WDM network design by ILP models based on flow aggregation

13 Pages·2006·0.39 MB·English

WDM network design by ILP models
based on flow aggregation
Massimo Tornatore, Guido Maier, Achille Pattavina
Abstract—Planning and optimization of WDM networks has proposed solutions can be classified into two main groups:
raised much interest among the research community in the heuristic methods and exact methods. The former return sub-
last years. Integer Linear Programming (ILP) is the most used
optimal solutions that in many cases are acceptable and have
exact method to perform this task and many studies have been
the advantage of requiring a limited computational effort. The
publishedconcerningthisissue.Unfortunatelymanyworkshave
shown that, even for small networks, the ILP formulations latter are much more computationally intensive and do not
can easily overwhelm the capabilities of today state-of-the-art scale well with the network size, being even not applicable
computing facilities. So in this article we focus our attention on in some cases. However since the exact methods are able to
ILP model computational efficiency in order to provide a more
identifytheabsoluteoptimalsolution,theyplayafundamental
effective tool in view of direct planning or other benchmarking
roleeitherasdirectplanningtoolsorasbenchmarkstovalidate
applications. Our formulation exploits flow aggregation and
consistsinanewILPformulationthatallowsustoreachoptimal and test heuristic methods.
solutions with less computational effort compared to other ILP The work we are presenting concerns exact methods to
approaches.Thisformulationappliestomultifibermeshnetworks plan and optimize multifiber WDM networks. In particular
with or without wavelength conversion. After presenting the
we focus on Integer Linear Programming (ILP), a widespread
formulation we discuss the results obtained in the optimization
technique to solve exact optimization: we propose a new
of case-study networks.
formulation of the optimization problem that we call source
formulation, in that it exploits the aggregation of all the flows
I. INTRODUCTION generated in a single source node [1]. Our source formulation
is equivalent to the well known flow formulation, but it
In recent optical networks the introduction of Wavelength
allows a relevant reduction of the number of variables and of
Division Multiplexing (WDM) technique has opened the road
constraints, thus sensibly diminishing computation time and
toanewparadigmoftransportinfrastructureevolutioncharac-
memory occupancy during optimization runs.
terized by high capacity and high reliability. On the switching
Thepapersummaryisasfollows.InsectionIIweintroduce
equipment side, Optical Cross Connects (OXC) systems have
our solution by presenting a short review of the literature
become available, beside the more mature Optical Add-Drop
regarding ILP applications to WDM optimization. In section
Multiplexers. This opened up the road to the possibility of
III the source formulation is presented and explained into
deployingcomplexWDMnetworksbasedonmeshtopologies,
details in the two versions for network with or without
while in the past single ring or overlaid multi-ring were the
wavelength conversion. Finally, in section IV results obtained
most used architectures for WDM. In order to transfer data
byapplyingthesourceformulationtocase-studynetworksare
between two nodes, an optical connection needs to be set up
shown and new the formulation is compared to the traditional
androutedattheopticallayerasinacircuit-switchednetwork.
flow and route formulations to point out the advantages of the
The increase in WDM complexity brought the need for
method we are proposing. An appendix is finally dedicated to
suitablenetworkplanningstrategiesintotheforeground.Prob-
show the equivalence of flow and source formulation.
lems such as optimal dimensioning, routing and resource
allocationforopticalconnectionsmustbecontinuouslysolved
by new and old operators, to plan new installations or to II. WDMNETWORKOPTIMIZATIONBYINTEGERLINEAR
update and expand the existing ones. These problems can no PROGRAMMING
longer be manually solved in complex network architectures,
Network design and planning is carried out with different
as it usually happened in the earlier experimental WDM
techniques according to the type of traffic the network has to
installations. Computer-aided planning tools and procedures
support.Weinvestigatethestatictrafficcaseinwhichaknown
are needed for the future which can determine how to utilize
setofpermanentconnectionrequestsisassignedaprioritothe
efficientlythenetworkresourcesinareasonablecomputational
network. The connections requested by the nodes at a given
time.
time to a WDM network all together form the offered traffic
Sincesomeyearsagoresearchonopticalnetworkshasbeen
matrix virtual topology (alias virtual topology). Each request
investigating design and optimization techniques. The various
is for one or more point-to-point optical circuits (lightpaths)
able to carry a given capacity from the source termination
A.Pattavina,GuidoMaierandM.TornatorearewithPolitecnicodiMilano,
Dept.ofElectronicsandInformation,ViaPonzio34-25-20133Milan,Italy. to the destination termination. We assume that all the WDM
E-mail:{pattavina,maier,tornator}@elet.polimi.it channels carry the same capacity. Lightpaths are routed and
ApreliminaryversionofthispaperhasbeenpresentedatINFOCOM2002
switched by the OXCs of the network and the two lightpath
Conference.WorkpartiallysupportedbytheEUNetworkofExcellence”E-
Photon/ONe+ terminations are located in the source and destination OXCs.
