza” of Rome, Italy, in 1977.
He was with the same University until 1991 whenhe moved to the Politecnico di Milano, Milan, Italy,where he is now Full Professor. He is author ofmore than 100 papers in the area of communicationsnetworks published in international journals andconference proceedings. He is the author of the bookSwitching Theory, Architectures and Performancein Broadband ATM Networks (Wiley). His main
research interests are in the areas of optical switching and networking, trafficmodeling, multilayer network design.
Dr. Pattavina has been Editor for Switching Architecture Performance of theIEEE TRANSACTIONS ON COMMUNICATIONS since 1994, and Editor-in-Chief ofthe European Transactions on Telecommunications since 2001. He is a SeniorMember of the IEEE Communications Society.
Andrea Concaro∗, Simone De Patre∗, Guido Maier∗, Massimo Tornatore†∗CoreCom, Via Colombo 81 - 20133 Milan, [email protected]†Department of Electronics and Information, Politecnico di Milano, Via Ponzio 34-35 - 20121 Milan, Italy
Abstract— In this paper1 we study the particular approachto planning the optical transport network under static trafficwhich consists in improving an initial solution by lightpathrerouting. Several different heuristic strategies to carry outthe rerouting process are proposed and described, includ-ing deterministic and stochastic algorithms. A performancecomparison among them is presented, based on case-studyexamples and considering: final fiber-number (cost function)value, execution time and convergence behavior. Results ofall the strategies are also compared with results obtained byinteger linear programming, evaluating of the possibility toheuristically obtain good suboptimal solutions. The two casesof unprotected connections and dedicated path-protection havebeen considered, with and without wavelength conversion.
I. INTRODUCTION
In the last few years an intense research activity has beenaddressed to the problem of optical network dimensioning.Operators are more and more frequently challenged bydesign problems, both to plan new installations and toupdate the existing ones, by the continuous growth of thedemand of bandwidth for new applications such as videoand multimedia streams and advanced digital services. Thisis particularly true for the Metropolitan Area Networks(MANs), where traffic aggregation is low and thus thebandwidth requirements of the single applications have ahigh impact on network performance. On the other hand,planning problems increase in complexity as the topologyof WDM networks evolves from ring to mesh, taking ad-vantage of the constant improvement of optical transmissionand switching technologies. The high connectivity of meshOptical Transport Networks (OTNs), relying upon OpticalCross Connects (OXCs) for switching, improves the band-width provisioning service, but requires careful planningto avoid useless capital expenditure. Planning is furthercomplicated by the need to equip the system with suitableprotection resources. High-speed optical connections (at 10Gbit/s or higher) are very vulnerable to failures: even a few-seconds outage means a huge waste of data. Survivability,that is the capability of the network of maintaining service
1Work partially supported by MIUR, Italy, under FIRB ProjectTANGO and Project WONDER.
continuity in presence of failures, from an attractive researchtopic has became an outstanding important planning issuefor every OTN.
Though dynamic traffic is becoming more and moreimportant and probably will eventually become dominantas the GMPLS/ASON (Generalized Multi-Protocol LabelSwitching, Automatically Switched Optical Network) ar-chitecture spreads pervasively, present optical-network op-erators have still to provide mostly permanent or semi-permanent optical circuits. In our paper we are thus goingto deal with OTN dimensioning in a static-traffic scenario.Given the physical topology and the set of Lightpath Con-nections (LCs) that must be setup (LC-layer topology),network capacity dimensioning and resource allocation aresolved simultaneously, minimizing a chosen cost function.An additional goal is to plan the spare capacity whennetwork survivability against a single link failure has tobe guaranteed. As a possible survivability technique, wehave chosen to refer to Dedicated path-protection (DPP),which is the simplest and fastest survivability scheme, notrequiring reconfiguration of transit OXCs upon a failure.Another advantage of DPP in a context of network planningunder static traffic is that channel assignment is completelyfailure independent.
In order to solve the planning problem outlined above,efficient dimensioning procedures are needed. The problemcan be tackled by using either an exact or a heuristic ap-proach. The former, namely the Integer Linear Programming(ILP), can find the optimal solution, but it can not beapplied to large networks due to its enormous complexity,increasing exponentially with network size. The latter doesnot guarantee the optimality of the obtained solution, but itis generally simpler and faster.
In this paper we wish to focus on the second approach,proposing a set of optimization strategies and comparingtheir performance by applying them to two realistic network-cases (NSFNET and EON). We are also going (when possi-ble) to compare the results of the heuristics to those providedby the ILP solution of the same design experiments. Aswe will summarize later on in the paper, many studieshave been published in the past on heuristic optimization
of the OTN. While most of such studies are focused onsingle specific methods, in this work we are proposing aflexible general scheme which is able to operate accordingto several heuristic methods, selecting one or another withjust few changes in the input-parameter set. This approachallows us to easily compare the most common methods,showing differences and commonalities both in their algo-rithmic structure and in their performance. We also havethe opportunity to observe and compare the computationaltime-complexity of the considered algorithms.
Sec. II describes how OTN is modelled for planning.Previous literature on heuristic optimization is reviewedin Sec. III; the section also defines the ILP formulationsthat are adopted in this paper. Sec. IV introduces to ourplanning method and describes the heuristic strategies wehave compared. Case-study analysis is reported in Sec. V,while the major results of this work are summarized in Sec.VI.
II. NETWORK MODEL
We consider multifiber optical networks. Each link of thenetwork is equipped with two independent sets of mono-directional fibers. Each fiber carries Nλ wavelengths of theWDM multiplex: Nλ is a pre-assigned design parameter,constant for every link of the network, and correspondingto a particular type of WDM transmission system that theoperator has chosen to deploy. The nodes are equipped withOptical Cross Connect (OXC). They are able to switchan incoming optical channel (a wavelength on a fiber)to an outgoing channel. We will consider OXCs with orwithout wavelength converters. In the former case, a nodecan convert the wavelength of an incoming lightpath to adifferent outgoing wavelength; in the latter case, the nodecan not change the wavelength of the incoming lightpath.
The LC-topology is known: connections are static andpoint-to-point (OXC-to-OXC) unidirectional. It is decidedin advance wether the network is unprotected or if allthe optical connections must be protected by DPP. In theunprotected case, a lightpath, i.e. a sequence of WDMchannels along a path on the physical topology, must beallocated for each connection request. If all the nodes ofthe network are equipped with wavelength converters, weare considering a VWP (Virtual Wavelength Path) network.If no node is equipped with wavelength converters, we areconsidering a WP (Wavelength Path) network. In the firstcase wavelengths can be assigned to a lightpath link-by-link, with possible intermediate wavelength changes; in thesecond case we must assign a single wavelength to the entirelightpath (i.e. the lightpath must satisfy the wavelengthcontinuity constraint). When DPP is adopted, two lightpathsmust be set up per connection-request, composing a workingand a protection pair (w/p pair): we have to route the w/ppair imposing that the two lightpaths must be link-disjoint
(they can not share a common physical link). In this way, incase of a (single) link failure, it will always be possible toreroute the interrupted working traffic on the spare capacity.
The variables of our model are the capacities of eachphysical link in terms of number of fibers and the costfunction is the total number of fibers F that will be deployedto satisfy all the requests of the LC topology (with orwithout protection). We are thus seeking the Routing andFiber and Wavelength Assignment (RFWA) solution for allthe lightpaths (or the w/p pairs) resulting in the smallestpossible F . The optimum RFWA is such that F is necessaryand sufficient to setup the LC-topology. With heuristicapproaches we are able to find RFWA solutions leading tovalues of F that are sufficient but not necessary, thoughhopefully not too much greater than the optimum F .
III. OTN OPTIMIZATION IN LITERATURE
As previously mentioned, this paper is mainly dedicatedto OTN heuristic design. We will however present also someILP-based results to benchmark the heuristic algorithms.Therefore in this section we are going to provide the“standard” ILP flow formulations for the unprotected andDPP scenarios, summarizing the ILP models presented inRefs. [8], [19], [24] to solve capacitated design in WDMnetworks. We will report only the case with full wavelengthconversion: the extension of the models to WP can be carriedon as explained in Refs. [22], [24], introducing a furtherterm of complexity, function of Nλ.
Let us consider the physical topology, modeled by thegraph G = G(N ,A) 2. Physical links are represented bythe undirected edges l ∈ A with |A| = L, while thenodes i ∈ N = {1, 2, ...N}, with |N | = N , representthe OXCs. Each link is equipped with a certain amountof unidirectional fibers in each of the two directions; fiberdirection is conventionally identified by the binary variable k(k = 0 for forward direction, k = 1 for backward direction).vc is the number of requested LCs having sc as source anddc as destination node, with c an index used to identify eachsource-destination node-couple requiring connectivity.
The (integer) variables involved in the unprotected flowformulation are:
• xl,k,c, flow variable indicating whether a WDM chan-nel on link l on a fiber having direction k has beenallocated to one of the connections requested by nodecouple c;
• Fl,k, capacity variable indicating the number of fiberson link l in direction k.
The following additional symbols are also defined:
• (l, k) identifies a “unidirectional link”, i.e. the set offibers of link l that are directed as indicated by k;
2The following formulations require that the topology is at least2-connected.
142
• I+
i is the set of “unidirectional links” having node i asone extreme and leaving the node; analogously, I−i isthe set of “unidirectional links” having node i as a oneextreme and pointing towards the node.
The cost function to be minimized is the total fiber number
min F = min∑(l,k)
Fl,k
The unprotected flow formulation is given by the followingset of equations
∑
(l,k)∈I+i
xl,k,c −∑
(l,k)∈I−i
xl,k,c =
⎧⎨⎩
vc if i = dc
−vc if i = sc
0 otherwise∀ i, c;
(1)
∑c
xi,k,c ≤ Wl · Fl,k ∀ (l, k); (2)
(3)
xl,k,c integer ∀ (l, k), c; (4)
Fl,k integer ∀ (l, k); (5)
Constraint (1) is a solenoidality constraint. Let us considerthe vc connections requested by c: the flow conservationcondition for vc in each node i states that vc leaving i mustbe equal to vc incident on i. In the source (destination) nodethe flow balance is satisfied by adapting the constraint to theborder condition: the total leaving (incoming) flow must beequal to vc. Constraint (2) ensures that the total number ofWDM channels allocated to spare and working lightpaths onthe unidirectional link (l, k) is bounded by the link capacity,given by the number of fibers Fl,k multiplied by the numberof wavelengths W . Constraints 4 and 5 enforce the integrityof the fiber number and flow unities.
