with Optical Bypassing
by
Poompat Saengudomlert
B.S.E., ElectricalEngineering, Princeton University (1996)
M.S., Electrical Engineering and Computer Science,
Massachusetts Institute of Technology (1998)
Submitted tothe Departmentof Electrical Engineeringand ComputerScience
inpartial ful llmentof the requirementsfor the degreeof
Doctor of Philosophy
atthe
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2002
c
Massachusetts Institute of Technology 2002. All rightsreserved.
Author...
Department of ElectricalEngineering and Computer Science
August 31, 2002
Certi ed by...
Eytan H. Modiano
AssistantProfessor, Department of Aeronautics and Astronautics
Thesis Supervisor
Certi ed by...
Robert G.Gallager
Professor, Department of ElectricalEngineering and Computer Science
Thesis Supervisor
Accepted by...
ArthurC. Smith
with Optical Bypassing
by
PoompatSaengudomlert
Submitted to theDepartment ofElectricalEngineering and ComputerScience
onAugust 31, 2002,inpartialful llmentof the
requirementsforthedegree of
Doctorof Philosophy
Abstract
We study the routing and wavelength assignment (RWA) problem in wavelength division
multi-plexing(WDM)networkswithnowavelengthconversion. Ina high-speedcorenetwork,thetraÆc
can be separated into two components. The rst is the aggregated traÆc from a largenumberof
small-rateusers. Each individualsessionisnotnecessarily static butthecombinedtraÆc streams
betweeneach pair of access nodesare approximately static. We support thistraÆc by static
pro-visioning of routes and wavelengths. In particular, we develop several o -line RWA algorithms
which use the minimumnumberof wavelengths to providel dedicated wavelength paths between
eachpair of accessnodesforbasicall-to-allconnectivity. Thetopologies weconsiderarearbitrary
tree,bidirectionalring,two-dimensionaltorus, and binary hypercube topologies. We observe that
wavelengthconvertersdonotdecreasethewavelengthrequirementtosupportthisuniformall-to-all
traÆc.
The second traÆc component containstraÆc sessions from a smallnumberof large-rate users
and cannot be wellapproximatedas static dueto insuÆcient aggregation. To supportthis traÆc
component, we performdynamicprovisioningof routesand wavelengths. Adoptinga nonblocking
formulation, we assume thatthe basic traÆcunit is a wavelength, and the traÆc matrix changes
fromtimeto timebutalwaysbelongsto agiven traÆcset. Morespeci cally,letN be thenumber
of access nodes, and k denote an integer vector [k
1 ;k
2 ;:::;k
N
]. We de ne the set of k-allowable
traÆc matrices to be such that, in each traÆc matrix, node i, 1 i N, can transmit at most
k
i
wavelengths and receive at most k
i
wavelengths. We develop several on-line RWA algorithms
whichcan supportallthek-allowable traÆcmatrices inarearrangeablynonblockingfashionwhile
usingclose to the minimumnumber of wavelengths and incurring few rearrangements of existing
lightpaths,ifany,foreachnewsessionrequest. Thetopologiesweconsiderarethesameasforstatic
provisioning. We observe that the number of lightpathrearrangements pernew sessionrequest is
e.g. networksize, butnoton thetraÆcvolumerepresentedbykas weincreasek bysome integer
factor.
Finally,webeginexploringanRWAprobleminwhichtraÆcisswitchedinbandsofwavelengths
ratherthanindividualwavelengths. Wepresentsomepreliminaryresultsbasedonthestartopology.
ThesisSupervisor: Eytan H.Modiano
Title: Assistant Professor,Department of Aeronautics and Astronautics
ThesisSupervisor: RobertG. Gallager
Firstandmostimportantly,Iwouldliketothankmytwothesisadvisors,ProfessorEytanModiano
andProfessorRobertGallager,formakingmylearningexperience atMITtruly memorable. Ifeel
extremelyfortunatetohavehadachancetoworkwiththem. Withouttheirguidanceandpatience,
thisthesis wouldnotbepossible.
Professor Vincent Chan has been very kind to me by spending a lot of time on various
dis-cussions. I am grateful for his support. I also want to thank my past advisors, Professor Muriel
Medard and Professor Amos Lapidoth, forteaching me how to do research in the early stages of
mygraduatestudy.
Several fellowgraduatestudentscreatedthefriendlyandhighlyintellectualatmospherearound
MITforme. Iwouldliketo recognizeseveralofthem: ChaleeAsavathiratham,RandallBerry,
Ser-enaChan,Li-WeiChen,AaronCohen,ToddColeman,AlvinFu,AngeliaGeary,Chi(Kyle)Guan,
Shane Haas, Tracey Ho, Ramesh Johari, Thierry Klein, Emre Koksal, Julius Kusuma, Nicholas
Laneman, Chunmei Liu, Thit Minn, Michael Neely, Asuman Ozdaglar, Natanael Peranginangin,
BrettSchein,KarinSigurd,JunSun,PoonsaengVisudhiphan,FrankWang, Hung-JenWang,Guy
Weichenberg, EdmundYeh, andWon Yoon.
My stay in Edgerton House was a lot of fun thanks to my roommates Yu-Han Chang and
Lawrence (Kai) Shih who made our apartment feel like home to me. In Edgerton House, I also
enjoyed the company of Stephane Bratu, Pei-Lin Hsiung, Eden Miller, Minghao Qi, Sarah Rhee,
andChatchaiUnahabhokha.
Mythree high-schoolclassmatesfrom Thailand,AriyaAkthakul, Chalee Asavathiratham,and
Poonsaeng Visudhiphan,joined me at MIT and gave me a lot of moral support for my research.
Besidesthem,IamgratefultoknowseveralotherThaistudentsatMIT:PongpunAmornvivat,
Su-tapaAmornvivat,SiwaphongBoonsalee,YotBoontongkong,SiriwutBuranapin,ViratChatdarong,
Nuwong Chollacoop,AcharawanChutarat,ThunyachateEkvetchavit, Kachaphol Harinsuit,Singh
Intrachooto,JenJootar, ThanisaraKiatbaramee,WatjanaLilaonitkul,PimpaLimthongkul,Manoj
Lohatepanont, Narintr Narisaranukul, Mukaya Panich, Joshua Pas, PitipornPhanaphat, Piyada
Phanaphat, Pradya Prempraneerach, Taweechai Pureetip, Thirapun Sanpakit, Thapanee
Siri-vadhanabhakdi, Paisarn Sonthikorn, Atiwong Suchato, Suchatvee Suwansawat, Isra Taulananda,
Thep-Throughoutmy study, I received constant love and supports from mybeloved family and my
1 Introduction 9
1.1 OpticalBypassing inAll-OpticalNetworks. . . 9
1.2 Recon gurableSwitchingNode Model . . . 11
1.3 Switching ofTraÆc inLargerGranularities . . . 13
1.4 CoreNetwork with AggregatedTraÆc . . . 14
1.5 Outlineofthe Thesis . . . 16
2 History and Problem Formulations 17 2.1 ExistingLiterature on theRWA Problem . . . 17
2.2 ThesisObjectives . . . 19
2.3 RWA ProblemforStatic TraÆc . . . 21
2.4 RWA ProblemforDynamic TraÆc . . . 22
3 RWA for Static l-Uniform TraÆc 24 3.1 ArbitraryTreeTopologies . . . 24
3.1.1 RegularTreeTopologies . . . 52
3.2 BidirectionalRingTopologies . . . 56
3.3 2DTorus Topologies . . . 61
3.4 BinaryHypercube Topologies . . . 63
3.5 ArbitraryTopologies . . . 68
3.5.1 Lower Boundon L s;l : theLinkCountingBound . . . 68
3.5.2 Lower Boundon L s;l : theCut SetBound . . . 69
3.5.3 UpperBound onW s;l : theEmbeddedTree Bound . . . 70
s;l s;l
4 RWA for Dynamic k-Allowable TraÆc 75
4.1 Star Topologies . . . 76
4.2 ArbitraryTreeTopologies . . . 83
4.3 Bidirectional RingTopologies . . . 91
4.3.1 RWA foraSingle-HubBidirectionalRing . . . 102
4.3.2 BidirectionalRingwithWavelength Converters . . . 104
4.4 2D TorusTopologies . . . 107
4.5 BinaryHypercubeTopologies . . . 115
4.6 ArbitraryTopologies . . . 116
4.6.1 LowerBound onL d;k : theLinkCountingBound . . . 116
4.6.2 LowerBound onL d;k : theCut SetBound . . . 117
4.6.3 UpperBoundon W d;k : theEmbeddedTreeBound . . . 118
4.6.4 UpperBoundon W d;k in termofL d;k : theGraphColoringBound . . . 121
5 Band/Wavelength RWA Problem 123 5.1 Reduction inOpticalSwitchesthroughBandSwitching . . . 124
5.2 Trade-O betweenOpticalSwitchesand Wavelengths . . . 128
5.3 AlternativeNetwork Architecture forBandSwitching . . . 129
5.4 TraÆc AggregationforBand Switching . . . 133
6 Conclusion and Directions for Future Research 136 A EÆcient Bipartite Matchings with Maximum Node Degree 2 139 B W d;k for Bidirectional Rings 143 B.1 Proofof W d;k =d3k=4e forN =3 . . . 144 B.2 Proofof W d;k =k forN =4 . . . 145 B.3 Proofof W d;k =d5k=3e forN =6 . . . 146
Introduction
1.1 Optical Bypassing in All-Optical Networks
Optical ber is a communication medium with a potential transmission bandwidth up to 25
THz [Gre93]. Practical networks employ wavelength division multiplexing (WDM) in which the
berbandwidthisdividedintomultiplefrequencybandsoftencalledwavelengths. Incurrent
prac-tical WDM systems, only a portion of the ber bandwidth is utilized. In addition, the highest
transmission rate over a single wavelength is 40 Gbps, whereas the total transmission rate over
multiplewavelengths inasingle beriscurrentlybeyond 1 Tbps[RS01 ].