We assume that the channels composing the lightpath (one ies the effects of imposing a constraint on the average delay
for each fiber it crosses) may have different wavelengths or seenbyasource-destinationpairandtheamountofprocessing
maybeallatthesamewavelength,accordingtotheavailability required at the nodes, while in Ref. [10] possible utilization
of the wavelength conversion function in the transit OXCs. of bounds derived from the two formulations by relaxation
To simplify, we have considered two extreme cases referring of the integer constraints are studied and compared. In other
to definitions introduced in [2]: the Virtual Wavelength Path works, the authors have selected as cost functions the number
(VWP) network case, in which all the OXC’s are able to ofwavelengths[11],[9]orthetotalnumberofWDMchannels
perform full wavelength conversion, and the Wavelength Path in the network [12], [13]. In Ref. [14] authors propose new
(WP) network case, in which no wavelength conversion is ILPformulations,whichtendtohaveintegeroptimalsolutions
allowed in the whole network and lightpaths are subject to even when the integrality constraints are relaxed, thereby
the “wavelength continuity” constraint, that is absent in the allowingtheproblemtobesolvedoptimallybyfastandhighly
VWP case. It’s important noting that wavelength assignment efficient linear (not integer) programming methods. In Ref.
to lightpaths in WP case is an NP-complete problem (it is [15] an exact linear formulation was presented for the logical
equivalent to the well-known graph-coloring problem) [3]. topology design problem with no wavelength converters. In
Today WDM networks are often designed in order to be Ref. [16] the authors have investigated the so called RWA-P,
resilienttofailuresthatmayoccurtoswitchingortransmission i.etheRWAproblemwhileallowingfordegradationofrouted
equipment. Though automatic lightpath protection is very signals by optical components.
importanttoday(giventhehighbit-ratesthataWDMchannel In optimization of multifiber WDM networks optimal al-
usually carries, e.g. 2.5 to 40 Gbit/s), this feature will not be location of fibers has also to be solved, thus complicating
covered in this work, for the reasons that will be explained the problem of lightpath set up into routing, fiber and wave-
later on. length assignment (RFWA). Solving RFWA becomes really
Static optimization of a WDM network can be summarized challenging even with relatively small networks, especially
as follows: given a static traffic matrix, find the optimum because routing and wavelength assignment is coupled to
values of a set of network variables that minimizes a given dimensioning. In this case a new set of variables representing
cost (or objective) function, under a set of constraints. The the number of fibers of each physical link must be considered
choiceofvariables,costfunctionandconstraintsgreatlyvaries in addition to the flow or the route variables defined above
fromcasetocase.InthepastmostofstudiesregardingWDM for the two formulations. This implies that RFWA has also to
networkplanningwereaimedatvirtualtopologyoptimization include the highly complex localization problem. The choice
with single-fiber WDM links [4], [5]. The cost function to be of complex cost functions such as those comprising node or
optimized was either the number of wavelengths necessary ductcostmakesthe achievementofILP optimalsolutionvery
to route the static traffic or the network load (the number challenging even for very small networks [17] (this is even
of channels routed on the most loaded link of the network) worse in the case of non-linear objective function that require
[6]. In Ref. [6] the authors introduce an ILP model based on integer non-linear programming [18]).
aggregated flows applied to virtual topology optimization. In When the problem becomes computationally impractical,
the work we are proposing the virtual topology optimization routeformulationbecomesmoreusefulthanflowformulation.
is accompanied by cost minimization of a multi-fiber physical If it is acceptable that RFWA is performed in a constrained
network: the number of fibers per link needed to support a way, then the solution complexity of the route formulation
preassigned traffic matrix is a variable of the problem to be can be controlled. For example, all the lightpaths can be
minimized,whiletheamountofwavelengthsperfiberispreset constrained to be routed along the first k shortest paths
[7]. connecting the source to the destination. Differently from the
WDMnetworkoptimizationbyILPhasbeenwidelystudied flowformulation,thecomplexityofwhichisstrictlydependent
in literature. We can subdivide research contributions in two on physical and virtual topologies, the complexity of the
groups according to the type of networks they are applied to: route formulation decreases with the number of paths that
• WDM networks with single-fiber links; can be employed to route the lightpaths. Multifiber network
• multifiber WDM networks. optimization with route formulation and constrained routing
In the first group the problem consists in optimal routing has been studied in Refs. [19], [20], [21], [9], [22], [23].
and wavelength assignment (RWA) of the lightpaths. This Beside route formulation with constrained routing, other
is a NP-complete problem, as it was demonstrated in Refs. methodstocontrolcomplexityhavebeenproposed.Apossibil-
[8], [3]. Two basic methods have been defined to model the ity is to stop the branch-and-bound algorithm (typically used
RWA problem: flow formulation(FF) and route formulation to solve ILP problems) after finding the first or a pre-definite
(RF) [9]. In the former the basic variables are the flows on number of integer solutions. Ref. [17] shows that acceptable
each link relative to each source-destination OXC pair; in results (though quite far from the optimal solution) can be
the latter the basic variables are the paths connecting each obtained when the branch-and-bound duration is fixed to 10
source-destination termination pair. Both these formulations minutes.Ref.[24]proposedthatthewholeRFWAproblemcan
have been employed to solve various sorts of problems and to besolvedasasequenceofsimplerproblems(e.g.firstrouting,
investigate different aspects of WDM networks. For example, then fiber assignment, and so on). Other possible approaches
inRef.[9]theoptimizationiscarriedoutinordertoemphasize are:exploitationoflagrangeanrelaxation[25],[23],relaxation
thedifferencebetweenWPandVWPscenarios.Ref.[6]stud- of integer constraints [19] and randomized routing [12].
FF (SF) solution, given a SF (FF) solution. In other words,
if the objective is to evaluate the number and the distribution
of the fibers in the network, we can simply apply SF in order
C C
to achieve the solution. Then, if we are interested also in the
S S
detailsoftheRWA(i.etheroutingandwavelengthassignment
of each connection request), we have to transform the SF
solution in a FF (or equivalent solution). This second step is
absolutely negligible from a complexity point of view: if the
(a) (b)
firstSFstepisalocalizationproblem,thesecondstep(needed
to transform the SF solution in detailed RFWA description)
Fig. 1. Example of three distinct source-destination commodities (a) and
the corresponding single source commodity (b), which will be exploited in has the same complexity of a mere max-flow algorithm (for
sourceformulation furtherdetailsrefertotheappendix).Soallthecomputational
times reported in the following are related to the SF step,
disregarding the possible following transformation.
