ITrends in Telecommunications TechnologiesTrends in Telecommunications TechnologiesEdited byChristos J. BourasIn-Tech intechweb.orgPublished by In-TehIn-TehOlajnica 19/2, 32000 Vukovar, CroatiaAbstractingandnon-proftuseofthematerialispermittedwithcredittothesource.Statementsand opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published articles. Publisher assumes no responsibility liability for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained inside. After this work has been published by the In-Teh, authors have the right to republish it, in whole or part, in any publication of which they are an author or editor, and the make other personal use of the work. 2010 In-tehwww.intechweb.orgAdditional copies can be obtained from: publication@intechweb.orgFirst published March 2010Printed in IndiaTechnical Editor: Goran BajacCover designed by Dino SmrekarTrends in Telecommunications Technologies,Edited by Christos J. Bourasp.cm.ISBN 978-953-307-072-8VPrefaceThisbookistheculminationofanefforttogatherworld-classscientists,engineersand educatorsengagedinthefeldsoftelecommunicationstomeetandpresenttheirlatest activities.Telecommunicationistheassistedtransmissionofsignalsoveradistancefor thepurposeofcommunication.Havingdrasticallytransformedthehumanwayofliving, telecommunications are considered the revolution of our times, and the catalyst for present and future technological and scientifc developments. Being a very active research feld, new advancesintelecommunicationsareconstantlychangingthelandscapeandintroducenew capabilitiesandenhancedwaysofcommunication.Literatureinthefeldisextensiveand constantlyenlarged.Thisbookthereforeintendstoincreasethedisseminationlevelofthe latest research advances and breakthroughs by making them available to a wide audience in a compact and friendly to the user way. It furthermore aims to provide a particularly good way for experts in one aspect of the feld to learn about advances made by their colleagues with different research interests. Themainfocusofthebookistheadvancesintelecommunicationsmodeling,policy,and technology.Inparticular,severalchaptersofthebookdealwithlow-levelnetworklayers andpresentissuesinopticalcommunicationtechnologyandopticalnetworks,including the deployment of optical hardware devices and the design of optical network architecture. Wirelessnetworkingisalsocovered,withafocusonWiFiandWiMAXtechnologies.The book also contains chapters that deal with transport issues, and namely protocols and policies for effcient and guaranteed transmission characteristics while transferring demanding data applications such as video. Finally, the book includes chapters that focus on the delivery of applications through common telecommunication channels such as the earth atmosphere. This book is useful for researchers working in the telecommunications feld, in order to read a compact gathering of some of the latest efforts in related areas. It is also useful for educators thatwishtogetanup-to-dateglimpseoftelecommunicationsresearchandpresentitin aneasilyunderstandableandconciseway.Itisfnallysuitablefortheengineersandother interested people that would beneft from an overview of ideas, experiments, algorithms and techniques that are presented throughout the book. VIBringingthisbooktothepublicationstageistherewardoftheeffortputbytheeditors, contributorsandreviewersofallthepresentedchapters.Wewouldthereforeliketo acknowledgethevaluablecontributionsofalltheauthorsofthechapterscontainedinthis book,whodecidedtooffertheirinsightsandexpertiseinordertohelpassemble,inour opinion, a highly useful and quality book. We would also like to thank the publishing house, which supported this effort and helped make the result available to a wide audience all over the world. Patras, 17th March, 2010 Christos J. Bouras Professor University of Patras and RACTIVIIContentsPreface V1. ANovelPFCCircuitforThree-phaseutilizingSingleSwitchingDevice 001KeijuMatsuiandMasaruHasegawa2. AdvancedModulationFormatsandMultiplexingTechniquesforOpticalTelecommunicationSystems 013GhafourAmouzadMahdirajiandAhmadFauziAbas3. ASurveyontheDesignofBinaryPulseCompressionCodeswithLowAutocorrelation 039MaryamAminNasrabadiandMohammadHassanBastani4. VirtualMulticast 063PetrHolubandEvaHladk5. TheAsymmetricalArchitectureofNewOpticalSwitchDevice 087MohammadSyuhaimiAb-RahmanandBoonchuanNg6. AdaptiveActiveQueueManagementforTCPFriendlyRateControl(TFRC)TraffcinHeterogeneousNetworks 109RahimRahmaniandChristerhlund7. Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 123SergeyDudinandMoonHoLee8. TelecommunicationPowerSystem:energysaving,renewablesourcesandenvironmentalmonitoring 145CarmineLubritto9. PropagationModelsandtheirApplicationsinDigitalTelevisionBroadcastNetworkDesignandImplementation 165ArmoogumV.,SoyjaudahK.M.S.,MohamudallyN.andFogartyT.10. InterferenceModelingforWirelessAdHocNetworks 185AltenisV.Lima-e-Lima,CarlosE.B.CruzPimentelandRenatoM.deMoraesVIII11. EnergySavingDrivesNewApproachestoTelecommunicationsPowerSystem 201RaisMiftakhutdinov12. DirectionalRoutingProtocolinWirelessMobileAdHocNetwork 235L.A.Latiff,N.Fisal,S.A.ArifnandA.AliAhmed13. FreeSpaceOpticalTechnologies 257DavideM.Forin,G.Incerti,G.M.TosiBeleff,A.L.J.Teixeira,L.N.Costa,P.S.DeBritoAndr,B.Geiger,E.LeitgebandF.Nadeem14. Novelmultipleaccessmodelsandtheirprobabilisticdescription 297DmitryOsipov15. Performanceanalysisofmulti-serverqueueingsystemoperatingundercontrolofarandomenvironment 317CheSoongKim,AlexanderDudin,ValentinaKlimenokandValentinaKhramova16. InterdomainQoSpathsfndingbasedonoverlaytopologyandQoSnegotiationapproach 345erbanGeorgicObrejaandEugenBorcoci17. DualLinearlyPolarizedMicrostripArrayAntenna 367M.S.RMohdShah,M.Z.AAbdulAziz,M.K.SuaidiandM.K.ARahim18. ATime-DelaySuppressionTechniqueforDigitalPWMControlCircuit 389YoichiIshizuka19. Layer2QualityofServiceArchitectures 399ChristosBouras,VaggelisKapoulas,VassilisPapapanagiotou,LeonidasPoulopoulos,DimitrisPrimpasandKostasStamos20. SecrecyonthePhysicalLayerinWirelessNetworks 413EduardA.Jorswieck,AnneWolf,andSabrinaGerbracht21. PerformanceAnalysisofTime-of-ArrivalMobilePositioninginWirelessCellularCDMANetworks 437M.A.Landolsi,A.H.Muqaibel,A.S.Al-Ahmari,H.-R.KhanandR.A.Al-Nimnim22. MobilityandHandoffManagementinWirelessNetworks 457JaydipSen23. GPSTotalElectronContent(TEC)PredictionatIonosphereLayerovertheEquatorialRegion 485NorsuzilaYaacob,MardinaAbdullahandMahamodIsmail24. PerformanceEvaluationMethodstoStudyIEEE802.11BroadbandWirelessNetworksunderthePCFMode 509VladimirVishnevskyandOlgaSemenova25. NextGenerationOpticalAccessNetworks:fromTDMtoWDM 537Ll.Gutierrez,P.Garfas,M.DeAndrade,C.Cervell-PastorandS.SallentIX26. BuildingenergyeffciencydesignfortelecommunicationbasestationsinGuangzhou 561YiChen,YufengZhangandQinglinMeng27. DynamicSpace-CodeMultipleAccess(DSCMA)System:ADoubleInterferenceCancellationMultipleAccessSchemeinWirelessCommunicationsSystem 583CheeKyunNg,NorKamariahNoordin,BorhanuddinMohdAli,andSudhanshuShekharJamuar28. VideoStreaminginEvolvingNetworksunderFuzzyLogicControl 613MartinFleury,EmmanuelJammeh,RouzbehRazavi,SandroMoironandMohammedGhanbari29. TheDevelopmentofCrosstalk-FreeSchedulingAlgorithmsforRoutinginOpticalMultistageInterconnectionNetworks 643MohamedOthman,andTgDianShahidaRajaMohdAuzar30. Traditionalfoatcharges:aretheysuitedtostationaryantimony-freeleadacidbatteries? 665T.M.PhuongNguyen,GuillaumeDillensegerChristianGlaizeandJeanAlzieu31. NeighborDiscovery:SecurityChallengesinWirelessAdhocandSensorNetworks 693MohammadSayadHaghighiandKamalMohamedpour32. NewTrendsinNetworkAnomalyDetection 715YasserYasamiandSaadatPourmozaffari33. AnEffcientEnergyAwareRoutingProtocolforRealTimeTraffcsinWirelessSensorNetworks 735AmirHosseinMohajerzadehandMohammadHosseinYaghmaee34. QualityofServiceDifferentiationinWiMAXNetworks 753PedroNeves,SusanaSargento,FranciscoFontes,ThomasM.BohnertandJooMonteiroANovelPFCCircuitforThree-phaseutilizingSingleSwitchingDevice 1ANovelPFCCircuitforThree-phaseutilizingSingleSwitchingDeviceKeijuMatsuiandMasaruHasegawaX A Novel PFC Circuit for Three-phaseutilizing Single Switching Device Keiju Matsui and Masaru Hasegawa ChubuUniversity Japan 1. Introduction Forconsumerorindustrialapplications,electricalappliancesusevarioustypesofrectifier, which give rise to distorted input current due their non linear characteristics. Problems are createdbythevariousharmonics,generatedinthepowersystem.Undersuch circumstances, IEC guideline was instituted ten and several years ago, and has recently been superseded.(JISC.2005).Withthespreadoftheuseofsuchnonlinearequipments,itis anticipatedthatwecannotavoidtheproblemsduetoharmonics.Withtherelatively increasedcapacityofindustryapplications,PWMrectifierscanbeexpectedtobeusedin three phase and single phase applications. (Takahashi. 1985, IEEJ Committee. 2000). Also in office environments, OA equipments, inverter type fluorescent lamps and inverter type air-conditionersarefrequentlyused,surelybringingharmonicproblemswiththem.Under suchconditions,variousnewtypePFCschemesarepresentedanddiscussed. (Takahashi.1900,Fujiwara.1991,Takeuchi2005).Methodsintendingtoimprovethecurrent towards a sinusoidal waveform by using switching devices will incur high cost performance andyettroublesomenoiseproblems.Certainapplicationsrequireaswitch-lessschemeto maintaintheelectromagneticenvironmentalstandards.(Yamamoto.2001,Takeuchi.2007). Also in the future, main stream methods will intend to achieve sinusoidal waveforms. From thinkingaboutresearchstreamuntilnow,moresimplifiedmethodorlowcostscheme would be discussed and developed in a similar manner also in the future. On the basis of the perceived requirements, in this paper, we propose and discuss a novel PFC circuit for three phase, employing a single switch in such a manner as to render the waveform as sinusoidal as possible. 2. Operational Principle 2.1 Prasad-Ziogas Circuit Figure1showsaconventionalcircuit,comprisingathree-phasecircuit,usingsingle switching device.(Prasad & Ziogas. 1991). The principle of operation is such that the three phase circuit is periodically shorted by a single switching device at a high frequency, so that theinputcurrentwaveformiscreatedinproportiontoinputvoltagewaveform.Theinput currentwaveformbecomes synchronizedwiththeinputvoltage,sothatthecircuit scheme 1TrendsinTelecommunicationsTechnologies 2 isconstructedasPFCcircuit.Inthispaper,thiscircuitisnamedthePZ(Prasad-Ziogas) circuit,oneoftheseindividualsbeingfamousforcontributionstowardpowerelectronics development. Fig. 1. Three-phase single switch PFC circuit by Prasad-Ziogas. Fig. 2. Equivalent circuit for Prasad-Ziogas. Figure2showstheequivalentcircuitofthePZcircuit.Thesecharacteristicsmaybe explainedasfollows;InFigure2(a),whenS1turns-on,theequivalentcircuitisestablished asshown,wheretheoperationwillbeexplainedasacurrent-discontinuousmodefor simplicity of circuit analysis. In Figure 2 (b), the terminal voltage across O and O' of fictional neutralpointcanbederivedfromFigure1,theamplitudebeingE0/6withanoperational frequency three times supply frequency, where E0 is the output dc link voltage. In Figure 2 (a), when S1 is turned on, circuit equation can be established as follows; dtdiL euu u (1) Theanalogousequationscanbedescribedalsoinphasevandphasew.FromEq.(1),the inputcurrentisincreasinginproportiontoamplitudeofeuatS1turn-on.(seeFigure3(b) and(c)).Whentheswitchisturned-off,theequivalentcircuitisestablishedasshownin Figure 2 (b), where, by analogy with the other phases, the equations become as follows; O O' eu iv iu iw Lu A S1 C1 C1 RoB C Vd iu eu eu>0 C1 O O' A Lu bS1 turn-off Lu iu eu eu>0 (a) S1 turn-on dtdiL v edtdiL v edtdiL v eww CO wvv BO vuu AO u (2) From these equations, it is clear that each phase current is decreasing in proportion to eu-vAO etc.Thesewaveformsare shownfortheS1-offperiodinFigure3.Ifthe currentwaveforms are decreasing, as shown by the dashed lines, the resultant current values could be obtained in proportion to the input voltage values. However, the terms for attenuation, such as eu-vAO, are nonlinear. (see Figure 4 showing vAO). Actual waveforms are attenuated by means of the termslikeeu-vAOetc.(Murphy.1985).If ev,foranexample,hasasmall value,thedegreeof attenuationmaybesmall,sothatagentlydecayingdashedlinewouldbeobtained,as shown. In this example case, however, the attenuation term is ev-vBO, so that the attenuation degree becomes severe. As a result, the sharply decaying solid line can be obtained, because of significant attenuation, producing nonlinear waveforms. Fig. 3. Input currentwaveforms at S1 switching. Fig. 4.Conceptual voltage waveform, vAO. (a) eu

