Reduced complexity in M/Ph/c/N queues

Transcription

1 Reduced complexity in M/Ph/c/N queues Alexandre Brandwajn, Thomas Begin To cite this version: Alexandre Brandwajn, Thomas Begin. Reduced complexity in M/Ph/c/N queues. [Research Report] RR-8303, INRIA. 2013, pp.15. <hal > HAL Id: hal Sumitted on 13 May 2013 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are pulished or not. The documents may come from teaching and research institutions in France or aroad, or from pulic or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, puliés ou non, émanant des étalissements d enseignement et de recherche français ou étrangers, des laoratoires pulics ou privés.

2 Reduced complexity in M/Ph/c/N queues Alexandre BRANDWAJN, Thomas BEGIN Prénom Nom N 8303! 01/05/2013!!! Project/Team!DANTE! ISSN

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4 Reduced complexity in M/Ph/c/N queues Alexandre Brandwjan 1, Thomas Begin 2 Project-Team DANTE Research Report!N! /05/ pages. Astract:!! A! large! numer! of! real/life! systems! can! e! viewed! as! instances! of! the! classical! M/G/c/N! queue.!! The! exact! analytical!solution!of!this!queueing!model!is!not!known,!and!a!frequently/used!approach!is!to!replace!the!general!service! time!distriution!y!a!phase/type!distriution.!!the!advantage!of!this!approach!is!that!the!resulting!m/ph/c/n!queue!can! e! descried! y! familiar! alance! equations.!! The! downside! is! that! the! size! of! the! resulting! state! space! suffers! from! the! dimensionality!curse,!i.e.,!exhiits!cominatorial!growth!as!the!numer!of!servers!and/or!phases!increases.!!!! To! circumvent! this! complexity! issue,! we! propose! to! use,! instead! of! the! classical! full! state! description,! a! reduced! state! description! in! which! the! state! of! only! one! server! is! represented! explicitly,! while! the! other! servers! are! accounted! for! through!their!rate!of!completions.!!the!accuracy!of!the!resulting!approximation!is!generally!good!and,!moreover,!tends!to! improve!as!the!numer!of!servers!in!the!system!increases.!its!computational!complexity!in!terms!of!the!numer!of!states! grows! only! linearly! in! the! numer! of! servers! and! phases,! thus! making! the! numerical! solution! of! such! queues! with! hundreds!of!servers!and!a!reasonale!numer!of!phases!computationally!affordale.!! Key2words:!Multiple!servers,!general!service,!finite!uffer,!M/Ph/c/N!queue,!reduced/state!approximation,!linear!complexity.!! 1 UCSC alex@soe.ucsc.edu 2 LIP Thomas.egin@ens-lyon.fr

5 Complexité réduite pour les files M/Ph/c/N Résumé :!! De! nomreux! systèmes! réels! peuvent! être! vus! comme! des! instantiations! de! la! file! classique! M/G/c/N.!! La! solution!analytique!exacte!de!ce!modèle!file!d attente!demeure!inconnu,!et!une!approche!fréquemment!employée!consiste! à!remplacer!la!distriution!générale!du!temps!de!service!par!une!distriution!de!type!phase.!!l avantage!de!cette!approche! est!que!la!file!m/ph/c/n!résultante!peut!être!décrite!par!des!équations!d équilires!familières.!!le!désavantage!est!que!la! taille!de!l espace!d état!résultant!souffre!du! dimensionality!curse,!i.e.,!il!croît!cominatoirement!lorsque!le!nomre!de! serveurs!et!/ou!de!phases!s accroît.!!!! Pour!pallier!ce!prolème!de!complexité,!nous!proposons!d utiliser,!à!la!place!de!la!description!d état!classique!complète,! une!description!d état!réduite!dans!laquelle!l état!d un!seul!serveur!est!représenté!explicitement,!les!autres!étant!pris!en! compte!par!leur!taux!de!fins!de!service.!!la!précision!de!l approximation!résultante!est!généralement!onne!et,!en!plus,!elle! tend!à!s améliorer!lorsque!le!nomre!de!serveurs!dans!le!système!s accroît.!sa!complexité!de!calcul!en!terme!de!nomre! d état!s accroît!seulement!linéairement!avec!le!nomre!de!serveurs!et!de!phases,!ce!qui!permet!la!résolution!numérique!de! ce!type!de!files!avec!des!centaines!de!serveurs!et!un!nomre!raisonnale!de!phases.! Mots clés :! Serveurs! multiples,! service! général,! tampon! fini,! file! M/Ph/c/N,! approximation! par! état! réduit,! complexité! linéaire