Undoubtedly the massive need for computational resources
We explain now the details of the source formulation, for
(i.e. processing time and memory occupation) represents the
which two different versions are reported related to networks
main obstacle to an efficient application of ILP in optical
with or without wavelength conversion capability.
network design. Constrained routing and the other simplifi-
cation techniques are able to overcome this limitation, but
A. Source formulation for VWP networks
the solution they produce is only an approximation of the
actual optimal network design. The great advantage of ILP First we consider a VWP network, provided with full
over heuristic methods is the ability to guarantee that the wavelengthconversionasdefinedinII.Thephysicaltopology
obtained solution is the absolute optimum value. Any of the is modeled by the graph G =G(N,A). Physical links are
above techniques aimed at reducing the computational burden represented by the undirected edges l ∈ A with |A| = L,
implies that the ILP approach loses its added value, even if while the nodes i ∈ N = {1,2,···N}, with |N| = N,
the approximated solutions may be close to the exact one. represent the OXCs. Each link is equipped with a certain
Our work develops and applies a new formulation of RFWA amount of unidirectional fibers in each of the two directions;
problem which is able to prune variable multiplicity without fiber direction is identified by the binary variable k. Finally,
introducing any approximation, thus preserving the added the virtual topology is represented by the set of known terms
value of mathematical programming. Ci,j,eachoneexpressingthenumberofconnectionsthatmust
beestablishedfromthesourcenodeitothedestinationnodej.
III. SOURCEFORMULATIONOFTHERFWAPROBLEM Unidirectionalpoint-to-pointconnectionsareconsidered(thus,
in the general case, C (cid:54)=C ).
Let us consider a multifiber WDM network environment i,j j,i
The variables in the source formulation are the following:
under static traffic, in which the number of wavelengths per
fiber W is given a priori, while the fiber numbers of each • xil,k is the number of WDM channels on link l on fibers
havingdirectionkwhichhavebeenallocatedtolightpaths
physical link are variables of the problem.
generated at node i;
Traditional ILP formulations based on flow or route para-
digm1 solve the RFWA problem managing source-destination • Fl,k is the number of fibers on link l with direction k.
commodity, that is to say that these formulations route static Itshouldbenotedthattheflowvariablesxil,k aredefinedin
connection requests identified by a source and a destination such a way that all the traffic originating from the same node
node on the graph representing the WDM network (see fig. and traveling on the same link in the same direction is rep-
1(a)). resented in an aggregated form, regardless of the destination.
In our proposal the ILP formulation will consider all the This is the main aspect that differentiates source from flow
connections originating from a single source OXC as a single formulation.
commodity (see fig.1(b)). Let us observe that single source The following additional symbols are defined:
commodityonlinkS−C assumesvalueequalto2,becauseon • (l,k) identifies the set of fibers of link l that are directed
thatlinktherearetwosource-destinationcommoditieshaving as indicated by k; for sake of clarity,in the followingwe
origininnodeS.Thankstothisnewmodel(thatfromnowon name (l,k) a “unidirectional link”;
wewillcallsourceformulationorSF),weareabletoprunethe • Ii+ is the set of “unidirectional links” having the node i
number of variables associated to traffic flows, thus reducing as one extreme and leaving the node; analogously, I− is
i
computational time and memory occupation compared to the the set of “unidirectional links” having the node i as a
flow formulation. one ex(cid:80)treme and pointing towards the node;
Inordertominimizethenumberoffibersneededtosupport • Si = jCi,j is the total number of requested connec-
a certain amount of traffic, source and flow formulations are tions having node i as source.
equivalent. In appendix we show the equivalence of these Nowwecandetailthesourceformulation.Thecostfunction
two formulations by describing how to obtain an equivalent to be minimized is the total fiber number
(cid:88)
1Fromnowontheflowformulationcasewillbeconsideredthemainterm min Fl,k
ofcomparison. (l,k)
Actuallythesourceformulationcanbeveryeasilyextended a c
to solve optimization problems based on the length metric.
The only change that must be made regards the cost function, s d
which becomes (cid:88)
min F ·p
l,k l
(l,k)
b e
where p is the geographical length of link l.
l
The set of constraints is the following
(cid:88)
xi =S ∀ i; (1)
l,k i
(l,k)∈I+
(cid:88)i (cid:88)
(a)
xi = xi −C ∀ (i,j),j (cid:54)=i; (2)
l,k l,k i,j a c
(l,k)∈I+ (l,k)∈I−
j j
(cid:88) s
xi ≤W ·F ∀ (l,k); (3)
l,k l,k
d
i
xi integer ∀ i,(l,k); (4)
l,k
F integer ∀(l,k); (5) b e
l,k
Constraint (1) is a solenoidality constraint which imposes
that the total flow (number of lightpaths) generated by node
i and exiting from it must be equal to the total number of
connection requests having node i as source. Note that the
solenoidality constraint is not applied on each node-pair (by
which a connection is requested) but on the aggregated traffic
(b)
relative to a source node: therefore it is not dependent on
destinations. a c a c
Constraint (2) is again a solenoidality constraint. It corre-
sponds to the following sequence. Let us take a node i. We s c d s c
+ +
express the flow conservation condition for each other node s s d
of the network j (cid:54)= i, considering only traffic having i as
sourcenode.Thisconditionstatesthatthetotalflowgenerated
b e b e
by i and leaving j is given by the total flow generated by i
and incident on j minus the number of requested connections (c)
having i as source and j as destination (C ).
i,j Fig. 2. The solenoidality constraint in flow formulation (a) and source
InFig.2weshowthedifferentapplicationofthesolenoidal- formulation(b).In(c),twoadmissiblesolutionsderivablefromtheprevious
ity constraint in the flow and source formulation cases using sourceformulationoutcome
twosimpleexamples.ThefirstexampleshowninFig.2arefers
to solenoidality constraint in the classical flow formulation. A
we have two entering flows and just one leaving due to the
singleconnectionrequesthasbeenroutedbetweensourcenode
flow which is dropped at that node).
s and destination node d through nodes a and c (dotted line).