0 (b) ev (c) ew 0 0 eu e eu eveeiv eu i S1 onS1 on 3.Inut current waveforms at S1 switching. S1 off iu iv iw vAO 32dV3dV0 ANovelPFCCircuitforThree-phaseutilizingSingleSwitchingDevice 3 isconstructedasPFCcircuit.Inthispaper,thiscircuitisnamedthePZ(Prasad-Ziogas) circuit,oneoftheseindividualsbeingfamousforcontributionstowardpowerelectronics development. Fig. 1. Three-phase single switch PFC circuit by Prasad-Ziogas. Fig. 2. Equivalent circuit for Prasad-Ziogas. Figure2showstheequivalentcircuitofthePZcircuit.Thesecharacteristicsmaybe explainedasfollows;InFigure2(a),whenS1turns-on,theequivalentcircuitisestablished asshown,wheretheoperationwillbeexplainedasacurrent-discontinuousmodefor simplicity of circuit analysis. In Figure 2 (b), the terminal voltage across O and O' of fictional neutralpointcanbederivedfromFigure1,theamplitudebeingE0/6withanoperational frequency three times supply frequency, where E0 is the output dc link voltage. In Figure 2 (a), when S1 is turned on, circuit equation can be established as follows; dtdiL euu u (1) Theanalogousequationscanbedescribedalsoinphasevandphasew.FromEq.(1),the inputcurrentisincreasinginproportiontoamplitudeofeuatS1turn-on.(seeFigure3(b) and(c)).Whentheswitchisturned-off,theequivalentcircuitisestablishedasshownin Figure 2 (b), where, by analogy with the other phases, the equations become as follows; O O' eu iv iu iw Lu A S1 C1 C1 RoB C Vd iu eu eu>0 C1 O O' A Lu bS1 turn-off Lu iu eu eu>0 (a) S1 turn-on dtdiL v edtdiL v edtdiL v eww CO wvv BO vuu AO u (2) From these equations, it is clear that each phase current is decreasing in proportion to eu-vAO etc.Thesewaveformsare shownfortheS1-offperiodinFigure3.Ifthe currentwaveforms are decreasing, as shown by the dashed lines, the resultant current values could be obtained in proportion to the input voltage values. However, the terms for attenuation, such as eu-vAO, are nonlinear. (see Figure 4 showing vAO). Actual waveforms are attenuated by means of the termslikeeu-vAOetc.(Murphy.1985).If ev,foranexample,hasasmall value,thedegreeof attenuationmaybesmall,sothatagentlydecayingdashedlinewouldbeobtained,as shown. In this example case, however, the attenuation term is ev-vBO, so that the attenuation degree becomes severe. As a result, the sharply decaying solid line can be obtained, because of significant attenuation, producing nonlinear waveforms. Fig. 3. Input currentwaveforms at S1 switching. Fig. 4.Conceptual voltage waveform, vAO. (a) eu