While processing WDM traÆc electronically at every network node may be technologically
feasible, it yieldsa very expensive network architecture. Electronic processing at every node was
adopted in the early days of communication networks, e.g. the ARPANET, when the cost of
transmissiondominatedthecostofprocessingat allthenetworknodes. However,forcurrent
high-speednetworks with optical transmissiontechnology,we expect thecost of electronic information
processing to dominate the cost of optical information transmission. Therefore, it is desirable to
eliminateunnecessary electronic processing in thenetwork. For example, considerthe scenario in
gure1-1. Thereare two sources,each sendingone wavelength worth oftraÆc to thedestination.
The wavelength from source1, denoted by
1
, can be combined withthe wavelength from source
2,denoted by
2
,using an optical multiplexerwithout anyelectronic processing. In thiscase, we
saythatthe traÆcsessionfromsource1 optically bypasses electronic processingat node 2.
source1 source2 destination
1
2
Figure1-1: Examplenetwork toillustrate opticalbypassing.
required by electronicprocessing. However, thisis notalwaysthe case. Suppose forexamplethat
each source in gure 1-1 transmits only half a wavelength worth of traÆc to the destination. It
followsthat, withopticalbypassingoftraÆcfromsource1at node2,i.e. nomultiplexingoftraÆc
at the subwavelength level, we needto utilize two wavelengths on thelinkfrom node 2 to node3.
If we use an electronic switch at node 2 to multiplex the two traÆc streams, then only a single
wavelength isrequired. Thus,optical bypassingmayrequiremore wavelengths whenthebypassed
traÆcsessionshave smallerratesthantherateofa singlewavelength. Despiteadditional required
wavelengths, thecostsavings fromtheeliminationof electronicprocessingcouldstillbeattractive
enough to justifyoptical bypassing.
Inan all-optical networkarchitecture,each traÆc sessionopticallybypasseselectronic
process-ing at all intermediate nodes, i.e. nodes that are neither the source nor the destination of that
session. Inother words,there isno electronicreceptionand retransmission ofdatapackets byany
intermediatenode. We shall concentrate on all-opticalnetwork architectures inthisthesis.
OpticalwavelengthchangersallowustochangethewavelengthofatraÆcsessionat
intermedi-ate nodes withoutelectronic processing. Sinceoptical wavelength changersarevery expensive,we
shallassumenoopticalwavelengthconversionexceptwhenexplicitlyindicated. Withthis
assump-tion, each optically bypassed traÆc session is subjected to the wavelength continuity constraint,
which dictates that the session must travel on the same wavelength on all links from the source
nodetothedestinationnode. ForagiventraÆcsession,de neitslightpathtobetherouteand the
wavelength usedto supportthatsession. There areusuallymultipleways to assignalightpath for
a given session. The problemof assigninglightpathsforall traÆcsessionsinthe network iscalled
therouting andwavelength assignment (RWA)problem, which isthemaintopic of thisthesis.
We have seen an example in which optical bypassing increases the required number of
wave-lengths ina berwhentheratesof bypassedtraÆcsessions aresmallerthanone wavelength unit.
constraint. Forexample, considerthescenario in gure 1-2. The rate of eachsession isone
wave-length. Without optical bypassing, the required number of wavelengths in each ber is equal to
themaximum link load which is two wavelengths inthis example. Onthe other hand, with
opti-calbypassing,weneed three wavelengths because each lightpath necessarilysharesa transmission
linkwith each of theother two lightpaths and thus needs a distinct wavelength. Notice that two
wavelengths suÆceinthisexampleifwavelengthchangersare employed.
on 1 session1 session2 session3 on 3 on 2
Figure 1-2: Increase in thenumberof wavelengthsdue tothewavelengthcontinuityconstraint.
In short, optical bypassing serves as an approach to reduce the cost of electronic processing
of information at the network nodes, but possibly at the cost of additional wavelengths. In fact,
optical bypassing can be viewed as a special case of the general trade-o between switching and
transmission costs in communication networks. What motivates us in this special case is the
potential of a signi cant reduction in switching cost with only a slight increase in transmission
cost.
1.2 Recon gurable Switching Node Model
Ourgenericmodelofarecon gurableswitchingnodeisillustratedin gure1-3. TraÆc sessionson
each input berareseparated byan optical demultiplexer(DMUX). The wavelengths (and hence
thetraÆcsessions) onthesamewavelength fromdi erentinput bersgo througha recon gurable
than one input(output) can be connectedto a single output (input). TraÆc sessions on di erent
wavelengths switched to the same output ber are combined by an optical multiplexer (MUX).
Some inputsessions are terminated or dropped to the end users or the subnetwork connected to
thisnetwork node. Similarly,some outputsessionsaretransmittedoraddedfromtheendusersor
thesubnetwork. ::: fullytunable transmitters . . . . . . . . . . . . . . . . . . . . . ::: . . . . . . . . . . . . . . . ::: W . . . 1 recon gurable switch switch ::: receivers fullytunable input bers output bers MUXs DMUXs switches wavelengthselective recon gurable recon gurable
Figure1-3: Recon gurableswitchingnodemodel.
Weshallassumethatopticaltransmittersandreceiversarefullytunable,i.e. asingletransmitter
(receiver) can be used to transmit (receive) on any wavelength in the ber. When possible, we
shall discuss how this assumption can be relaxed. The use of tunable transmitters and receivers
requires additionaloptical switches inorder to guide thetransmitted and received wavelengths to
appropriateoptical switches, asillustratedin gure 1-3.
Certain wavelengths may be used to provide dedicated static connections. For these
wave-lengths, thetransmitters and receivers neednot be tunable. In addition,we can replace
recon g-urable wavelength selective switches with xed wavelength selective switches. Figure 1-4shows a
recon gurable switching node model inwhich a subset of wavelengths, denoted by
V+1 , ...,
W ,
. . . . . . . . . . . . input bers . . . . . . . . . W V +1 . . . . . . . . . V 1 . . . . . . . . . . . . switches wavelengthselective recon gurable recon gurable switch ::: switch ::: . . . . . . . . . . . . ::: ::: ::: ::: xed wavelength selective DMUXs . . . output bers . . . . . . . . . MUXs recon gurable switches receivers fullytunable fullytunable transmitters non-tunable transmitters non-tunable receivers
Figure 1-4: Recon gurable switching node model in which wavelengths
V+1 , ...,
W
are used for
dedicatedstaticconnections.
1.3 Switching of TraÆc in Larger Granularities
As the amount of traÆc among network nodes increases, it is more eÆcient to switch traÆc in
largerandlargertraÆcunits. Morespeci cally,weexpectto switchtraÆcinunitsofwavelengths,
bands of wavelengths, bers, bundles of bers, and so on. At each increment of the traÆc unit,
there is a potential cost saving from bypassing the processing of traÆc in the smaller unit at
wavelengths. Ifweexpecta bandof wavelengths to bea commontraÆc unit,then we can bypass
wavelength-level optical MUXs and DMUXs, i.e. use onlyband-leveloptical MUXs and DMUXs,
possibly at the price of more bands of wavelengths. Notice that, for di erent increments of the
traÆcunit,thelogicalproblemsofhowto eÆcientlybypass theprocessingoftraÆcinthesmaller
unit aresimilar. Themaindi erences lie rst inacommontraÆc unit,andsecond inanavailable
switchingtechnologyforthatunit. ByanappropriatescalingofthecommontraÆcunit,asolution
for the bypassing problem with one common traÆc unit may be used for the bypassing problem
withanother commontraÆcunit.
However,thetrade-o sbetweenthereductioninswitchingcostandtheincreaseinwavelengths
can di er greatly in the bypassing problems with di erent common traÆc units. For example,
when a wavelength is a common traÆcunit,optical bypassing ofelectronic processing can o er a
signi cant savingin switching cost at a relatively smallprice of more wavelengths. On the other
hand, when a bandof wavelengths is a common traÆc unit,bypassingof wavelength-level optical
processing can reduce wavelength-level optical MUXs, DMUXs, and recon gurable switches but
may ormay notjustify a priceof more bands ofwavelengths. Thedetailednature ofthese
trade-o s are beyond the scope of this thesis. For the most part, we shall concentrate on the cases in
which a wavelength is a common traÆc unit and investigate how to switch wavelengths of traÆc
eÆciently. In thelastpart ofthisthesis, we shallexplore howto eÆcientlyswitch traÆcinbands
of wavelengths.