It is worth noting that the source formulation does not
The flows associated to this connection are represented by a
return a detailed mapping of routing (i.e. a path for each
solidarrowintheroundwindowsthatmagnifythesituationin
single connection request), even if it optimally assigns the
nodes s, c and d: in the source (destination) the node leaving
number of fibers needed to support the traffic; a second step2
(entering) flow is equal to the offered traffic (i.e. a traffic
must be used to identify the routing of the connections. In
unit), while in the intermediate nodes the leaving flows equal
other words the source formulation loses the information of
the entering flows (e.g., in node c the leaving arrow has a
the routing of each single connection due to the aggregation
correspondent entering arrow).
of flows on the basic variable. Let us refer to Fig. 2c: over
Fig. 2b refers to solenoidality constraint working in source
the source formulation outcome shown in Fig. 2b, we can
formulation case: a simple network case with two connection
map two distinct (yet admissible) routing assignments (RA):
requests (between s and c and s and d) is shown. At the
in a first RA the two connections are routed on the two paths
source node s, the sum of the leaving flows is enforced to be
s−a−c−d and s−b−e−c, while a second admissible
equal to the sum of the traffics to be routed towards all the
RA could be s−b−e−c−d and s−a−c.
destinations (in this example two traffic units, one destined to
Thecapacityconstraint(3)allowsustodimensionthephys-
node c, the other destined to node d). In the other nodes the
ical network capacity. In order to ensure a feasible resource
sum of entering flows equals the sum of leaving flows plus
the traffic that is dropped at that node (e.g node c in Fig. 2b 2forfurtherdetailsseetheappendix
allocation it imposes that on each link the sum of flows The set of constraints is modified as shown below:
generated by all the nodes is smaller than the product of the (cid:88)
xi =s ∀ (i,λ); (6)
number of fibers by the number of wavelengths per fiber. The l,k,λ i,λ
remaining constraints (4 and 5) enforce variable integrity. (l,k)∈I+
(cid:88)i
Let us now discuss the source formulation complexity for
s =S ∀ i; (7)
i,λ i
a VWP network. Table I shows the relations expressing the
(cid:88) λ (cid:88)
total number of variables and constraints as functions of
xi = xi −c ∀ (i,j,λ),j (cid:54)=i;
the physical topology size and the number of node pairs l,k,λ l,k,λ i,j,λ
requiringconnections.Thecorrespondingrelationsfortheflow (l,k)∈Ij− (l,k)∈Ij+
(8)
and route formulation are reported for comparison (symbols
reported in Table I have been previously described, except
(cid:88)
for R that represents the mean number of possible alternative c =C ∀ (i,j) i(cid:54)=j; (9)
i,j,λ i,j
routes between two nodes in the network). In the table,
(cid:88)λ
C is the number of source-destination node-pairs requiring
xi ≤F ∀ (l,k,λ); (10)
connections, that is upper-bounded by the number of node l,k,λ l,k
i
pairs of the virtual topology C =N(N −1).
xi integer ∀ (i,l,k,λ);
The number of variables of the source formulation grows l,k,λ
with the product of the number of links by the number of Fl,k integer ∀ (l,k);
nodes. In the flow formulation it grows instead with the s integer ∀ (i,λ);
i,λ
product of the number of links by the number of node pairs
c integer ∀ (i,j,λ), i(cid:54)=j;
i,j,λ
requestingconnections.So,fromthevariablenumberpointof
view, source formulation should be more efficient than flow The solenoidality constraints are split into the sets (6) and
formulation under the condition C >N, that is presumably a (8) in order to impose flow conservation independently for
common situation in real networks. If there is at least one eachwavelength.Alsothecapacityconstraint(10)ismodified.
lightpath requested by each node pair, then we could set The new constraints (7) and (9) express the distribution of the
C = N(N −1) (the maximal value that could be achieved total number of connections among the different wavelengths
by C). Source formulation in this case allows a reduction forasourcenodeandforasource-destinationpairrespectively.
of the number of variables by a factor N compared to flow TableIcompares thecomplexityof sourceand flowformu-
formulation. The same order of reduction is obtained on lations also in the WP case. In both formulations the number
the number of constraints, whose complexity decreases from of constraints and variables increases linearly with W. It is
NC((cid:39)N3)toN2.Thepreviouscomparisonisfocusedonthe important to notice that the increase of W in the WP scenario
difference between flow and source formulations. As far the isaccompaniednotonlybyagrowthofvariableandconstraint
routeformulationwithoutconstrainedroutingisconcerned,we numbers,butalsobytheextensionofrangeofpossiblevalues
caneasilynoticethatthevariablesnumberisdependentonthe thatthevariablecantake.Althoughnotdirectlyarguablefrom
term C·R, i.e. the total number of possible alternative paths the table, this has a great impact on computational time and
for each node pair requiring connections in a network. This memory requirement.