0 (b) ev (c) ew 0 0 eu e eu eveeiv eu i S1 onS1 on 3.Inut current waveforms at S1 switching. S1 off iu iv iw vAO 32dV3dV0 TrendsinTelecommunicationsTechnologies 4 Figure 4 shows conceptual waveform as vAO. When S1 is turned-off, the corresponding diode conducts.DependingonwhethertheamplitudeofvAO=2Vd/3orVd/3,whereVdisthe output voltage, the degree of attenuation at S1 turn-off is varied. Fig. 5. Explanation of distorted input current waveform in conventional method. Figure 5 shows the operational waveforms for Figure 1 from circuit simulation. From these figures,thereasonsforwaveformdistortionintheconventionalinputcurrentcanbe explainedtoacertainextent.Fromthephasevoltage,eu,inFigure5(a),theinputcurrent waveform, iu, appears as in Figure 5 (b), using single device switching. It can be found that the envelope of a six stepped waveform vAO appears and the distortion of iu is generated as inFigure5(b).Thetermeu-vAOin(2)appearsasanenvelopeoftheappliedvoltageacross the input inductor in Figure 5 (c). From equation vL=Ludiu/dt, it can be seen that the integral ofvLbecomestheinputcurrent,iu,sothattheimprovementschemeforinputcurrent waveformcanbedeterminedfromobservingtheinductorvoltagewave,vL,toacertain extent. 2.2 Operation Principle of the Proposed CircuitFigure6showsoneoftheproposedtypesof,three-phase,singleswitchconverter.Inthis paper,wewilldiscusstheboosttypeconverter.Inthefuture,however,itmaybepossible thatabucktypeconvertercouldberealizedunderadequatediscussion.Thus,thispaper titledoesnotrestricttheconcepttotheboosttypeconverter.Thecircuitconfiguration originatesfromtheabovementionedPrasad-Ziogascircuit.Thenotablefeatureisthat severalelectrolyticcapacitorsareparallel-connectedtorectifyingdiodes.Bymeansofthis configuration, theinputvoltagecircuitisalwaysconnectedtoeither dc outputbus,so that continuityandimprovement ofthe inputcurrentcanberealized. In suchaway,aboosted dcvoltage,utilizingthePFCscheme,canbeobtainedincomparisontotheconventional circuit.Thecircuitoperationcanberoughlydividedintosixperiods,whereeachperiodis 60 degrees. From the operation waveforms in Figure 7 and the operational periods shown in Figure 8, the circuit operation can be discussed. To simplify the analysis of the operation, we willassumeaunitypowerfactorofphase,u,wherefundamentalvoltageandcurrent components are almost synchronized with each other. iu (c) vL 200V 100A 300V 0 0 0 0 0 0 (a) (b) eu iu vL Fig. 6. Proposed circuit configuration. Fig. 7. Waveforms for proposed circuit. eu ewev Lu R0Cu C0 S1 A O vs Cx vd Lsw (a) (c) (d) (e) (f) (g) (h) (i) (j) eu iu iv iw vcu vcx vAO iDu icu vs 200V 00 00 0 0 0 0 0 0 00 00 0 0 0 0 0 0 100A 100A 100A 400V 400V 400V 100A 100A 1.0kV t0t1 t2 t3t4 t5 t6 (b) ANovelPFCCircuitforThree-phaseutilizingSingleSwitchingDevice 5 Figure 4 shows conceptual waveform as vAO. When S1 is turned-off, the corresponding diode conducts.DependingonwhethertheamplitudeofvAO=2Vd/3orVd/3,whereVdisthe output voltage, the degree of attenuation at S1 turn-off is varied. Fig. 5. Explanation of distorted input current waveform in conventional method. Figure 5 shows the operational waveforms for Figure 1 from circuit simulation. From these figures,thereasonsforwaveformdistortionintheconventionalinputcurrentcanbe explainedtoacertainextent.Fromthephasevoltage,eu,inFigure5(a),theinputcurrent waveform, iu, appears as in Figure 5 (b), using single device switching. It can be found that the envelope of a six stepped waveform vAO appears and the distortion of iu is generated as inFigure5(b).Thetermeu-vAOin(2)appearsasanenvelopeoftheappliedvoltageacross the input inductor in Figure 5 (c). From equation vL=Ludiu/dt, it can be seen that the integral ofvLbecomestheinputcurrent,iu,sothattheimprovementschemeforinputcurrent waveformcanbedeterminedfromobservingtheinductorvoltagewave,vL,toacertain extent. 2.2 Operation Principle of the Proposed CircuitFigure6showsoneoftheproposedtypesof,three-phase,singleswitchconverter.Inthis paper,wewilldiscusstheboosttypeconverter.Inthefuture,however,itmaybepossible thatabucktypeconvertercouldberealizedunderadequatediscussion.Thus,thispaper titledoesnotrestricttheconcepttotheboosttypeconverter.Thecircuitconfiguration originatesfromtheabovementionedPrasad-Ziogascircuit.Thenotablefeatureisthat severalelectrolyticcapacitorsareparallel-connectedtorectifyingdiodes.Bymeansofthis configuration, theinputvoltagecircuitisalwaysconnectedtoeither dc outputbus,so that continuityandimprovement ofthe inputcurrentcanberealized. In suchaway,aboosted dcvoltage,utilizingthePFCscheme,canbeobtainedincomparisontotheconventional circuit.Thecircuitoperationcanberoughlydividedintosixperiods,whereeachperiodis 60 degrees. From the operation waveforms in Figure 7 and the operational periods shown in Figure 8, the circuit operation can be discussed. To simplify the analysis of the operation, we willassumeaunitypowerfactorofphase,u,wherefundamentalvoltageandcurrent components are almost synchronized with each other. iu (c) vL 200V 100A 300V 0 0 0 0 0 0 (a) (b) eu iu vL Fig. 6. Proposed circuit configuration. Fig. 7. Waveforms for proposed circuit. eu ewev Lu R0Cu C0 S1 A O vs Cx vd Lsw (a) (c) (d) (e) (f) (g) (h) (i) (j) eu iu iv iw vcu vcx vAO iDu icu vs 200V 00 00 0 0 0 0 0 0 00 00 0 0 0 0 0 0 100A 100A 100A 400V 400V 400V 100A 100A 1.0kV t0t1 t2 t3t4 t5 t6 (b) TrendsinTelecommunicationsTechnologies 6 (a) period (t0 . 0 twith the finite state space{0,..., }. WThe sojourn timeof theMarkov chain tv ,> . 0 t inthestatev hasanexponentialdistributionwiththeparameter v v . = . . 0 WAfterthissojourntimeexpires,withprobabilitypk v v' ( , ),theprocessvt,> . 0 t transitsto thestatev' ,andk sessions,= . . 0 1 k arriveintothesystem.Theintensitiesofjumpsfrom onestateintoanother,whichareaccompaniedbyanarrivalofk sessions,arecombined intothematrices. = . . 0 1kDk ofsize+ + ( 1) ( 1) W W .Thematrixgeneratingfunctionof these matrices is= + . ' 's0 1( ) 1 D z D D z z . The matrix(1) Dis the infinitesimal generator of the processv . > . 0t tThe stationary distribution vectoroof this process satisfies the equations = . = . 0 o o (1) 1 D e Hereandinthesequel0 isthezerorowvectorande isthecolumn vectorofappropriatesizeconsistingof1s.Incasethedimensionalityofthevectorisnot clearfromthecontext,itisindicatedasalowerindex,e.g. We denotestheunitcolumn vector of dimensionality= + 1 W W .The average intensity(fundamental rate) of theMAPis defined as = . o1D e The variancevof intervals between session arrivals is calculated as = . o1 1 202 ( ) v D e the squared coefficient varcof variation is calculated by = . o102 ( ) 1varc D e whilethecorrelationcoefficient corc ofintervalsbetweensuccessivegrouparrivals isgiven by = / . o1 1 1 20 1 0( ( ) ( ) )corc D D D v e Formore information about the MAP , its special cases and propertiesand related research see(Fisher&Meier-Hellstern,1993),(Lucantoni,1991)andthesurveypaperbyS. Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 125 means of a token mechanism. The number of tokens, which defines the maximal number of flowsthatcanbeadmittedintothesystemsimultaneously,isveryimportantcontrol parameter.Ifthisnumberissmall,thechannelmaybeunderutilized.Ifthisnumberistoo large,thechannelmaybecomecongested.Averagedeliveringtimeandjittermayincrease essentiallyandGradeofServicebecomesbad.So,theproblemofdefiningtheoptimal numberoftokensisofpracticalimportanceandnon-trivial.In(Kistetal.,2005), performancemeasuresoftheSAPOR schemeofroutinginIP networksareevaluatedby means of computer simulation.Analogoussituationalsonaturallyarisesinmodelinginformationretrievinginrelational databaseswhere,besidestheCPUanddiscmemory,someadditional"threads"or "connections"shouldbeprovidedtostarttheusersapplicationprocessing.Inthis interpretation, sessionmeans applicationwhile requestsarequeriestobe processedwithin this application and tokens are threads or connections.In the paper (Lee et al., 2007), the Markovian queueing model with a finite buffer that suits for analytical performance evaluation and capacity planning of theSAPORrouting scheme aswellasformodellingtheotherreal-worldsystemswithtimedistributedarrivalof requestsinasessionisconsidered.Tothebestofourknowledge,suchkindofqueueing models was not considered and investigated in literature previously. In (Lee et al., 2007), the problem of the system throughput maximization subject to restriction of the loss probability forrequestsfromacceptedsessionsissolved.Inthepaper(Kimetal.,2009),theanalysis givenin(Leeetal.,2007)isextendedinthreedirections.InsteadofthestationaryPoisson arrival process of sessions, the Markov Arrival Process (MAP) is considered. It allows catching theeffectofcorrelationofflowofsessions.Thepresentednumericalresultsshowthatthe correlationhasprofoundeffectonthesystemperformancemeasures.Theseconddirection is consideration of the Phase type (PH) service process instead of an exponential service time distributionassumedin(Leeetal.,2007).BecausePH typedistributionsaresuitablefor fitting an arbitrary distribution, this allows to take into account the service time distribution andvarianceofthistimeinparticular,carefully.Thethirddirectionofextensionisthe followingone.Itisassumedin(Leeetal.,2007),thattheloss(duetoabufferoverflow)of therequestfromtheacceptedsessionnevercauseslossofawholesessionitself.More realisticassumptioninsomesituationsisthatthesessionmightbelost(terminates connectionaheadofschedule).E.g.,itcanhappenifthepercentageoflostvoiceorvideo packets (and quality of speech or movie) becomes unacceptable for the user. To take such a possibility into account in some extent, it is assumed in this paper that the loss of a request from the admitted session, with fixed probability, leads to the loss of a session to which this requestbelongs.Influenceof thisprobabilityisnumericallyinvestigated inthepaper(Kim et al., 2009).Inthepresentpaper,themodificationofmodelfrom(Kimetal.,2009)tothecaseofan infinitebufferisunderstudy.Incontrasttothemodelwithafinitebufferconsideredin (Kimetal.,2009)wheretheproblemofthethroughputmaximizationwassolvedunder constraintontheprobabilityofthelossofarequestfromanacceptedsession,herewedo nothavesuchaloss.So,theproblemofthethroughputmaximizationissolvedunder constraintontheaveragesojourntimeofrequestsfromtheacceptedsessions.Insection2, themathematicalmodelisdescribedindetail.Stabilitycondition,whichisnotrequiredin the model (Kim et al., 2009) with a finite state space but is very important in the model with aninfinitebufferspace,isderivedinasimpleform.Thisconditioncreatesanadditional constraintinmaximizationproblem.Thesteadystatejointdistributionofthenumberof sessions and requests in the system is analyzed by means of the matrix analytical technique andexpressionsforthemainperformancemeasuresofthesystemaregiveninsection3. Section4isdevotedtoconsiderationoftherequestandthesessionsojourntime distribution. Section 5 contains numerical illustrations and their short discussion and section 6 concludes the paper. 2. Mathematical model Weconsiderasingleserverqueueingsystemwithabufferofaninfinitecapacity.The requestsarrivetothesysteminsessions.SessionsarriveaccordingtotheMarkovArrival Process. Sessions arrival in theMAPis directed by an irreducible continuous time Markov chainvt,> . 0 t with thefinitestate space{0,..., }. WThesojourn time of theMarkovchain tv ,> . 0 t inthestatev hasanexponentialdistributionwiththeparameter v v . = . . 0 WAfterthissojourntimeexpires,withprobabilitypk v v' ( , ),theprocessvt,> . 0 t transitsto thestatev' ,andk sessions,= . . 0 1 k arriveintothesystem.Theintensitiesofjumpsfrom onestateintoanother,whichareaccompaniedbyanarrivalofk sessions,arecombined intothematrices. = . . 0 1kDk ofsize+ + ( 1) ( 1) W W .Thematrixgeneratingfunctionof these matrices is= + . ' 's0 1( ) 1 D z D D z z . The matrix(1) Dis the infinitesimal generator of the processv . > . 0t tThe stationary distribution vectoroof this process satisfies the equations = . = . 0 o o (1) 1 D e Hereandinthesequel0 isthezerorowvectorande isthecolumn vectorofappropriatesizeconsistingof1s.Incasethedimensionalityofthevectorisnot clearfromthecontext,itisindicatedasalowerindex,e.g. We denotestheunitcolumn vector of dimensionality= + 1 W W .The average intensity(fundamental rate) of theMAPis defined as = . o1D e The variancevof intervals between session arrivals is calculated as = . o1 1 202 ( ) v D e the squared coefficient varcof variation is calculated by = . o102 ( ) 1varc D e whilethecorrelationcoefficient corc ofintervalsbetweensuccessivegrouparrivals isgiven by = / . o1 1 1 20 1 0( ( ) ( ) )corc D D D v e Formoreinformation about the MAP ,its specialcasesandpropertiesandrelatedresearch see(Fisher&Meier-Hellstern,1993),(Lucantoni,1991)andthesurveypaperbyS. TrendsinTelecommunicationsTechnologies 126 Chakravarthy(Chakravarthy,2001).UsefulnessoftheMAP inmodeling telecommunicationsystemsismentionedin(Heyman&Lucantoni,2003),(Klemmetal., 2003).Note,thattheproblemofconstructingthe. MAP whichfitswellarealarrival process,isnotverysimple.However,thisproblemhaspracticalimportanceandis intensivelysolving.Forrelevantreferencesandthefittingalgorithmssee,e.g.,(Heyman& Lucantoni, 2003), (Klemm et al., 2003), (Asmussen et al., 1996) and (Panchenko & Buchholz, 2007).Following (Kist et al., 2005) , we assume that admission of sessions (they are called flows in (Kist et al., 2005)and called threads, connections, sessions, exchanges, windows, etc. in different real-worldapplications)isrestrictedbymeansoftokens.Thetotalnumberofavailable tokens is assumed to be. > . 1 KKFurther we consider the numberKas a control parameter and solve the corresponding optimization problem.Ifthereisnotokenavailableatasessionarrivalepochthesessionisrejected.Itleavesthe system forever. If the number of available tokens at the session arrival epoch is positive this session is admitted into the system and the number of available tokens decreases by one. We assume that one request of a session arrives at the session arrival epoch and if it meets free server, it occupies the server and is processed. If the server is busy, the request moves to the bufferandlateritispickedupfortheserviceaccordingtotheFirstCame-FirstServed discipline.After admission of the session, the next request of this session can arrive into the system in anexponentiallydistributedwiththeparameter time.Thenumberofrequestsinthe session has the geometrical distribution with the parameteru u . < < . 0 1i.e., probability that thesessionconsistsofk requestsisequaltou u . > .1(1 ) 1kk Theaveragesizeofthe session is equal to 1(1 ) u . Iftheexponentiallydistributedwiththeparameter timesincearrivaloftheprevious request of a session expires and new request does not arrive, it means that the arrival of the session is finished. The token, which was obtained by this session upon arrival, is returned to the pool of available tokens. The requests of this session, which stay in the system at the epochofreturningthetoken,mustbecompletelyprocessedbythesystem.Whenthelast request is served, the sojourn time of the session in the system is considered finished.TheservicetimeofarequestisassumedhavingPHdistribution.Itmeansthefollowing. Requests service time is governed by the directing process, 0, n >t twhich is the continuous time Markov chain with the state space. . . 1 { M}The initial state of the process0tt n . > .at theepochofstartingtheserviceisdeterminedbytheprobabilisticrow-vector = . .1( )M | | | .Thetransitionsoftheprocess0tt n . > . thatdonotleadtotheservice completion,aredefinedbytheirreduciblematrixS ofsizeM M .Theintensitiesof transitions,whichleadtotheservicecompletion,aredefinedbythecolumnvector 0S = S e .Theservicetimedistributionfunctionhastheform( ) 1SxB x e = e | .Laplace-Stieltjestransform 0( )sxe dB xofthisdistributionfunctionis 10( ) sI S . S | Theaverage service time is given by 11( ) b S = e | . The matrix 0S + S |is assumed to be irreducible. The moredetaileddescriptionofthePH -typedistributionanditspartialcasescanbefound, e.g.,inthebook(Neuts,1981).UsefulnessofPHdistributionindescriptionofservice processintelecommunicationnetworksisstated,e.g.,in(Pattavina&Parini,2005)and (Riska et al., 2002).It is intuitively clear that the described mechanism of arrivals restriction by means of tokens isreasonable.Attheexpenseofsomesessionsrejection,itallowstodecreasethesojourn timeandjitterforadmittedsessions.Thisisimportantinmodelingreal-worldsystems becausethequalityoftransmissionofacceptedinformationunitsshouldsatisfyimposed requirements of Quality of Service. Quantitative analysis of advantages and shortcomings of this mechanism as well as optimal choice of the number of tokens requires calculation of the main performance measures of the system under the fixed valueKof tokens in the system. Thesemeasurescanbe calculatedbasedonthe knowledgeof stationary distributionof the random process describing dynamics of the system under study. 3. Stationary distribution of the system states Let us assume that the number. > . 1 K Kof tokens is fixed and let- tibe the total number of requests in the system,0ti > . - tk bethenumberofsessionshavingtokenforadmissiontothesystem, 0tk K = . . - tv and tn bethestatesofthedirectingprocessesoftheMAP arrival process andPHservice process,0 1t tW M v n = . . = . . at the epoch0 t t . > . Note that when0ti = .i.e., requests are absent in the system, the value of the component tn .whichdescribesthestateoftheservicedirectingprocess,isnotdefined.Toavoidspecial treatment of this situation, without loss of generality, we assume that if the server becomes idle the state of the component tnis chosen randomly according to the probabilistic vector |and is kept until the next service beginning moment.Itisobviousthatthefour-dimensionalprocess0t t t t t{ik } t c v n = . . . . > . istheirreducible regular continuous time Markov chain.Let us enumerate the states of this Markov chain in lexicographic order and refer to( ) i k .as macro-state consisting of 1( 1) M W M = +states( ) 0 1 i k W M v n v n . . . . = . . = . . For the use in the sequel, introduce the following notation:-(1 ) u u += . = .1 1 1M M MI I I + +I = . I = . I = : -0 1Kdiag{ K}C = . . . isthediagonalmatrixwiththediagonalentries 0 1 { K} . . . .1K M KC IC= :