The definition of an ILP model in a WDM network withdedicated path protection is a well-known problem: to theset of constraints of the unprotected formulation, we mustadd constraints deriving from the link disjointness condition.These can be easily set, provided that the basic flow variableis enriched with a new index, increasing the descriptiondetail of the flows. If vc > 1, we add an auxiliary index thaving values between 1 and vc; the flow variables become:
• xl,k,c,t, boolean variable indicating whether a WDMchannel on link l on a fiber having direction k hasbeen allocated to the t-th connection requested by nodecouple c.
We further introduce the following symbol:
• (c, t), identifying a single connection request.
The set of constraints is the following∑
(l,k)∈I+i
xl,k,c,t −∑
(l,k)∈I−i
xl,k,c,t =
⎧⎨⎩
2 if i = sc
−2 if i = dc
0 otherwise∀ i, (c, t); (6)
∑(c,t)
xl,k,c,t ≤ W · Fl,k ∀ (l, k); (7)
∑k
xl,k,c,t ≤ 1 ∀ l, (c, t); (8)
xl,k,c,t binary ∀ (l, k), (c, t); (9)
Fl,k integer ∀ (l, k); (10)
Constraint (6) enforces the flow solenoidality, as in theunprotected case. A slight difference exists in the source(destination) node of the connection request (c, t), in whichthe outgoing (incoming) flow must be equal to 2. This is dueto the fact that a w/p pair, instead of a single lightpath, isassociated to the connection request (note that the distinctionbetween working and protection lightpaths is irrelevant).Constraints concerning dimensioning (7) are simple exten-sions of the corresponding constraints in the unprotectedcase. Constraint (8) stems from link-disjointness condition:no more than one lightpath associated to connection request(c, t) can exist on the same link, in both the oppositedirections.
Solving the above set of equations for realistic networks isreally difficult, since the number of variables and constraintstends to explode with the network size (OTN planning withcapacitation is well-known to be a NP-hard problem). Inorder to keep computational time and memory occupation atreasonable levels we have actually obtained the results thatwill be presented in Sec. V by exploiting slightly different,but more efficient formulations compared to the “standard”flow formulation presented above. Such formulations, whosedetailed description is outside the scope of this paper, arereported in Refs. [22], [23].
We will concentrate in the rest of the paper on the heuris-tic approach to OTN design, as an alternative to the ILP. Weare going to consider some heuristic optimization strategies,all based on the concept of lightpath rerouting: given aninitial RFWA solution (i.e. a feasible accommodation of allthe requested lightpaths on the physical topology) we try toimprove it by rerouting the optical connections on alternativepaths, subject to the constraint that all the LC-requests mustbe satisfied. This kind of approach to the problem, despitebeing a very simple and direct solution method, can achievegood results, as we are going to verify by case-studies.
We can find in literature a lot of examples of heuristiclightpath-rerouting. In [24] the total number of used wave-lengths is minimized by rerouting lightpaths that crosses
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the most-loaded links, while in [3] the connections, initiallyrouted on shortest paths, are rerouted on alternative shortestpaths, if this operation can decrease the load on the mostcongested link. In [1], where multifiber networks are consid-ered, new configurations are iteratively searched by modify-ing the initial routing and/or wavelength assignment, tryingto lower the cost of requested fibers. Ref. [17] considers anSDH-over-WDM network: the rerouting method is utilizedto find a good LC-topology for the WDM layer, given therequests for SDH circuits. The SDH connection-requests arerouted on a sequence of lightpaths (multi-hop routing) tryingto minimize the number of LC-requests. In order to achievethis purpose, the optimization algorithm tries to reroute theSDH channels initially carried by a lightpath on the residualcapacity of other lightpaths.
All the above-cited techniques can be classified as de-terministic heuristics, since rerouting attempts are carriedout according to a fixed lightpath-sorting rule, acceptingrerouting each time it leads to an improvement of thecost function. Due to this feature, deterministic methodsare affected by the problem of getting trapped in localminima, which can not be overcome with a “down-slope”-only decision-rule.
Other heuristic rerouting approaches do exist, which canbe classified as stochastic and are able to avoid local-minimatrapping. Simulated annealing [15] is a well-known generaloptimization method which stochastically simulates the slowcooling process of a physical system. The temperature of thesystem is lowered by small steps until the system “freezes”and no further changes occur. To apply simulated annealing,the system is initialized with a particular configuration. Anew configuration is constructed by imposing a randomdisplacement. If the energy (the cost function) of this newstate is lower than that of the previous one, the changeis accepted unconditionally and the system is updated. Ifthe energy is greater, the new configuration is acceptedprobabilistically. This procedure allows the system to moveconsistently towards lower energy states, yet still “jumping”out of local minima due to the probabilistic acceptance ofsome upward moves. In [21] simulated annealing is appliedto OTN with the purpose of mapping a regular virtualtopology on a given physical topology. In [20] simulatedannealing is utilized to find a good virtual topology, andthen a flow-deviation algorithm optimizes traffic routing.Simulated annealing and other stochastic algorithms (ran-dom sampling, local search, threshold accepting, tabu searchand genetic algorithms) are applied in [10] to ATM networkdimensioning. Other stochastic algorithms have been appliedto optical networks, such as Monte Carlo [7] and geneticalgorithms [2], [14]. Stochastic and deterministic heuristicapproaches to OTN planning are compared in Refs. [2],[7], proving how a random component can often be usefulto obtain good suboptimal results. Similar comparisons are
presented in [16], where the objective of optimization isregular topology for packet-switched optical networks.
IV. HEURISTIC PLANNING METHOD
The heuristic approach to static-OTN design we havedeveloped is based on the rerouting concept. As previouslymentioned, it is divided into two steps:
Step 1 feasible RFWA-solution evaluation;Step 2 improvement by rerouting.
In the whole design procedure the network state has beenrepresented by a Multifiber Layered Graph (MLG). This isderived from the layered graph, introduced for mono-fibernetworks [9], and extended in [18] to multifiber networks.
A. Step one: greedy resource allocation
Let us provide a synthesis overview of the first step, whichis not the actual objective of this paper. A feasible RFWAsolution is obtained with the following technique. Startingfrom the idle physical topology, all the connection requestsof the LC-layer topology are set up in sequence one afterthe other until all have been satisfied. Each link initiallycontains a number of fibers so large to be considered infinite:in this way the existence of a solution is guaranteed. Theconnection requests of the LC-layer topology are initiallysorted according to a “balanced” sorting rule: node-pairswith greatest topological distance and largest amount ofconnection-requests are served with the highest priority. Thetechnique is greedy, as each requests is satisfied regardlessof all the others, allocating resources to a lightpath (unpro-tected case) or a w/p pair (DPP case).
In the unprotected case, RWFA is carried out on a singlelightpath by adopting the prioritized multi-criteria approach,extensively described in [11]. The set of criteria adoptedin this work was identified in [11] as the one leading inmost of the cases to the best greedy allocation (with thelowest number of fibers). It comprises, with decreasingpriority: “Shortest Path Routing” (SPR), “First Fit” fiberselection (FFF), “First Fit” wavelength assignment (FFW),“Least-Loaded Routing” (LLR) [4], [18]. The length metricadopted for routing is the number (minimum hop - mH)of the crossed links. With the “First Fit” criterion, fibers(wavelengths) are sorted in the same way in all the links(fibers) of the network and the first available is alwayschosen according to this sorting. To implement the aboveheuristic RFWA criteria, suitable weights are assigned to theMLG arcs and the Dijkstra algorithm is used to find the routeconnecting the source to the destination on the MLG withthe least total weight. We shall note that RFWA is performedin an unconstrained mode, that is all the possible routes onthe MLG connecting source to destination are scanned insetting up a new lightpath.
When DPP must be supported, the RFWA of the workinglightpath is coupled to the RFWA of the protection lightpath
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of the same w/p pair by the route-diversity constraint. Thereare two main techniques to find two link-disjoint pathsconnecting two nodes of a mesh topology. The simplestone is called two-step search. The shortest path is found(e.g. applying the Dijkstra algorithm) and it is allocated tothe working lightpath. Then the shortest-path algorithm isrun again to route the protection lightpath, which will beassigned, in this way, the second-shortest path. The two-stepapproach (greedy, because of the sequential computation ofthe two paths) in some special cases [12] may fail to find aw/p pair solution even if such solution actually exists. Bhan-dari [5], [6] proposed to overcome such a limitation by aone-step search, in which the two link-disjoint paths are notrouted separately, but they are jointly routed by performinga suitable algorithm (the modified Dijkstra algorithm) in adirected graph. Besides being able to solve trap-networks,it also finds the actual minimum-length cycle connectingtwo nodes. We applied the one-step technique in the DPPcase by adapting the Bhandari algorithm to the MLG. Inparticular, a set of rules has been added to the modifiedDijkstra algorithm to control the edge inversion operation[6] in the MLG environment with and without wavelengthconversion and to support LLR.
B. Step two: rerouting procedures
We are now going to discuss more in depth the secondplanning step. The basic purpose of this step is to modify thenon-optimized network solution found by greedy resourceallocation. The chosen cost function, which in our caseis the total number of fibers, is decreased by reroutingsome connections, under the constraint of preserving con-nectivity of the LC-layer topology. Qualitatively, some kindof iteration selects one specific fiber of the network at atime. Lightpaths or w/p pairs crossing the selected fiber aretentatively rerouted on the other fibers of the network. Iffree resources are enough to allow rerouting, the selectedfiber is removed. Otherwise, everything remains unchanged.Iteration is used to reach at least a local minimum of the costfunction, i.e. such that further improvement are impossibleby applying the same heuristic procedure.
The order by which fibers are selected and the itera-tion control depends on the particular optimization strategyadopted to solve the problem. In this work we have testeddifferent heuristic strategies, belonging both to the determin-istic and the stochastic class, which we are going to presentin details. Since most strategies have been implementedas variations of a common procedure, it is convenient todescribe them starting from their common elements. Let usdefine some basic actions that will be used many times inthe procedures. In the following, we consider a fiber f onwhich a set Sf of lf lightpaths (which can be working orprotection in the DPP case) are routed.