meansthatthenumberofvariablestendstogrowveryquickly Theadvantageofthesourceformulationcanbeevaluatedin
withnetworkconnectivityanddimension,sothat,forexample asimplewaybyconsideringafully-connectedvirtualtopology
in our case study-networks, flow formulation is modelized by inwhichC =N(N−1).Undersuchassumptionthedominant
a lower number of variables than route formulation. termofthenumberofvariablesis2W·L·N and2WL·N2 for
source and flow formulation, respectively. As for the number
B. Source formulation for WP networks ofconstraints,thetwodominanttermsareW·N2 andW·N3,
Thesourceformulationcanbeextendedtonetworkswithout respectively.
wavelength conversion capability. ILP complexity in the WP Finally,weshallmentionalimitationofthesourceformula-
case grows with the number of wavelengths per fiber W tion.Unfortunately,thisformulationcannotbeextendedtoop-
andconstraintsbecomemorecomplicatedbecausewavelength timize path-protected WDM networks. In fact path protection
continuity has to be imposed on the lightpaths. Nevertheless requires to route lightpaths under the link-disjoint constraint,
the advantages of the source over flow formulation are still so that a working lightpath can not share any physical link
relevant. with its protection lightpath. The basic variables xi contains
l,k
The cost function is the same as in the VWP case III-A. information concerning all the connections having the same
A new index λ ∈ {1,2,...W} must be added to identify source node aggregated together. No explicit reference can be
the wavelength of the WDM channels, in order to impose inferred regarding lightpaths having the same source and the
the wavelength continuity constraint along a lightpath. Flow same destination, so that the link-disjoint constraint can not
variablesdefinedintheVWPcasearetransformed:xi now beenforced.Anywayotherprotectiontechniques,suchaslink
l,k,λ
indicates the number of WDM channels having wavelength protection, could be planned using source formulation. In fact
λ which on the “unidirectional link” (l,k) carry lightpaths an approach to link protection consists in providing for each
generated at node i. The known terms S and C have to be link (i.e for all its fibers) an alternative route in order to face
i i,j
split, originating the new variables s and c . linkfailure:suchafeaturedoesn’tneedinformationrelatedto
i,λ i,j,λ
TABLEI
COMPARISONONCONSTRAINTANDVARIABLENUMBERSBETWEENSOURCEANDFLOWFORMULATIONS.
formulation constraints variables
VWPsource 2L+N2 2L(1+N)
VWPflow 2L+N·C 2L(1+C)
VWProute 2L+C C·R+2L
WPsource W(2L+N2)+C+N W(N+2L·N+C)+2L
WPflow W(2L+(N−1)·C)+C WC(1+2L)+2L
WProute C·(W +1)+2L·W C·R·W +C·W +2L
TABLEII
ILPVARIABLESANDCONSTRAINTSFORNSFNETANDEONINTHE
VWPCASE.
network/formul. constraints variables
NSFNET/source 240 660
NSFNET/flow 1556 4796
NSFNET/route 152 14604
EON/source 439 1560
EON/flow 6576 26754
(a) (b) EON/route 420 ∼=3·108
Fig. 3. Physical topologies of two case-study networks: (a) NSFNET and
(b)EON.
using the aggregation of flows. Data are computed using the
relations reported in Table I except the number of variables in
trafficdestinationnode.Asourceformulationbasedmodelfor
routeformulation,whichneedsasinputvariablealladmissible
link protection in both dedicated and shared cases is currently
pathsinthenetworkbetweennodesrequiringconnections.So,
under study.
inordertorunanoptimizationbasedonnon-constrainedroute
formulation we have precomputed all the possible alternative
IV. CASESTUDIESANDRESULTCOMPARISON paths using a greedy routine: our algorithm takes about five
In this section we present and discuss the results obtained hours to compute the 14604 paths in the NSFNET network,
by ILP optimization exploiting source formulation on two while in the EON, due to its greater dimension, our algorithm
case-study network types in comparison with results obtained takesabouttenhourstocomputeabout1.57·105pathsbetween
usingtraditionalfloworrouteformulations.Well-knownmesh asinglenodeandalltheothernodestakenasdestinations.The
networks are considered first, that is the National Science huge number of variables in this last case induced us not to
Foundation Network (NSFNET) and the European Optical proceed on EON optimization based on unconstrained route
Network (EON). Then a class of networks called “wheel formulation.
networks” are considered, in which the variation of the con- For the WP case, a comparison can be done taking into
nectivity index [2], [11] defines a set of topologies ranging account route formulation complexity. In this latter case, as
from the ring to the full-mesh network. wecanarguefromFig.4,thenumberofconstraintsissmaller
than in source formulation, but the number of variables grows
rapidly. In fact it is associated to the number of all the
A. NSFNET and EON
possible routes connecting each node couple, that increases
DataregardingthephysicaltopologyofNSFNETandEON, exponentially with network dimension and in particular it is
represented in Fig. 3a and 3b, have been taken from Ref. influenced by the connectivity index of the network. In the
[21]3 and Ref. [26], respectively. NSFNET has 14 nodes and following we will see how the increase in network dimension
22 links, while EON has 19 nodes and 39 links. The virtual and connectivity will affect ILP model performance.
topologies are based on the static (symmetric) traffic matrices To solve the ILP problems we used the software tool
derived from real traffic measurements which are reported CPLEX 6.5 based on the branch-and-bound method [27].
in the same references. The two traffic matrices comprise As hardware platform a workstation equipped with a 1 GHz
360and1380unidirectionalconnectionrequestsforNSFNET processor was used. The available memory (physical RAM +
and EON, respectively, while the distinct node pairs requiring swap) amounted to 900 MByte.
connections 4 are respectively 108 and 342. Both VWP and
Before the presentation of numerical results, it is crucial
WP cases have been analyzed.
to remember that the the source and the flow formulation are
Table II shows the number of variables and constraints that
equivalent (see appendix). This equivalence is confirmed in
are involved in the ILP problem applied to the two networks
all the network cases in which both formulations succeed in
in the VWP case. They clearly show the advantage achieved
finding the optimum value: this value in fact results to be the
same in the two formulations.