6 1. INTRODUCTION Alargenumerofreal.lifesystems(suchascallcenters,multi.coreprocessors,etc)canemodeledasinstancesoftheclassical M/G/c/Nqueue(i.e.anM/G/cqueuewithamaximumofNrequestsinthesystem)ifthepatternofrequestarrivalsisrelatively wellehavedandcanerepresentedyaquasi.poissonprocess.theexactanalyticalsolutionofthisqueueingmodelisnot known,andexistingapproximationsareeitherdifficulttoevaluatecomputationally[hok78,miy86,citesparkimura]orfailto capturethepotentiallyimportantdependenceoftheperformanceofsuchaqueueingsystemonhigher.orderpropertiesofthe servicetimedistriution[gou96,smi03].acriticalreviewofseveraloftheseapproximationscanefoundinthepapersy Kimura[KIM93,KIM96].AmorerecentreferenceistheworkySmith[SMI03]whichproposesanapproximationfortheloss proailityasedonlyonthefirsttwomomentsoftheservicetime. Outsidesimulation,afrequently.usedapproachistoreplacethegeneralservicetimedistriutionyaphase.typedistriution, asitisknownthatanydistriutioncaneapproximatedaritrarilycloselyyadistriutionofthelattertype[joh88].the oviousadvantageofthisapproachisthat,insteadystate,theresultingm/ph/c/nqueuecanedescriedyfamiliaralance equations.generallyspeaking,thesealanceequationscaneotainedusingoneoftwopossilestatedescriptionsinvolving thecurrentnumerofrequestinthesystemandavectortorepresentthestateoftheservers.thefirstoneisthevectorofthe currentnumerofserversineachphaseoftheserviceprocess.thesecondpossiledescriptionisthevectorofthecurrent phasesforeachserver(notethattheserversareassumedtoehomogenousuttheyarenotsynchronized.)thislatterstate descriptionisalwayslessthriftythanthefirstoneandrarely,ifever,used.bothdescriptionsexhiitcominatorialgrowthas thenumerofphasesandthenumerofserversgrow. Severalmethods(e.g,matrixgeometric,directiteration[SEE86,RAM85a,RAM85,LAT93,LAT94])caneusedtosolvethese equationsnumerically.aslongasthenumerofserversandservicephasesremainssmallthesemethodsworkfine.however, itisalsoknownthatthesizeofthesystemofequationstoesolvedsuffersfromwhathaseentermedthe dimensionality curse, in that the numer of states grows cominatorially as the numer of servers and phases increase. Thus, for larger numersofservers,thesemethodsecomeimpractical,andthereisaclearneedforanapproachthatwouldhandlelarger numersofservers(say,hundreds)withareasonalenumerofservicetimephases. OurgoalinthispaperistoproposeadifferentapproachtotheapproximateevaluationoftheM/Ph/c/Nqueue.Ourapproach isasedonareducedstatedescriptiontocircumventtheexplosionofthenumerofstatesdiscussedaove.inthefollowing section,wedescrieinmoredetailthequeueingsystemconsideredandweintroducethereducedstatedescription.insection 3,wepresentnumericalresultsillustratingtheaccuracyoftheproposedapproximation,aswellasthesavingsinthesizeofthe statespace.section4concludesthispaper. 2. MODEL,/STATE/DESCRIPTION/AND/SOLUTION/ ConsidertheM/Ph/c/NqueuerepresentedinFigure1.Thetimesetweenarrivalsareassumedtomemoryless(quasi.Poisson) andtheservicetimesarerepresentedasaphase.typedistriutionwithatotalof phases.thereare c homogenousservers inoursystemandtheufferspaceisrestrictedtoamaximumof N requestsinthesystems(queuedandinservice.)we assumethat N > c,sinceotherwisetherewouldenoqueueuilduppossile.wealsoassumethattherateofarrivalsand theparametersoftheserviceprocessmaydependonthecurrentnumerofrequestsinthesystem,denotedy n.thistype of state dependence is useful, in particular, to represent arrivals from a finite numer of exponential sources and service processwhichvarieswiththeworkload.thedetailednotationusedinourpaperisgivenintale1. Weconsiderthestationaryehaviorofsuchaqueue.Asmentionedintheintroduction,thestateofoursystemcouldefully descriedythetotalcurrentnumerofrequests inthesystemandthenumersofrequestsineachphaseoftheservice processor,alternatively,ythecurrenttotalnumerofrequestsandthecurrentphaseofeachserver.insteadofsuchafull state description, we propose to use a reduced state description in which we select one server among the c servers and descrie the system y the total numer of requests and the current phase of the selected server, (n,i). For n < c, with proaility (c n) / c theselectedservermayeidle,inwhichcaseweusethevalue i = 0 todenoteitsidlestate.