• FTr - Temporary reroute f . Let us consider the unpro-tected case. The following steps are performed
1) The current RFWA of all the lf lightpaths isstored in a separate database
2) Each lightpath of Sf is deallocated by freeing allthe MLG arcs it occupies
3) Fiber f is disabled by preventing any subsequentoccupation of the MLG arcs belonging to it (e.g.setting their weights to infinite)
4) A new RFWA on the residual MLG is attemptedfor each connection request having a lightpathbelonging to Sf
In the DPP case, the above steps may be described inthe same way, only considering in steps 1 and 2, insteadof lightpaths, the w/p pairs with a lightpath crossing f .
• FBl - Block f . This conditional function returns NO ifall the connection requests affected by a FTr functioncould be routed successfully on the residual MLG andYES otherwise.
• FRe - Remove f . All the MLG arcs belonging to f arepermanently disabled, and f is removed from the listof the existing fibers. Moreover, a register SCS is setto TRUE, indicating that at least an FTr function hassuccessfully terminated.
• FRs - Restore f . The RFWAs saved in step 1 of anFTr is restored on the MLG and arcs belonging to fare re-enabled for future usage.
Fig. 1 shows how the above functions are combined tocompose the core procedure. At the beginning of suchprocedure all the existing fibers are numbered from 1 to F ∗.f is used as a local index to scan the existing-fiber set. Eachfiber is processed by FTr and the subsequent functions FBl,FRe and FRs, provided that two conditional functions Yand X result to be TRUE. The definitions of such functionsdepend upon the specific heuristic strategy and will be givenshortly below.
Our spectrum of optimization strategies covers the follow-ing alternatives: Idle-Busy (IB), Busy-Idle (BI), RandomOrder (RO), Random Polarized (RP), Shortest Path (SP),Simulated Annealing (SA).
The first four strategies have similar structures, which areall together represented by the flow-chart of Fig. 2. In Startthe network solution found by the greedy planning step isconsidered. The first action performed is the removal ofany residual empty fiber, by executing FRe on all the fibershaving of = 0, where of is the number of WDM channelsof fiber f allocated to working or spare lightpaths. After thisaction, common to all the strategies, the procedure branches.One of the alternative branches is executed, according tothe optimization strategy chosen for the particular planningexperiment. In all the cases, residual empty fibers are againremoved at the end; then, the result in terms of RFWA forall the lightpaths and network dimensioning is returned in
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f = f + 1f > F*
Label fibersfrom 1 to F* Yf = 1 X(of,k) Temporary
reroute f
Block f
Remove f
Restore f
TRUE
FALSE YES
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Fig. 1. Flow chart of the core procedure.
FALSE
k = 0
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i = 0
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i = N - 1
k = k + 1
k = V(i)
TRUETRUE
FALSE
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IB BI RP
RO
TRUE TRUE
SP SA
Fig. 2. General flow chart with the various heuristic strategies (IB, BI.RO and RP detailed here).
Stop.The IB strategy [18] is deterministic. Fibers are selected
for rerouting from the idlest to the busiest, beginning fromthose carrying only one lightpath and ending with thosewith just one WDM channel free. In the core procedure,the conditional function X(of , k) is: of ≤ k ∧ of > 0(Y , not needed, is set to the constant TRUE). The strategyhas the advantage of trying to eliminate first fibers that areeasy to free, since FTr initially involves few connections.Beside that, saturation of the free capacity due to successfulrerouting is gradually distributed in the k cycle, avoidinghigh blocking probability at the beginning. A fiber hasthe chance of being selected multiple times with differentvalues of k. In fact, for each k increment, non-empty fibershaving up to k (and not simply k) busy channels areconsidered. There are good chances that at the end of thek-cycle a suboptimal solution has been reached. Note thatX(of , k) prevents empty fibers from being selected: thisfurther condition leaves more spare capacity for reroutingduring the k-cycle, thus reducing blocking probability.
The BI strategy, again deterministic, is almost the dualof the previous one: fiber selection-order is from the most
to the least loaded. In the core procedure, the conditionalfunction X(of , k) is of = k (Y is set to TRUE): onlyfibers having exactly k busy channels are considered foreach k increment. In order to compensate this restriction,the entire k cycle is repeated several times until FTr is nomore possible for any fiber (FBl always detects a block): theoptimization terminates when the register SCS is detectedto remain FALSE after the end of the inner cycle. Therationale of BI is that deallocation of very loaded fibers isattempted at the beginning, when chances of blocking arelow due to a relatively high amount of unused capacity.
The first stochastic strategy we have considered is theRO: fibers are considered for selection in a random order ofoccupation. At the beginning of the procedure, the functionRandom V randomly inserts the integer numbers from 1 toNλ − 1 into the (Nλ − 1)-element vector V. The vector isthen scanned by the index i, selecting a value k = V(i)per iteration. Fibers are selected according to the conditionX(of , k): of = k (Y is set to TRUE). As in the BI strategy,the entire k cycle is repeated many times until FTr is nomore possible for any fiber (exploiting the check on SCSas termination condition).
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The other stochastic strategy detailed in Fig. 2 is RP.Fiber selection occurs exactly as in IB, setting X(of , k) to:of ≤ k∧of > 0. The selected fiber is however processed bythe core procedure or skipped according to a random choice.Eval p(k) at the beginning of the k cycle sets a probabilitythreshold p(k) given by the expression
p(k) = p0 + (1 − p0)k − 1
Nλ − 2where p0 (0 < p0 < 1) is a preset parameter. Inside the coreprocedure, the Y condition is defined as follows:Y - x < p(k), where x a random variable uniformly
distributed between 0 and 1.The strategy is random and polarized since the thresholdp(k) increases with k, making the probability of processingp(k) low at the beginning of the k-cycle and steadilyincreasing with k up to 1 when k = Nλ−1. The effect is thatthe deterministic order of fiber processing is perturbed by arandom fluctuation with an amplitude high for idle and lowfor busy fibers. This strategy can be regarded as an hybridbetween the deterministic IB and the stochastic approach.It should be noted that RP has also common aspects withsimulated annealing, since p(k) plays the role of a systemtemperature: however, it should be actually considered a“simulated melting”, as the temperature increases.
Another deterministic strategy is SP. Its flow-chart is notworth to be represented, since it is a mere iteration of IB.The only change to IB is a modification of the FBl conditionadding a constraint on the length of the rerouted lightpaths.Let us suppose that c is a connection of the set of thosethat must be tentatively rerouted. The topological distance(the length of the shortest path, in hop) between source anddestination is Hc, while the minimum-length cycle betweenthe two nodes has length (in hop) Cc. If n is the IB-iterationcounter, going from 1 to nmax +1, in the unprotected case,a possible newly-rerouted lightpath is constrained to have alength (in hops) less than or equal to Hc + n − 1, while inthe DPP case, the sum of the lengths (in hops) of the newly-rerouted working and protection lightpaths is constrained tobe less or equal to Cc + n − 1. The conditional functionFBl in the SP strategy returns NO if all the connectionrequests affected by a FTr function could be carried outsuccessfully on the residual MLG, subject to the abovelength constraints. Otherwise, YES is the result. The use ofn allows to progressively relax the constraint up to a prefixedvalue nmax, while controlling the number of iterations. TheSP technique has the advantage of limiting the amountof resources that can be used by lightpath rerouting, thusslowing the saturation the unused capacity of the network.
The SA strategy is the application of the well-knownSimulated Annealing method to our planning problem. Sincea flow-chart would be too complex to represent, we willexplain the SA strategy in words. In SA, differently fromall the other strategies, fibers can be not only removed but
also added to the network: this gives the chance of escapingfrom local minima. The control temperature of the annealingprocess in our case is represented by the probability p ofaccepting fiber addition. p is initially set to a high value p0
(0 ≤ p0 ≤ 1). Then a fiber-processing cycle begins. NF
fibers are processed in each iteration, as described below.At the end of an iteration, if the current value of p is lowerthan a prefixed threshold pTh, the SA phase terminates; else,the new current value of p is set to p · ∆ and the cycleiteration is repeated again, processing NF fibers. ∆ is thepreset cooling-rate annealing parameter.
The NF fibers processed in a cycle iteration are randomlychosen. Let us describe what happens when a given f fiberhas been selected. First of all, the current network stateS is saved and the current fiber number F ∗ is computed.Then, FTr is performed on f and the FBl condition isevaluated. If no block is detected, all the empty fibers ofthe network (including f ) are permanently removed and thenetwork state consequently updated. Processing begins againby randomly choosing a new fiber. If block is detected, arandom choice is taken. With probability 1 − p, f is leftin place and no further action is taken: S is restored and anew fiber is selected. With probability p, instead, a fiber-addition procedure is performed. This latter comprises thefollowing steps. First, one fiber is added to any link ofthe network. Then FTr is performed on f again: this timethere can obviously be no block. After lightpath have beenrerouted, all the empty fibers of the network (including f )are removed and the total number of fiber F ∗∗ is evaluatedagain. If F ∗∗−F ∗ > FT , being FT another preset annealingparameter, fiber addition is considered unacceptably large: Sis restored, returning to the network state as it was before thebeginning of processing of f , and processing of a new fiberbegins. Else, the network state is updated by permanentlyaccepting the added fibers and the newly-routed lightpathsbefore beginning to process another fiber.
V. RESULTS
We are now going to compare the various heuristicplanning procedures and ILP optimization on the basisof numerical results. We have considered two well-knownrealistic networks for case-study: the National Science Foun-dation Network (NSFNET, 14 nodes and 44 links) andthe European Optical Network (EON, 19 nodes and 78links). Their physical topology is shown in Fig. 3: as itclearly appears, EON is much more densely-connected thanNSFNET. The LC-layer topologies used for the planningexperiments have been derived from the static (symmetric)traffic matrices based on real traffic measurements which arereported in Refs. [13], [19]. The two LC-layer topologiescomprise 360 and 1380 unidirectional connection requestsfor NSFNET and EON, respectively.
Stochastic strategies benefit from the advantage over thedeterministic approach that re-executing the dimensioning
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(a)
(b)
Fig. 3. Physical topologies of NSFNET (a) and EON (b) networks.
TABLE IFINAL FIBER NUMBER OBTAINED BY PLANNING NSFNET IN
THE UNPROTECTED CASE.