1.4 Core Network with Aggregated TraÆc
We shall focus our attention on the design of a high-speedcore network that interconnects
sub-networks using electronic switches at the access nodes. Figure 1-5 shows an example of such a
core network. Withrespectto thecore network, the accessnodes actasentry and exitpoints for
traÆc from individualend users in the subnetworks. The core network may have nodes that are
notaccessnodesbutareusedto switchtraÆc. Electronicswitchesattheaccessnodescan beused
to aggregateand deaggregate small-rate traÆcsessions from individualendusersin subnetworks.
network linkand a subnetwork linkon agiven lightpath.
corenetworklink
corenetworknodewith
noelectronicswitch
corenetworkaccessnode
withelectronicswitch
atsubnetworkinterface subnetworklink subnetworknode subnetwork4 subnetwork2 subnetwork1 subnetwork3 subnetwork5
Figure1-5: Acorenetworkinterconnectingsubnetworksthroughelectronicswitchesattheaccessnodes.
In the core network, each traÆc session is transmitted from one access node, referred to as
thesourcenode,to anotheraccess node, referredto asthe destinationnode. A single sessionmay
resultfromtraÆcaggregationofalargenumberofsmall-ratesessionsinasubnetwork. Inthiscase,
we expect each traÆc session to be somewhat static and shall provide its route and wavelength
ina static fashion. Onthe other hand, a single sessionmay result from traÆc aggregation of few
large-rate sessions or even from a single large-rate session in a subnetwork. In this case, sessions
might have short lifetimes, so it is necessary to change routes and wavelengths in a dynamic
fashion. For thepurposeof RWA algorithm designs,we can considerstatic provisioningof routes
and wavelengths as if we were to support static traÆc sessions. Throughout the thesis, we shall
usetheterms static traÆcand dynamictraÆcto referto thecases inwhichweperformstaticand
dynamicprovisioning of routesand wavelengths respectively, even though each supported session
isnotstatic understatic provisioning.
Weshalladoptall-opticalnetworkarchitecturesandaimtodevelopRWAalgorithmsto support
and dynamictraÆc models, we assume thateach sessionhasa rate equal to one wavelength unit.
This assumption is reasonable for the design of high-speed core networks in which each pair of
subnetworks have multiple wavelengths of traÆc to communicate. In addition, this assumption
allows us to neglect the additional wavelengths required for optical bypassing due to the traÆc
sessions whoseratesaresmaller thana wavelength.
1.5 Outline of the Thesis
The remaining parts of this thesis are organized as follows. Chapter 2 brie y discusses existing
literatureon theRWAprobleminWDMnetworks. Italsostatesourthesisobjectivesandpresents
our problem formulations. Chapter 3 discussesstatic RWA and presents our RWA algorithms for
static traÆc. Chapter 4 discusses dynamic RWA and presents our RWA algorithms for dynamic
traÆc. Chapter5 exploresfurtherreductioninswitchingcostbyperformingswitchinginbandsof
wavelengths insteadofinwavelengths. Finally,chapter 6summarizesourachievementsandpoints
History and Problem Formulations
2.1 Existing Literature on the RWA Problem
Several papersinvestigatetheroutingandwavelengthassignment(RWA)problemina wavelength
divisionmultiplexing(WDM) network under thewavelength continuityconstraint. A
comprehen-sive overview of di erent problem formulations and solution approaches taken by researchers is
availablein [YB97, ZJM00 ]. We can categorizeexisting results into two groupsbased on whether
staticordynamicprovisioningofroutesand wavelengthsisperformed. Forstaticprovisioning,the
traÆctobesupportedisassumedknownand xedovertime. Thegoalisoftentominimizethe
num-berofwavelengthsusedinthenetwork[BM96,RS96 ]. Alternatively,ifthenumberofwavelengthsis
xedinadvance,onegoalistomaximizethenumberofsupportedtraÆcsessionsaccordingtosome
knownand xedtraÆcdemands[CGK92,RS95,ZA95,CB96 ]. Theseproblemscanbeformulated
asmixedintegerlinearprogramming(ILP)problems[RS95 ,ZA95 ,BM96,CB96 ,RS96], whichare
knownto be NP-complete[CGK92]. Consequently,theRWA problemsarefrequentlydividedinto
twosteps,the rstforroutingand thesecondforwavelengthassignment. Thesetwostepsarethen
solved separately and suboptimally. In some cases, partialroutingdecisions aremade at the time
of wavelength assignment. For example, an RWA algorithm may assign a few routes in advance
foreach sessionwiththe nalchoiceto bemadeat thetimeofwavelengthassignment[RS95 ]. For
some regular topologies and speci c traÆc, e.g. all-to-all uniform traÆc in the bidirectionalring
topology, the overall RWA problem can be solved to obtainclosed form solutions [Elr93, Wil96].
Dynamicprovisioningof routesand wavelengths givesus exibilityinsupportingtraÆc which
may change overtime through sessionarrivals and sessiondepartures. To modeldynamic traÆc,
session arrivals can be assumed to form stochastic processes [Bir96, SAS96]. In addition, session
lifetimesarestochastic. Thegoal isusuallyto developanon-line RWAalgorithm whichminimizes
the average blockingprobabilityfora new sessionrequest given a xednumberof wavelengths in
the network. We refer to this type of problem formulation as the blocking formulation. Due to
thecomplexityincomputingblockingprobabilities,some approximationsaremadeto simplifythe
analysis. For example, session arrivals on di erent links are assumed to be independent [Bir96 ,
BH96 ], or correlated among adjacent linksin the same fashion throughout the network [SAS96].
Based on such approximations, several dynamicRWA heuristicsaredeveloped [LS99,ZRP00].
Anothertypeofproblemformulation,referredto asthenonblocking formulation,assumesprior
knowledge of the set of all the traÆc matrices, or equivalently the traÆc demands, to be
sup-ported [Pan92, Ger+99 , NLM02 ]. In [Ger+99 ], the set of traÆc matrices is characterized by the
maximum linkloadinthenetwork. In[Pan92,NLM02 ],thesetof traÆcmatrices ischaracterized
by the number of tunable transmitters and tunable receivers at each end node. A new sessionis
said to be allowable if its arrivalresults in a traÆc matrixwhich is stillin the set of supportable
traÆc. Thegoalisusuallytodevelopanon-lineRWAalgorithmwhichdoesnotblockanyallowable
sessionand uses theminimumnumberofwavelengths.
Ifweallowsomeexistinglightpathsto berearrangedinordertosupportanewsession,the
cor-respondingRWAalgorithm issaid to be rearrangeably nonblocking. 1
Ifwe allowno rearrangement
of any existing lightpath in order to support a new session, the corresponding RWA algorithm is
said to bewide-sense nonblocking. Note thatifan RWA algorithm iswide-sensenonblocking, itis
also rearrangeably nonblocking. Therefore, for thesame setof traÆc matrices,the required
num-berof wavelengths ishigherforawide-sensenonblockingRWAalgorithm thanforarearrangeably
1
The terminology comes from standard de nitions in switching theory. A switching network is rearrangeably
nonblocking ifany allowable sessioncan be supported, possibly after some rearrangementsof existing sessions. A
switching network is wide-sense nonblocking if any allowable session can be supportedwithout rearrangement of
existing sessions provided that all the existing sessions have been routed according to some algorithm. Finally,
a switching network is strict-sense nonblocking if any allowable session can be supported without rearrangement
of existing sessions. Notice that, ina strict-sense nonblocking network, we cansupport each allowable session by
choosinganyoftheroutesavailableatthetime. Byde nition,astrict-sensenonblockingnetworkisalsowide-sense
SwitchingoftraÆcinmultiplelevelsofgranularityappearsinseveralinvestigationsonthetraÆc
grooming problem. In thetraÆc groomingproblem, theobjective is to eÆcientlyaggregate
small-rate traÆc sessions onto wavelengths using electronic switches and to perform optical bypassing
to minimize the cost of electronic switches [BM00, CM00 , GRS00 ]. Similar problems exist for
larger levels of traÆc granularity. In particular, as traÆc demands increase, we expect to reduce
the switching cost further by switching traÆc in bands of wavelengths instead of in wavelengths
whenit isappropriate. In thiscase, thecost savings comefrom thereduction of optical switching
resources. For convenience, we shall refer to a switch whose basic traÆc unit is a wavelength as
a wavelength switch. Accordingly, we shall refer to a switch whose basic traÆc unit is a bandof
wavelengthsasabandswitch. Inaddition,weshallreferto theRWAproblemwithwavelengthsand
bandsofwavelengths asthetwo levels oftraÆc granularityastheband/wavelength RWA problem.
