Queueing Networks: A Fundamental Approach

813 Pages·2011·6.175 MB·English

by  Nico M. Van Dijk (auth.), Richard J. Boucherie, Nico M. van Dijk (eds.)

International Series in Operations
Research & Management Science
Volume 154
SeriesEditor:
FrederickS.Hillier
StanfordUniversity,CA,USA
SpecialEditorialConsultant:
CamilleC.Price
StephenF.Austin,StateUniversity,TX,USA
Forfurthervolumes:
http://www.springer.com/series/6161

Richard J. Boucherie • Nico M. van Dijk
Editors
Queueing Networks
A Fundamental Approach
123
Editors
Richard J. Boucherie Nico M. van Dijk
Departement of Applied Mathematics Faculty of Economics and Business
University of Twente University of Amsterdam
Stochastic OR Group Roetersstraat 11
PO Box 217 1018 WB Amsterdam
7500 AE Enschede The Netherlands
The Netherlands [email protected]
[email protected]
ISSN 0884-8289
ISBN 978-1-4419-6471-7 e-ISBN 978-1-4419-6472-4
DOI 10.1007/978-1-4419-6472-4
Springer New York Dordrecht Heidelberg London
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Preface
TheoriginofqueueingtheoryanditsapplicationtracesbacktoErlang’shistorical
workfortelephonynetworksasrecentlycelebratedbythe ErlangCentennial,100
Years of Queueing, Copenhagen, recalling his first paper in 1909. Ever since, the
simplicityandfundamentalflavourofErlang’sfamousexpressions,suchashisloss
formulaforanincomingcallin a circuitswitchedsystem tobe lost,hasremained
intriguing. It has motivated the development of results with similar elegance and
expression power for various systems modeling congestion and competition over
resources.
Asecondmilestonewasthestepofqueueingtheoryintoqueueingnetworksas
motivated by the first so-called product form results for assembly type networks
in manufacturing in the nineteen fifties (R.R.P. Jackson 1954, J.R. Jackson 1957,
and E. Koenigsberg 1958, 1959). These results revealed that the queue lengths at
nodesofanetwork,wherecustomersrouteamongthenodesuponservicecomple-
tion in equilibrium can be regarded as independent random variables, that is, the
equilibriumdistributionofthenetworkofnodesfactorizesover(isaproductof)the
marginalequilibriumdistributionsof the individualnodesas if in isolation. These
networksarenowadaysreferredtoasJacksonnetworks.
A third milestone was inspired by the rapid developmentof computer systems
andbroughttheattentionforservicedisciplinessuchastheProcessorSharingdis-
ciplineintroducedbyKleinrockin1967.Morecomplicatedmultiservernodesand
service disciplines such as First-Come-First-Served, Last-Come-First-Served and
ProcessorSharing,andtheirmixingwithinanetworkhaveledtoasurgeintheoret-
icaldevelopmentsandawideapplicabilityofqueuingtheory.
Queueingnetworkshave obtained their place in both theoryand practice. New
technologicaldevelopmentssuchasInternetandwirelesscommunications,butalso
advancementsin existing applications such as manufacturingand productionsys-
tems,publictransportation,logistics,andhealthcare,havetriggeredmanytheoreti-
calandpracticalresults.
Queueing network theory has focused on both the analysis of complex nodes,
and the interaction between nodes in networks. This handbook aims to highlight
fundamental,methodologicaland computationalaspects of networksof queuesto
v
vi Preface
provide insight and unify results that can be applied in a more general manner.
Severaltopics thatare closely related are treated from the perspectiveof different
authors to also provide different intuition that, in our opinion, is of fundamental
importancetoappreciatetheresultsfornetworksofqueues.Ofcourse,applications
ofmodernqueueingnetworksaremanifold.Theseareillustratedintheconcluding
chaptersofthishandbook.Thehandbookisorganizedinfiveparts.
Part1. Exact analyticalresults,chapters1–7
Product form expressions for the equilibrium distribution of networks are by far
leading and have been most dominant in the literature on exact analytical results
forqueueingnetworks.Inrecentyears,featuressuchasbatchrouting,negativecus-
tomers and signals have been introduced to enhance the modeling power of this
class of networks.A unified theoryfrom differentperspectivesis containedin the
firstpartofthishandbook.Topicsinclude
acharacterizationofproductformsbyphysicalbalanceconceptsandsimpletraf-
•
ficflowequations,
classesofserviceandqueuedisciplinessuchasInvariantDisciplinesandOrder
•
Independentqueuesthatallowaproductform,
aunifieddescriptionofproductformsfordiscretetimequeueingnetworks,
•
insightsforinsensitivityfromtheclassicalErlanglossmodeluptoGeneralised
•
SemiMarkovProcessesandpartiallyinsensitivenetworks,
aggregationanddecompositionresultsthatallowsubnetworkstobeaggregated
•
intosinglenodestoreducecomputationalburden.