Nλ IB BI RO RP SP SA ILP2 380 380 380 380 380 380 3744 199 198 198 200 199 197 188-192
procedure different results may be obtained. We have ex-ploited this property for NSFNET: being this network lessconnected than EON, the repetition of dimensioning doesnot take too much computational time. Therefore, resultsconcerning RO, RP and SA have been obtained for theNSFNET by repeating the second planning step twice andconsidering the best result. The following values have beenused for strategies requiring preset parameters (see Sec. IV-B): p0 = 0.2 for RP; nmax = 6 for SP; p0 = .5, ∆ = 0.95,pTh = 0.01 and NF = 10 for SA.
A detailed list of the results is displayed for the NSFNETonly by the two tables I and II. Results are given in terms
TABLE IIFINAL FIBER NUMBER OBTAINED BY PLANNING NSFNET IN
THE DPP CASE.
Nλ IB BI RO RP SP SA ILP2 995 995 995 995 995 994 9834 505 507 505 505 506 505 492
of values of the cost function, which is the total numberof fibers F deployed in the dimensioned networks, andthey have been grouped by survivability feature (unprotectedconnections or DPP required for each connection). Insideeach table, the WP and VWP cases are displayed, providinga row for each value of the number of wavelengths perfiber Nλ. A bold number in a column indicates that thecorresponding strategy reached the best result among all theconsidered heuristic optimization strategies. A dash tells thatthe corresponding heuristic strategy could not be applied dueto memory (900 Mbyte) exhaustion.
ILP optimization has been carried out by exploiting thecommercial software CPLEX, implementing the branch-and-bound algorithm. When a single value in normal typeappears in the ILP column, the optimization ended, findingthe optimum. In all the other cases, optimization was in-terrupted for memory exhaustion or after 3 days. In someof these cases, a single number in italic indicates that theonly solution obtained was achieved relaxing all the integrityconstraints on the link capacity. In some other cases, twovalues are reported: the one in italic is the lower boundestimated by CPLEX, while the one in normal type is thebest non-relaxed solution found, not guaranteed to be theactual optimum. A dash with no numbers tells that CPLEXwas not even able to setup the optimization session.
Fig. 4(a) gives a synthetic overview of the strategycomparison. Each column, referring to a particular heuristicstrategy, is subdivided in four rectangles. The hight of a rect-angle referring to NSFNET is obtained by counting in tableI the number of occurrences of a bold entry in the column ofthe strategy (similarly for rectangles referring to EON). Thetotal hight of the bar is the total number of times the strategywas able to find the best heuristic F in the experimentscarried out on both the networks. As expected, SA is onaverage the most reliable strategy. It is remarkable, however,that it has been overtaken by a deterministic strategy in some
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of the experiments. The best deterministic strategy is IB,followed by SP (which overtakes IB in the DPP-EON case).The other two stochastic strategies, RO and RP, are not asgood as the best deterministic competitors: the comparisonbetween these two proves that a deterministic random-choicepolarization (in favor of less loaded fibers) is an advantage.The BI approach resulted to be inappropriate in all the cases:clearly, too many lightpaths rerouted at the beginning ofthe cycle too quickly saturate the unused network capacity,preventing further rerouting. It should be noted that BIfiber selection is worse than random selection (RO). Theimprovement due to repetition for the stochastic strategiesclearly appears by comparing the rectangles correspondingto the NSFNET and the EON. The lack of repetition makesstochastic strategies less performing than deterministic (e.g.SA vs. SP in the DPP-EON case). It can happen howeverthat non-repeated SA is nevertheless the best strategy (e.g.in the unprotected-EON VWP case). Performance relationsremain similar to those shown in the figure, if we count thenumber of times each strategy achieved the worst (insteadof the best) result, with the only exception that IB comesout to be slightly worse than SP.
Another synthetic view of the gathered data is given byFig. 4(b), in which we show the differences between theheuristic approaches and the ILP method, in terms of F andfor different values of Nλ. We have plotted the DPP-NSFVWP case, since a complete set of ILP results was available.∆min (∆max) is the difference (in absolute terms) betweenthe best (worst) heuristic F and F found by ILP. When twovalues appear in the ILP column, the non-relaxed result isemployed. The differences are also plot in percent, usingthe ILP value of F for normalization. The best heuristicsuboptima are never more than 11 fibers greater than theILP result; the distance between the best and worst heuristicis also no greater than 3 fibers. The decreasing behavior of∆min and ∆max seems not to be very meaningful, sinceit is not confirmed by the other experiments concerningEON and/or different protection and conversion conditions.The increasing trend of %∆min and %∆max is due tothe descent of the optimum F , which is roughly inverselyproportional to Nλ (constant offered traffic). Averaging onall the experiments, including EON (considering only casesin which ILP returns an integer solution), we have: ∆min =8.2 and ∆max = 11. Moreover, if the two best heuristicstrategies (SA and IB) are used, the maximum differencefrom ILP is 16%, while the mean difference is 5%: thesedata prove the validity of heuristic optimization as a practicaldesign procedure for OTN. It should further noted that insome unprotected cases (WP NSFNET with Nλ = 32, WPEON with Nλ = 8, VWP EON with Nλ = 64), ILP wasable only to find a worse integer solution than heuristics(these cases have been obviously excluded from the ILP-heuristic comparison).
IB BI RO RP SP SA0
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Fig. 4. (a) Number of times each strategy gives the best result.(b) Comparison between ILP and heuristic solutions.
The comparison between the heuristic strategies presentedabove can be completed by considering their computingtimes and their convergence behaviors. Fig. 5 displays theexecution times of the second planning phase (heuristic op-timization) for NSFNET, VWP, in the unprotected and DPPcases (all the strategies are run on a 1-GHz-clock computer).For all the strategies, there is a strong dependence of theexecution time on Nλ. By curve-fitting, we have verifiedthat the execution time is O[N2
λ] for unprotected and O[N3λ]
for DPP. The difference is probably due to the complexity ofthe routing algorithms in the two cases when implementedon the MLG. Execution time in the VWP case resulted tobe higher than in the WP case for all the strategies and forany value of Nλ, probably due to the increased complexityof the MLG in the VWP case, in which the vertical arcsrepresenting wavelength conversions must be added to thegraph. A quantitative evaluation of complexity is howeverstill under study and will probably give an explanation tothese differences. A vertical comparison between the curvesof the graphs points out the differences in computation timebetween the various strategies. Strategies having of ≤ k in
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Fig. 5. Comparison between execution times in NSFNET VWPunprotected (a) and with DPP (b).
the X condition (IB, SP and RP) have a higher executiontime than the others (except SA), but achieve better results,showing the trade-off between accuracy and computationalcomplexity. SA displays times similar to those of IB andRP: this is a consequence of the choice of the values givento the SA parameters (p0, ∆, pTh and NF ), taken preciselywith the criterion of having execution times of the sameorder of magnitude for all the strategies. Finally, we shallpoint out that SP computation time is roughly 6 times thatof IB, according to the assignment: n = 6. Thus, SP andIB accuracy is similar, but the former requires much highertime than the latter.
In Fig. 6 we show how the strategies convergence to theirfinal solutions during their execution-time, considering asexample the optimization of the NSFNET in the DPP andVWP case, with Nλ = 8. The horizontal coordinate has beennormalized to the total execution duration of each strategy.The fastest converging strategy, IB, does most of the workat the beginning, when reallocating low-loaded fibers iseasier; at the opposite, we have BI technique, which ismore successful in deallocating fibers after some iterations.
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Fig. 6. Convergence behavior of (a) IB, RO and RP and (b) SAin DPP-NSF (VWP case, Nλ = 8).
RP follows the behavior of IB but more slowly, due to therandomization of the reallocation acceptance decisions. SAwas plotted separately in Fig. 6(b): its irregular trace resultsfrom the ability of adding fibers. It should be noted that SAhas hit the best already in the middle of the execution, butit jumps out because a still high temperature, to return to itat the end of the process.
VI. CONCLUSION
For OTN, the possibility of obtaining an exact solution tothe NP-hard problem of RFWA of static traffic demands withcapacity dimensioning is severely limited by computationalcomplexity. As we have shown, ILP applied to realisticnetwork examples and solved with standard computingequipment in many cases does not provide any results or isnot able to guarantee optimality: problems are exacerbatedby wavelength continuity (WP cases), by dedicated pathprotection and by high values of Nλ.
The heuristic approach proves to be an acceptably reliablealternative when ILP does not make it. We have proposeda heuristic method based on finding a first greedy but
150
feasible solution which is subsequently improved by light-path rerouting. We have presented six heuristic strategiesto perform this task, the most accurate ones of which areable to contain the distance to the ILP solution below the16% of this solution itself (5% on average). Result-analysisshows that the best compromise between computationalcomplexity and reliability is achieved with the deterministicIB and the stochastic SA strategies. IB is however simpler toimplement, it does not require special procedural parametersto be preset and it usually converges rapidly to the finalsolution. Stochastic heuristics gain a clear advantage overdeterministic ones only when there is the possibility, as inSA, of adding fibers beside pruning. In practice, however,the problem of local-minima trapping is not so important,and it can be conjectured that SA would be able to solve itin most of the cases only with execution times much greaterthan those of IB. Finally, we have shown that in same casesconstrained rerouting (SP strategy) can improve IB accuracyat the cost of an execution-time extension.
As a final remark we shall point out that a theoretical anal-ysis of complexity of the compared strategies is currentlyunder study.
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[2] M. Ali, B. Ramamurthy, and J. S. Deogun. Routing andWavelength Assignment (RWA) with Power ConsiderationsIn All-Optical Wavelength-Routed Networks. In GlobalTelecommunications Conference - Globecom’99, pages 1433–1437, 1999.
[3] S. Baroni and P. Bayvel. Wavelength Requirements inArbitrarily Connected Wavelength-Routed Optical Networks.IEEE Journal of Lightwave Technology, 15(2):242–251, Feb.1997.
[4] S. Baroni, P. Bayvel, R. J. Gibbens, and S. K. Korotky.Analysis and design of resilient multifiber wavelength-routedoptical transport networks. IEEE Journal of LightwaveTechnology, 17:743–758, May 1999.
[5] R. Bhandari. Optimal Physical Diversity Algorithms andSurvivable Networks. In Computers and Communications,1997. Second IEEE Symposium on, pages 433–441, 1997.