Despite theirsimilarities,there aresome fundamentaldi erences betweenthetraÆc grooming
problemandtheband/wavelengthRWAproblem. Sincewe stilloperateintheopticaldomain, the
wavelengthcontinuity constraint appliesat theinterfacebetween abandswitch and a wavelength
switch,whereasthereisnosuchconstraintattheelectronicinterface. Inaddition,thecoststructure
of an optical switch is di erent from that of an electronic switch. More speci cally, the cost
of an electronic switch primarily depends on the total input traÆc rate, while the cost of an
opticalswitch may onlydependon thetotal numberof inputports. Forexample, withpromising
microelectromechanical system (MEMS) technologies, an optical switch can be constructed from
a set oftinymirrors used to re ect traÆc streamsin theform of light beams from inputports to
outputports [RS01 ]. Such anoptical switch can be usedasa bandswitch ora wavelengthswitch
withoutsigni cantcost di erence.
2.2 Thesis Objectives
Inthisthesis,we considerthe RWA problemina WDM networkunder thewavelength continuity
constraint for bothstatic and dynamic traÆc. By static traÆc, we refer to static provisioningof
routesandwavelengthsfortraÆcsessions. Inahigh-speedcorenetwork,suchstaticprovisioningof
Bycarefullychoosingthelocationsofaccessnodesandthesizesoftheircorrespondingsubnetworks,
wemaybeabletoformacorenetworksuchthataggregatedtraÆcstreamsamongtheaccessnodes
aresomewhat uniform. Suchuniformityof traÆcmaynotbe achievable inpractice. Nevertheless,
we are interested in the case of providing one or a few wavelength paths between each pair of
accessnodesforbasicall-to-allconnectivity. Inaddition,havingthesededicated wavelengthpaths
between all pairs of nodes can simplify network operationssince most small-rate sessions can be
supported on dedicated paths and there is rarelya need to recon gure the switching nodes as a
resultof asmalltraÆcchange. We viewthisstaticprovisioningofroutesandwavelengths asifwe
wereto support static uniformall-to-all traÆc. Our goal is to develop an o -line RWA algorithm
whichuses theminimumnumberofwavelengths forstatic uniformall-to-all traÆc.
On the other hand, by dynamic traÆc, we refer to dynamicprovisioningof routes and
wave-lengths for traÆc sessions. In a high-speed core network, dynamic provisioning of routes and
wavelengths can be used to support traÆc streams which cannot be well approximated as static
dueto insuÆcientaggregation. Adoptingthe nonblockingformulation,we assume that thetraÆc
matrixchanges from timeto timebutalways belongsto a known traÆcset. Our goalis to design
an on-line RWA algorithmswhich can supportall the traÆcmatrices intheknown traÆcset ina
rearrangeably nonblockingfashionwhileusingtheminimumnumberof wavelengths and incurring
few rearrangementsof existinglightpaths, ifany,foreach traÆcchange.
Instead of trying to solve the RWA problem foran arbitrarilygiven network topology,we aim
to investigate what topological properties contribute to good network architectures. To do so,
we formulate RWA problems in a tractable fashion so that eÆcient solutions can be analytically
derived. Itisourhopethatsomeoftheanalyticaltechniquesdevelopedinthisthesiscancontribute
to greaterunderstanding of network architectures. To buildan analyticalframework, we consider
a few speci c topologies including an arbitrary tree, a bidirectional ring, a two-dimensional (2D)
torus, and a binary hypercube. Notice that these topologies are listed from the least densely
connected to themostdenselyconnected.
In the last part of this thesis, we perform preliminary study of the band/wavelength RWA
probleminaWDMnetworkunderthewavelengthcontinuityconstraint. Ourgoalistounderstand
top-level network nodes switch traÆc in bands of wavelengths and the lower-level network nodes
switchtraÆc inwavelengths.
Intheremaining sections,weformulate indetailthestatic and thedynamicRWAproblemsof
interestin thisthesis.
2.3 RWA Problem for Static TraÆc
Thissection formulates theRWA problem forstatic traÆc. This problem isinvestigatedin detail
in chapter 3. Consider an all-optical WDM network with no optical wavelength conversion. In
any given network topology, assumethat adjacent nodes areconnected bytwo bers,one ineach
direction. Assumealsothatall berscontainthesamenumberofwavelengths,i.e. WDMchannels.
We shall refer to a network node which sources and sinks traÆc as an end node. Let N be the
number of end nodes in the network. In the context of a core network, an end node corresponds
to an accessnode. Noticethat there may be some network nodeswhichare notendnodes, e.g. a
switchinghubnode inthestar topology.
De ne l-uniform traÆc to be static traÆc inwhich each endnode transmitsl wavelengths to,
andreceivesl wavelengths from,eachof theother endnodes. 2
Notethatl-uniformtraÆcrequires
l(N 1) transmitters and l(N 1) receivers at each end node. Since the traÆc is static, these
transmittersandreceiversneednotbetunable. Moreover,ateachswitchingnode,wecanuse xed
opticalswitches insteadof recon gurableoptical switches. The RWA problemforl-uniformtraÆc
isgiven below.
Problem 1 (O -LineRWA forl-Uniform TraÆc) ForagivennetworktopologywithN end
nodes, let W
s;l
denote the minimum numberof wavelengths which, if provided in each ber, can
support l-uniformtraÆc withno wavelength conversion. We want to ndthevalue of W
s;l and a
corresponding o -lineRWA algorithm.
In the above problem formulation, we model a traÆc stream which is the aggregation of a
large number of small-rate sessions as being static. The uniformity of static traÆc may not be
2
WereservethetermstransmitandreceivefortheendnodeswhichsourceandsinktraÆcsessions. Intermediate
supporting1-uniformtraÆc is an interesting problem in how to provideminimaloptical all-to-all
connectivity amongtheendnodes.
2.4 RWA Problem for Dynamic TraÆc
In this section, we formulate the RWA problem for dynamic traÆc. We shall investigate this
problemin chapter 4. As inthe RWA problem forstatic traÆc, we consider an all-optical WDM
networkwith N endnodesand no opticalwavelengthconversion. Assumethatnode i,1iN,
is equipped withk
i
fully tunabletransmitters and k
i
fully tunable receivers. At anytime, node i
can transmit at most k
i
wavelengths and receive at mostk
i
wavelengths. Such a traÆc matrixis
said to belong to a set of k-allowable traÆc, where k = [k
1 ;k
2 ;:::;k
N
]. Assume that each traÆc
sessionhasa rate ofone wavelength. WemodeldynamictraÆcasa session-by-sessionarrivaland
departure process inwhich sessions arrive and depart one at a time. Inother words, a transition
from one traÆc matrix to another is a result of either a single session arrival or a single session
departure.
Anewsessionrequestis allowableiftheresultanttraÆcmatrixisstillinthesetofk-allowable
traÆc. The de nitionimplies that, for each allowable session request, there is a free transmitter
at thesourcenode anda freereceiverat thedestination node. We want to designarearrangeably
nonblocking RWA algorithm which can assign a lightpath to any allowable session, perhapsafter
some rearrangements of existing lightpaths. Our algorithms will be centralized in nature. We
assume that traÆc does not change too frequently and the RWA algorithms always have correct
knowledgeoftheRWAinthenetwork. Inaddition,weassumethere issuÆcienttimeforlightpath
rearrangementsbetween consecutive transitionsof thetraÆcmatrix.
Problem 2 (On-Line RWA for k-Allowable TraÆc) For a given network topologywith N
end nodes, let W
d;k
denote the minimumnumber of wavelengths which, if providedin each ber,
cansupportdynamick-allowabletraÆcinarearrangeablynonblockingfashionwithnowavelength
conversion. We want to ndthe valueof W
d;k
and a corresponding on-line RWA algorithm which
bythegivennumberoffullytunabletransmittersandreceiversink. Inpractice,pasttraÆchistory
maysuggestthatweneedtoprovidenetworkresourcesonlyforastrictsubsetofk-allowabletraÆc.
Nevertheless,we shallconcentrate onsupportingtheentirek-allowabletraÆc set. Itis clearthat,
for any network, the value of W
d;k
is an upper bound on the minimum number of wavelengths
requiredto supportanystrict subsetof k-allowabletraÆc.
To establish some connection between static and dynamic traÆc, consider l-uniform traÆc.
When all the k
i
's are equal to l(N 1), l-uniformtraÆc belongsto theset of k-allowable traÆc.
Itfollows thatW
s;l W
d;k
inthiscase. Inaddition,a givendynamicRWA algorithm canbe used
to support l-uniform traÆc. However, the number of wavelengths used by the algorithm will be
RWA for Static l-Uniform TraÆc
In this chapter, we study the routing and wavelength assignment (RWA) problem for l-uniform
traÆc. Inl-uniformtraÆc,eachendnodetransmitslwavelengthstoandreceiveslwavelengthsfrom
each ofthe otherendnodes. Whileourgoal includesunderstanding arbitrarymeshtopologies,we
solvetheRWAprobleminafewspecialcases. Thespeci ctopologiesweshallconsiderarearbitrary
tree topologies, a bidirectional ring, a two-dimensional (2D) torus, and a binary hypercube. Let
W
s;l
denote the minimum number of wavelengths which, if provided in each ber, can support
l-uniform traÆc with no wavelength conversion. In the future, we aim to extend our analytical
techniquesto obtaina good boundon thevalue ofW
s;l
foranygiven topology.