3ThereportedNSFNETtopologyisactuallytheNSFNETT1backbone[6]
We have already shown the advantage of source formula-
withtheadditionofoneextralink.
4Eachnodepaircanrequiremorethanoneconnection. tion versus flow and route formulation in terms of variable
NSFNET WP TABLEV
1,2 105 VWPEONOPTIMIZATION:COMPUTATIONALTIMEANDgapBETWEEN
source
1 105 frloouwte INTEGERSOLUTIONFOUNDANDLOWERBOUNDRETURNEDBYBRANCH
ANDBOUNDALGORITHM
bles 8 104
a
ari EON sourceform. flowform.
er of v 6 104 W time gap time gap
Numb 4 104 24 21.2mh 0.306%% 135.5.6hh 1.02%%
2 104 8 2.2h 0.87% 20.3h 2.5%
16 2.3h 1.74% 19h 6.2%
0 32 1.7h 9% 46h 15%
W=2 W=4 W=8
Number of wavelengths, W
NSFNET WP
1,2 104 available memory is filled up before the optimal solution can
source
1 104 frloouwte be found. In this cases CPLEX returns the best but non-
optimal solution that branch-and-bound has been able to find
aints 8000 and forces the execution to quit. These cases are identified by
er of constr 6000 mtheeaosuurte-osfh-mowemloonrygtiatgha(sOt.aOk.eMn.t)oafinldlutphemceommopryu.taTthioisnainltteigmeer
b
um 4000 solution, forced to be returned because of the limited amount
N
of memory, is associated to the so-called gap parameter
2000
that expresses the percentage difference between the integer
0 solution found and the minimal possible value the solution
2 4 8
Number of wavelengths, W could reach (i.e. a lower bound returned by branch and bound
Fig.4. ILPvariablesandconstraintsforNSFNETintheWPcase. algorithm).Thisparameterreturnsanestimationofthequality
ofthenon-optimalintegersolutionfoundintermsofmaximal
TABLEIII possible distance from the optimum. From tables III and IV
VWPNSFNETOPTIMIZATION:COMPUTATIONALTIME. we can see that the out-of-memory event is less frequent with
the source formulation than with flow formulation. Moreover
W sourceform. flowform. routeform.
source formulation always requires a smaller memory amount
2 27m 1.5h 1.6h
and a shorter run duration than the other two formulations.
4 55m 3.7h 1.8h
8 36s 26m 1.5h The gap in run duration between source and the other two
16 3m 9h 3h tends to increase with the W parameter. This is probably due
32 19m 7,9h 6h
to the extension of the range of the possible values that the
variable can take.
Table V shows resource occupation comparison between
and constraint numbers. It is important to see how much
source and flow formulation in EON network (the route for-
this advantage affects the actual computational performance
mulation is not feasible due to the huge number of variables);
of ILP. Tables III and IV display computational time and
again computational times in bold are associated to runs suc-
memory occupation measurements of NSFNET optimization
ceeding in finding optimal values. We will show here the gap
in the VWP case (s, m, h and d stand for seconds, minutes,
parameter to compare the quality of integer solution found:
hours and days respectively, while MB stands for mega-byte).
neither source formulation nor flow formulation succeeds
Computationaltimesinboldareassociatedtorunssucceeding
in demonstrating the optimality of returned integer solution
in finding optimal values.
except for W =2, but quality of source formulation solutions
To clearly understand the reported data, a particular aspect
is evidently better than quality of flow solutions. So O.O.M
of ILP must be clarified. The branch-and-bound algorithm
event happens in all the optimizations, while for W = 2 the
progressively occupies memory with its data structure while
amount of occupied memory is about 10 MB in both cases.
it is running. When the optimal solution is found, the algo-
NSFNEToptimizationintheWPscenariotakesaverylong
rithm stops and the computational time and the final memory
time with the hardware we employed. In some cases it was
occupation can be measured. In some cases, however, all the
too long to wait either for the optimal result or for an out-
of-memory event. Thus in table VI we have reported the time
TABLEIV necessary to fill up of the first 100 MB of memory. The case
VWPNSFNETOPTIMIZATION:MEMORYOCCUPATION. with W ≥ 16 has proved to be too complex to be solved in
a reasonable time and therefore it has been omitted (except
W sourceform. flowform routeform. for W = 16 in the source formulation). The speed of the
2 0,39MB 1,3MB 33MB
branch-and-bound algorithm applied to the flow formulation
4 O.O.M O.O.M. O.O.M.
decreases dramatically for high values of W. Although in the
8 5MB 42MB 432MB
16 47MB O.O.M. 852MB source formulation the speed does not decrease so much, the
32 180MB O.O.M. 750MB model simplification allows a significant computational time
Source vs Flow Source vs Flow
12 30
M 10 NSFNET VWP M 25 NSFNET VWP
EON VWP e EON VWP
ce, 8 enc 20
differen 6 er differ 15
Total fiber 24 Percent fib 150
0
0
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
Number of wavelengths, W Number of wavelengths, W
(a) (b)
Fig.5. Source-flowcomparisononthefinalnumberoffibersintheVWPcase,asabsolute(a)andpercentrelativedifference(b).