[6] R. Bhandari. Survivable networks, algorithms for diverserouting. Kluwer Academic Publishers, 1999.
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[8] B. V. Caenegem, W. V. Parys, F. D. Turck, and P. M.Deemester. Dimensioning of survivable WDM networks.IEEE Journal on Selected Areas in Communications, pages1146–1157, Sept 1998.
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[12] D. A. Dunn, W. D. Grover, and M. H. Gregor. Comparisonof k-shortest paths and maximum flow routing for networkfavility restoration. IEEE Journal on Selected Areas inCommunications, pages 88–89, 1994.
[13] A. Fumagalli, I. Cerutti, M. Tacca, F. Masetti, R. Jagannathan,and S. Alagar. Survivable Networks Based on OptimalRouting and WDM Self-Healing Rings. In Proceedings, IEEEINFOCOM ’99, 1999.
[14] R. Inkret, B. Mikac, and I. Podnar. A Heuristic Approach toWavelength Assignment in All-Optical Networks. In Mediter-ranean Electrotechnical Conference, 1998. MELECON 98,volume 2, pages 759 –763, 1998.
[15] S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi. Optimisa-tion by Simulated Annealing. Science, 220(4598):671–680,13 May 1983.
[16] O. Komolafe, D. Harle, and D. Cotter. Next generationoptical network design and modelling, chapter : “A studyon the efficacy of regular virtual topology design heuristicsfor optical packet switching”. (A. Bianco and F. Neri editors)Kluwer Academic Publisher, 2003.
[17] V. R. Konda and T. Y. Chow. Algorithm for Traffic Groomingin Optical Networks to Minimize the Number of Transceivers.In IEEE Workshop on High Performance Switching andRouting, pages 218–221, 2001.
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122 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 1, JANUARY 2002
Traffic Grooming in an Optical WDM Mesh NetworkKeyao Zhu, Student Member, IEEE,and Biswanath Mukherjee, Member, IEEE
Abstract—In wavelength-division multiplexing (WDM) opticalnetworks, the bandwidth request of a traffic stream can be muchlower than the capacity of a lightpath. Efficiently groominglow-speed connections onto high-capacity lightpaths will im-prove the network throughput and reduce the network cost. InWDM/SONET ring networks, it has been shown in the opticalnetwork literature that by carefully grooming the low-speedconnection and using wavelength-division multiplexer (OADM)to perform the optical bypass at intermediate nodes, electronicADMs can be saved and network cost will be reduced. In this study,we investigate the traffic-grooming problem in a WDM-basedoptical mesh topology network. Our objective is to improve thenetwork throughput. We study the node architecture for a WDMmesh network with traffic-grooming capability. A mathematicalformulation of the traffic-grooming problem is presented in thisstudy and several fast heuristics are also proposed and evaluated.
Index Terms—Integer linear program, lightpath, mesh net-work, optical network, traffic grooming, wavelength-divisionmultiplexing.
I. INTRODUCTION
F IBER OPTICS and wavelength-division multiplexing(WDM) are promising technologies that are expected
to satisfy the drastically increasing bandwidth requirementsof the Internet. WDM is an approach that can exploit thehuge optoelectronic bandwidth mismatch by requiring thateach end-user’s equipment operate only at electronic rate, butmultiple WDM channels from different end-users may be mul-tiplexed on the same fiber [1]. In a wavelength-routed WDMnetwork, a “lightpath” may be established from a source nodeto a destination node and it may span multiple fiber links [2].In an all-optical network, the lightpath may remain entirely inthe optical domain, optically bypassing the intermediate nodes.Using wavelength-routing switches (WRSs) [1] at intermediatenodes, and via appropriate routing and wavelength assignment(RWA), a lightpath can create logical (or virtual) neighbors outof nodes that are geographically far apart in the network.
Assigning network resources (e.g., wavelengths, trans-ceivers) to successfully carry the connection requests (light-paths) is well known as the routing and wavelength assignmentproblem [1], [3]. It is also known as a lightpath-provisioningproblem [4]. A number of RWA studies have been reported inthe optical networking literature, either based on static traffic
Manuscript received February 15, 2001; revised July 31, 2001. This workwas supported in part by the National Science Foundation (NSF) under GrantsNCR-9508239 and ANI-9805285. This paper was presented in part at the IEEEInternational Conference on Communications (ICC’01) at Helsinki, Finland, inJune 2001.
The authors are with the Department of Computer Science, University of Cali-fornia, Davis, CA 95616 USA (e-mail: [email protected]; [email protected]).
Publisher Item Identifier S 0733-8716(02)00156-7.
(a)
(b)
Fig. 1. Illustrative example of traffic grooming.
demands [1]–[7] or based on dynamic traffic demands [8]–[10].Most previous studies have assumed that a connection requestsbandwidth for an entire lightpath channel. In this study, weassume the bandwidth of connection requests can be somefraction of the lightpath capacity, which makes the problemmore practical.
We investigate the problem of how to “groom” low-speedconnection requests to high-capacity lightpaths efficiently. Thetraffic-grooming problem has been studied on the SONET ringtopology. References [11]–[16] reported some previous work onthe traffic-grooming problem on the WDM SONET ring net-works. The objective function in these studies is to minimize thetotal network cost, measured in terms of the number of SONETadd–drop multiplexers (ADMs). In this paper, we use irregularmesh WDM networks as our study topologies and assume that aconnection requests a bandwidth that is a fraction of the wave-length capacity.
Fig. 1 shows an illustrative example of traffic grooming in aWDM mesh network. Fig. 1(a) shows a small six-node network.Each fiber has two wavelength channels. The capacity of eachwavelength channel is OC-48, i.e., approx. 2.5 Gb/s.1 Each nodeis equipped with a tunable transmitter and a tunable receiver,both of which can be tuned to any wavelength. There are three
1Note that the bandwidth of an OC-n channel is approximatelyn � 51:84
ZHU AND MUKHERJEE: TRAFFIC GROOMING IN AN OPTICAL WDM MESH NETWORK 123
connection requests: (0, 2) with bandwidth requirement OC-12;(2, 4) with bandwidth requirement OC-12; and (0, 4) with band-width requirement OC-3. Two lightpaths have already been setup to carry these three connections, as shown in Fig. 1(a). Be-cause of the resource limitations (transmitter in node 0 and re-ceiver in node 4 are busy), we cannot set up a lightpath directlyfrom node 0 to node 4; thus, connection 3 has to be carried bythe spare capacity of the two existing lightpaths, as shown inFig. 1(b). Different connection requests between the same nodepair can be either groomed on the same lightpath, whichdirectly joins , using various multiplexing techniques, orrouted separately through different virtual paths. A connectionmay traverse multiple lightpaths if no resources are available toset up a lightpath between the source and destination directly.
We investigate the node architecture for the WDM optical net-work to support traffic-grooming capability. We study an opticalwide-area WDM network which utilizes a grooming-capableoptical node architecture, so that a group of lightpaths can beset up to optimally carry the low-speed connection requests.
We formulate the traffic-grooming problem in a mesh net-work as an optimization problem with the following objectivefunction: for a given traffic matrix set and network resources,maximize the (weighted) network throughput. The mathemat-ical formulation is presented for static traffic demands. Severalsimple provisioning algorithms, i.e., heuristics, are also pro-posed and their performance is compared. Finally, we show howto extend the mathematical formulation to accommodate othernetwork optimization criteria.
II. GENERAL PROBLEM STATEMENT
The problem of grooming low-speed traffic requests to high-bandwidth wavelength channels on a given physical topology(fiber network) is formally stated below. We are given the fol-lowing inputs to the problem.
1) A physical topology consisting of aweighted unidirectional graph, where is the set ofnetwork nodes and is the set of physical links,connecting the nodes. Nodes correspond to networknodes and links correspond to the fibers between nodes.Though links are unidirectional, we assume that there arean equal number of fibers joining two nodes in differentdirections. Links are assigned weights, which may cor-respond to the physical distance between nodes. In thisstudy, we assume that all links have the same weight 1,which corresponds to the fiber hop distance. A networknode is assumed to be equipped with awavelength-routing switch (WRS), where denotesthe number of incoming fiber links to node. 2
2) Number of wavelength channels carried by each fiber. Capacity of a wavelength .
3) A set of traffic matrices, where . Eachtraffic matrix in the traffic-matrix set represents one par-ticular group of low-speed connection requests betweenthe nodes of the network. For example, if is OC-48,
2For any nodei, the number of incoming fiber links is equal to the number ofoutgoing fiber links.
there may exist four traffic matrices: an OC-1 traffic ma-trix, an OC-3 traffic matrix, an OC-12 traffic matrix, andan OC-48 traffic matrix.
4) The number of lasers (transmitters) and filters (re-ceivers) at each node. Note that the transceivercan be either wavelength-tunable or part of a fixed-tunedarray.
Our goals are to determine the following.
1) A virtual topology . The nodes of thevirtual topology correspond to the nodes in the physicaltopology. A link between nodesand corresponds to aunidirectional lightpath set up between node pair .
2) Routing connection requests on the virtual topology toeither minimize the total network cost or maximize totalthroughput. In this study, we consider maximizing totalthroughput.
III. N ODE ARCHITECTURE
To carry connection requests in a WDM network, lightpathconnections may be established between pairs of nodes. A con-nection request may traverse through one or more lightpathsbefore it reaches the destination. Two important functionalitiesmust be supported by the WDM network nodes: one is wave-length routing and the other is multiplexing and demultiplexing.A WRS in [1] and [3] provides the wavelength-routing capa-bility to the WDM network nodes. Optical multiplexer/demul-tiplexer can multiplex/demultiplex several wavelengths to thesame fiber link. Low-speed connection requests will be multi-plexed on the same lightpath channel by using an electronic-do-main TDM-based multiplexing technique. Figs. 2 and 3 showtwo sample node architectures in a WDM optical network.