Let L
s;l
denote the minimum number of wavelengths in a berrequired to support l-uniform
traÆc given full wavelength conversion at all network nodes. It is clear that L
s;l W
s;l
for any
given topology. We shallseethat, inall thenetwork topologiesforwhichwe can obtaintheclosed
form expressions forW
s;l and L
s;l
, we can performRWA eÆcientlyto achieve W
s;l =L
s;l
without
anywavelength converterinthenetwork.
3.1 Arbitrary Tree Topologies
In thissection,we solvethe RWA problem forl-uniformtraÆcinan arbitrary tree topology. In a
given tree topology, we assume there are N >2 end nodes which are the leaf nodes of the tree. 1
WedescribeatreebyasetofnodesN and asetofbidirectionallinksT. ForthepurposeofRWA,
1
we can assume that each non-leaf node has degree at least 3. Note that if a non-leaf node has
degree less than 3, thenit can be removed from the tree withoutchanging the RWA problem, as
illustrated in gure 3-1. Since there is a unique route for each traÆc session, there is no routing
problemin a tree topology. Thus, we only have to perform wavelength assignment (WA) in the
RWA problem.
leafnode non-leafnode non-leaf
nodewith
degree2
modi edtreewhosenon-leaf
nodeshavedegree atleast3
Figure 3-1: Removal of anon-leafnodewithdegree lessthan3.
Let us consider the WA problem for 1-uniform traÆc. The results are later extended, in a
straightforwardmanner,to l-uniformtraÆc. LetL
s;1
denote theminimumnumberofwavelengths
which, ifprovided ineach ber, can support1-uniform traÆc given fullwavelength conversion at
all nodes. Each link einthe tree correspondsto a cutwhichseparates theN endnodes into two
sets, denoted by N
e;1
and N
e;2
. The amount of traÆc (in wavelengths)on a beracross link e is
equalto jN e;1 jjN e;2 j. Let w
denote themaximumtraÆc overall the bers. Clearly,L
s;1 is equal to w ,asgiven below. L s;1 = w = max e2T jN e;1 jjN e;2 j (3.1) Let W s;1
denote the minimum number of wavelengths which, if provided in each ber, can
support 1-uniform traÆc with no wavelength conversion. We shall show that W
s;1 is bounded by W s;1 w , which implies W s;1 = L s;1 = w . We do so by constructing a WA algorithm.
Figure 3-2 illustrates an example scenario in which a greedy WA algorithm fails to support
1-uniform traÆc using w
wavelengths. In this example, inspectionshows that w
= 2. Note that
thesamewavelengthisassignedto theoppositelydirectedsessionsbetweenthesamepairofnodes,
2
Since weassume thateachlinkconsists oftwo bers,oneineachdirection,theindegree andthe outdegreeof
1 1
(2,1) and wavelength
2
to sessions (1,3) and (3,1), neither
1 nor
2
can be assigned to support
session(2,3). It follows that more thanw
=2 wavelengths arerequired. Therefore, this example
scenario tellsusthat thedesignofa WA algorithm usingw
wavelengths isnottrivial. Figure3-2
also demonstrates that, in order to use the minimum number of wavelengths, we may need to
supporttheoppositelydirected sessionsbetween thesame pair ofnodeson di erentwavelengths.
(1,2) (2,1) (1,3) (3,1) (2,3) (1,2)on 1 (2,1)on 1 (1,3)on 2 (3,1)on 2 (2,3)noton 1 or 2 1 2 1 2 1 or 2 node1 cannot use w =2 node2 node3 sequenceof sessions forWA
(i;j)denotesasession fromnodeitonodej. corresponding
sequenceofWAsteps
Figure3-2: An examplein which agreedy approach requiresmorethan w
wavelengths.
We now derive afew usefulproperties relatedto theminimumnumberofwavelengths w
. Let
e
denote thelink associated withw
. Note that there may be multiple choices fore
. Theexact
choicedoesnotmatterinthefollowingdiscussion. Weshallrefertoe
asthebottleneck linksinceit
isthelinkwiththemaximumtraÆcona ber. Linke
separatestheleafnodesintotwosets N
e ;1 and N e ;2
. Without loss of generality, choose N
e ;1 such that jN e ;1 j jN e ;2 j. Since we assume
there aremore thantwo leaf nodes, N
e
;2
must contain multipleleaf nodes. De ne the bottleneck
nodev
to be theendpoint ofe
oppositeto N
e
;1
,i.e. thesubtreeconnectedto v
bye
contains
all theleaf nodes inN
e
;1
,asillustratedin gure3-3a.
We shall refer to each subtree connected to v
as a top-level subtree. Note that a top-level
subtree can be asingle node. Figure 3-3bshowsthe top-level subtreesassociated withthe tree in
gure 3-3a. Let d be the degree of v . Since v isa non-leafnode, d
3. It follows that there
are d 3 top-level subtrees. Let S i , 1 i d
, denote the set of all the leaf nodes in top-level subtree i, and x
i = jS
i j.
The followinglemma providesusefulproperties ofthetop-levelsubtreesconnectedto v
aswellas
boundson theminimumnumberofwavelengths w
top-level subtree3 top-level subtree2 top-level subtree1 e v 5leafnodes 2leafnodes e N e ;1 N e ;2 w =30 v (a) (b)
Figure3-3: Thebottleneck link e
andthebottleneck nodev
.
Lemma1 Number the top-level subtree connected to the bottleneck node v
by the bottleneck link
e
as top-level subtree1, and the rest of the top-level subtrees from 2 to d , where d isthe degree of v . Then, 1. x i x 1 N=2 for all 1id , and
2. the minimum number of wavelengths w is bounded by 1 d 1 1 d N 2 w N 2 4 : Proof:
1. De ne f(x) = x(N x). Note that f(x
i
) is the traÆc (in wavelengths) carried on each of
the two bers between the bottlenecknode v
and top-level subtree i. By the de nitionof
e , f(x 1 ) f(x i ) for all 2 i d
. We now prove that x
i x 1 for all 2 i d using
contradiction. Assumethatx
i >x
1
forsome i6=1. Sinced 3,itfollowsthatx 1 +x i <N, yielding x i <N x 1
. Asillustrated in gure3-4, f(x)isconcave and symmetricaround the
maximumvalueatx=N=2. Thus,therelationx
1 <x i <N x 1 impliesthatf(x i )>f(x 1 ), yielding a contradiction.
Since top-level subtree 1 containsall theleaf nodes inN
e ;1 and jN e ;1 jjN e ;2 j, it follows that x 1 N=2. We concludethat x i x 1 N=2 forall 1id .
2. Note that f(x) = x(N x) has the maximum value of N 2 =4 at x = N=2, as shown in gure3-4. Sincew =f(x 1 ),itisclearthatw N 2
w x 1 N=2 N x N x 1 N 2 =4 Figure 3-4: Graph of f(x)=x(N x). w =f(x 1 )isan increasingfunctionof x 1 for0<x 1 <N=2. Thus,w isminimizedwhen x 1
takesthelowestpossiblevaluewhichisequal to dN=d e. It follows that w f( dN=d e) f( N=d );
which isthe desiredlowerbound. 2
Before describing our WA algorithm, we describe some of the ideas behind it. De ne a local
session to be a traÆc session whose source and destination are in the same top-level subtree.
Accordingly, a non-local session has its source and its destination in di erent top-level subtrees.
Notethat anon-localsessionhasto travel throughthebottlenecknode v
,whereasalocalsession
does not have to travel all the way to v
and back to its destination, i.e. each session never uses
thesame link twice inoppositedirections.
Our WA algorithm rst assigns wavelengths to all of the non-local sessions. It then assigns
wavelengths to all thelocalsessions ineach top-level subtree. Consider top-level subtree 1. Since
thereareintotalx
1
(N 1)localandnon-localsessionstransmittedfromnodesinthissubtreewhile
there areonlyx
1 (N x
1
)wavelengthsavailable,itisclearthatweneedto reusesome wavelengths
previously assigned to non-local sessions to support local sessions. Such wavelength reuse is the
cause ofthe maincomplexityinthedesignof an eÆcient WA algorithm.
Letn
i;j
denoteleaf nodej inS
i ,where 1id and 1jx i . Withrespectto n i;j ,de ne
a reusable wavelength to be a wavelength usedby n
i;j
to receive a non-local session(from a node
ina di erent top-level subtree),butnot usedbyn
i;j
1
withrespectto n
i;j
. Thefollowinglemmastatesabasicpropertyofnon-localsessionsandreusable
wavelengths.
(a) type-1reusablewavelength v n i;j n i;j 0 non-local receive session session transmit non-local 1 n i;j v non-local receive session 1 1 1 1
Localsessionsareshownasthicklines.
n i;j 0 1 isnotusedto
transmitanynon-local
session fromthis
top-levelsubtree.
(b)type-2reusablewavelength
Figure3-5: Reusable wavelength
1
withrespect to noden
i;j .