TABLEVI NSFNET WP
WPNSFNETOPTIMIZATION:TIMEREQUIREDTOFILLUP100MBOF 12 25
MEMORY. M 10 20 Pe
rc
1W2486 sour16c58.e.772hhdfhorm. flow61102–fhdohrm. rou1t30e99.–3fhhohrm. otal fiber difference 2468 percent difference 51105 ent fiber difference
T Absolute difference M
decrement compared to flow formulation. The route formula-
0 0
tion, despite the great difference on the number of variables,
0 5 10 15 20 25 30 35
has comparable results regarding computational performance;
Number of wavelengths, W
this is probably due to the structure of route variables that
make simpler to set the wavelength continuity constraints. Fig.6. Source-flowcomparisononthefinalnumberoffibersintheWPcase
Now we are going to compare the source and the flow forNSFNET.
formulationsonthebasisofthefinalvalueofthecostfunction.
In all the cases in which, for both formulations, the branch-
case for W = 2 and W = 8, in accordance with table IV5.
and-bound ends up before an out-of-memory event, the final
It should be noted that sub-optimal solutions with the flow
values obtained are coincident, thus proving the equivalence
formulation can be up to 18% worse than the corresponding
ofsourceandflowformulation.Inalltheothercases,thebest
solutions produced by the source formulation.
integer solution returned by the SF is smaller than or equal to
the FF best integer solution. Furthermore, the gap parameter InFig.6∆M and(cid:178)M aredisplayedforNSFNETintheWP
associated integer solutions is always smaller for SF than for case.Inthiscasethestrongincreaseofvariableandconstraint
FF, showing that SF optimization runs are able to get closer number with W causes a relevant increase of the differences
to lower bound of the problem. between the two formulations. It is worth noting that both SF
We focus our analysis on the performance comparison be- orFFareunabletoreachtheendofoptimizationruns(except
tween flow and source formulation. The following parameters for the W = 2 case). We evaluate the quality of the integer
are introduced: solution not only comparing its numerical value (as shown in
Fig.6),butalsocomparingthegapparameterassociatedtothe
• Msource(Mflow): total fiber number returned by ILP integer solution: for W=2,4,8,16 the value of the gap is equal
based on the source (flow) formulation;
to 0%, 1,9%, 5.5% and 12% for SF and 0,3%, 2,4%, 8%, and
• ∆M: difference Mflow−Msource; 30% for FF. In conclusion, integer solutions provided by SF
• (cid:178)M: percent relative difference 100 · (Mflow − outperform solutions provided by FF as far as the numerical
M )/M ;
source source values,computationaltimesandproximitytothelowerbound
Wewillshowresultsconcerningthetwocase-studynetworks.
are concerned.
The difference between the two formulations obtained in
Up to this point of the paper we have described multifiber
the VWP scenario are represented in Fig. 5a (absolute values)
WDM network optimization having the total number of fibers
andinFig.5b(percent)asfunctionsoftheparameterW.The
absolute difference is on average greater for the EON which
5in the case W = 4, SF and FF return the same integer solution, which
hasalargernumberof nodes andlinks. Convergencebetween
is very likely to be the optimal, but the both of them fail in proving the
the two formulations occurs, for example, in the NSFNET optimalityofintegersolutionfound
Source vs Flow
as cost function. This interpretation of network cost is called
5
hop metric and it models a situation in which all the fibers of
thenetworkhavethesamecost.Howeverinrealnetworksthe P NSFNET VWP
4
cost of a link also depends on its geographical length, which ce EON VWP
n
for example determines the number of optical line amplifiers ere 3
that must be installed. Measuring the cost of a fiber in this diff
situationbecomesmuchmorecomplicatedandthehopmetric h 2
gt
is not appropriate any more. Another simple alternative is the n
e
length metric, which assigns a cost to each fiber proportional nt l 1
e
to the geographical length of the link it belongs to. Although c
er 0
still not completely realistic (e.g. it does not take into account P
that the cost of the duct should be shared by all the fibers of -1
a link), it could be useful in many situations (e.g. when the 0 10 20 30 40 50 60 70
cost of optical line amplifiers is an important issue). Clearly, Number of wavelengths, W
the hop metric can be regarded as a particular case of length
metric in which all the links have unity length6. Fig. 7. Source-flow comparison on the final total fiber length in the VWP
case,aspercentrelativedifference.
We have tested source formulation based on length metric
on NSFNET and EON in the VWP case. In a fashion similar TABLEVII
to hop metric, the following parameters have been defined WHEELNETWORKSOPTIMIZATION:RESULTSONFIBERNUMBER
• Psource(Pflow): total fiber length returned by ILP based W α=0.29 α=0.43 α=0.57 α=0.71 α=1
on the source (flow) formulation;
2 64 44 40 38 38
• ∆P: difference Pflow−Psource; 4 32 24 23 23 23
• (cid:178)P: percent relative difference 100 · (Pflow − 8 16 15 15 15 15
16 14 10 10 10 10
P )/P ;
source source 32 8 8 8 8 8
Link lengths p were assigned for the two networks according
l
to Refs. [21], [26].