The node architecture is composed of two components: WRSand access station. The WRS performs wavelength routingand wavelength multiplexing/demultiplexing. The accessstation performs local traffic adding/dropping and low-speedtraffic-grooming functionalities. WRS is composed of anoptical crossconnect (OXC), network control and managementunit (NC&M), and optical multiplexer/demultiplexer. In theNC&M, the network-to-network interface (NNI) will configurethe OXC and exchange control messages with peer nodes ona dedicated wavelength channel (shown as wavelength 0 inFigs. 2 and 3). The network-to-user interface (NUI) will com-municate with the NNI and exchange control information withthe user-to-network interface (UNI), the control component ofthe access station. The OXC provides wavelength-switchingfunctionality. As shown in Fig. 2, each fiber has three wave-lengths. Wavelength 0 is used as a control channel for theNC&M to exchange control messages between network nodes.Other wavelengths are used to transmit data traffic.
In Fig. 2, each access station is equipped with some trans-mitters and receivers (transceivers). Traffic originated from theaccess station is sent out as an optical signal on one wavelengthchannel by a transmitter. Traffic destined to the access stationis converted from an optical signal to electronic data by a re-ceiver. Both tunable transceivers and fixed transceivers could beused in a WDM network. A tunable transceiver can be tuned be-tween different wavelengths so that it can send out (or receive)
124 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 1, JANUARY 2002
Fig. 2. Node architecture 1: IP over WDM.
an optical signal on any free wavelength in its tuning range. Afixed transceiver can only emit (or receive) an optical signalson one wavelength. To explore all of the wavelength channelson a fiber, a set of fixed transceivers, one per wavelength, isgrouped together to form a transceiver array. The size of a fixedtransceiver array can be equal to or smaller than the number ofwavelengths on a fiber, and the number of tranceiver arrays canbe equal to or smaller than the number of fibers joining a node.
The access station in Fig. 2 provides a flexible, software-based bandwidth-provisioning capability to the network. Mul-tiplexing low-speed connections to high-capacity lightpaths isdone by the MPLS/IP router using a software-based queuingscheme. The advantages of this model are that: 1) it providesflexible bandwidth granularity for the traffic requests and 2)this MPLS/IP-over-WDM model has much less overhead thanthe SONET-over-WDM model, shown in Fig. 3. A potentialdisadvantage of this model is that the processing speed of theMPLS/IP router may not be fast enough compared with the vastamount of the bandwidth provided by the optical fiber link.
In Fig. 3, each access station is equipped with severalSONET add–drop multiplexers (ADMs). Each SONET ADMhas the ability to separate a high-rate SONET signal intolower rate components [13]. In order for a node to transmitor receive traffic on a wavelength, the wavelength must beadded or dropped at the node and a SONET ADM must beused. Generally, each SONET ADM is equipped with a fixedtransceiver and operates only on one wavelength as shown inFig. 3. The digital crossconnect (DCS) can interconnect thelow-speed traffic streams between the access station and theADMs. A low-speed traffic stream on one wavelength can beeither dropped to the local client (IP router, ATM switch, etc.)or switched to another ADM and sent out on another wave-length. Fig. 3 presents a SONET-over-WDM node architecture.
SONET components (ADM, DCS, etc.) and SONET framingschemes can provide TDM-based fast multiplexing/demulit-plexing capability, compared with the software-based schemein Fig. 3. The disadvantage of this approach is the high cost ofSONET components, such as ADM and DCS. In reality, bothkinds of access stations may be used together to be connectedwith an OXC in order to have a multiservice platform foraccessing an OXC in a carrier’s network.
IV. M ATHEMATICAL (ILP) FORMULATION OF THE
TRAFFIC-GROOMING PROBLEM
The traffic-grooming problem in a mesh network with statictraffic pattern turns out to be an integer linear program (ILP).We make the following assumptions in our study.
1) The network is a single-fiber irregular mesh network, i.e.,there is at most one fiber link between each node pair.
2) The wavelength-routing switches in the network nodesdo not have wavelength conversion capability. A light-path connection must be set up on the same wavelengthchannel if it traverses through several fibers. An exten-sion of this problem to include wavelength conversion isstraightforward and it actually makes the problem simplerin terms of the number of variables and equations.
3) The transceivers in a network node are tunable to anywavelength on the fiber.
4) A connection request cannot be divided into several lowerspeed connections and routed separately from the sourceto the destination. The data traffic on a connection requestshould always follow the same route.
5) Each node has unlimited multiplexing/demultiplexing ca-pability and time-slot interchange capability. This means
ZHU AND MUKHERJEE: TRAFFIC GROOMING IN AN OPTICAL WDM MESH NETWORK 125
Fig. 3. Node architecture 2: SONET over WDM.
that the access station of a network node can multiplex/de-multiplex as many low-speed traffic streams to a lightpathas needed, as long as the aggregated traffic does not ex-ceed the lightpath capacity.3
A. Multihop Traffic Grooming
In this section, we assume that a connection can traverse mul-tiple lightpaths before it reaches the destination. So, a connec-tion may be groomed with different connections on differentlightpaths. By extending the work in [1] and [6], we formulatethe problem as an optimization problem. We will use the fol-lowing notation in our mathematical formulation.
and endpoints of a physical fiber link that might occur ina lightpath.
and originating and terminating nodes for a lightpath. Alightpath may traverse single or multiple physicalfiber links.
and source and destination of the end-to-end traffic re-quest. The end-to-end traffic may traverse througha single or multiple lightpaths. Fig. 4 shows how anend-to-end connection request may be carried.granularity of low-speed traffic requests. We assume
, which means that traffic de-mands between node pairs can be any of OC-1,OC-3, OC-12 and OC-48.
3This may only be true for the software-based provisioning scheme in Fig. 2,which may support virtual-circuit connections. The grooming capability of thenode architecture in Fig. 3 is limited by the number of output ports of SONETADM and the size and the functionality of DCS.
Fig. 4. Illustrative example of a fiber link, a lightpath, and a connectionrequest.
index of OC- traffic request for any given node pair. For example, if there are ten OC-1 requests
between node pair , then .
• Given:number of nodes in the network.number of wavelengths per fiber. We assume all of thefibers in the network carry the same number of wave-lengths.number of fibers interconnecting nodeand node .
for node pair which is not physically adjacentto each other. if and only if thereexists a direct physical fiber link between nodesand
.wavelength on fiber . .number of transmitters at node.number of receivers at node. Note that, in this set ofILP formulation, we assume all the nodes are equippedwith tunable transceivers, which can be tuned to any of
wavelengths.
126 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 1, JANUARY 2002
capacity of each channel (wavelength).traffic matrix set. , where can be any al-lowed low-speed streams, 1, 3, 12, etc. In our study,
. is the number of OC- con-nection requests between node pair .
• Variables:— Virtual topology:
• number of lightpaths from nodeto nodein virtual topology. does not imply that
.• number of lightpaths from nodeto node
on wavelength . Note that, if , thelightpaths between nodeand on wavelength
may take different paths.— Physical topology route:
• number of lightpaths between nodesrouted through fiber link on wavelength
.— Traffic route:
• : The th OC- low-speed traffic requestfrom node to node employing lightpathas an intermediate virtual link.
• if the th OC- low-speed con-nection request from nodeto node has beensuccessfully routed; otherwise, .
• Optimize: Maximize the total successfully-routedlow-speed traffic.
(1)
• Constraints:— On virtual-topology connection variables
(2)
(3)
(4)
(5)
— On physical route variables
if (6)
(7)
(8)
(9)
(10)
(11)
(12)
— On virtual-topology traffic variables
(13)
(14)
if (15)
(16)
(17)
(18)
(19)
• Explanation of equations: The above equations are basedon principles of conservation of flows and resources(transceivers, wavelengths, etc.).
— Equation (1) shows the optimization objective func-tion.
— Equations (2), (3) ensure that the number of lightpathsbetween node pair is less than or equal to thenumber of transmitters at nodeand the number ofreceivers at node.
— Equation (4) shows that the lightpaths betweenare composed of lightpaths on different wavelengthsbetween node pair . Note that the value of canbe greater than 1. For example, in Fig. 1, two lightpathson the same wavelengthcan be set up between node0 and node 5. One follows route (0, 1, 2, 5), while theother follows route (0, 3, 4, 5).
— Equations (6)–(10) are the multicommodity equations(flow conservation) that account for the routing ofa lightpath from its origin to its termination. Theflow-conservation equations have been formulated intwo different ways [5]: i) disaggregate formulationand ii) aggregate formulation. In the disaggregateformulation, every - (or - ) pair corresponds to acommodity, while in the aggregate formulation, all thetraffic that is “sourced” from node(or node ) whichcorresponds to the same commodity, regardless ofthe traffic’s destination. We employ the disaggregateformulation for the flow-conservation equations sinceit properly describes the traffic requests betweendifferent node pairs. Note that (6)–(10) employ thewavelength-continuity constraint on the lightpathroute.
• Equation (6) ensures that, for an intermediatenode of lightpath on wavelength , thenumber of incoming lightpath streams is equal tothe number of outgoing lightpath streams.
ZHU AND MUKHERJEE: TRAFFIC GROOMING IN AN OPTICAL WDM MESH NETWORK 127
• Equation (7) ensures that, for the origin nodeof lightpath on wavelength , the numberof incoming lightpath streams is 0.
• Equation (8) ensures that, for the terminationnode of lightpath on wavelength , thenumber of outgoing lightpath streams is 0.
• Equation (9) ensures that, for the origin nodeof lightpath on wavelength , the numberof outgoing lightpath streams is equal to the totalnumber of lightpaths between node pair onwavelength .
• Equation (10) ensures that, for the terminationnode of lightpath on wavelength , thenumber of incoming lightpath streams is equal tothe total number of lightpaths between node pair
on wavelength .— Equations (11), (12) ensure that wavelengthon one
fiber link can only be present in at most onelightpath in the virtual topology.
— Equations (13)–(19) are responsible for the routing oflow-speed traffic requests on the virtual topology, andthey take into account the fact that the aggregate trafficflowing through lightpaths cannot exceed the overalwavelength (channel) capacity.
B. Single-Hop Traffic Grooming
In this section, we assume that a connection can only tra-verse a single lightpath, i.e., only end-to-end traffic groomingis allowed. The formulation of the single-hop traffic groomingproblem is similar to the formulation of the multihop traffic-grooming problem, which was presented in the previous sec-tion, except for routing of connection requests on the virtualtopology. We only present the different part as follows.
• On virtual-topology traffic variables
(20)
(21)
C. Formulation Extension for Fixed-Transceiver Array
The mathematical formulations in the previous two sectionsare based on the assumption that the transceivers in a networknode are tunable to any wavelength. If fixed-transceiver arraysare used at every network node and ifdenotes the number offixed-transceiver arrays used at each node, we can easily extendour formulation as follows.