7 1. INTRODUCTION Alargenumerofreal.lifesystems(suchascallcenters,multi.coreprocessors,etc)canemodeledasinstancesoftheclassical M/G/c/Nqueue(i.e.anM/G/cqueuewithamaximumofNrequestsinthesystem)ifthepatternofrequestarrivalsisrelatively wellehavedandcanerepresentedyaquasi.poissonprocess.theexactanalyticalsolutionofthisqueueingmodelisnot known,andexistingapproximationsareeitherdifficulttoevaluatecomputationally[hok78,miy86,citesparkimura]orfailto capturethepotentiallyimportantdependenceoftheperformanceofsuchaqueueingsystemonhigher.orderpropertiesofthe servicetimedistriution[gou96,smi03].acriticalreviewofseveraloftheseapproximationscanefoundinthepapersy Kimura[KIM93,KIM96].AmorerecentreferenceistheworkySmith[SMI03]whichproposesanapproximationfortheloss proailityasedonlyonthefirsttwomomentsoftheservicetime. Outsidesimulation,afrequently.usedapproachistoreplacethegeneralservicetimedistriutionyaphase.typedistriution, asitisknownthatanydistriutioncaneapproximatedaritrarilycloselyyadistriutionofthelattertype[joh88].the oviousadvantageofthisapproachisthat,insteadystate,theresultingm/ph/c/nqueuecanedescriedyfamiliaralance equations.generallyspeaking,thesealanceequationscaneotainedusingoneoftwopossilestatedescriptionsinvolving thecurrentnumerofrequestinthesystemandavectortorepresentthestateoftheservers.thefirstoneisthevectorofthe currentnumerofserversineachphaseoftheserviceprocess.thesecondpossiledescriptionisthevectorofthecurrent phasesforeachserver(notethattheserversareassumedtoehomogenousuttheyarenotsynchronized.)thislatterstate descriptionisalwayslessthriftythanthefirstoneandrarely,ifever,used.bothdescriptionsexhiitcominatorialgrowthas thenumerofphasesandthenumerofserversgrow. Severalmethods(e.g,matrixgeometric,directiteration[SEE86,RAM85a,RAM85,LAT93,LAT94])caneusedtosolvethese equationsnumerically.aslongasthenumerofserversandservicephasesremainssmallthesemethodsworkfine.however, itisalsoknownthatthesizeofthesystemofequationstoesolvedsuffersfromwhathaseentermedthe dimensionality curse, in that the numer of states grows cominatorially as the numer of servers and phases increase. Thus, for larger numersofservers,thesemethodsecomeimpractical,andthereisaclearneedforanapproachthatwouldhandlelarger numersofservers(say,hundreds)withareasonalenumerofservicetimephases. OurgoalinthispaperistoproposeadifferentapproachtotheapproximateevaluationoftheM/Ph/c/Nqueue.Ourapproach isasedonareducedstatedescriptiontocircumventtheexplosionofthenumerofstatesdiscussedaove.inthefollowing section,wedescrieinmoredetailthequeueingsystemconsideredandweintroducethereducedstatedescription.insection 3,wepresentnumericalresultsillustratingtheaccuracyoftheproposedapproximation,aswellasthesavingsinthesizeofthe statespace.section4concludesthispaper. 2. MODEL,/STATE/DESCRIPTION/AND/SOLUTION/ ConsidertheM/Ph/c/NqueuerepresentedinFigure1.Thetimesetweenarrivalsareassumedtomemoryless(quasi.Poisson) andtheservicetimesarerepresentedasaphase.typedistriutionwithatotalof phases.thereare c homogenousservers inoursystemandtheufferspaceisrestrictedtoamaximumof N requestsinthesystems(queuedandinservice.)we assumethat N > c,sinceotherwisetherewouldenoqueueuilduppossile.wealsoassumethattherateofarrivalsand theparametersoftheserviceprocessmaydependonthecurrentnumerofrequestsinthesystem,denotedy n.thistype of state dependence is useful, in particular, to represent arrivals from a finite numer of exponential sources and service processwhichvarieswiththeworkload.thedetailednotationusedinourpaperisgivenintale1. Weconsiderthestationaryehaviorofsuchaqueue.Asmentionedintheintroduction,thestateofoursystemcouldefully descriedythetotalcurrentnumerofrequests inthesystemandthenumersofrequestsineachphaseoftheservice processor,alternatively,ythecurrenttotalnumerofrequestsandthecurrentphaseofeachserver.insteadofsuchafull state description, we propose to use a reduced state description in which we select one server among the c servers and descrie the system y the total numer of requests and the current phase of the selected server, (n,i). For n < c, with proaility (c n) / c theselectedservermayeidle,inwhichcaseweusethevalue i = 0 todenoteitsidlestate.

