DUAL PROCESSES TO SOLVE SINGLE SERVER SYSTEMS August 2004

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1 DUAL ROCESSES TO SOLVE SINGLE SERVER SYSTEMS As 4 Al Kr Gerrdo Rbo Del Mrcs Rdll J Swf ss Ksfy olly Lm Absrc A ew roch s sed o deerme he rse robbly f of he clsscl ee sysems: M/M/ M/M// d M/M// wh csrohes Ths ew solo mehod ses dl rocesses rdomzo d lce h comborcs The mehod revels h he rse robbly f for M/M// d M/M// wh csrohes hve he sme mhemcl form Mhemcs Sbec Clssfco s: 6K5 6J7 Keywords: Dl rocesses; Sle server ee sysem; Trse robbly f; M/M/; Lce h comborcs; Rdomzo; M/M//; Csrohes I r o d c o For over ffy yers deerm ew mehods o ob he rse robbly f of he clsscl sle server ee sysems hs cred d med he eres of heorss d rcoers le I hs rcle ew smle h roch h combes dl rocesses rdomzo d lce h comborcs s sed o ob he rse robbly f of hree sle server ee sysems Seco cos bcrod merl d some mor resls coec he rso robbly f of brh-deh rocess d s dl rocess The rse robbly f of he clsscl M/M/ ee sysem re re-derved s dl rocesses d he refleco rcle seco 3 I seco 4 he rse robbly f of he M/M// sysem re deermed d formled erms of dl rocesses The solo mehod reles o lce h comborcs sed of he rdol eevle roch Seco 4 lso cos eres forml for co he mber of lce hs o from se o se ses cofed wh ve horzol sr The dl rocess roch eerlzes o cer o brh-deh rocesses I seco 5 he rse robbly f of M/M// sysem wh csrohes s deermed The lyss s srrsly reled o sbly modfed solo of he M/M// sysem s descrbed seco 4 I hs wy he dl rocess/rdomzo/lce h comborcs roch my fy he comlced lyss of fd rse robbly f for well ow Mrov ee sysems Dl rocesses Cosder eerl brh-deh rocess hv rso brh res for d rso deh res for 3 s show he se re rso drm Fre All of hese res re ssmed o be oeve mbers The se sce my be fe or coble Derme of Mhemcs Clfor Se olyechc Uversy omo CA 9768 USA INRIA IRISA Cms Uversre de Bele 354 RENNES Cedex Frce

2 Dl rocesses To Solve Sle Server Sysems Or eres s o deerme he rse robbly f where 3 For fe or coble se sce wh formly boded rso res s deermed by solv he Kolmoorov bcwrd or forwrd eos see Bhchry d Wymre 99 or Gross d rrs 985 The resl s sysem of dfferel eos h my be wre mrx form s Q Q where s he mrx of rso robbly f d Q s he rso re mrx I he fe dmesol cse he mrx form of he bcwrd eo s well osed owever he fe dmesol cse we ssme hroho hs Fre

3 Dl rocesses To Solve Sle Server Sysems rcle h he res d 3 re formly boded Ths rees h he brh-deh sysem hs e rso robbly solo cf Aderso 99 The solo of he Kolmoorov bcwrd eo my he be wre see Bhchry d Wymre 99 s Q e The dl rocess of he eerl brh-deh rocess of Fre hs se re rso drm s show Fre Fre The rse robbly f of hs dl rocess re deoed by By rooso 3 o e 69 of Aderso 99 he follow relosh holds bewee eerl brhdeh sysem d s dl Theorem If d re he rse robbly f of he Mrov rocesses corresod o Fre d resecvely he [ ] d [ ] for ll ses 3 wh he coveo f > - The roof of hs resl s cely reseed Aderso 99 d deeds o he forwrd d bcwrd eos d some lebrc smlfcos Coseely f he rse robbly f eher he orl brh-deh rocess or dl sysem re ow he he rse robbly f he oher sysem re s well Corollry If d re he rse robbly f corresod o he Mrov rocesses deced Fres d resecvely he [ ] for ll ses 3 wh he coveo f > -

4 Dl rocesses To Solve Sle Server Sysems For he eerl brh-deh rocess Fre Mrov ch clled he ssoced rdomzed ch Fre 3 s dered r r r r Fre 3 where r for 3 d r rovded h b b b b 3 wh b chose sch h s b < d by coveo The follow heorem clled rdomzo or formzo s well-ow resl sed rmrly for he mercl como of he rso robbly f of Mrov rocess cf Gross d rrs 985 I les for he reced brh deh rocesses of Fre d more eerlly for y Mrov rocess wh coble se sce hv formly boded dol rso res he Q mrx Theorem Rdomzo Sose Mrov rocess o coble se sce hs rso re mrx Q wh s b < he he rso robbly f my be wre s where b e b for 3! s he -se rso robbly of he ssoced rdomzed Mrov ch I shold be oed h s comleely deermed oce s ow For exmles of solv for by fd see Bohm e l 997 Ch e l 4 Gree e l 3 Kr e l 997 d Leesdro e l 993 I s lso ecessry o rdomze he dl rocess Fre ob b b e!