These productformresults encompassa numberof intriguingaspects thatare not
onlymostusefulforpracticalpurposesbutalsoindicateavarietyofopenproblems
whichremaintobetackled.
Part2. Monotonicityand comparisonresults,chapters 8–9
Exact(productform)resultsareonlyavailableforalimitedclassofnetworks.These
exactresults,however,mayalsobeinvokedtoobtainboundsforperformancemea-
suresforintractablequeueingnetworks.Twobasicapproachescanbeidentified:
stochasticmonotonicityandorderingresultsbasedontheorderingofthegener-
•
atorsoftheprocesses,
comparisonresultsandexpliciterrorboundsbasedonanunderlyingMarkovre-
•
wardstructurewhichleadstoorderingofexpectationsofperformancemeasures.
There is a clear trade-off for applying either of these two approaches. Stochastic
monotonicity yields stronger results such as with non-exponential service times.
Preface vii
TheMarkovrewardapproachin turnisapplicableunderlessstringentconditions,
particularlywithmorecomplexstructuresasina queueingnetwork.Theseresults
arenotonlyoftheoreticalandqualitativeinterestbythemselves,butalsomotivate
thederivationofexactanalyticalresultstoenablebounds.
Part3. Diffusionand fluid results,chapters 10–12
Limitingregimesoftenallow foramenableexpressionsfor performancemeasures
in systems that are otherwise intractable. Two particular regimes are of interest:
thefluidregimeandthediffusionregimethatareillustratedthroughthefollowing
topics:
fluidlimitsforanalysisofsystemstability,
•
diffusionapproximationformulti-serversystems,
•
systemfedbyGaussiantraffictomodelvariationinthearrivalprocess.
•
Thesetopicsillustrate arichclassofsystemsthatmaybeanalyzedin thelimiting
regimeandidentifyanimportantareaofcurrentresearch.
Part4. Computationaland approximateresults,chapters 13–15
Practical applications such as in manufacturing, computer performance and com-
munications rapidly prove to be beyond analytical solvability due to e.g. non-
exponential service times, capacity constraints, synchronization or prioritization.
Numericallyexactorapproximateapproachesforaveragesordistributionsofper-
formancemeasureshavebeendevelopedinliterature.Anillustrationisprovidedvia
thefollowingtopics:
MVA(meanvalueanalysis)andQNA(queueingnetworkanalyzer)focusingon
•
mean and variance of performancemeasuressuch as queuelength and sojourn
times,
numericalapproximationofresponsetimedistributions
•
approximatedecompositionresultsforlargeopenqueueingnetworks.
•
The numerical approach to performance analysis is a lively research community
thatconsiderablycontributestothesuccessofqueueingtheoryinapplicationsasit
allowsforexplicitnumericalresultsforperformancemeasures.
viii Preface
Part5. Selected applications,chapters 16–18
Applicationsofqueueingnetworksaremanifold.Toillustratetheapplicationpower
ofqueueingtheory,somespecialapplicationareasandtheirspecificqueueingnet-
workaspectsareenlightened:
lossnetworksasoriginatingfromcircuitswitchedtelecommunicationsapplica-
•
tions,
capacitysharingasoriginatingfrompacketswitchingindatanetworks,
•
hospitallogistics.
•
Thefirsttwoapplicationshaveatheoreticalnatureastheyillustrateatypicalclassof
queueingnetworks.Thelastapplicationillustratesatypicalapproachforapplication
ofqueueingtheoryinapracticalenvironment.
Despite the fundamental theoretical flavour of this book, it is to be kept in mind
that the area of queueing theory would not have existed and would not have pro-
gressedsostronglyhaditnotbeendrivenbyapplicationareasthatledtothevarious
fundamentalquestions.Theintertwinedprogressoftheoryandpracticewillremain
to be most intriguingand will continueto be the basis of furtherdevelopmentsin
queueingtheory.Youarehighlyinvitedtostepin.
Acknowledgments
We wouldliketo expresssomewordsofgratitude.Firstofall,we aremostgrate-
fultoallauthorsinvolvedfortheirpositivereactionsandenthusiasmrightfromthe
start,fortheircooperationandfortheirinvaluablecontributions.We wishtoapol-
ogizetothoseauthorswhohadalreadysubmittedatanearlystagebutwhohadto
wait for such a long time until its final publication.We like to apologize to some
otherauthorsforour persistently hamperingontheir shoulders.We hopethatyou
canappreciateourintentionandfinaloutcome.