8 Let p(n,i) ethesteady.stateproailitycorrespondingtothisreducedstatedescription.thegeneralalanceequationfor c < n < N isgiveny p(n,i)[λ(n)+ µ i + v(n,i)] = p(n 1,i)λ(n 1)+ p(n, j)µ j q ji + p(n +1, j)µ j ˆq j σ i + p(n +1,i)ν(n +1,i). (1) j=1 Thecorrespondingalanceequationsforothervaluesof n aregivenintheappendix. Notethat,forsimplicity,weomitinthealanceequationsthepossiledependenceon n fortheparametersoftheservice process.intheaoveequation, ν(n,i) denotestheconditionalrateofdepartures(requestcompletions)yserversotherthan theselectedservergiventhecurrentstate (n,i).clearly,weneedawaytodetermine ν(n,i) forourreducedstateequations toeofuse. We denote y u(n) the overall departure rate from the set of c servers given that the current numer of requests in the systemis n.using,forexample,thesecondfullstatedescriptionmentionedaove,thisconditionalcompletionsratecane expressedas u(n) = p( i n) µ ii ˆq ik, (2) i c k=1 j=1 where p( i n) istheconditionalproailityofthecurrentservicephaseofeachofthe c serversgiventhecurrentnumerof requestinthesystem.thefirstsuminformula(2)isoverallpossilesetsofserverphases. Thesteady.stateproailitythatthereare n requestsinthesystem,denotedy p(n),caneexpressedas p(n) = p(n, i) (3) i=0 or,alternatively,computedas p(n) = 1 n λ(k 1), n = 0,1,..., N (4) G u(k) k=1 G isanormalizingconstantsuchthat p(n) =1. N n=0 Letω(n) etherateofcompletionsfortheselectedserver.usingourreducedstatedescription,wehave ω(n) = p(n,i)µ i ˆq i / p(n). (5) i=1 Sincetheserversarehomogenous,for n c wemusthave u(n) = cω(n). (6) Recallthat,toealetosolvethealanceequations,weneedtheconditionalratesofdeparture ν(n,i).weusethefollowing intuitiveapproximation ν(n,i) u(n) ω(n). (7)

9 For n c, the aove approximation amounts to assuming ν(n,i) (c 1)ω(n). Essentially, we assume that the rate of completionsforserversotherthantheselectedserverexhiitslittledependenceonthecurrentservicephaseforthelatter. Thecorrespondingalanceequationsanddetailsofthecomputationof ν(n,i) forothervaluesof n aregivenintheappendix. Thus, together with equations (3), (5), (6) and (7) we get a system of equations for p(n,i) which can e solved in several differentways.inournumericalexamples,weuseasimplefixed.pointiteration. Clearly,thesizeofthestate.space (n,i) andhencethenumerofequationstosolveisingeneralfarsmallerthanwitheither ofthetwofullstatedescriptionsmentionedearlier.additionallyandimportantly,itgrowsonlylinearlywiththenumerof serversandthenumerofphases,whilethecomplexityofthefullstatedescriptiongrowscominatorially. c'servers$ µ 1 λ σ 1 σ 2 q 12 µ 2 ˆ q 2 ˆ q 1 Numer$of$requests$limited$to$N$$ σ q 1 µ ˆ q The$M/Ph/c/N'queue$ The$phase$distriu5on$with$$ phases$for$service$5mes$ Figure1 M/Ph/c/Nqueuewithoutstatedependencies Numerofphasesfortheservicetimedistriution c Numerofservers N Bufferspace,i.e.maximumofrequestsinthesystems(queuedandinservice.) n Totalcurrentnumerofrequestsinthesystem; n = 0,..., N λ(n) Rateofrequestsarrivalsgiventhecurrentnumerofrequestsinthesystem σ i (n) Proailitythatserviceofarequeststartsinphase i, i =1,..., given n µ i (n) Completionrateforphase i ofserviceprocess q ji (n) Proailitythatserviceprocesscontinuesinphase j uponcompletionofphase i, j,i =1,..., ˆq i (n) Proaility that service process ends (request departs the system) upon completion of phase i, i =1,..., m s Meanservicetimeofarequest c s s s Coefficientofvariationofrequestservicetime Skewnessofarequestservicetime