Fig. 7 displays the percent relative difference between the WPcase,however,heuristicmethodsaremuchfasterthanILP,
total fiber lengths obtained applying the source and the flow even when source formulation is adopted. It can be noticed in
formulation. The same conclusions drawn for the hop metric Fig. 8b that heuristic has been the only possible approach to
canbeextendedtothesenewoptimizationexperiments.Source obtain a result with W = 32, given the hardware limitations
formulation performs better in all the cases in which an out- of our workstation. Concluding, SF has provided a useful
of-memoryeventoccurs;otherwise,theresultsarecoincident, benchmark to evaluate the performance of a heuristic strategy
butsourceformulationconvergesmorerapidly(computational on NSFNET and EON, that can be considered two significant
timesareomittedforbrevity).Itisworthnotingthatthelength test-case networks.
metric results in more solutions found for the considered set
of W with respect to hop metric case: this is because there
B. Wheel networks
are less tie-breaks in B&B algorithm than in the hop metric
Inordertoshowtheeffectivenessofoursourceformulation,
case, where the weight assigned to each link is the same.
we have performed optimization experiments also on the set
In confirmation of this observation, the values of the gap
of 8-node “wheel networks” shown in Fig. 9. This network
parameter for the length metric are always smaller than those
class defines topologies with increasing connectivity degree,
reported for hop metric in Table V.
starting from the ring network and ending with a full mesh
FinallyweshowacomparisonbetweentheILPoptimization
network. This is obtained by increasing the number of edges
carried out by source formulation and the optimization by
with respect to initial ring topology, so that the connectivity
the heuristic approach described in Ref. [28]. Let us consider
index α (i.e. the ratio between the number of links in the
NSFNET and the hop metric. In Fig. 8 ILP and heuristic final
considered network and the number of link in the full-mesh
results are displayed in the VWP (Fig. 8a) and WP case (Fig.
networkcase)assumesthevalues0.29(ring),0.43,0.57,0.71,
8b).
1 (full-mesh). Again, we have assumed different values of
Thecomparisonshowsthattheresultsofthetwotechniques
W, that is W = {2,4,8,16,32}. This new class of network
are quite close: the heuristic approach is able to provide good
topologies will allow us to better appreciate the behavior of
sub-optimal results, but only the exact approach allows to
SF with respect to FF and RF, while varying one crucial
reach the absolute optimum (or to come closer to it when
network parameter, the connectivity index. We assume that
limitationsonmemoryorcomputationaltimepreventsbranch-
offered traffic is uniform and equal to one connection request
and-bound to converge). As far as the computational time
for each node couple.
is concerned, we have noticed that heuristic and source-
The numerical results obtained by SF for these networks
formulation ILP behave similarly for VWP networks. In the
are summarized in Table VII. Let us now analyze in detail
these results. SF, FF and RF all lead to the optimum solution
6Neitherhopnorlengthmetrictakethenodecostintoaccount.Node-cost
optimizationissuesarenotcoveredbythispaper. for α = 0.29, 0.43, 0.57 and all values of W, for α=0.71
NSFNET VWP NSFNET WP
400 400
350 source form. 350 source form
M 300 heuristic M 300 heuristic
er, er,
b 250 b 250
m m
u u
n 200 n 200
er er
al fib 150 al fib 150
ot 100 ot 100
T T
50 50
0 0
0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35
Number of wavelengths, W Number of wavelengths, W
(a) (b)
Fig.8. NSFNETtotalfibernumberoptimizedbyILPsourceformulationandbyadeterministicheuristic,intheVWP(a)andWP(b)cases.
α=0.29 α=0.43 α=0.57, α=0.71 α=1
Fig.9. Wheelnetworkswithdifferentconnectivitydegrees.
TABLEVIII
with W = 2,16,32 and for α=1 with W = 16,32 (the
“WHEEL”NETWORKWITHα=1:RESULTSONFIBERNUMBER
correspondingnumericalvaluesareinboldinTableVII.When
all the three formulations succeed in finding optimal values, W SF FF RF
SF takes sensibly lower computational times than the other 2 38 39 39
two formulations (see the corresponding values in Fig. 10, 4 23 25 28
8 15 15 17
where we have drawn the computational times for each value
16 10 10 13
of connectivity index).
Moreover SF allows us to obtain optimal values also in
some network cases in which FF and RF fail; in fact for α= SF, due to the exponential increase of admissible paths in the
0.71 with W = 4,8 and α=1 with W = 16 SF succeeds full-meshed case.
in finding optimal values that are not reached by FF and RF
Concluding, Fig. 10 clearly shows the difference in compu-
(thesevaluesarereportedinitalicstyle).Forthesethreecases,
tational time among the three approaches: SF outperforms FF
Fig. 10 campares the computational times SF takes to reach and RF for all the values of W8. In particular, for increasing
the optimal value with the computational times required by
valuesofα,alargernumberofvariablesandconstraintsresults
FF and RF to reach their best integer values under O.O.M.
inlargercomputationaltimes,butSFkeepsreturningthebetter
condition.
results.
Finally in the cases α=1 with W = 2,4,8 the three
formulations fail in proving the optimality of the best integer
V. CONCLUSIONS
solution returned, so the computational times refer all to
O.O.M cases (time needed to occupy the whole memory). We have presented and discussed a novel formulation,
So, to better appreciate the performance of SF in full-mesh calledsourceformulation,tosolvestatic-trafficWDMnetwork
network, in Table VIII we have reported the best integer optimization by ILP. This formulation has been defined for
solutions reached by the three formulations before O.O.M multifiber networks with or without wavelength conversion
event7. The results obtained by SF are strong candidates to capability supporting unidirectional unprotected optical con-
be the effective optimal values, because they are equal to the nections. Thanks to the source formulation, we are able to
optimal values for α=0.71. FF shows worse results than SF substantially prune the multiplicity of both variables and
on fiber number, but in particular RF returns results far from
8Wehaveomittedtoreporttheringnetworkcase,whichisrapidlysolved
by each of the three approaches. Anyway a deeper analysis of this simple
7ExceptforW =32forwhichallthethreeapproachesfindtheoptimum caseshowsthatB&Balgorithmexploresalowernumberofnodesinthecase
solution ofSFcomparedtoFFandRF.