- 11K MR diag{ K} I = . . : Iis an identity matrix,Ois a zero matrix;- +| | | | ||I I || ||I I = . = : || || ||I I\ . \ . 0 1 0 00 0 1 02 20 0 0 10 0 0 0O O O O O O O O O O O A EO O O K K Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 127 Chakravarthy(Chakravarthy,2001).UsefulnessoftheMAP inmodeling telecommunicationsystemsismentionedin(Heyman&Lucantoni,2003),(Klemmetal., 2003).Note,thattheproblemofconstructingthe. MAP whichfitswellarealarrival process,isnotverysimple.However,thisproblemhaspracticalimportanceandis intensivelysolving.Forrelevantreferencesandthefittingalgorithmssee,e.g.,(Heyman& Lucantoni, 2003), (Klemm et al., 2003), (Asmussen et al., 1996) and (Panchenko & Buchholz, 2007).Following (Kist et al., 2005) , we assume that admission of sessions (they are called flows in (Kist et al., 2005)and called threads, connections, sessions, exchanges, windows, etc. in different real-worldapplications)isrestrictedbymeansoftokens.Thetotalnumberofavailable tokens is assumed to be. > . 1 KKFurther we consider the numberKas a control parameter and solve the corresponding optimization problem.Ifthereisnotokenavailableatasessionarrivalepochthesessionisrejected.Itleavesthe system forever. If the number of available tokens at the session arrival epoch is positive this session is admitted into the system and the number of available tokens decreases by one. We assume that one request of a session arrives at the session arrival epoch and if it meets free server, it occupies the server and is processed. If the server is busy, the request moves to the bufferandlateritispickedupfortheserviceaccordingtotheFirstCame-FirstServed discipline.After admission of the session, the next request of this session can arrive into the system in anexponentiallydistributedwiththeparameter time.Thenumberofrequestsinthe session has the geometrical distribution with the parameteru u . < < . 0 1i.e., probability that thesessionconsistsofk requestsisequaltou u . > .1(1 ) 1kk Theaveragesizeofthe session is equal to 1(1 ) u . Iftheexponentiallydistributedwiththeparameter timesincearrivaloftheprevious request of a session expires and new request does not arrive, it means that the arrival of the session is finished. The token, which was obtained by this session upon arrival, is returned to the pool of available tokens. The requests of this session, which stay in the system at the epochofreturningthetoken,mustbecompletelyprocessedbythesystem.Whenthelast request is served, the sojourn time of the session in the system is considered finished.TheservicetimeofarequestisassumedhavingPHdistribution.Itmeansthefollowing. Requests service time is governed by the directing process, 0, n >t twhich is the continuous time Markov chain with the state space. . . 1 { M}The initial state of the process0tt n . > .at theepochofstartingtheserviceisdeterminedbytheprobabilisticrow-vector = . .1( )M | | | .Thetransitionsoftheprocess0tt n . > . thatdonotleadtotheservice completion,aredefinedbytheirreduciblematrixS ofsizeM M .Theintensitiesof transitions,whichleadtotheservicecompletion,aredefinedbythecolumnvector 0S = S e .Theservicetimedistributionfunctionhastheform( ) 1SxB x e = e | .Laplace-Stieltjestransform 0( )sxe dB xofthisdistributionfunctionis 10( ) sI S . S | Theaverage service time is given by 11( ) b S = e | . The matrix 0S + S |is assumed to be irreducible. The moredetaileddescriptionofthePH -typedistributionanditspartialcasescanbefound, e.g.,inthebook(Neuts,1981).UsefulnessofPHdistributionindescriptionofservice processintelecommunicationnetworksisstated,e.g.,in(Pattavina&Parini,2005)and (Riska et al., 2002).It is intuitively clear that the described mechanism of arrivals restriction by means of tokens isreasonable.Attheexpenseofsomesessionsrejection,itallowstodecreasethesojourn timeandjitterforadmittedsessions.Thisisimportantinmodelingreal-worldsystems becausethequalityoftransmissionofacceptedinformationunitsshouldsatisfyimposed requirements of Quality of Service. Quantitative analysis of advantages and shortcomings of this mechanism as well as optimal choice of the number of tokens requires calculation of the main performance measures of the system under the fixed valueKof tokens in the system. Thesemeasurescanbe calculatedbasedon the knowledgeofstationarydistributionofthe random process describing dynamics of the system under study. 3. Stationary distribution of the system states Let us assume that the number. > . 1 K Kof tokens is fixed and let- tibe the total number of requests in the system,0ti > . - tk bethenumberofsessionshavingtokenforadmissiontothesystem, 0tk K = . . - tv and tn bethestatesofthedirectingprocessesoftheMAP arrival process andPHservice process,0 1t tW M v n = . . = . . at the epoch0 t t . > . Note that when0ti = .i.e., requests are absent in the system, the value of the component tn .whichdescribesthestateoftheservicedirectingprocess,isnotdefined.Toavoidspecial treatment of this situation, without loss of generality, we assume that if the server becomes idle the state of the component tnis chosen randomly according to the probabilistic vector |and is kept until the next service beginning moment.Itisobviousthatthefour-dimensionalprocess0t t t t t{ik } t c v n = . . . . > . istheirreducible regular continuous time Markov chain.Let us enumerate the states of this Markov chain in lexicographic order and refer to( ) i k .as macro-state consisting of 1( 1) M W M = +states( ) 0 1 i k W M v n v n . . . . = . . = . . For the use in the sequel, introduce the following notation:-(1 ) u u += . = .1 1 1M M MI I I + +I = . I = . I = : -0 1Kdiag{ K}C = . . . isthediagonalmatrixwiththediagonalentries 0 1 { K} . . . .1K M KC IC= :

- 11K MR diag{ K} I = . . : Iis an identity matrix,Ois a zero matrix;- +| | | | ||I I || ||I I = . = : || || ||I I\ . \ . 0 1 0 00 0 1 02 20 0 0 10 0 0 0O O O O O O O O O O O A EO O O K K TrendsinTelecommunicationsTechnologies 128 - I| | | | ||I I || ||I = . = : || || || I I\ . \ . 10 0 0 02 0 0 0 020 0 0 0( 1) 0 0 0 1O O O O O O O O A EO O O K K - i jo . is Kronecker delta, i jo . is equal to 1, ifi j =and equal to 0 otherwise; -is the symbol of Kronecker product of matrices; -is the symbol of Kronecker sum of matrices; - Tbdenotes transposed vectorb . LetQ bethegeneratoroftheMarkovchain0t t c . > . withblocks i jQ.consistingof intensities( )i j k kQ' . .oftheMarkovchain0t t c . > . transitionsfromthemacro-state( ) i k . to the macro-state( ) 0 j k k k K ' ' . .. = . .The diagonal entries of the matrix i iQ. are negative and the modulusofthediagonalentryof( )i i k kQ. .definesthetotalintensityofleavingthe correspondingstate( ) i k v n . . . oftheMarkovchain.Theblock0i jQ i j. . .> . hasdimension 1 1K K .where 1 1( 1) K K M = + . Lemma 1. GeneratorQhas the three block diagonal structure: 0 0 01 0 21 0 2Q Q O O Q Q Q O QO Q Q Q .| | | |=| | |\ . where non-zero blocks i jQ. are defined by 0 0 1 0 1 K M MQ A I D I E D I. += + + . 1 1 0 1( )K MQ A I D S E D I+= + + . 0 1 K MQ C E D I + += + .2 1 1 0 K WQ I I+ += . S | Proof of the lemma consists of analysis of the Markov chain0t t c .> .transitions during the infinitesimal interval of time and further assembling the corresponding transition intensities into the matrix blocks. Value is the intensity of a token releasing due to the finish of the session arrival, + is the intensity of a new request in the session arrival.Let us investigate the Markov chain0t t c .> .defined by the generatorQ.To this end, at first we should derive conditions under which this Markov chain is ergodic (positive recurrent). Theorem1.Markovchain0t t t t t{ik } t c v n = . . . . > . isergodicifandonlyifthefollowing inequality is fulfilled: 11 1 10 0K Kk W k Wk kk D u ++ += => A = + . x e x e(1) whereuis the average service rate defined by 1 11( ) b S u = = e | and 0( )K = . . x x x isthevectorofthestationarydistributionofthesystem0 MAP M K / //with theMAParrival process, defined by the matrices 0Dand 1Dand the average service rate .Proof.Itfollowsfrom(Neuts,1981)thattheergodicityconditionoftheMarkovchain 0t t t t t{ik } t c v n = . . . . > .is the fulfillment of inequality

2 0Q Q > . y e y e(2) where the row vectoryis solution to the system of linear algebraic equations of form

0 1 2( ) 1 Q Q Q + + = . = . y ye 0 (3) It is easy to verify that 0 1 2 ( 1)( 1) 0( )M K WQ Q Q B I I S+ ++ + = + + . S | whereBisthegeneratoroftheMarkovchain,whichdescribesbehaviorofthe 0 MAP M K ' ' 'system with theMAParrivalprocess defined by matrices 0Dand 1Dand average service rate : 1 00 100 2 2(1)D D O O OI D I D O OI D I O O BO O O K I D K I | | | | | = . | | |\ . According to the definition, vectorxsatisfies equations 1 B = . = . x xe 0(4) By direct substitution into (3), it can be verified that the vector. ywhich is solution to theQueueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 129 - I| | | | ||I I || ||I = . = : || || || I I\ . \ . 10 0 0 02 0 0 0 020 0 0 0( 1) 0 0 0 1O O O O O O O O A EO O O K K - i jo . is Kronecker delta, i jo . is equal to 1, ifi j =and equal to 0 otherwise; -is the symbol of Kronecker product of matrices; -is the symbol of Kronecker sum of matrices; - Tbdenotes transposed vectorb . LetQ bethegeneratoroftheMarkovchain0t t c . > . withblocks i jQ.consistingof intensities( )i j k kQ' . .oftheMarkovchain0t t c . > . transitionsfromthemacro-state( ) i k . to the macro-state( ) 0 j k k k K ' ' . .. = . .The diagonal entries of the matrix i iQ. are negative and the modulusofthediagonalentryof( )i i k kQ. .definesthetotalintensityofleavingthe correspondingstate( ) i k v n . . . oftheMarkovchain.Theblock0i jQ i j. . .> . hasdimension 1 1K K .where 1 1( 1) K K M = + . Lemma 1. GeneratorQhas the three block diagonal structure: 0 0 01 0 21 0 2Q Q O O Q Q Q O QO Q Q Q .| | | |=| | |\ . where non-zero blocks i jQ. are defined by 0 0 1 0 1 K M MQ A I D I E D I. += + + . 1 1 0 1( )K MQ A I D S E D I+= + + . 0 1 K MQ C E D I + += + .2 1 1 0 K WQ I I+ += . S | Proof of the lemma consists of analysis of the Markov chain0t t c .> .transitions during the infinitesimal interval of time and further assembling the corresponding transition intensities into the matrix blocks. Value is the intensity of a token releasing due to the finish of the session arrival, + is the intensity of a new request in the session arrival.Let us investigate the Markov chain0t t c .> .defined by the generatorQ.To this end, at first we should derive conditions under which this Markov chain is ergodic (positive recurrent). Theorem1.Markovchain0t t t t t{ik } t c v n = . . . . > . isergodicifandonlyifthefollowing inequality is fulfilled: 11 1 10 0K Kk W k Wk kk D u ++ += => A = + . x e x e(1) whereuis the average service rate defined by 1 11( ) b S u = = e | and 0( )K = . . x x x isthevectorofthestationarydistributionofthesystem0 MAP M K / //with theMAParrival process, defined by the matrices 0Dand 1Dand the average service rate .Proof.Itfollowsfrom(Neuts,1981)thattheergodicityconditionoftheMarkovchain 0t t t t t{ik } t c v n = . . . . > .is the fulfillment of inequality

2 0Q Q > . y e y e(2) where the row vectoryis solution to the system of linear algebraic equations of form