10 p(n,i) Proailitythatthereare nrequestsinthesystemandthecurrentphaseoftheserviceprocessis i p(n) u(n) ω(n) ν(n,i) Marginalproailitythatthereare n requestsinthesystem Overalldepartureratefromthesetof c serversgiventhatthecurrentnumerofrequestsinthe systemis n Departureratefromtheselectedservergiventhatthecurrentnumerofrequestsinthesystemis n (whentheserverisnotidle) Departureratefromserversotherthantheselectedservergiventhatthecurrentnumerofrequests inthesystemis n andthecurrentphaseoftheserviceprocessattheselectedserveris i Tale1 Notationusedinthispaper Inthenextsectionwestudytheehavioroftheproposedreducedstatedescriptionintermsofaccuracyandcomputational complexity. 3. NUMERICAL/RESULTS/ SinceperformancemeasuressuchasthemeannumerofrequestsinthesysteminanM/Ph/cqueueareknowntodependon the shape of the service time distriution (and not only its first two moments) [GUP07, WHI80, WOL77], we organize our explorationasfollows.weconsiderseveralsetsofvaluesforthenumerofservers c andthemaximumnumerofrequests inthesystem N.Wealsoconsiderseveralsetsofvaluesforthecoefficientofvariation c s oftheservicetime,andweuild severaldistriutionswithdifferenthigher.orderproperties,viz.skewness. Touildsuchdistriutionswekeepthemeanservicetime m s at1,andweusethealgorithmyboioet*al.[bob05].the valuesofskewness s s exploredrangefrom c s to100.recallthattheskewnessofarandomvariale S isconsideredtoea measure of the asymmetry of the underlying distriution and is defined as s s = E[(S m s) 3 ] Var[S] 3/2 where Var[S] denotes the varianceof S.Generally,largervalueof s s correspondtolonger.taileddistriutions.thealgorithmusedaimstoproducea phase.typedistriutionwiththeminimumnumerofphasestomatchthespecifiedfirstthreemomentsofthedistriution.in ournumericalexamples,thenumerofphasesvariesetween2and13(mostfrequentlyaround4). Theperformancemetricsconsideredincludethemeannumerofrequestsinthesystem(relativeerror,inFigures2and3),the lossproaility(asoluteerror,infigure4)andtheoverallshapeofthesteady.stateproailitydistriution p(n) infigure5. We define the percentage relative error of our reduced state description versus the actual values as the ratio 100 (approximate actual) / actual.theactualvaluesareotainedfromanumericalsolutionofthefull.descriptionalance equationsforthenumerofservers c <128,andydiscrete.eventsimulationforlargervaluesof c.inthesimulationrunswe use7independentreplicationswith10,000,000completionsperreplication.inallexamplesinthispaperweconsiderasimple Poissonarrivalprocesswithrate λ.notethatwiththeexceptionoffigure5,eachfigurecorrespondstohundredsofdata pointsexplored,andthesurfacesshownareotainedusinganinterpolationfromsetsofscattereddatapoints.

11 skewness skewness impossile area coefficient of variation impossile area coefficient of variation 15 Fig.2a Relativeerrorsoftheapproximatesolutionforthe meannumerofrequestsinm/ph/c/nqueuewith c = 8, N = c +10 and λ = 0.8 c Exp.1 Fig.2 Relativeerrorsoftheapproximatesolutionforthe meannumerofrequestsinm/ph/c/nqueuewith c = 64, N = c,and λ =1.2 c Exp.10 InFigure2a,weshowtheaccuracyoftheproposedmethodforanM/Ph/c/Nqueuewith c = 8, N = c +10 andofferedload λ = 0.8 c (i.e.,moderatelyhighutilization)forarangeofvaluesofthecoefficientofvariation c s andskewness s s ofthe servicetimedistriution.wenotethattherelativeerrorinthemeannumerofrequestsinthesystemisgenerallysmall. around1%.itcanreacharound6%forsomedistriutions,whichcorrespond(inthiscase)tolargervaluesof c (say,over5) andanarrowandofskewnessvalues,viz.smallskewness.notethatinanon.negativedistriutionwithagivencoefficientof variation c s the skewness s s must e greater than a certain value (see Appendix). This is the reason ehind the white impossilearea andinfigure2. Figure2illustratestheaccuracyofourmethodforalargernumerofservers c = 64 withaandsignificantlylargerqueueing room N = c,i.e., N = 306,andtheofferedof λ =1.2 c.inthissettherelativeerrorinthemeannumerofrequests inthesystemremainsunder5%(andmostfarless)forcoefficientsofvariation c s notexceeding5.forlargervaluesof c s,the relative errors can e larger, attaining 15% for a coefficient of variation of 10 and a small set of values of the skeweness, although,overall,evenforsuchlargervaluesof c s therelativeerrorsremainwellelow5%. s