Lemma2 In any giventop-level subtree,
1. all the non-local sessionsarereceived on distinctwavelengths,
2. all the non-local sessionsaretransmitted on distinctwavelengths, and
3. any two reusablewavelengths with respect tothe same node orwith respect todi erent nodes
in the subtreearedistinct.
Proof:
1. Consider top-level subtree i, where 1 i d
. Any pair of non-local sessions which are
received inthistop-levelsubtree musttraverse the berfromthe bottlenecknode v
to
top-level subtree i. It follows that their wavelengths must be distinct, orelse there would be a
wavelength collisionon this ber.
2. The proof is identical to that of statement 1, except that we considera pair of transmitted
non-localsessionsand thelink fromtop-level subtreeito v
.
3. Since any pair of reusablewavelengths are used to receive two non-local sessions, it follows
i;j
which is also used by a di erent node in the same top-level subtree (i.e. top-level subtree i) to
transmitanon-localsession. Forexample,in gure3-5a,withrespectton
i;j ,
1
isatype-1reusable
wavelength. Inaddition,withrespectton
i;j
,de neatype-1local nodetobeadi erentnodeinthe
same top-level subtreewhich transmitsa non-localsessionon a reusablewavelength (with respect
to n
i;j
). Forexample, in gure3-5a, withrespect to n
i;j ,n
i;j 0
isa type-1local node.
Withrespect to n
i;j
,de ne a type-2reusable wavelengthto be a reusablewavelength which is
nottype-1,i.e. itisnotusedbyanyothernodeinthesametop-levelsubtreetotransmitanon-local
session. For example, in gure 3-5b, with respect to n
i;j ,
1
is a type-2 reusable wavelength. In
addition,withrespectto n
i;j
,de neatype-2 local nodetobeadi erentnodeinthesametop-level
subtreewhichisnottype-1,i.e. itdoesnottransmitanon-localsessiononanyreusablewavelength
(with respect to n
i;j
). For example, in gure 3-5b, with respect to n
i;j , if n
i;j 0
does not use any
reusable wavelength (with respect to n
i;j
) to transmit a non-local session, then n
i;j
0 is a type-2
localnode.
Notice that, by the above de nitions, with respect to any given node n
i;j , each node n i;j 0 , j 0
6=j,is eitheratype-1ortype-2localnode. Thefollowinglemmaindicatesone possiblestrategy
ofassigningwavelengthstothelocalsessionstransmittedfromn
i;j
usingreusablewavelengthswith
respectto n
i;j
Lemma 3 With respect to node n
i;j
, we have the following properties.
1. Noden
i;j
can transmitalocalsessiontotype-1localnoden
i;j 0
ona type-1reusablewavelength
(with respect ton
i;j
)which is used by n
i;j
0 totransmit a non-local session.
2. Node n
i;j
can transmit a local session to type-2 local node n
i;j
0 on any type-2 reusable
wave-length (with respect ton
i;j ).
Proof:
1. Figure 3-5a illustrates statement 1 of thelemma. Let
1
denote the reusablewavelength of
interest. Letr denote thenon-localsessionreceivedbyn
i;j on
1
. Lettdenotethenon-local
sessiontransmittedbyn
i;j 0 on
1
. Let ldenote thelocalsessionon
1 from n i;j to n i;j 0. We
show below that these three sessions never share a ber, and thus there is no wavelength
Since all the bers used by r are directed away from the bottleneck node v while all the
bers used by t are directed towards v
, r and t never use the same ber. We now show
that r andlneverusethesame ber. Weproceedbycontradiction. Assume that bere, i.e.
unidirectional link e, is used by bothr and l. Since e is used by r,e is necessarily directed
away from v
and towards n
i;j
. If e is also used by l, then l must have traversed the link
whichcontains eintheoppositedirection,i.e. towards v
,sincethere isa uniquepathfrom
n
i;j
to the starting point of e. This contradicts thefact that no local sessionuses the same
link twiceintheoppositedirections.
Similar arguments showthat tand l neverusethe same ber.
2. Figure 3-5b illustrates statement 2 of the lemma. The proof is identical to the proof for
statement 1 that r and lneverusethesame ber. We shallnotrepeatthedetailshere. 2
Lemma3suggeststhefollowingmethodofassigningwavelengthsto thelocalsessions. Consider
thelocalsessionstransmittedfromnode n
i;j intop-levelsubtreeS i . Therearex i 1suchsessions. Let P (1) i;j and P (2) i;j
be the sets of type-1 and type-2 local nodes with respect to n
i;j
respectively.
Notice that type-1 local nodes (with respect to n
i;j
) have associated with them distinct reusable
wavelengths (with respect to n
i;j
). From statement 1 of lemma 3, n
i;j
can use a distinct type-1
reusablewavelength (with respectto n
i;j
) to transmit a localsession to each type-1 local node in
P (1)
i;j
. It remains to provide wavelengths for the local sessions to type-2 local nodes (with respect
to n
i;j ).
We shall show shortlyinourWA algorithm that itis alwayspossibleto assign wavelengths to
thenon-local sessionssothat there are at leastjP (2)
i;j
j type-2reusablewavelengths with respectto
eachnoden
i;j
inthetree. GivenjP (2)
i;j
jtype-2reusablewavelengths withrespectton
i;j
,statement
2of lemma 3 impliesthatn
i;j
can use adistinct type-2reusablewavelength (with respectto n
i;j )
to transmita localsessionto each type-2localnode inP (2)
i;j .
We repeat the same process for all the leaf nodes. From statement 3 of lemma 2, since all
the reusable wavelengths (with respect to the same node or with respect to di erent nodes) in
eachtop-levelsubtreearedistinct,di erentlocalsessions(transmittedfromthesamenodeorfrom
In conclusion, the condition that there are at least jP
i;j
j type-2 reusable wavelengths with
respect to each node n
i;j
is a suÆcient condition for the WA of all the local sessions to exist.
We state thisconclusion formally in the followinglemma, which is later used to develop our WA
algorithm.
Lemma 4 If there are at least jP (2)
i;j
j type-2 reusable wavelengths with respect to node n
i;j for all 1 i d and 1 j x i
, then we can assign wavelengths to all the local sessions as follows.
Consider the local sessions transmitted from n
i;j in S
i .
1. Totransmitalocalsessiontoatype-1localnodeinP (1)
i;j ,n
i;j
usesatype-1reusablewavelength
(with respect ton
i;j
)which is used by that node to transmit a non-local session.
2. To transmit a local session to a type-2 local node in P (2)
i;j , n
i;j
uses a distinct type-2 reusable
wavelength (with respect to n
i;j ).
Our WA algorithm operates in three phases. In phase 1, we assign wavelength bands each
of which is used by the non-local sessions from one top-level subtree to another. In phase 2, we
performWAforindividualnon-localsessionsbasedonthewavelengthbandsobtainedfromphase1.
The goalofphase2 isto assignwavelengthsinsuch awaythatenoughtype-1and type-2reusable
wavelengths exist tosupportalllocaltraÆc. Finally,inphase 3,we performWA forlocalsessions
independentlyineach top-levelsubtree. ThefollowingisourWA algorithmfor1-uniformtraÆcin
an arbitrarytreetopology. Thealgorithmusesw
wavelengths ineach ber. We shallreferto this
algorithm astheo -line tree WA algorithm.
Algorithm 1 (O -LineTree WA Algorithm) (Use w
wavelengths ineach ber.)
Number the top-level subtrees so that the numbers of leaf nodes, denoted by x
1 ;:::;x d , satisfy x 1 x 2 :::x d . Note thatw =x 1 (N x 1 ).
Phase 1: Assign the wavelength band for the non-local sessions from one top-level subtree to
anotherasfollows. Forconvenience, let
(i;i 0
)
denotethewavelengthbandforthenon-localsessions
from S i to S i 0. Note that (i;i 0 ) contains x i x i
0 wavelengths. Figure 3-6 speci es the wavelength
bandsbetweenallpairsoftop-levelsubtrees. Toobtainwavelengthband
(i;i 0 ) ,wherei<i 0 ,follow
thediagramin gure3-6a. Thereared
1rowsof wavelengthbands. Inrowi,1id
i;i+1 , ...,
i;d
. For example, the wavelength band
(1;3)
contains 6 wavelengths with indices10
to 15. On the other hand, to obtain wavelength band
(i;i 0
)
, where i 0
< i, follow the diagram in
gure 3-6b. There are d
1 rows of wavelength bands. In row i 0 , 1 i 0 d 1, we assign
consecutive wavelengths starting from wavelength w
(from right to left) to wavelength bands
i 0 +1;i 0, ..., d ;i 0
. Forexample, thewavelengthband
(4;2)
contains3wavelengths with indices10
to 12. Althougha speci c exampleisillustrated,thegeneral scheme shouldbe clear.