0 1 2( ) 1 Q Q Q + + = . = . y ye 0 (3) It is easy to verify that 0 1 2 ( 1)( 1) 0( )M K WQ Q Q B I I S+ ++ + = + + . S | whereBisthegeneratoroftheMarkovchain,whichdescribesbehaviorofthe 0 MAP M K ' ' ' system with theMAP arrivalprocessdefinedbymatrices 0D and 1Dand average service rate : 1 00 100 2 2(1)D D O O OI D I D O OI D I O O BO O O K I D K I | | | | | = . | | |\ . According to the definition, vectorxsatisfies equations 1 B = . = . x xe 0(4) By direct substitution into (3), it can be verified that the vector. ywhich is solution to theTrendsinTelecommunicationsTechnologies 130 system (3), can be represented in the form= y x v , wherevis the unique solution of the system of linear algebraic equations 0( ) 1 S + = . = . S e 0 v | v (5) Bysubstitutingvector= y x v intoinequality(2),aftersometransformationsweget inequality (1). Theorem 1 is proven.Inwhatfollowsweassumethatcondition(1)isfulfilled.Thenthefollowinglimits (stationary probabilities) exist: ( ) lim 0 0 0 1t t t tti k P{i i k k } i k K W M t v n v v n n v n. . . = = . = . = . = . > . = . . = . . = . . Let us combine these probabilities into the row-vectors ( ) ( ( 1) ( 2) ( )) i k i k i k i k M v t v t v t v . . = . . . . . . . . . . . . . t( ) ( ( 0) ( 1) ( )) i k i k i k i k W . = . . . . . . . . . . t t t t( ( 0) ( 1) ( )) 0ii i i K i = . . . . . . . > . t t t t The following statement directly stems from the results in (Neuts, 1981).Theorem 2. The stationary probability vectors0i i .> . tare calculated by 00iiR i = .> . t t where the matrixRis the minimal non-negative solution to the equation 22 1 0R Q RQ Q O + + = . and the vector 0tis the unique solution to the system of linear algebraic equations 10 0 0 2 0( ) ( ) 1 Q RQ I R . + = . = . e 0 t t Havingstationaryprobabilityvectors0i i .> . t beencomputed,wecancalculatedifferent performance measures of the system. Some of them are given in the following statements.Corollary1.Theprobabilitydistributionofthenumberofrequestsinthesystemis computed by lim 0t itP{i i} i= = .> . e t The average numberLof requests in the system is computed by 200( )iiL i R I R== = .e e t t The probability distribution of the number of sessions in the system is computed by ( ) 100lim ( ) ( ) ( ) 0kttiP{k k} i k I R k K== = . = . = . .e e e t t wherethecolumnvector ( ) ke hasallzeroentriesexceptthek thone,whichisequalto1, 0 k K = . . The average numberZof sessions in the system is computed by ( ) 101 0 1( ) ( ) ( )K Kkk i kZ k i k I R k= = == . = . e e e t t Thedistributionfunction( ) R t ofatime,duringwhicharrivalsfromanarbitrarysession occur, is computed by 1(1 ) 110( )( ) (1 ) 1( 1)t ly t llyR t e dy el u u u == = . ! The average numberTof requests processed by the system at unit of time (throughput) is computed by 10 0 ( 1)( 1) 01 0 0 1( )( ) ( ) ( )K W MK Wi kT i k R I Rnv nt v n+ += = = == . . . = .S e S t Remark 1. In contrast to the model with a finite buffer, see (Lee et al., 2007) and (Kim et al., 2009), where the arriving session can be rejected not only due to the tokens absence but also duetothebufferoverloading,distributionofthenumberofsessionsinthemodelunder study does not depend on the number of requests in the system. It is defined by formula lim 0t ktP{k k} k K= = . = . . x e where the vectors0k k K . = . . xare the entries of the vector 0( )K = . . x x xwhich satisfies the system(5).However,distribution( ) 0 0 i k i k K . .> . = . . t doesnothavemultiplicativeform because the number of requests in the system depends on the number of sessions currently presenting in the system.Remark 2. It can be verified that the considered model with the infinite buffer has the steady statedistributionoftheprocess0t t t t t{ik } t c v n = . . . . > . coincidingwiththesteadystate distributionofthequeueingmodelofthe1 MAP PH / / typewiththephaseservicetime distributionhavingirreduciblerepresentation( ) S . | andtheMAP arrivalprocessdefined by the matrices 0Dand 1Dhaving the form Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 131 system (3), can be represented in the form= y x v , wherevis the unique solution of the system of linear algebraic equations 0( ) 1 S + = . = . S e 0 v | v (5) Bysubstitutingvector= y x v intoinequality(2),aftersometransformationsweget inequality (1). Theorem 1 is proven.Inwhatfollowsweassumethatcondition(1)isfulfilled.Thenthefollowinglimits (stationary probabilities) exist: ( ) lim 0 0 0 1t t t tti k P{i i k k } i k K W M t v n v v n n v n. . . = = . = . = . = . > . = . . = . . = . . Let us combine these probabilities into the row-vectors ( ) ( ( 1) ( 2) ( )) i k i k i k i k M v t v t v t v . . = . . . . . . . . . . . . . t( ) ( ( 0) ( 1) ( )) i k i k i k i k W . = . . . . . . . . . . t t t t( ( 0) ( 1) ( )) 0ii i i K i = . . . . . . . > . t t t t The following statement directly stems from the results in (Neuts, 1981).Theorem 2. The stationary probability vectors0i i .> . tare calculated by 00iiR i = .> . t t where the matrixRis the minimal non-negative solution to the equation 22 1 0R Q RQ Q O + + = . and the vector 0tis the unique solution to the system of linear algebraic equations 10 0 0 2 0( ) ( ) 1 Q RQ I R . + = . = . e 0 t t Havingstationaryprobabilityvectors0i i .> . t beencomputed,wecancalculatedifferent performance measures of the system. Some of them are given in the following statements.Corollary1.Theprobabilitydistributionofthenumberofrequestsinthesystemis computed by lim 0t itP{i i} i= = .> . e t The average numberLof requests in the system is computed by 200( )iiL i R I R== = .e e t t The probability distribution of the number of sessions in the system is computed by ( ) 100lim ( ) ( ) ( ) 0kttiP{k k} i k I R k K== = . = . = . .e e e t t wherethecolumnvector ( ) ke hasallzeroentriesexceptthek thone,whichisequalto1, 0 k K = . . The average numberZof sessions in the system is computed by ( ) 101 0 1( ) ( ) ( )K Kkk i kZ k i k I R k= = == . = . e e e t t Thedistributionfunction( ) R t ofatime,duringwhicharrivalsfromanarbitrarysession occur, is computed by 1(1 ) 110( )( ) (1 ) 1( 1)t ly t llyR t e dy el u u u == = . ! The average numberTof requests processed by the system at unit of time (throughput) is computed by 10 0 ( 1)( 1) 01 0 0 1( )( ) ( ) ( )K W MK Wi kT i k R I Rnv nt v n+ += = = == . . . = .S e S t Remark 1. In contrast to the model with a finite buffer, see (Lee et al., 2007) and (Kim et al., 2009), where the arriving session can be rejected not only due to the tokens absence but also duetothebufferoverloading,distributionofthenumberofsessionsinthemodelunder study does not depend on the number of requests in the system. It is defined by formula lim 0t ktP{k k} k K= = . = . . x e where the vectors0k k K . = . . xare the entries of the vector 0( )K = . . x x xwhich satisfies the system(5).However,distribution( ) 0 0 i k i k K . .> . = . . t doesnothavemultiplicativeform because the number of requests in the system depends on the number of sessions currently presenting in the system.Remark 2. It can be verified that the considered model with the infinite buffer has the steady statedistributionoftheprocess0t t t t t{ik } t c v n = . . . . > . coincidingwiththesteadystate distributionofthequeueingmodelofthe1 MAP PH / / typewiththephaseservicetime distributionhavingirreduciblerepresentation( ) S . | andtheMAP arrivalprocessdefined by the matrices 0Dand 1Dhaving the form TrendsinTelecommunicationsTechnologies 132 + + +| | | | || || || = . = . || || ||\ . \ . 0 10 10 0 10 2 2 0 2(1)D O O O O O D O O OI D I O O O O I D O OI D I O O O I O O D DO O O K I D K I O O O OK I It is easy to verify that the fundamental rate of thisMAPis equal toAwhich is defined in (1). So, stability condition (1) is intuitively clear: the average service rate should exceed the averagearrivalrate.Notethatthefirstsummandinexpression 11 1 10 0K Kk W k Wk kk D ++ += =A = + . x e x e fortherateA representstherateofrequestsfromalready acceptedsessions,i.e.,therateofrequestswhoarenotthefirstinasession.Thesecond summand is the rate of the sessions arrival.Theorem 2. The probability ( ) lossbPof an arbitrary session rejection upon arrival is computed by( ) 1 10( )( )loss Mb KiD I DP i K == . = .e x e t The probability ( ) losscPof an arbitrary request rejection upon arrival is computed by ( ) 1 10( )( )loss Mc KiD I DP i K== . =A Ae x e t where 1 KD A = A + . x e

Proofofformulaforprobability ( ) lossbP accountsthatthesessionisrejecteduponarrivalif and only if the number of sessions in the system at this epoch is equal toK . So 1( ) 1 0010 0( )( )( )( )( )( )Mloss M ib KiMi ki KD ID IP i Ki k D I=== =. = = . .. eeettt Rejection of a request can occur only if this request is the first in a session and the number of sessions in the system at this session arrival epoch is equal toK . So 1( ) 1 0010 0( )( )( )( )( )Mloss M ic KiMi ki KD ID IP i Ki kD k I I =+== =. = = . .A. + eeettt 4. Distribution of the sojourn times Let( )bV x . ( )cV x and ( )( )acV x bedistributionfunctionsofthesojourntimeofanarbitrary session, an arbitrary request, which is the first in a session, and an arbitrary request from the admitted session, which is not the first in this session, in the system under study, and( )bv s .( )cv sand ( )( )acv s . 0 Re s > .be their Laplace-Stieltjes transforms (LSTs): ( ) ( )0 0 0( ) ( ) ( ) ( ) ( ) ( )a a sx sx sxb b c c c cv s e dV x v s e dV x v s e dV x = . = . = . Formulae for calculation of the LSTs( )cv sand ( )( )acv sare the following: 1 1 10 1 0 1 ( 1)( 1) 0 011( ) [ ( ) ( ) ( )( (( ) ))( ( ) ) ]ic M i M K Wiv s D I sI S D I sI S sI S + +== + =e S e S S t | t |1 10 1 0 0 01[ ( ) ( ) ( ( ) )MD I sI S R sI S = + e S S t | t |1 1 10 1 ( 1)( 1) 0( ( ( ) )) ( )( (( ) ))]M K WI R sI S D I sI S + + . S e S |( ) 1011 01( ) [ (0 ) ( )( )Kac Kkk iv s k k sI Sk i k+ +== == . +.e Set |t 1 11 0 01( )( (( ) ))( ( ) ) ]iWii k sI S sI S +=+ . .e S S t | Formulaefortheaveragesojourntime cV ofanarbitraryrequest,whichisthefirstina session, the average sojourn time cV- of an arbitrary non-rejected request, which is the first inasession,andtheaveragesojourntime ( ) acV ofanarbitraryrequestfromtheadmitted session, which is not the first in this session, are as follows: 10 1 1 1 ( 1)( 1) 111[ ( ) ( )(( ( ) ) )]c M i M K WiV D I b D I S ib+ +== + + =e e e e t t2 1 1 01 1 1 ( 1)( 1)[( ( ) )( ) ( ) ( )( ( ) ]M M K WI R I R D I b R I R D I S + += + + . e e et ( )1cc lossbVVP-= . 11 1 1( ) 1 11 0[ (0 ) ( )(( ( ) ) )]( )KWa k ic Kk ik k b i k S ibVk i k+ += =+= =. + . += .. e e e eet tt Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 133 + + +| | | | || || || = . = . || || ||\ . \ . 0 10 10 0 10 2 2 0 2(1)D O O O O O D O O OI D I O O O O I D O OI D I O O O I O O D DO O O K I D K I O O O OK I It is easy to verify that the fundamental rate of thisMAPis equal toAwhich is defined in (1). So, stability condition (1) is intuitively clear: the average service rate should exceed the averagearrivalrate.Notethatthefirstsummandinexpression 11 1 10 0K Kk W k Wk kk D ++ += =A = + . x e x e fortherateA representstherateofrequestsfromalready acceptedsessions,i.e.,therateofrequestswhoarenotthefirstinasession.Thesecond summand is the rate of the sessions arrival.Theorem 2. The probability ( ) lossbPof an arbitrary session rejection upon arrival is computed by( ) 1 10( )( )loss Mb KiD I DP i K == . = .e x e t The probability ( ) losscPof an arbitrary request rejection upon arrival is computed by ( ) 1 10( )( )loss Mc KiD I DP i K== . =A Ae x e t where 1 KD A = A + . x e