12 offered load (as a function of c) offered load (as a function of c) numer of servers numer of servers Fig.3a Relativeerrorsoftheapproximatesolutionforthe meannumerofrequestsinm/ph/c/nqueuewith N = c +10, c s = 5 and s s = 6 Exp.15 Fig.3 Relativeerrorsoftheapproximatesolutionforthe meannumerofrequestsinm/ph/c/nqueuewith N = 4 c + 20, c s = 5 and s s =15 Exp offered load (as a function of c) numer of servers Fig.3c Relativeerrorsoftheapproximatesolutionforthe meannumerofrequestsinm/ph/c/nqueuewith N = 4 c + 20 c s = 5 and s s = 60 Exp.14 InFigure3a,westudytherelativeerrorinthemeannumerofrequestsinthesystemasafunctionofthenumerofservers andoftheofferedloadformaximumtotalnumerinthesystem N = c +10.Thevaluesofthecoefficientofvariation c s = 5 andoftheskewness s s = 6 wereselectedonpurposesincetheycorrespondtoanotparticularlyfavoralecase,ascaneseen

13 fromfigure2a.weoservethattherelativeerrorstayselow4%andclearlytendstodecreaseasthenumerofservers increases. InFigure3,weshowanalogousresultsforalargerufferspace N = 4 c + 20.Again,onpurpose,weselectaservicetime distriutionthatcorrespondstolargerrelativeerrorsinfigure2,viz. c s = 5 and s s =15.Here,too,wenotethattherelative errorsforthemeannumerofrequestsinthesystemdecreaseasthenumerofserversincreases,rangingfromaround10% for c = 32 tonomorethan2%for256servers. Figure3cillustrateswhatseemstoetypicalaccuracyoftheproposedmethodfor c s = 5.Theexampleshowncorrespondsto skewness s s = 60 anduffercapacity N = 4 c + 20.Noticethattherelativeerrorforthemeannumerofrequestsinthe systemiselow10%for16servers,andecomesvirtuallynegligilewhenthenumerofserversexceeds64. Examiningthereasonsfordeviationsetweentheexactvaluesandthoseproducedyourapproach,wenotethattheonly approximationinourreducedstatesolutionisinthecomputationoftheconditionalcompletionratesof other serversgiven thenumerofrequestinthesystemandthecurrentphaseoftheselectedserver ν(n,i) u(n) ω(n) (ourformula(7)).itis notsurprisingthenthatthedeviationstendtodecreaseandvanishasthenumerofserversincreases.indeed,theknowledge ofthecurrentphaseofjustoneoutofmanyserversdoesnotconveymuchknowledgeaoutthestateoftheotherservers offered load (as a function of c) offered load (as a function of c) numer of servers numer of servers 0 Fig.4a AsoluteerrorsforthelossproailityinM/Ph/c/N queuewith N = c +10, c s = 5 and s s = 6 Exp.15 Fig.4 AsoluteerrorsforthelossproailityinM/Ph/c/N queuewith N = 4 c + 20 c s = 5 and s s = 60 Exp.14 Figure 4 shows the asolute deviation etween the actual loss proaility and that otained using our reduced state approximation. The results in Figure 4a correspond to queue with a small uffer size of N =10 + c. As in Figure 3a, we selected c s = 5 and s s = 6.Wenotethattheasolutedeviationremainselow0.04,andtendstodecreaseasthenumerof serversincreases(allotherthingseingequal.) Figure4illustratesthedeviationinthelossproailityinthecaseofamuchlargeruffer,i.e., N = c.AsinFigure3c, theselecteddistriutioncorrespondsto c s = 5 and s s = 60.Here,thedeviationremainselow0.03anddecreasesrapidly withthenumerofservers.

14 Itisworthwhilementioningthatthedeviationinthelossproailityisgenerallymuchsmallerifthecoefficientofvariationof theservicedistriutiondoesnotexceed,say,3. Since our method produces (approximate) values for p(n), the steady.state proaility that there are n requests in the system,asanexample,wecompareinfigure5theactualandtheapproximatevaluesforthisproaility.here,weusethe samevaluesofthecoefficientofvariationandskewnessasinfigures3cand4for c = 32 serversandofferedload λ = 0.8 c. We oserve that the shape of the p(n) distriution is well reproduced. Clearly, the agreement etween actual and approximate values may e less perfect in some cases, ut, overall, the shape of the distriution tends to e reproduced correctly full state description reduced state description 0.04 p(n) n Fig.5 Comparisonetweenactualandapproximatevaluesforsteady.stateproaility p(n) inm/ph/c/n*queuewith c = 32, λ = 0.8 c, N = 4 c + 20, c s = 5 and s s = 60 Exp14 Wenowconsidertheissueofcomputationalcomplexityoftheproposedapproach.Asmentionedintheintroduction,there areessentiallytwopossilefullstatedescriptionsforanm/ph/cqueue.thefirstdescriptionconsistsofthenumerofrequests andthevectorofthecurrentnumerofserversineachphaseoftheserviceprocess.thetotalnumerofstatesforthis c 1 + n 1% descriptionisgiveny1+ $ ' + (N c +1) + c 1 % $ '.Theseconddescriptioninvolvesthecurrentnumerofrequests n & c & n=1 inthesystemandthevectorofthecurrentphasesforeachserver.thetotalnumerofstatesinthisstatedescriptionisgiven c 1! c y1+ $ n% & n + (N c +1) c.thenumerofstatesintheproposedreducedstatedescriptionisgiveny n=1 1+ (c 1)( +1)+ (N c +1).