Wedeeplyoweourgratefulnesstotheeditor-in-chiefDr.F.S.Hillier.Haditnot
been for his stimulation, patience and confidencein its finalization, we would not
havebeenabletocompletethisbooksuccessfully.
ThesupportbytheDutchOrganisationforScientificResearch(NWO)bywhich
theoutcomeofthisbookcouldberealizedhasbeenhighlyappreciated.
Enschede, RichardJ.Boucherie
Amsterdam, NicoM.VanDijk
June2010
Contents
1 OnPracticalProductFormCharacterizations ................... 1
NicoM.vanDijk
A:ProductForms:SingleStationHierarchy ...................... 2
1.1 Introduction.............................................. 2
1.2 ProductForms:ThreeBalances.............................. 4
1.2.1 StationBalance:B-DorErlang-Engsetsystems ........ 4
1.2.2 Classbalance:Coordinateconvexproperty(CCP) ...... 6
1.2.3 JobLocalBalance:Necessity........................ 15
1.2.4 LCFS-precase:Nonexponential ..................... 19
1.2.5 SymmetricDisciplinesandJob-Local-balance(JLB) .... 22
1.3 InvariantDisciplinesandJLB ............................... 25
1.3.1 InvarianceCondition............................... 25
1.3.2 Serviceinvariantexamples.......................... 28
1.3.3 Ageneralizedsymmetricinsensitivityresult ........... 32
1.4 Anapplication,literaturediscussionandhierarchyreview ....... 35
1.4.1 AnM Gcc+mapplication......................... 35
| | |
1.4.2 Literaturediscussion............................... 37
1.4.3 Ahierarchyreview ................................ 39
B:ProductForms:TandemandClusterStructures ................ 41
1.5 TandemQueues........................................... 41
1.5.1 Introduction ...................................... 41
1.5.2 ProductFormTandemQueues....................... 45
1.5.3 Serviceexamples.................................. 49
1.5.4 Blockingexamples ................................ 51
1.5.5 Mixedexamples................................... 54
1.6 Jacksonianclusters ........................................ 56
1.6.1 AJacksoncluster.................................. 56
1.6.2 ArestrictedJacksoncluster ......................... 58
1.6.3 Aconservativeproductformprotocol................. 60
1.7 Productformboundsfornetworksofrestrictedclusters ......... 62
1.7.1 Instructivetandemextension ........................ 63
ix
x Contents
1.7.2 AJacksonTandem ................................ 66
1.7.3 Anestedcase..................................... 68
1.7.4 Furtherillustrativeexamples ........................ 69
1.7.5 AnOptimalDesignApplication ..................... 73
1.8 Ahospitalapplication...................................... 73
1.8.1 Motivation ....................................... 73
1.8.2 Modelformulation ................................ 74
1.8.3 Boundsandapplication............................. 76
1.9 Evaluation ............................................... 77
1.9.1 Literature ........................................ 77
1.9.2 ReviewPartB .................................... 79
1.9.3 Someremainingquestions .......................... 80
References..................................................... 81
2 OrderIndependentQueues ................................... 85
A.E.Krzesinski
2.1 Introduction.............................................. 85
2.2 TheOIQueue ............................................ 87
2.2.1 TheDefinitionofanOIQueue ...................... 88
2.2.2 TheImplicationsoftheOIConditions ................ 88
2.2.3 TheStationaryDistribution ......................... 89
2.2.4 ModelsCoveredbytheOIClass..................... 92
2.3 NumericalTechniquesfortheOIQueue ...................... 95
2.3.1 AggregatingtheStateSpace ........................ 95
2.3.2 ThePerformanceMeasures:theMSCCCQueue........ 96
2.4 TheOILossQueue........................................102
2.4.1 TheStationaryDistribution .........................103
2.4.2 ThePerformanceMeasures:theMSCCCLossQueue ...105
2.4.3 OILossNetworks .................................109
2.5 OIApplications...........................................110
2.5.1 MultiportedMemory...............................110
2.5.2 AMessagingCard.................................110
2.5.3 MultilayerWindowFlowControl ....................111
2.5.4 MachineSchedulingModel .........................111
2.5.5 BlockedCallsCleared .............................111
2.5.6 BlockedCallsQueued .............................112
2.5.7 BlockedCallsQueuedwithSourceRejection ..........113
2.5.8 LocalandLongDistanceCalls ......................113
2.5.9 LocalandTransitCalls.............................114
2.5.10 HierarchicalTreeNetworks .........................115
2.5.11 LocalandExternalNetworks........................115
2.5.12 TransitCallsamongNetworks.......................116
2.6 AnAlgorithmtoComputethePerformanceMeasuresofthe
MSCCC .................................................118
References.....................................................119