Proofofformulaforprobability ( ) lossbP accountsthatthesessionisrejecteduponarrivalif and only if the number of sessions in the system at this epoch is equal toK . So 1( ) 1 0010 0( )( )( )( )( )( )Mloss M ib KiMi ki KD ID IP i Ki k D I=== =. = = . .. eeettt Rejection of a request can occur only if this request is the first in a session and the number of sessions in the system at this session arrival epoch is equal toK . So 1( ) 1 0010 0( )( )( )( )( )Mloss M ic KiMi ki KD ID IP i Ki kD k I I =+== =. = = . .A. + eeettt 4. Distribution of the sojourn times Let( )bV x . ( )cV x and ( )( )acV x bedistributionfunctionsofthesojourntimeofanarbitrary session, an arbitrary request, which is the first in a session, and an arbitrary request from the admitted session, which is not the first in this session, in the system under study, and( )bv s .( )cv sand ( )( )acv s . 0 Re s > .be their Laplace-Stieltjes transforms (LSTs): ( ) ( )0 0 0( ) ( ) ( ) ( ) ( ) ( )a a sx sx sxb b c c c cv s e dV x v s e dV x v s e dV x = . = . = . Formulae for calculation of the LSTs( )cv sand ( )( )acv sare the following: 1 1 10 1 0 1 ( 1)( 1) 0 011( ) [ ( ) ( ) ( )( (( ) ))( ( ) ) ]ic M i M K Wiv s D I sI S D I sI S sI S + +== + =e S e S S t | t |1 10 1 0 0 01[ ( ) ( ) ( ( ) )MD I sI S R sI S = + e S S t | t |1 1 10 1 ( 1)( 1) 0( ( ( ) )) ( )( (( ) ))]M K WI R sI S D I sI S + + . S e S |( ) 1011 01( ) [ (0 ) ( )( )Kac Kkk iv s k k sI Sk i k+ +== == . +.e Set |t 1 11 0 01( )( (( ) ))( ( ) ) ]iWii k sI S sI S +=+ . .e S S t | Formulaefortheaveragesojourntime cV ofanarbitraryrequest,whichisthefirstina session, the average sojourn time cV- of an arbitrary non-rejected request, which is the first inasession,andtheaveragesojourntime ( ) acV ofanarbitraryrequestfromtheadmitted session, which is not the first in this session, are as follows: 10 1 1 1 ( 1)( 1) 111[ ( ) ( )(( ( ) ) )]c M i M K WiV D I b D I S ib+ +== + + =e e e e t t2 1 1 01 1 1 ( 1)( 1)[( ( ) )( ) ( ) ( )( ( ) ]M M K WI R I R D I b R I R D I S + += + + . e e et ( )1cc lossbVVP-= . 11 1 1( ) 1 11 0[ (0 ) ( )(( ( ) ) )]( )KWa k ic Kk ik k b i k S ibVk i k+ += =+= =. + . += .. e e e eet tt TrendsinTelecommunicationsTechnologies 134 If the service time distribution is exponential, expression for the average sojourn time cVof an arbitrary arriving request, which is the first in a session, becomes simpler: 2 01 1( )cV b I R D= . et DerivationofformulaforcalculationoftheLST( )bv s ismoreinvolved.Recallthatthe sojourn time of an arbitrary session in the system lasts since the epoch of the session arrival intothe systemuntilthemomentwhenthearrival ofasessionisfinishedandallrequests, which belong to this session, leave the system. We will derive expression for the LST( )bv sbymeansofthemethodofcollectivemarks(methodofadditionalevent,methodof catastrophes),forreferencessee,e.g.,(Kasten&Runnenburg,1956)and(Danzig,1955).To this end, we interpret the variablesas the intensity of some virtual stationary Poisson flow of catastrophes. So,( )bv shas meaning of probability that no one catastrophe arrives during the sojourn time of an arbitrary session.Wewilltaganarbitrarysessionandwillkeeptrackofitsstayinginthesystem.Let ( ) v s i l k v n . . . . .be the probability that catastrophe will not arrive during the rest of the tagged sessionsojourntimeinthesystemconditionalthat,atthegivenmoment,thenumberof sessionsprocessedinthesystemisequalto1 k k K . = . . thenumberofrequestsisequalto 0 i i .> . thelast(intheorderofarrival)requestofataggedsessionhaspositionnumber 0 l l i .= . .in the system, and the states of the processes0t t t v n . .> .are v n . . Position number 0 means that currently there is no one request of the tagged session in the system.Itfollowsfromtheformulaoftotalprobabilitythatifwewillhavefunctions( ) v s i l k v n . . . . .been calculated the Laplace-Stieltjes transform( )bv scan be computed by 1( ) (1)0 0 0 1 01( ) ( ) ( 1 1 1 )K W M Wlossb bi kv s P i k p v s i i kv v vv n vt v n v n ' .' = = = = =' = + . . . . + . + . + . . . (6) Thesystemoflinearalgebraicequationsforfunctions( ) v s i l k v n . . . . . isderivedbymeansof formula of total probability in the following form: (1)0( ) [ ((1 ) ( 1 1 )Wk Kv s i l k p v s i l kv v vvv n o v n' . .'=' . . . . . = . + . . + . . + (7) (0)0( )) ( )Wk Kv s i l k p v s i l kv v vvo v n v n' . .' =' ' + . . . . . + . . . . . + 0 0 0 01(1 )( ) [ ( 1 1 )(1 ) ( 1 0 ) ]Mi l lv s i l k v s i kn nno | v n o v n o' . . .'=' ' + . . . . . + . . . . . +S01(1 ) ( ) ( ) ( 1 1 )MiS v s i l k v s i i kn nno v n v n+' . .' =' + . . . . . + . + . + . . . + ( 1) ( 1 ) ( 1) ( 1 ) k v s i l k k v s i l k v n v n+ + . + . . . . + . . . . . + 1 1 10 0 0 0[(( ) ) ( ( ) ) (1 ) ]]ll lsI S sI Sn o o . .+ + S S |1( ) 0 0 1 0 1 s S k l i i k K W Mv n n v n. + + . = . .> . = . . = . . = . . Let us explain formula (7) in brief. The denominator of the right hand side of (7) is equal to thetotalintensityoftheeventswhichcanhappenafterthearbitrarytimemoment: catastrophe arrival, transition of the directing process of the , MAPtransition of the directing process of thePHservice process, and expiring the time till the moment of possible request arrival from sessions already admitted into the system. The first term in the square brackets in(7)correspondstothecasewhenanewsessionarrives. Thesecondtermcorrespondsto thecasewhentransitionofthedirectingprocessoftheMAP occurswithoutnewsession generation. The third term corresponds to the case when service completion takes place. The fourth term corresponds to the case when the transition of the directing process of thePHservice process occurs without the service completion. The fifth term corresponds to the case when the new request of the tagged session arrives into the system. In this case, the position of the last request of the tagged session in the system is reinstalled fromlto1 i + .The sixth termcorrespondstothecasewhenthenewrequestfromanothersession,whichwas alreadyadmittedtothesystem,arrives.Theseventhtermcorrespondstothecasewhen some non-tagged session terminates arrivals. The eighth term corresponds to the case when the expected new request of the tagged session does not arrive into the system and arrival of requests of the tagged session is stopped. This session will not more counted as arriving into thesystemandthetaggedrequestfinishesitssojourntimewhenthelastrequest,whois currentlythel thinthesystem,willleavethesystem.Number 10(( ) ) sI Sn S definesthe probability that catastrophe will not arrive during the residual service time conditional that the directing process of thePHservice is currently in the staten.The number 10( ) sI S S definesprobabilitythatcatastrophewillnotarriveduringtheservicetimeofanarbitrary request.Let us introduce column vectors ( ) ( ( 1) ( ))Ts i l k v s i l k v s i l k M v v v . . . . = . . . . . . . . . . . . . v( ) ( ( 0) ( ))Ts i l k s i l k s i l k W . . . = . . . . . . . . . . . v v v( ) ( ( 1) ( ))Ts i l s i l s i l K . . = . . . . . . . . . v v v( ) ( ( 0) ( )) ( ) ( ( 0) ( 1) )T Ts i s i s i i s s s . = . . . . . . . = . . . . . v v v v v v System (7) of linear algebraic equations can be rewritten to the matrix form as 0 01 1 1 ( ) ( ) ( 1 ) ( 1 1)(1 ) ( 1 0)l li i i i i i i isI s i l s i l s i l v s iQ Q Q Qo o. .. . + . . . . + . + . + . . + . . + v v v(8) 11 1 1( 1) 0 0( 1 1) (( ) )( ( ) ) 0 0l TK K W KMI s i i sI S sI S l i i + ++ I . + . + + = . = . .> . v e S S 0 | where 1 0 0 1 0 0( )(1 ) (( )) ( ) 0K i M K M ii iA I D S E D I I D I iQo o. .. = + + + . > . Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 135 If the service time distribution is exponential, expression for the average sojourn time cVof an arbitrary arriving request, which is the first in a session, becomes simpler: 2 01 1( )cV b I R D= . et DerivationofformulaforcalculationoftheLST( )bv s ismoreinvolved.Recallthatthe sojourn time of an arbitrary session in the system lasts since the epoch of the session arrival intothe systemuntilthemomentwhenthearrival ofasessionisfinishedandallrequests, which belong to this session, leave the system. We will derive expression for the LST( )bv sbymeansofthemethodofcollectivemarks(methodofadditionalevent,methodof catastrophes),forreferencessee,e.g.,(Kasten&Runnenburg,1956)and(Danzig,1955).To this end, we interpret the variablesas the intensity of some virtual stationary Poisson flow of catastrophes. So,( )bv shas meaning of probability that no one catastrophe arrives during the sojourn time of an arbitrary session.Wewilltaganarbitrarysessionandwillkeeptrackofitsstayinginthesystem.Let ( ) v s i l k v n . . . . .be the probability that catastrophe will not arrive during the rest of the tagged sessionsojourntimeinthesystemconditionalthat,atthegivenmoment,thenumberof sessionsprocessedinthesystemisequalto1 k k K . = . . thenumberofrequestsisequalto 0 i i .> . thelast(intheorderofarrival)requestofataggedsessionhaspositionnumber 0 l l i .= . .in the system, and the states of the processes0t t t v n . .> .are v n . . Position number 0 means that currently there is no one request of the tagged session in the system.Itfollowsfromtheformulaoftotalprobabilitythatifwewillhavefunctions( ) v s i l k v n . . . . .been calculated the Laplace-Stieltjes transform( )bv scan be computed by 1( ) (1)0 0 0 1 01( ) ( ) ( 1 1 1 )K W M Wlossb bi kv s P i k p v s i i kv v vv n vt v n v n ' .' = = = = =' = + . . . . + . + . + . . . (6) Thesystemoflinearalgebraicequationsforfunctions( ) v s i l k v n . . . . . isderivedbymeansof formula of total probability in the following form: (1)0( ) [ ((1 ) ( 1 1 )Wk Kv s i l k p v s i l kv v vvv n o v n' . .'=' . . . . . = . + . . + . . + (7) (0)0( )) ( )Wk Kv s i l k p v s i l kv v vvo v n v n' . .' =' ' + . . . . . + . . . . . + 0 0 0 01(1 )( ) [ ( 1 1 )(1 ) ( 1 0 ) ]Mi l lv s i l k v s i kn nno | v n o v n o' . . .'=' ' + . . . . . + . . . . . +S01(1 ) ( ) ( ) ( 1 1 )MiS v s i l k v s i i kn nno v n v n+' . .' =' + . . . . . + . + . + . . . + ( 1) ( 1 ) ( 1) ( 1 ) k v s i l k k v s i l k v n v n+ + . + . . . . + . . . . . + 1 1 10 0 0 0[(( ) ) ( ( ) ) (1 ) ]]ll lsI S sI Sn o o . .+ + S S |1( ) 0 0 1 0 1 s S k l i i k K W Mv n n v n. + + . = . .> . = . . = . . = . . Let us explain formula (7) in brief. The denominator of the right hand side of (7) is equal to thetotalintensityoftheeventswhichcanhappenafterthearbitrarytimemoment: catastrophe arrival, transition of the directing process of the , MAPtransition of the directing process of thePHservice process, and expiring the time till the moment of possible request arrival from sessions already admitted into the system. The first term in the square brackets in(7)correspondstothecasewhenanewsessionarrives. Thesecondtermcorrespondsto thecasewhentransitionofthedirectingprocessoftheMAP occurswithoutnewsession generation. The third term corresponds to the case when service completion takes place. The fourth term corresponds to the case when the transition of the directing process of thePHservice process occurs without the service completion. The fifth term corresponds to the case when the new request of the tagged session arrives into the system. In this case, the position of the last request of the tagged session in the system is reinstalled fromlto1 i + .The sixth termcorrespondstothecasewhenthenewrequestfromanothersession,whichwas alreadyadmittedtothesystem,arrives.Theseventhtermcorrespondstothecasewhen some non-tagged session terminates arrivals. The eighth term corresponds to the case when the expected new request of the tagged session does not arrive into the system and arrival of requests of the tagged session is stopped. This session will not more counted as arriving into thesystemandthetaggedrequestfinishesitssojourntimewhenthelastrequest,whois currentlythel thinthesystem,willleavethesystem.Number 10(( ) ) sI Sn S definesthe probability that catastrophe will not arrive during the residual service time conditional that the directing process of thePHservice is currently in the staten.The number 10( ) sI S S definesprobabilitythatcatastrophewillnotarriveduringtheservicetimeofanarbitrary request.Let us introduce column vectors ( ) ( ( 1) ( ))Ts i l k v s i l k v s i l k M v v v . . . . = . . . . . . . . . . . . . v( ) ( ( 0) ( ))Ts i l k s i l k s i l k W . . . = . . . . . . . . . . . v v v( ) ( ( 1) ( ))Ts i l s i l s i l K . . = . . . . . . . . . v v v( ) ( ( 0) ( )) ( ) ( ( 0) ( 1) )T Ts i s i s i i s s s . = . . . . . . . = . . . . . v v v v v v System (7) of linear algebraic equations can be rewritten to the matrix form as 0 01 1 1 ( ) ( ) ( 1 ) ( 1 1)(1 ) ( 1 0)l li i i i i i i isI s i l s i l s i l v s iQ Q Q Qo o. .. . + . . . . + . + . + . . + . . + v v v(8) 11 1 1( 1) 0 0( 1 1) (( ) )( ( ) ) 0 0l TK K W KMI s i i sI S sI S l i i + ++ I . + . + + = . = . .> . v e S S 0 | where 1 0 0 1 0 0( )(1 ) (( )) ( ) 0K i M K M ii iA I D S E D I I D I iQo o. .. = + + + . > . TrendsinTelecommunicationsTechnologies 136 1 11( ) 0K K Mi iC E D I iQ+ +. + = + .> .( 1) 01 0 1 0K Wi iI i OQ Q+. .= .> . = . S | Let us introduce notation:( ) s Ois three block diagonal matrix with non-zero blocks ( ) max 0 1 1 0i j s j { i } i i i.O . = . . .+ .> .defined by( )1 1 31 ( ) ( )ii i i i ii i i is I sI DQ Q. + . . . O = . O = .( ) ( )1 1 21i ii i Ki iD D IQ+. +. +O = + I . Herethematrix ( )1iD ofsize( 1) ( 2) i i + + isobtainedfromtheidentitymatrix 1 iI +by means of supplementing from the right by the column 1Ti + . 0The matrix ( )2iDof the same size has the last column consisting of 1s and other columns consisting of 0s. The matrix ( )3iDof size( 1) i i + isobtainedfromtheidentitymatrix iI bymeansofsupplementingfrom above by the row(1 0 0) . . . . Vector( ) s Bis defined by0( ) ( ( ) ( ) )TNs s s = . . . B B Bwhere 11 1 11 0 1 0 0( ) ( ( ) (( ) ) ( )i K M K W K Ws sI S sI S sI S + += . . . . e e e e S e e S S | B + .> .1 1 11 0 0(( ) )( ( ) ) ) 0i TK WsI S sI S i e e S S | Using this notation we can rewrite the system (7) to the form ( ) ( ) ( )Ts s s O + = . v 0 B(9) Itcanbeverifiedthatthediagonalentriesofthematrix( ) s O dominateinallrowsofthis matrix. So the inverse matrix exists. Thus we proved the following assertion.Theorem3.Thevector( ) s v consistingofconditionalLaplace-StieltjestransformsLST( ) 0 0 1 v s i l k l i i k K v n . . . . . .= . .> . = . . 0 1 W M v n = . . = . .is calculated by