15 Numer of states (logarithmic scale) full state description reduced state description Numer of states (logarithmic scale) full state description reduced state description Numer of servers Numer of phases in the service process Fig.6a Comparisonetweenthenumerofstatesinthefull andthereducedstatedescriptionform/ph/c/nqueuewith = 4 and N = 4 c + 20 Fig.6 Comparisonetweenthenumerofstatesinthefull andthereduced.statedescriptionform/ph/c/nqueuewith c = 64 and N = 4 c + 20 Fig.6c Evolutionofthenumerofstatesinourreduced.statedescriptionasafunctionof and c Figure6comparesthecomputationalcomplexityinthenumerofstatesetweenthefirstfulldescriptionandourreduced statedescription.wedonotincludethesecondfullstatedescriptioninourfiguresincethenumerofstatesissystematically higher than for the first one.it is ovious from Figure 6 that, as the numer of servers and phases increases, there is a differenceofmanyordersofmagnitudeetweenthecomplexityofthefullstatedescriptionandourproposedreducedstate description.evenwitharelativelysmallnumerofphases(say4),thecomplexityofthefullstatedescriptionresultsinaout 13,000,000stateswhilethereducedstatedescriptioninvolveslessthan1,500stateswithonly64servers(seeFigure6a).The

16 differencegetsevenmoredramaticforlargernumersofphases.with8phasesand64servers,thereareover3trillionstates inthefulldescriptioncomparedwithlessthan3200statesforourmethod,amountingto9ordersofmagnitudedifference(see Figure6). Figure6cshowsthenumerofstatesinourreducedstatedescriptionforawiderangeofvaluesofthenumerofserversand thenumerofphases.thegraduallinearincreaseinthecomplexityisselfevident. Thenextsectionpresentstheconclusionsofthispaper. 4. CONCLUSION/ Thispaperpresentsanapproach,elievedtoenovel,tothesolutionofaqueueingsystemwithmultiplehomogenousservers, quasi.generalservicetimes(phase.typedistriutions),quasi.poissonarrivalsandalimitedufferspace(queueingroom.)the proposedapproachusesareducedstatedescriptioninwhichthestateofonlyoneserverisrepresentedexplicitlywhilethe otherserversareaccountedforthroughtheirrateofcompletions. Theresultingaccuracyisgenerallygoodand,importantly,tendstoimproveasthenumerserversinthesystemincreases.This conclusionissupportedyalargenumerofdatapointsonlyasmallfractionofwhichisshowninthispaper. Intheclassicalstatedescriptionuseduntilnowforthistypeofqueueingsystem,thenumerofstatesgrowscominatorially, makingtheprolemintractaleforlargernumersofserversand/orphases.bycontrast,thecomputationalcomplexityin termsofthenumerofstatesinourreducedstatedescriptiongrowsonlylinearlyinthenumerofserversandphases.this, forthefirsttime,putsprolemswithhundredsofserversandseveralphaseswithineasyreachofafastnumericalsolution. Future work includes extension of our approach to the case of unrestricted queueing room, as well as quasi.general times etweenrequestarrivals. REFERENCES/ [BOB05]Boio,A.,Horvath,A.,andTelek,M.,Matchingthreemomentswithminimalacyclicphasetypedistriutions, Stochastic*Models,Vol.21,2005,pp [GOU96]Gouweleeuw,F.N.,andTijms,H.,Asimpleheuristicforufferdesigninfinite.capacityqueues.European*Journal*of* Operational*Research,Vol.88(3),1996,pp [GUP07]Gupta,V.,Harchol.Balter,M.,Dai,J.andZwart,B.,Theeffectofhighermomentsofjosizedistriutiononthe performanceofanm/g/s*queueingsystem,performance*evaluation*review,vol.35(2),2007,pp [HOK78]Hokstad,P.,ApproximationsfortheM/G/mQueue,Operations*Research,Vol.26(3),1978,pp [JOH88]Johnson,M.A.,andTaaffe,M.R,Thedensenessofphasedistriutions.School*of*Industrial*Engineering,Purdue University [KIM93]Kimura,T.,EquivalencerelationsintheapproximationsfortheM/G/s/s+rqueue,Mathematical*and*computer* modeling,vol.31(10),2000,pp [KIM96]Kimura,T.,Atransform.freeapproximationforthefinitecapacityM/G/squeue,Operations*Research,Vol.44(6),1996, pp [LAT93]Latouche,G.andRamaswami,V,Alogarithmicreductionalgorithmforquasi.irth.and.deathprocesses,Journal*of* Applied*Proaility.Vol.30,1993,pp [LAT94]Latouche,G.,Newton siterationfornon.linearequationsinmarkovchains,ima*journal*of*numerical*analysis,vol.14,