1 3 5 7 9 11 13 15 17 x 4 =1 x 3 =2 x 2 =3 x 1 =3 v e w =18 top-levelsubtrees indices (1;2) wavelength (1;3) (1;4) (2;3) top-levelsubtree2 transmitted from (2;4) top-levelsubtree1 transmitted from (3;4) top-levelsubtree3 transmitted from
(a) wavelengthbands
(i;i 0 ) wherei<i 0 top-levelsubtree1 top-levelsubtree2 top-levelsubtree3 transmitted to transmitted to transmitted to (b)wavelengthbands (i;i 0 ) wherei>i 0 (3;1) (2;1) (3;2) (4;1) (4;2) (4;3)
Figure3-6: Phase 1 oftheo -line treeWA algorithm.
We shall show that, in each top-level subtree, the assigned receive wavelength bands do not
overlap, i.e. thereis nowavelengthcollisionbetweentwonon-localreceive sessionsintwodi erent
bands. In addition, the assigned transmit wavelength bands do not overlap. As a result, there
is no wavelength collisionamong the non-local transmit sessionsand among thenon-localreceive
sessionsineach top-levelsubtree.
Asanexampleto showhowtheschemeworks,considertwowavelengthbands
(1;4) and
(2;4)
fornon-localreceivesessionsintop-levelsubtree4. Thehighestwavelengthindexin
(2;4) ,denoted by + (2;4) ,isx 2 x 3 +x 2 x 4
. Thelowestwavelengthindexin
(1;4) ,denotedby (1;4) ,isx 1 x 2 +x 1 x 3 +1. Sincex 1 x 2 :::x d ,itfollows that x 1 x 2 x 2 x 3 andx 1 x 3 x 2 x 4 . Thus,
(1;4) = x 1 x 2 +x 1 x 3 +1 > x 2 x 3 +x 2 x 4 = (2;4) :
It follows thata non-local sessioninwavelengthband
(1;4)
and a non-localsessioninwavelength
band
(2;4)
neversharethesamewavelengthandthereforedonotcollide. Acompletegeneralproof
is given later intheproof ofalgorithm correctness.
Phase 2: In this phase, we assign wavelengths to individual non-local sessions based on the
wavelength bands obtained from phase 1. Our goal is to assign wavelengths so that there are at
leastjP (2)
i;j
jtype-2reusablewavelengthswithrespectto node n
i;j forall1id and 1jx i , assuggested bylemma 4.
We rstperformpartialWA asfollows. Foreach wavelengthband
(i;j) containingx i x j
wave-lengths (used forthenon-localsessions from S
i to S
j
), we breakthe bandup into x
j
subbandsof
x
i
contiguous wavelengths. The rst subband isassigned to be receive wavelengths fornode n
j;1 .
The second subbandis assigned to be receive wavelengths for n
j;2
,and soon. Forexample, based
ontheexamplein gure3-6,intop-levelsubtree1,node n
1;1
receivesthreenon-localsessionsfrom
top-levelsubtree 2on thesubbandof
(2;1)
containingwavelengths 10,11,and 12. Noticethatwe
have not speci ed which node in top-level subtree 2 uses a speci c wavelength (10, 11, or 12) to
transmit to n
1;1
. Figure 3-7 illustrates the result of the partial WA in top-level subtree 1. Note
that thepartialWA also speci es thesubbandsused by thenodesin S
1
to transmit to each node
inS
i 0
,i 0
6=1,as shownin gure3-7b. Forexample, in
(1;2)
, wavelengths 1,2, and 3 areused for
thenon-localsessionsfrom S
1 to n
2;1 .
It remainsto specifythe sourcenodes forspeci c wavelengths in eachsubband, i.e. lling the
empty slots in each subband in gure 3-7b with n
1;1 , n
1;2
, and n
1;3
. Such speci cations in
top-level subtree1 can be doneindependentlyfrom thesimilarspeci cations inall theother top-level
subtrees sincethelightpathscorrespondingtoeach subbandtraverse thesame setof bersoutside
top-level subtree 1. In other words,the WA outside top-level subtree1 looks thesame regardless
of how we lltheemptyslotsin gure 3-7b. Furthermore, suchspeci cationsyield,foreach node,
the corresponding type-1 and type-2 reusable wavelengths together with type-1 and type-2 local
1 2 3 4 5 6 7 8 9 10 11 12 14 16 17 18 indices wavelengths sessiononspeci c receivinganon-local nodesinS 1 n 1;1 n 1;1 n 1;1 n 1;1 n 1;1 n 1;1 n 1;2 n 1;2 n 1;2 n 1;2 n 1;2 n 1;2 n 1;3 n 1;3 n 1;3 n 1;3 n 1;3 n 1;3 (2;1) 13 15 (4;1) (fromS 4 ) (3;1) (fromS 2 ) (fromS 3 )
(a)nodesreceivingnon-localsessionsfromS
i 0,i 0 6=1,toS 1 onspeci cwavelengths nodesinS1 transmittinganon-local sessiononspeci c wavelengths ton 2;1 ton 2;2 ton 2;3 ton 3;1 ton 3;2 ton 4;1 (1;2) (1;3) (1;4) (toS 2 ) (toS 3 ) (toS 4 )
(b)nodestransmittingnon-localsessionsfromS1 toS
i 0,i
0
6=1,onspeci cwavelengths(tobeassigned)
Figure3-7: TheresultofthepartialWAinphase2oftheo -linetreeWAalgorithmfortop-levelsubtree
1in gure 3-6.
Assume fornowthat theset of wavelengths used to receive and to transmit non-local sessions
arethesame ina given top-levelsubtree i. (Thisis thecase fortop-level subtree1 and anyother
subtreei withx
i =x
1
. However, the assumptiondoes notalwayshold, e.g. top-levelsubtree 3 in
gure 3-6.) We show below how to assign the source nodes in each wavelength subband so that
jP (2)
i;j
j= 0, i.e. no type-2 local node with respect to n
i;j
, for each node n
i;j in S
i
. Note that the
conditionjP (2)
i;j
j=0 yieldsthesuÆcient conditioninlemma 4 for theWA of all the local sessions
intop-levelsubtreeito exist,i.e. thereareatleastjP (2)
i;j
jtype-2reusablewavelengthswithrespect
toeachnoden i;j inS i . ThegoaljP (2) i;j j=0isequivalenttojP (1) i;j j=x i
1. Thatis,wemustensure
that,eachnoden
i;j 0 ,j 0 6=j,inS i
transmitsatleastonenon-localsessionononeofthewavelengths
usedbyn
i;j
to receive non-local sessions.
We can visualize theproblem of assigning thesource nodesin each subband using a bipartite
graph. Weconsidereachtop-levelsubtreeseparately. Fortop-levelsubtreei,constructapartialWA
bipartitegraphdenotedby(V
1 ;V
2
;E) asfollows. The setV
1
containstheN x
i
leafnodesoutside
top-levelsubtreei,i.e. fn
i 0 ;j 0 :i 0 6=i;1j 0 x i 0 g. ThesetV 2 isequaltoS i ,i.e. fn i;j :1jx i g.
In theset of edges E, an edge joins n
i 0 ;j 0 in V 1 and n i;j inV 2
foreach wavelength that is used to
receive a non-localsessionfrom anode inS
i byn i 0 ;j 0,
and is usedto receivea non-local sessionby
n
i;j
. Theremaybe multipleedgesbetweenthesamepair ofnodes. Forexample, gure3-8ashows
thepartialWA bipartitegraph speci edby thepartialWA intop-levelsubtree 1 in gure 3-7. In
particular,the edge between n
2;1 inV 1 and n 1;1 in V 2
1 2;1 1;1 edges betweenn 2;3 inV 1 and n 1;3 inV 2
correspond to wavelengths8 and 9which areusedbothto
receive a non-localsessionfrom S
1 byn
2;3
and to receivea non-local sessionbyn
1;3 . n 2;1 n 2;2 n2;3 n3;1 n 3;2 n 4;1 n 1;1 n1;2 n1;3 n 2;1 n 2;2 n2;3 n3;1 n 3;2 n 4;1 n 1;1 n1;2 n1;3 n 2;1 n 2;2 n2;3 n3;1 n 3;2 n 4;1 n 1;1 n1;2 n1;3 n 2;1 n 2;2 n2;3 n3;1 n 3;2 n 4;1 n 1;1 n1;2 n1;3 V 1
(a)partialWAbipartitegraph
fortop-levelsubtree1
Eachedgedislabelledwithadistinctwavelengthindex.
V 1 V 2 10 13 16 6 1 2 4 9 11 14 17 3 5 7 12 15 18 8
Eachedgecorresponds
toadistinct
wavelength.
isincidenttoallthenodesinV
1 andV
2 (b)partitioningofEsothateachsubset subsetE 1 forn 1;1 subsetE 2 forn 1;2 subsetE 3 forn 1;3 V 1 V 1 V 2 V 2 V 2
Figure3-8: Partial WA bipartitegraphfor top-levelsubtree 1in gure 3-6.
From the assumption that, in top-level subtree i, the set of non-localtransmit wavelengths is
equaltothesetofnon-localreceivewavelengths,itfollowsthateverynon-localtransmitwavelength
correspondsone-to-one to an edge inthe partialWA bipartite graph. Since each node n
i 0 ;j 0 inV 1 receives x i
non-local sessions from S
i , each node n i 0 ;j 0 has degree x i
. Since each node n
i;j in V
2
receives N x
i
non-local sessions,each node n
i;j inV
2
has degree N x
i
. In addition, there are
intotal x
i
(N x
i
)edges inthepartialWA bipartitegraph.