1( ) ( ) ( ) s s s= O . v B(10) Corollary 2. The average sojourn time bVof an arbitrary session is calculated by 1100 0( ) ( 1 1 1)( )KMb si kD I s i i kV i ks == = c . + . + . += . ' .cvt where column vectors ( 1 1 1)0s i i ks sc . + . + . += c'v are calculated as the blocks of the vector ( )0d ss ds ='v defined by10 0( ) ( )(0)( (0))s sd s d sds ds= =' = O ' + .vvB where 1(0) (0) = O . v e Corollary 3. The average sojourn time ( ) acceptbVof an arbitrary admitted session is calculated by( )( )1accept bb lossbVVP= . where ( ) lossbPis probability of an arbitrary session rejection upon arrival. 5. Optimization problem and numerical examples Itisobviousthatthemostimportantfromeconomicalpointofviewcharacteristicofthe considered model is the throughputTof the system because it defines the profit earned by information transmission. If the numberKthat restricts the number of sessions, which can be served in the system simultaneously, increases the throughputTof the system increases andtheprobability ( ) lossbP ofanarbitrarysessionrejectionuponarrivaldecreases.So,it seemstobereasonabletoincreasethenumberK asmuchaspossibleuntilstability condition (1) is violated. However, such performance measures as the average sojourn time ofanarbitraryrequestandjitterarealsoveryimportantbecausetheyshouldfit requirementsofQualityofService.Theseperformancemeasuresbecomeworseifthe numberK grows.Evidently,itdoesnotmakesensetoadmittoomanysessionsintothe system simultaneously and provide bad Quality of Service (average sojourn time and jitter) for them. So, the system manager should decide how many sessions can be allowed to enter thesystemsimultaneouslytofitrequirementsofQualityofServiceandtoreachthe maximally possible throughput.Thus, one should solve, e.g., the following non-trivial optimization problem: ( ) max T T K = (11) subject to constraints (1) and cV V-s .(12) whereVis the maximal admissible value of the sojourn time of the first request from non-rejected session and is assumed to be fixed in advance.Thisoptimizationproblemcanbeeasysolvedbymeansofcomputer,basedonpresented aboveexpressionsforthemainperformancemeasuresofthesystem,bymeansof enumeration,i.e.,increasingthevalueK untilconstraints(1)and(12)areviolated.The Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 137 1 11( ) 0K K Mi iC E D I iQ+ +. + = + .> .( 1) 01 0 1 0K Wi iI i OQ Q+. .= .> . = . S | Let us introduce notation:( ) s Ois three block diagonal matrix with non-zero blocks ( ) max 0 1 1 0i j s j { i } i i i.O . = . . .+ .> .defined by( )1 1 31 ( ) ( )ii i i i ii i i is I sI DQ Q. + . . . O = . O = .( ) ( )1 1 21i ii i Ki iD D IQ+. +. +O = + I . Herethematrix ( )1iD ofsize( 1) ( 2) i i + + isobtainedfromtheidentitymatrix 1 iI +by means of supplementing from the right by the column 1Ti + . 0The matrix ( )2iDof the same size has the last column consisting of 1s and other columns consisting of 0s. The matrix ( )3iDof size( 1) i i + isobtainedfromtheidentitymatrix iI bymeansofsupplementingfrom above by the row(1 0 0) . . . . Vector( ) s Bis defined by0( ) ( ( ) ( ) )TNs s s = . . . B B Bwhere 11 1 11 0 1 0 0( ) ( ( ) (( ) ) ( )i K M K W K Ws sI S sI S sI S + += . . . . e e e e S e e S S | B + .> .1 1 11 0 0(( ) )( ( ) ) ) 0i TK WsI S sI S i e e S S | Using this notation we can rewrite the system (7) to the form ( ) ( ) ( )Ts s s O + = . v 0 B(9) Itcanbeverifiedthatthediagonalentriesofthematrix( ) s O dominateinallrowsofthis matrix. So the inverse matrix exists. Thus we proved the following assertion.Theorem3.Thevector( ) s v consistingofconditionalLaplace-StieltjestransformsLST( ) 0 0 1 v s i l k l i i k K v n . . . . . .= . .> . = . . 0 1 W M v n = . . = . .is calculated by