17 1994,pp [MIY86]Miyazawa,M.,ApproximationoftheQueue.LengthDistriutionofanM/GI/sQueueytheBasicEquations,Journal*of* Applied*Proaility,Vol.23(2),1986,pp [RAM85a]Ramaswami,V.,andLucantoni,D.M.,Stationarywaitingtimedistriutioninqueueswithphasetypeserviceandin quasi.irth.and.death.processes,stochastic*models,vol.1,1985,pp [RAM85]Ramaswami,V.,andLucantoni,D.M.,Algorithmsforthemulti.serverqueuewithphasetypeservice,*Stochastic* Models,*Vol.1,1985,pp [SEE86]Seelen,L.P.,AnAlgorithmforPh/Ph/cQueues,European*Journal*of*the*Operations*Research*Society,Vol.23,1986, pp [SMI03]Smith,J.M.,M/*G/c/Klockingproailitymodelsandsystemperformance,Performance*Evaluation,Vol.52(4),2003, pp [WHI80]Whitt,W.,TheeffectofvariailityintheGI/G/squeue,Journal*of*Applied*Proaility,Vol.17(4),1980,pp [WOL77]Wolff,R.W.,TheEffectofServiceTimeRegularityonSystemPerformance,Computer*Performance,NorthHolland, 1977,pp APPENDIX/ A.1/Balance/equations// Wegiveherethealanceequationsforthecasesnotexplicitlytreatedintheodyofthepaper,startingwiththecase 0 < n < c. p(n,i)[λ(n)+ µ i + v(n,i)] = p(n 1, 0)λ(n 1)σ i / (c n +1)+ p(n 1,i)λ(n 1)*(c n) / (c n +1)+ p(n, j)µ j q ji + p(n +1,i)ν(n +1,i), j=1 i =1,..., p(n, 0)[λ(n)+ v(n, 0)] = p(n +1,i)µ i ˆq i + p(n +1, 0)ν(n +1, 0)+ p(n 1, 0)λ(n 1)(c n) / (c n +1). i=1 Note that we use the notation p(n, 0) to denote the case when the selected server is idle, which, in our model, can only happenif n < c.notealsothatwehave ν(n =1,i 0) = 0. Inthecase n = 0,wecanonlyhave p(0, 0)[λ(0)] = p(1,i)µ i ˆq i +p(1, 0)ν(1, 0). i=1 For n = c wehave p(c,i)[λ(c)+ µ i + v(c,i)] = p(c 1, 0)λ(c 1)σ i + p(c 1,i)λ(c 1)+ p(c, j)µ j q ji + p(c +1,i)v(c +1,i) + p(c +1, j)µ j ˆq j σ i, i =1,..., j=1 j=1.

18 Finally,for n = N weotain p(n,i)[µ i +ν(n,i)] = p(n 1,i)λ(N 1)+ p(n, j)µ j q ji, i =1,...,. i 1 j=1 Theconditionalrateofcompletionsfortheselectedserverwhenitisnotidleandthereare n requestsinthesystemisgiveny ω(n) = p(n,i)µ i ˆq i / p(n, j), and can also e expressed as ω(n) = u(n) / min(n,c). As efore, we approximate the i=1 j=1 conditionalratesofdeparture ν(n,i) for i =1,,,, y ν(n,i) u(n) ω(n).for 0 < n < c,weuse ν(n, 0) u(n). A.2/Constraints/on/skewness/ Let Z eanon.negativerandomvariale,anddenotey m i itsi.thmoment E[Z i ] andy n i itsi.thnormalizedmoment.we have n 2 = m 2 / m i 2 and n 3 = m 3 / (m 1 m 2 ).Wemusthave(see[Osogami]) n 3 n 2.Itfollowstheskewnessof Z,denotedy s Z,mustsatisfytherelationship s z m 1 (c Z 1/ c Z ), where c Z = (m 2 / m 1 2 1) 1/2 and s Z = (m 3 3m 1 m 2 + 2m 1 3 ) / (m 2 m 1 2 ) 3/2. c Z is the coefficient of variation of the random variale Z.Inourcase, m 1 =1 sothatwemusthave s s (c s 1/ c s ).

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