• On virtual-topology connection variables
(22)
(23)
(24)
(25)
(a)
(b)
Fig. 5. (a) A six-node network and (b) a 15-node network.
The other parts of the formulations in the previous two sec-tions are still the same. Equations (22), (23) ensure that thenumber of lightpaths between node pair on wavelengthis less than or equal to the number of transmitters at nodeandthe number of receivers at nodeon the wavelength (everyfixed-transceiver array only has one transceiver on each wave-length).
D. Computational Complexity
It is well known that the RWA optimization problem isNP-complete [1]. If we assume that each connection requestrequires the full capacity of a lightpath, the traffic groomingproblem we are studying becomes the standard RWA optimiza-tion problem. It is easy to see that the traffic-grooming problemin a mesh network is also a NP-complete problem since theRWA optimization problem is NP-complete. As the number ofvariables and equations increases exponentially with the size ofnetwork and the number of wavelengths on each fiber, we use asmall network topology as an example for obtaining ILP result.For large networks, we will use heuristic approaches.
V. ILLUSTRATIVE NUMERICAL RESULT FROM ILPFORMULATIONS
This section presents numerical examples of thetraffic-grooming problem using Fig. 5(a) as our physicaltopology. The traffic matrices are randomly generated. As an
128 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 1, JANUARY 2002
TABLE ITRAFFIC MATRIX OF OC-1 CONNECTION REQUESTS
TABLE IITRAFFIC MATRIX OF OC-3 CONNECTION REQUESTS
example, we allow the traffic demand to be any one of OC-1,OC-3, and OC-12. The traffic matrices are generated as follows:1) the number of OC-1 connection requests between each nodepair is generated as a uniformly distributed random numberbetween 0 and 16; 2) the number of OC-3 connection requestsbetween each node pair is generated as a uniformly distributedrandom number between 0 and 8; and 3) the number of OC-12connection requests between each node pair is generated as auniformly distributed random number between 0 and 2. Thesethree traffic matrices are shown in Tables I–III, and the totaltraffic demand turns out to be the equivalent of OC-988. Thecapacity of each wavelength (channel) is OC-48.
Table IV shows the corresponding result for the networkthroughput obtained from a commercial ILP solver, “CPLEX”,based on different network resource parameters. In Table IV,
denotes the number of transceivers and denotes thenumber of wavelengths. In the single-hop case, we only allowa connection to transverse a single lightpath, which means thatonly end-to-end traffic-grooming (multiplexing) is allowed. Inthe multihop case, a connection is allowed to traverse multiplelightpaths, i.e., a connection can be dropped at intermediatenodes and groomed with other low-speed connections ondifferent lightpaths before it reaches its destination. Fig. 1(b)shows a multihop grooming case, where connection 3 traversedtwo lightpaths; it was groomed with connection 1 on lightpath(0, 2) and with connection 2 on lightpath (2, 4). As expected, themultihop case leads to higher throughput than the single-hopcase.
We can see from Table IV that, when the number of tunabletransceivers at each node is increased from 3 to 5, the networkthroughput increases significantly, both in the multihop case andin the single-hop case. But when the number of tunable trans-ceivers at each node increases from 5 to 7, network throughputdoes not improve. This is because there are not enough wave-lengths to setup more lightpaths to carry the blocked connectionrequests. Some illustrative results of transceiver and wavelengthutilization for the multihop case are shown in Tables V and VI.
TABLE IIITRAFFIC MATRIX OF OC-12 CONNECTION REQUESTS
In multihop case, when the transceiver is not a limited re-source compared with wavelength, more short lightpaths may beset up to carry connections through multiple lightpaths, but thisscenario is less likely in the single-hop case. This is shown inTable IV where and . When, if multihop grooming is allowed, the network throughput is
100%; otherwise, some connections get blocked. In the mul-tihop case, 29 lightpaths are established compared with 28 light-paths in the single-hop case.
Tables V and VI show some results for the node transceiverutilization and link wavelength utilization for the multihopcase. When the number of transceivers is increased (from 3 to5), the overall wavelength utilization is increased, as shown inTable VI. This is because more lightpaths have been establishedto carry the connection requests, shown in Table IV. As wementioned, when most of the links have fully utilized theavailable wavelengths, increasing the number of transceivers(from 5 to 7) will not help to improve the network throughputand will result in lower transceiver utilization, shown in Table V( and ).
Table VII shows the virtual topology and the lightpath ca-pacity utilization for the multihop case, when and. As we can see, most of the lightpaths have high capacity uti-
lization (above 90%). There are some node pairs ((0, 1), (1, 3),etc.) which have multiple lightpaths set up, though the aggregatetraffic between them can be carried by a single lightpath. Theextra lightpaths are used to carry multihop connection traffic.
In the ILP formulation, we treat the low-speed connectionrequests separately. The results from the ILP solutions showthat, if there is a lightpath set up between , the low-speedconnections between tend to be packed on this lightpathchannel directly. Based on this observation, we propose twosimple heuristic algorithms for solving the traffic-groomingproblem in large networks.
VI. HEURISTIC APPROACH
The optimization problem of traffic grooming is NP-com-plete. It can be partitioned into the following four subproblems,which are not necessarily independent.
1) Determine a virtual topology, i.e., determine the numberof lightpaths between any node pair.
2) Route the lightpaths over the physical topology.3) Assign wavelengths optimally to the lightpaths.4) Route the low-speed connection requests on the virtual
topology.
ZHU AND MUKHERJEE: TRAFFIC GROOMING IN AN OPTICAL WDM MESH NETWORK 129
TABLE IVTHROUGHPUT ANDNUMBER OF LIGHTPATHS ESTABLISHED (TOTAL TRAFFIC DEMAND IS OC-988)
The routing and wavelength assignment problem (RWA) hasreceived a lot of attention in the WDM networking literature.The current well-known routing approaches are fixed routing,fixed-alternate routing, and adaptive routing [10].
In fixed routing, the connections are always routed through apredefined fixed route for a given source-destination pair. Oneexample of such an approach is fixed shortest path routing. Theshortest path route for each source-destination pair is calculatedoffline using standard shortest path algorithms, such as Dijk-stra’s algorithm. If there are not enough resources to satisfy aconnection request, the connection gets blocked.
In fixed-alternate routing, multiple fixed routes are consid-ered when a connection request comes. In this approach, eachnode in the network is required to maintain a routing table thatcontains an ordered list of a number of fixed routes to each des-tination node. For example, these routes can be the first shortestpath, the second shortest path, etc. When a connection requestcomes, the source node attempts to establish the connectionon each of the routes from the routing table in sequence, untilthe connection is successfully established. Since fixed-alternaterouting provides simplicity of control for setting up and tearingdown connections, it is also widely used in the dynamic con-nection-provisioning case. It has been shown that, for certainnetworks, having as few as two alternate routes provides signif-icantly lower blocking than having full wavelength conversionat each node with fixed routing [17].
In adaptive routing, the route from a source node to a des-tination node is chosen dynamically, depending on the currentnetwork state. The current network state is determined by theset of all connections that are currently in progress [10]. For ex-ample, when a connection request arrives, the current shortestpath between the source and the destination is calculated basedon the available resources in the network; then, the connectionis established through the route. If a connection gets blockedin the adaptive-routing approach, it will also be blocked in thefixed-alternate routing approach. Since each time a new connec-tion request comes to a node, the route needs to be calculatedbased on the current network state, adaptive routing will requiremore computation and a longer response time than fixed-alter-nate routing, but it is also more flexible than fixed-alternaterouting.
In our heuristics, we will use adaptive routing.
B. Wavelength Assignment
Once the route has been chosen for each lightpath, the numberof lightpaths going through a physical fiber link defines thecongestion on that particular link. With the wavelength-conti-nuity constraint, we need to assign wavelengths to each light-path such that any two lightpaths passing through the same phys-ical link are assigned different wavelengths.4 Ten wavelength-assignment approaches have been compared in [10], and all ofthem were found to perform similarly. We will use one simpleapproach, first-fit (FF). In FF, all wavelengths are numbered.When searching for an available wavelength, a lower numbered
4We assume a single-fiber network system. There is only one fiber in eachdirection if two nodes are connected.
130 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 1, JANUARY 2002
TABLE VIIRESULT: VIRTUAL TOPOLOGY AND LIGHTPATH UTILIZATION (MULTIHOP CASE WITH T = 5 AND W = 3)
wavelength is considered before a higher numbered wavelength.The first available wavelength is then selected. The idea behindthis simple scheme is that it tries to pack all of the in-use wave-lengths toward the low end of the wavelength space.
C. Heuristics
We propose two heuristic algorithms for the traffic-groomingproblem. Let denote the aggregate traffic between nodepair and , which has not been successfully carried. Letdenote one connection request betweenand , which has notbeen successfully carried yet. Letdenote the wavelength ca-pacity.
• Maximizing Single-Hop Traffic(MST). The basic idea ofthis heuristic is introduced in [1] for the traditional vir-tual-topology design problem. This simple heuristic at-tempts to establish lightpaths between source-destinationpairs with the highest values, subject to constraintson the number of transceivers at the two end nodes andthe availability of a wavelength in the path connecting thetwo end nodes. The connection requests betweenandwill be carried on the new established lightpath as much aspossible. If there is enough capacity in the network, everyconnection will traverse a single lightpath hop. If thereare not enough resources to establish enough lightpaths,the algorithm will try to carry the blocked connection re-quests using currently available spare capacity of the vir-tual topology. The pseudocode for this heuristic follows.
Construct virtual topology :Sort all of the node pairsaccording to the sum of uncarriedtraffic request betweenand put them into a list in de-cending order .Try to setup a lightpath betweenthe first node pair in usingfirst-fit wavelength assignmentand shortest-path routing, sub-ject to the wavelength and tran-ceiver constraints. If it fails,delete from ; otherwise, let
Statisfy all of the connection re-quests which can be carried through
single lightpath hop, and update thevirtual topology network state .Route the remaining connection re-quests based on the current virtualtopology network state, in the de-cending order of the connections’bandwidth requirement .