We next partition the set of edges, or equivalently the set of non-local transmit wavelengths
fromS
i
,intox
i
subsetseachwithN x
i
edges. Eachsubsetofwavelengthsarethenusedbysome
node n i;j in S i (or equivalentlyV 2 ) to transmit its N x i
non-local sessions to the N x
i nodes
in V
1
. Thus, it is necessary that each subset of edges contains N x
i
edges and is incident on
all the nodes inV
1
, orelse there would be a node in V
1
notreachable from S
i
in some subset of
wavelengths. To achieve the goal of having jP (2) i;j j = 0 for each n i;j in S i , we require in addition
thateach subsetofedgesisincidentto allthenodesinV
2
. Toseewhythisadditionalrequirement
is a suÆcient condition forour goal, considera given node n
i;j in S
i
. Since every subset of edges
is incident on n
i;j
,it follows that each of the other nodesin S
i
transmits a non-local sessionon a
wavelength usedbyn
i;j
to n
i;j
. Therefore,with respectto each n
i;j inS
i
,there is notype-2 localnode, i.e. jP
i;j j=0.
Therefore,wewantto partitionE into x
i
subsetsofN x
i
edgessothateachsubsetisincident
to all the nodes in V
1
and V
2
. We shall show that this partitioning problem can be solved by
reducing it to a bipartite matching problem. For example, for the partial WA bipartite graph
in gure 3-8a, gure 3-8b shows one possible partitioning of E such that each subset of edges is
incident to all thenodesinV
1 and V
2 .
Asmentionedabove, afterthe partitionofE,we assignthewavelengths correspondingto each
subsetof E to eachn
i;j inS
i
to transmitits non-localsessions. Forexample, accordingto gure
3-8b, we assign subsets E 1 , E 2 , and E 3 to n 1;1 , n 1;2 , and n 1;3 respectively. Node n 1;1 transmits its
non-local sessions on wavelengths 1, 6, 8, 10, 13, and 16 to n
2;1 , n 2;2 , n 2;3 , n 3;1 , n 3;2 , and n 4;1
respectively. To complete theexample in gure 3-7, we specifythe source nodes ineach transmit
subband based on the partitioning of E in gure 3-8b. The nal result of phase 2 for top-level
subtree1 isshown in gure3-9b.
n1;1n1;2n1;3n1;2n1;3n1;1n1;3n1;1n1;2n1;1n1;2n1;3n1;1n1;2n1;3n1;1n1;2n1;3 1 2 3 4 5 6 7 8 9 10 11 12 14 16 17 18 wavelength indices wavelengths sessiononspeci c receivinganon-local nodesinS 1 n 1;1 n 1;1 n 1;1 n 1;1 n 1;1 n 1;1 n 1;2 n 1;2 n 1;2 n 1;2 n 1;2 n 1;2 n 1;3 n 1;3 n 1;3 n 1;3 n 1;3 n 1;3 (4;1) (3;1) (2;1) 13 15 fromS 4 fromS 3 fromS 2
(a)nodesreceivingnon-localsessionsfromS
i 0,i 0 6=1,toS1 onspeci cwavelengths sessiononspeci c wavelengths ton 2;1 ton 2;2 ton 2;3 ton 3;1 ton 3;2 ton 4;1 (1;3) (1;4) receivinganon-local nodesinS1 (1;2) toS2 toS3 toS4
(b)nodestransmittingnon-localsessionsfromS
1 toS i 0,i 0 6=1,onspeci cwavelengths
Figure 3-9: The nal result of phase 2 of the o -line tree WA algorithm for top-level subtree 1 in
gure3-6.
Itremainsto considerthetop-levelsubtreeswhich donotsatisfythepreviousassumption that
theset of non-local transmit wavelengths is equal to theset of non-local receive wavelengths. As
anexample, considertop-levelsubtree 3basedon thesameexamplein gure3-6. The wavelength
bandsused fornon-local sessionsto and from top-levelsubtree 3 areshown in gure 3-10. Notice
inS 3 . (4;3) (1;3) (2;3) (3;2) (3;1) (3;4) top-levelsubtree3 transmittedfrom wavelengthbands indices wavelength 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 transmittedto top-levelsubtree3 wavelengthbands
Figure3-10: Wavelengthbandsto andfromtop-levelsubtree 3 in gure3-6.
TheresultofthepartialWAisshownin gure3-11. FromthegivenpartialWA,wecancreate
the partial WA bipartite graph for top-level subtree 3 in the same fashion as we have done for
top-level subtree 1. This partial WA bipartite graph is the bipartite graph shown in gure 3-12a
butwithonlythesolidlinesasits edges. Notethatonlythenon-localtransmitwavelengths which
are also the non-local receive wavelengths in top-level subtree 3 correspond to the edges in the
partialWAbipartitegraph. Forexample,thesolidedgesin gure3-12acorrespondto wavelengths
1, 2, 4, 5, 6, 13, 14, 15, 17, and 18 which are both non-local transmit wavelengths and non-local
receive wavelengths in top-level subtree 3. However, the non-local transmit wavelengths 7, 8, 9,
and 16 do notcorrespond to anyedgeinthe partialWA bipartitegraph.
1 2 3 4 5 6 7 8 9 10 11 12 14 16 17 18 wavelength indices 13 15 wavelengths sessiononspeci c receivinganon-local nodesinS 3 n 3;1 (2;3) (1;3) (4;3) n 3;1 n 3;1 n 3;1 n 3;1 n 3;1 n 3;1 n 3;2 n 3;2 n 3;2 n 3;2 n 3;2 n 3;2 n 3;2
fromS2 fromS1 fromS4
(a)nodesreceivingnon-localsessionsfromS
i 0,i 0 6=3,toS3onspeci cwavelengths nodesinS3 transmittinganon-local sessiononspeci c wavelengths ton 4;1 ton 1;1 ton 1;2 ton 1;3 ton 2;1 ton 2;2 ton 2;3 (3;4) (3;1) (3;2)
(b)nodestransmittingnon-localsessionsfromS
3 toS
i 0,i
0
6=3,onspeci cwavelengths(tobeassigned) toS 4 toS 1 toS 2
Figure 3-11: The result of the partial WA in phase 2 of the o -line tree WA algorithm for top-level
n 1;1 n 4;1 n1;2 n1;3 n 2;1 n 2;2 n2;3 n 1;1 n 4;1 n1;2 n1;3 n 2;1 n 2;2 n2;3 n 1;1 n 4;1 n1;2 n1;3 n 2;1 n 2;2 n2;3 n3;1 n 3;2 n3;1 n 3;2 n3;1 n 3;2 Eachedgecorresponds
toadistinct
wavelength.
fortop-levelsubtree3 (a)partialWAbipartitegraph
isshownasadashedline. type-2reusablewavelength Anedgeinvolvinga subsetE 0 2 forn3;2 subsetE 0 1 forn3;1 4 5 6 13 14 15 16 17 1 2 18 (b)partitioningofE 0
sothateachsubset
isincidenttoallthenodesinV
1 andV 2 8 7 9 V 1 V2 V 1 V2 V 1 V2
Figure3-12: Partial WAbipartitegraphfortop-levelsubtree 3 in gure 3-6.
In general,given top-levelsubtree iwhich doesnot satisfythe assumptionthat thesetof
non-localtransmitwavelengths isequalto theset ofnon-localreceive wavelengths,we canperformthe
partialWA and construct the partial WA bipartite graph, as we have done for top-level subtree
3 in gure 3-12a. Since some non-local transmit wavelengths will not be present in the partial
WAbipartitegraph,wecannotpartitiontheedgesto assignthenon-localtransmitwavelengths to
eachnode n
i;j in S
i
aswe have doneearlierfortop-levelsubtree1. Toovercome thisdiÆculty,we
pair up in a one-to-one fashion each type-2 reusablewavelength with respectto some node inS
i ,
i.e. a non-local receive wavelength which isnot a non-localtransmit wavelength, with a non-local
transmitwavelengthwhichisnota non-localreceive wavelength. Ifk isthetotalnumberof type-2
reusablewavelengths intop-level subtreei, thereare k!waysto do thispairing. However, inwhat
follows,itdoesnotmatter whichwaythepairingiscarried out. Forexample, fortop-level subtree
3 in gure 3-11a, we can pair up type-2 reusable wavelengths 3, 10, 11, and 12 with non-local
transmitwavelengths 7,8,9, and 16respectively.
We modifythesetofedges E inthepartialWA bipartitegraphasfollows. To create anewset
ofedges,denotedbyE 0
,weregardeachtype-2reusablewavelengthasbeingequivalenttoitspaired
value, i.e. a non-local transmitwavelength. As before, an edgejoins n
i 0 ;j 0 in V 1 and n i;j inV 2 for
each wavelength that is usedbothto receive a non-local sessionfrom S
i byn i 0 ;j 0 and to receive a non-localsessionbyn i;j .