1( ) ( ) ( ) s s s= O . v B(10) Corollary 2. The average sojourn time bVof an arbitrary session is calculated by 1100 0( ) ( 1 1 1)( )KMb si kD I s i i kV i ks == = c . + . + . += . ' .cvt where column vectors ( 1 1 1)0s i i ks sc . + . + . += c'v are calculated as the blocks of the vector ( )0d ss ds ='v defined by10 0( ) ( )(0)( (0))s sd s d sds ds= =' = O ' + .vvB where 1(0) (0) = O . v e Corollary 3. The average sojourn time ( ) acceptbVof an arbitrary admitted session is calculated by( )( )1accept bb lossbVVP= . where ( ) lossbPis probability of an arbitrary session rejection upon arrival. 5. Optimization problem and numerical examples Itisobviousthatthemostimportantfromeconomicalpointofviewcharacteristicofthe considered model is the throughputTof the system because it defines the profit earned by information transmission. If the numberKthat restricts the number of sessions, which can be served in the system simultaneously, increases the throughputTof the system increases andtheprobability ( ) lossbP ofanarbitrarysessionrejectionuponarrivaldecreases.So,it seemstobereasonabletoincreasethenumberK asmuchaspossibleuntilstability condition (1) is violated. However, such performance measures as the average sojourn time ofanarbitraryrequestandjitterarealsoveryimportantbecausetheyshouldfit requirementsofQualityofService.Theseperformancemeasuresbecomeworseifthe numberK grows.Evidently,itdoesnotmakesensetoadmittoomanysessionsintothe system simultaneously and provide bad Quality of Service (average sojourn time and jitter) for them. So, the system manager should decide how many sessions can be allowed to enter thesystemsimultaneouslytofitrequirementsofQualityofServiceandtoreachthe maximally possible throughput.Thus, one should solve, e.g., the following non-trivial optimization problem: ( ) max T T K = (11) subject to constraints (1) and cV V-s .(12) whereVis the maximal admissible value of the sojourn time of the first request from non-rejected session and is assumed to be fixed in advance.Thisoptimizationproblemcanbeeasysolvedbymeansofcomputer,basedonpresented aboveexpressionsforthemainperformancemeasuresofthesystem,bymeansof enumeration,i.e.,increasingthevalueK untilconstraints(1)and(12)areviolated.The TrendsinTelecommunicationsTechnologies 138 optimalvalueofK intheoptimizationproblem(1),(11),(12)willbedenotedbyK-.Corresponding computer program allows to validate the feasibility of such an optimization algorithmandtoillustratethedependenciesofthesystemcharacteristicsonthesystem parameters and the value ofK.In what follows several illustrative examples are presented.Beforetostartdescriptionoftheseexamples,letusmentionthatnumerousexperiments showthatthefamousLittlesformulaholdsgoodforthesystemunderstudyintheform cL V = . whereL istheaveragenumberofrequestsinthesystemand cV istheaverage sojourn time of an arbitrary request which is the first in a session. 5.1. Dependence of probabilities lossbPof an arbitrary session loss and losscPof an arbitrary request loss on the numberKof tokens and correlation in the sessions arrival process Theexperimenthastwogoals.Oneistoillustratequantitativelythedependenceof probabilities lossbP ofanarbitrarysessionlossand losscP ofanarbitraryrequestlossonthe numberKof tokens. The second goal is to show that for several different arrival processes havingthesameaverageratebutdifferentcorrelationthisdependenceisquitedifferent. This explains the importance of consideration of the model with theMAParrival process of sessions, which can be essentially correlated in real telecommunication networks, instead of analysis of simpler model with the stationary Poisson arrival process of sessions.Weconsidersixdifferent MAPshavingthesamefundamentalrate1 = . ThefirstMAPis the stationary Poisson arrival process. Variationcoefficient of inter-arrival times is equal to 1.FourotherMAPshavethevariationcoefficientequalto2butdifferentcoefficientsof correlation of successive intervals between sessions arrival. These fourMAPs are described as follows.-MAP ( IPP Interrupted PoissonProcess ) flow with correlation coefficient equal to 0 is defined by the matrices0 10 4 0 16 0 24 0 0 01 3 69 4 68 1 0 0 01 3 1 3 270 100 2 167 2 0D D . . . | | | | ||= . . . : = . || ||. . . .\ . \ . -MAP flow with correlation coefficient equal to 0.1 is defined by the matrices0 12 66 0 12 0 12 2 3 0 08 0 040 13 0 5 0 08 0 09 0 18 0 020 14 0 08 0 32 0 5 0 01 0 04D D . . . . . . | | | | ||= . . . : = . . . . || ||. . . . . .\ . \ . -MAP flow with correlation coefficient equal to 0.2 is defined by the matrices0 13 16 0 12 0 12 2 84 0 06 0 020 1 0 45 0 09 0 02 0 21 0 030 12 0 11 0 39 0 02 0 04 0 1D D . . . . . . | | | | ||= . . . : = . . . . || ||. . . . . .\ . \ . -MAP flow with correlation coefficient equal to 0.3 is defined by the matrices0 15 11 0 08 0 07 4 85 0 09 0 020 029 0 446 0 04 0 007 0 333 0 0370 06 0 08 0 35 0 0 05 0 16D D . . . . . . | | | | ||= . . . : = . . . . || ||. . . . .\ . \ . -ThesixthMAPhascorrelationcoefficientequalto-0.16andthesquared correlation coefficient equal to 1.896. It is defined by the matrices 0 13 607 0 0 347 3 260 0 617 0 478 0 139D D . . .| | | |= : = . || . . .\ . \ . The service time distribution is Erlangian of order 2 with intensity of the phase equal to 16. The rest of the parameters are the following:2 = . 0 9 u = . . Figures1and2illustratethedependenciesofprobability lossbP ofanarbitrarysessionloss and losscP ofanarbitraryrequestlossonthenumberK oftokensforthelistedabove differentMAP s with the same fundamental rate but the different correlation.

Fig. 1. Dependence of probability lossbPof arbitrary session loss on the number of tokensK Fig. 2. Dependence of probability of an arbitrary request loss on the number of tokensK One can pay attention that the curves corresponding to the differentMAP s terminate at the different points, e.g., the curve corresponding to the stationary Poisson process terminates at thepoint5 K = . thecurvecorrespondingtotheMAP shavingcorrelationcoefficient0.3 terminates at the point12 K = .The reason of termination is that the stationary distribution existence condition violates forKlarger than 5 and 12 correspondingly.Itisworthtomention,thatthepreviousanalysisofdifferentqueueswiththeBatch Markovian Arrival Process given in many papers shows that usually the stability condition depends on the average arrival rate, but does not depend on correlation. It the model under study, stability condition (1) depends on correlation as well. This has the clear explanation: stabilityconditionincludesthestationarydistributionofthecorresponding0 MAP M K / //Queueswithsessionarrivalsasmodelsforoptimizingthetraffccontrolintelecommunicationnetworks 139 optimalvalueofK intheoptimizationproblem(1),(11),(12)willbedenotedbyK-.Corresponding computer program allows to validate the feasibility of such an optimization algorithmandtoillustratethedependenciesofthesystemcharacteristicsonthesystem parameters and the value ofK.In what follows several illustrative examples are presented.Beforetostartdescriptionoftheseexamples,letusmentionthatnumerousexperiments showthatthefamousLittlesformulaholdsgoodforthesystemunderstudyintheform cL V = . whereL istheaveragenumberofrequestsinthesystemand cV istheaverage sojourn time of an arbitrary request which is the first in a session. 5.1. Dependence of probabilities lossbPof an arbitrary session loss and losscPof an arbitrary request loss on the numberKof tokens and correlation in the sessions arrival process Theexperimenthastwogoals.Oneistoillustratequantitativelythedependenceof probabilities lossbP ofanarbitrarysessionlossand losscP ofanarbitraryrequestlossonthe numberKof tokens. The second goal is to show that for several different arrival processes havingthesameaverageratebutdifferentcorrelationthisdependenceisquitedifferent. This explains the importance of consideration of the model with theMAParrival process of sessions, which can be essentially correlated in real telecommunication networks, instead of analysis of simpler model with the stationary Poisson arrival process of sessions.Weconsidersixdifferent MAPshavingthesamefundamentalrate1 = . ThefirstMAPis the stationary Poisson arrival process. Variationcoefficient of inter-arrival times is equal to 1.FourotherMAPshavethevariationcoefficientequalto2butdifferentcoefficientsof correlation of successive intervals between sessions arrival. These fourMAPs are described as follows.-MAP ( IPP Interrupted PoissonProcess ) flow with correlation coefficient equal to 0 is defined by the matrices0 10 4 0 16 0 24 0 0 01 3 69 4 68 1 0 0 01 3 1 3 270 100 2 167 2 0D D . . . | | | | ||= . . . : = . || ||. . . .\ . \ . -MAP flow with correlation coefficient equal to 0.1 is defined by the matrices0 12 66 0 12 0 12 2 3 0 08 0 040 13 0 5 0 08 0 09 0 18 0 020 14 0 08 0 32 0 5 0 01 0 04D D . . . . . . | | | | ||= . . . : = . . . . || ||. . . . . .\ . \ . -MAP flow with correlation coefficient equal to 0.2 is defined by the matrices0 13 16 0 12 0 12 2 84 0 06 0 020 1 0 45 0 09 0 02 0 21 0 030 12 0 11 0 39 0 02 0 04 0 1D D . . . . . . | | | | ||= . . . : = . . . . || ||. . . . . .\ . \ . -MAP flow with correlation coefficient equal to 0.3 is defined by the matrices0 15 11 0 08 0 07 4 85 0 09 0 020 029 0 446 0 04 0 007 0 333 0 0370 06 0 08 0 35 0 0 05 0 16D D . . . . . . | | | | ||= . . . : = . . . . || ||. . . . .\ . \ . -ThesixthMAPhascorrelationcoefficientequalto-0.16andthesquared correlation coefficient equal to 1.896. It is defined by the matrices 0 13 607 0 0 347 3 260 0 617 0 478 0 139D D . . .| | | |= : = . || . . .\ . \ . The service time distribution is Erlangian of order 2 with intensity of the phase equal to 16. The rest of the parameters are the following:2 = . 0 9 u = . . Figures1and2illustratethedependenciesofprobability lossbP ofanarbitrarysessionloss and losscP ofanarbitraryrequestlossonthenumberK oftokensforthelistedabove differentMAP s with the same fundamental rate but the different correlation.

Fig. 1. Dependence of probability lossbPof arbitrary session loss on the number of tokensK Fig. 2. Dependence of probability of an arbitrary request loss on the number of tokensK One can pay attention that the curves corresponding to the differentMAP s terminate at the different points, e.g., the curve corresponding to the stationary Poisson process terminates at thepoint5 K = . thecurvecorrespondingtotheMAP shavingcorrelationcoefficient0.3 terminates at the point12 K = .The reason of termination is that the stationary distribution existence condition violates forKlarger than 5 and 12 correspondingly.Itisworthtomention,thatthepreviousanalysisofdifferentqueueswiththeBatch Markovian Arrival Process given in many papers shows that usually the stability condition depends on the average arrival rate, but does not depend on correlation. It the model under study, stability condition (1) depends on correlation as well. This has the clear explanation: stabilityconditionincludesthestationarydistributionofthecorresponding0 MAP M K / //TrendsinTelecommunicationsTechnologies 140 queueingsystemwhichdescribesthebehaviorofthenumberofbusytokens.Asitis illustrated in (Klimenok et al., 2005), this distribution essentially depends on the correlation in the arrival process.Conclusionthatcanbemadebasedonthesenumericalresultsisthefollowing:higher correlation of the sessions arrival process implies highervalue of lossbPand losscPbut larger numberofsessionswhichcanbesimultaneouslyprocessedinthesystemwithout overloading the system.IPPprocess violates this rule a bit. This is well known very special kindofarrivalprocess.Ithaszerocorrelation.Intervalswherearrivalsoccurmoreorless intensively alternate with time periods when no arrivals are possible. Such irregular arrivals make theIPPviolating the conclusion made above. Note that the system with the negative correlationinthearrivalprocesshascharacteristicsclosetocharacteristicsofthesystem withthestationaryPoissonprocess.Whilethemoreorlessstrongpositivecorrelation changes these characteristics essentially. 5.2. Dependence of the throughput of the system on the number of tokens and correlation in the sessions arrival process Letusconsiderthesamesystemasinthepreviousexperimentandconsideroptimization problem (11), (12) where the limiting value of the average sojourn time for the first request innon-rejectedsessionisassumedtobe40 V Figure3illustratesthedependenceofthe throughputTof the system on the number of tokensKAs it is expected, the throughput T istheincreasingfunctionofK forallarrivalprocesses.However,theshapeofthis function depends on the correlation in the sessions arrival process. The lines corresponding to the differentMAP s terminate when condition (12) is not hold true. So, as it is seen from Figure3theoptimalvalueKoftokensisequalto5whenthearrivalprocessisthe stationary Poisson or has the negative correlation or is equal to 0.1 and is equal to 6 for the rest of the arrival processes.ItisseenfromFigures1-3thatpositivecorrelationhasthenegativeimpactonthesystem performance. Although the number of simultaneously processed requests can be larger, loss probability is higher and the throughput of the system is lesser.Dependence of the average sojourn time cV for the first request in non-rejected sessions on the number of tokens in these examples is presented on Figure 4.

Fig. 3. Dependence of the throughputTof the system on the number of tokens under restriction40cV Fig.4. Dependence of cV- onKIt is seen that the average sojourn time cV- sharply increases when the number of tokensKapproachesthevalue5 K = or6 K = . dependingoncorrelationinthearrivalprocess.For themodelwiththestationaryPoissonarrivalprocessstationarydistributiondoesnotexist for6 K = . 5.3. Dependence of the optimal number of tokens on the session size, arrival and service rates The goal of this experiment is to illustrate the dependence of the optimal number of tokens on the session size, average arrival and average service rates.Firstly,letasclarifytheimpactofthesessionsize.WeassumethattheMAP processof sessions is defined by the matrices 0 16 88 0 0008 6 8 0 07920 0008 0 22 0 016 0 2032D D . . . .| | | |= : = . ||. . . .\ . \ . ThisMAP hastheaveragerateequalto1.37,