• Maximizing Resource Utilization(MRU). Letdenote the hop distance on physical topology betweennode pair . Define as the con-nection resource utilization value, which represents theaverage traffic per wavelength link. This quantity showshow efficiently the resources have been used to carrythe traffic requests. This heuristic tries to establish thelightpaths between the node pairs with the maximumresource utilization values. When no lightpath can be setup, the remaining blocked traffic requests will be routedon the virtual topology based on their connection resourceutilization value , where denotesa blocked connection request, and denotes thehop distance betweenand on the virtual topology. Thepseudocode for this heuristic follows the same steps asthe pseudocode of MST, except that the node pairs andblocked connections are sorted based on their resourceutilization values.
Both heuristic algorithms have two stages. Based on our ob-servations from the ILP results, we find that packing differentconnections between the same node pair within the same ex-isting lightpath, which directly joins the end points, is a very ef-ficient grooming scheme. In the first stage, the algorithms try toestablish lightpaths as much as possible to satisfy the aggregateend-to-end connection requests. If there are enough resourcesin the network, every connection request will be successfullyrouted through a single lightpath hop. This will minimize thetraffic delay. In the second stage, the spare capacity of the cur-rently established lightpath channels is used to carry the connec-tion requests blocked in the first stage, and the algorithms givesingle-hop groomable connections high priority to be satisfied.
D. Heuristic Results and Comparison
Table VIII shows a comparison between the results obtainedfrom ILP solver and the heuristic algorithms for the six-nodenetwork in Fig. 5(a). We can observe that the MST and MRUheuristic algorithms show reasonable performance whencompared with the results obtained from the ILP solver. Theheuristic approaches have much less computation complexitythan the ILP approach. The two proposed algorithms are
ZHU AND MUKHERJEE: TRAFFIC GROOMING IN AN OPTICAL WDM MESH NETWORK 131
TABLE VIIITHROUGHPUTRESULTSCOMPARISONBETWEEN ILP AND HEURISTIC ALGORITHMS (TOTAL TRAFFIC DEMAND IS OC-988)
Fig. 6. Network throughput versus number of wavelengths for the networktopology in Fig. 5(b) with ten tunable transceivers at each node.
Fig. 7. Network throughput versus number of tunable transceivers for thenetwork topology in Fig. 5(b) with ten wavelengths on each fiber link.
relatively simple and straightforward; by using other RWAalgorithms instead of adaptive routing and first-fit wavelengthassignment, it may be possible to develop other elaborateheuristic algorithms to achieve better performance.
Figs. 6–8 show the results from the two heuristic algorithms,when applied to the larger network topology in Fig. 5(b). Thetraffic matrices follow the same distribution as we mentioned inSection V.
In Fig. 6, we plot the network throughput versus thenumber of wavelengths on every fiber link when each node is
Fig. 8. Network throughput versus number of wavelengths (size of fixedtransceiver array) for the network topology in Fig. 5(b) with 12 tunabletransceivers at each node.
equipped with ten tunable transceivers. We observe that theMRU heuristic performs better than the MST algorithm withrespect to network throughput. Since the number of tunabletransceivers at each node is limited (10 in this case), whenthe number of wavelengths on each fiber link reaches a certainvalue (16 in this case), increasing the number of wavelengthsdoes not help to increase the network throughput.
In Fig. 7, we plot the network throughput versus the numberof transceivers at every node when each fiber link carries tenwavelengths. We compare the performance of the two heuristicalgorithms. The results show that, when the number of trans-ceivers is small ( 7 in this case), MST performs better thanMRU. This is because MRU tries to utilize wavelengths effi-ciently. When the number of transceivers is small, wavelengthis not the limiting resource in the network any more. So maxi-mizing wavelength utilization will not help to improve the per-formance.
Fig. 8 compares the performance using tunable transceiverand fixed transceiver in every network node. Each node isequipped with 12 transceivers if we use a tunable transceiver.Each node is equipped with one transceiver array if we usefixed transceivers and the size of the transceiver array is equalto the number of wavelengths supported by every fiber link.The results in Fig. 8 indicate that, when nodes are equippedwith the same number of transceivers, the tunable-transceiverarchitecture has better performance. For the fixed-transceivercase, MST performs better than MRU.
132 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 1, JANUARY 2002
VII. M ATHEMATICAL FORMULATION EXTENSION FOROTHER
OPTIMIZATION CRITERIA
In this section, we show how to extend our ILP formulationsto handle different optimization criteria for the traffic-groomingproblem.
A. Extension for Network Revenue Model
It has been shown earlier that the low-speed connectionrequests between the same node pair tend to be packed togetheron to the same lightpath channel. The connections, whichcan be carried by a single lightpath channel are more likelyto be satisfied than the connections which have to traversemultiple lightpaths, when they have the same bandwidthrequirement and the optimization objective is to maximizenetwork throughput. To make the problem more realistic, it isreasonable for us to assume that two connection requests mayhave different priority, even if they have the same bandwidthrequirement. This is because different connection requestsmay have different end-node distance, quality-of-servicerequirement, etc. A connection’s priority can be represented bya “weight” associated with it. In this section, we assume thatthe weight is determined by the bandwidth requirement andend-node distance of the connection request. For a given net-work topology and traffic demand, the objective is to maximizethe weighted network throughput. Let denote the weight ofconnection denote the end-node distance of connection,and denote the bandwidth requirement of connection. Theconnection’s weight function is defined as
(26)
where and is measured by the shortest path dis-tance of the connection’s end nodes on the physical topology.Equation (26) is called the power-law cost function [18]. It isused to study the actual tariffs demanded by communicationsservices for high-speed telecommunication channels, and thereis effectively a “quantity discount” (controlled by) in that ca-pacity cost (per unit of channel capacity) decreases as the ca-pacity increases. Thus, the network’s weighted throughput be-comes
(27)
where if connection requesthas been satisfied; other-wise , and the total number of connection requests is.
can also be called “network revenue”. We can easily modifyour ILP formulation to optimize . The only part of the equa-tion which should be modified is shown as follows:
• Optimize: Maximize network revenue
(28)
where denotes the distance between node pair .
B. Illustrative Results
In this section, we show some illustrative results to optimizenetwork revenue using our ILP formulation extension. We use
TABLE IXRESULTS OFCOMPARISONBETWEEN REVENUE MODEL AND NETWORK
THROUGHPUTMODEL
the same network topology and traffic matrix set as in Section V.In (28), is measured by the shortest path hop distance be-tween node and on the physical topology, and is equal to0.8.
Table IX compares the results between the two optimizationmodels. In Table IX, denotes the number of tunable trans-ceivers per node and denotes the number of wavelengths perfiber link. Multihop grooming is allowed in both models. It isshown that, in the revenue model, when and ,the maximal achievable revenue is 83.7%, and 72.4% of trafficrequests have been satisfied to achieve the revenue, while themaximal achievable traffic load the network can carry is 74.7%.In revenue model, we find that if there is a lightpath set up be-tween , it may first be used to carry some long multihopconnections (with higher weight) which will traverse this light-path as an intermediate hop. Thus, some connections directlybetween may be blocked. This means that packing dif-ferent connections between the same node pair within the sameexisting lightpath, which directly joins the end points, is not agood grooming scheme any more. We find that, because of thequantity-discount parameterin (26), lower speed connectionsare more likely to be satisfied than higher speed connection re-quests. It is obvious that different heuristics are needed basedon the different optimization criteria.
VIII. C ONCLUSION
This study was devoted to the traffic-grooming problem ina WDM mesh network. We studied the architecture of a nodewith grooming capability. We presented the ILP formulation fortraffic-grooming in such a WDM mesh network. We comparedthe performance of the single-hop grooming approach andmultihop grooming approach on a small six-node network withrandomly generated traffic pattern. Results from ILP showedthat the end-to-end aggregate traffic between the same node pairtends to be groomed on to the same lightpath channel, whichdirectly joins the end points, if the optimization objective is tomaximize the network throughput. Two heuristic approacheswere also proposed for solving the traffic-grooming problemin large networks. We compared the performance of these twoheuristic algorithms, MST and MRU, with different networkresource parameters. The comparison results showed thatMRU performs better if tunable transceivers are used and MSTperforms better if fixed transceivers are used. We extended theoptimization problem to a network-revenue model and founda different grooming scheme, which can be used to designan efficient heuristic algorithm on network-revenue model.
ZHU AND MUKHERJEE: TRAFFIC GROOMING IN AN OPTICAL WDM MESH NETWORK 133
We showed that, with proper extension, our ILP mathematicalmodel can be used to examine good grooming schemes fordifferent models. These schemes can be used to design efficientheuristic algorithms, which are practical for large and realisticnetworks.
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Keyao Zhu (S’98) received the B.S. degree fromPeking University, Beijing, China, in 1998, and theM.S. degree from the University of California, Davis,in March 2001. Currently, he is a Ph.D. student inthe Computer Science Department, University ofCalifornia, Davis.
He works as a Research Assistant in the networkslaboratory at the University of California, Davis. Hisresearch interests include wavelength-routed WDMnetwork design and analysis and WDM network pro-tection and restoration.
Biswanath Mukherjee (S’84–M’87) received theB.Tech. (Hons.) degree from the Indian Instituteof Technology, Kharagpur, India, in 1980, and thePh.D. degree from the University of Washington,Seattle, in June 1987.
At the University of Washington, he held aGTE Teaching Fellowship and a General ElectricFoundation Fellowship. In July 1987, he joined theUniversity of California, Davis, where he has beena Professor of Computer Science since July 1995and Chairman of the Computer Science Department
from September 1997 to June 2000. He is author of the textbookOptical Com-munication Networks(New York: McGraw-Hill, 1997), a book which receivedthe Association of American Publishers, Inc.’s 1997 Honorable Mention inComputer Science. His research interests include lightwave networks, networksecurity, and wireless networks.
Dr. Mukherjee is a co-winner of paper awards presented at the 1991 andthe 1994 National Computer Security Conferences. He serves or has served onthe editorial boards of the IEEE/ACM TRANSACTIONS ONNETWORKING, IEEENetwork, ACM/Baltzer Wireless Information Networks(WINET), Journal ofHigh-Speed Networks, Photonic Network Communications, andOptical Net-work Magazine. He also served as Editor-at-Large for optical networking andcommunications for the IEEE Communications Society. He served as the Tech-nical Program Chair of the IEEE INFOCOM’96 Conference.
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