5 Dl rocesses To Solve Sle Server Sysems where below s he -se rso robbly of he Mrov Ch show he drm - ρ ρ ρ 3 where ρ for 3 d b s reer h or el o he b b sremm of he bsole vle of he dol res of Q Noe he ssmo h he rso res Q re formly boded mles commo vle b for he rdomzos of he rocesses er boh Fres d c be fod 3 The M/M/ Qee Sysem Fre 4 We ow der he clsscl sle server ee sysem M/M/ see Fre 5 The brh or rrvl res d he deh or servce res re osve s rereseed by d resecvely The se sce of rl mbers rereses he mber of csomers he ee y me rereses he rse robbly f for 3 he follow se re rso drm Fre 5 The ssoced rdomzed Mrov ch s show he follow se rso robbly drm Fre 6

6 Dl rocesses To Solve Sle Server Sysems wh d Sose re he rse robbly f of he dl M/M/ rocess Fre 7 The rdomzed Mrov ch of Fre 7 s show Fre By Rdomzo wh b Fre 8 he rso robbly f re e [ ]! I order o ob he rse robbles for he M/M/ sysem we clcle for The dve of he dl rocess roch s he fcly co smle hs Fre 8 s comred o smle hs Fre 6 see Kr e l 997 To fd he smle hs Fre 8 o from se o se -ses eed o be coed These smle hs my be vslzed for exmle s lce h Fre 9 where s ssmed here h Fre 9 x x

7 Dl rocesses To Solve Sle Server Sysems Noe h d d - d - where s he mber of wrd ses d d s he mber of dowwrd ses Solv for d d ves d d so mber of smle hs from o -ses d To co he mber of smle hs o from o Fre 8 -ses he ssoced lce hs re coed owever some lce hs re o relzble s smle h s see Fre 8 For exmle he blc lce h from o drw Fre s sch h x x Fre Forely hese yes of lce hs my be coed by mes of oe-o-oe corresodece wh lce hs o from sr o - o deso -ses show Fre s he dshed/blc lce h Ths s he well ow refleco rcle see Mohy 979 or Nry 979 whch leds o mber of smle hs Fre 8 from o -ses

8 Dl rocesses To Solve Sle Server Sysems Therefore for Noe hs exresso els whe < s he sl coveos cocer boml coeffces By Theorem Rdomzo [ ]! e d by Theorem [ ] [ ] e! [ ]! for d d [ ] [ ] e! for The reced exressos re exlc solos of he rse robbly f for he clsscl M/M/ ee sysem These resls re evle o exressos er Bohm e l 997 Kr e l 997 d Leesdro e l 993; see Lm 4 for verfco of hs evlece The dl rocess wh

9 Dl rocesses To Solve Sle Server Sysems rdomzo roch rovdes smlfco of he lyss reseed Kr e l 997 d Leesdro e l 993 Noe he reced exressos my be wre rdomzed form s [ ]! e where where s he -se rso robbly from o o Fre 6 where d Noe by or coveos wo of he boml coeffces wh hs smmd vsh for y ve d Ad where s he -se rso robbly from o o Fre 6 where Alerve exressos for erms of fe smmo over from - o - my be fod by s Corollry lce of Theorem Dels of hs lerve develome for M/M/ my be fod Lm 4 The Corollry roch s lso demosred he ex seco o fd he rse robbly f of he M/M// ee sysem

10 Dl rocesses To Solve Sle Server Sysems 4 The M/M// Qee Sysem We ow der he clsscl sle server ee sysem hv fe ccy M/M// see Fre The brh or rrvl res d he deh or servce res re osve s rereseed by d resecvely The se sce { } rereses he mber of csomers he ee y me rereses he rse robbly f for 3 he follow se re rso drm 3 - The ssoced rdomzed Mrov ch s show he follow se rso robbly drm wh d 3 Le deoe he rse robbly f of he dl M/M// rocess hv rso re drm Fre 3 Fre Fre Fre 3 The rdomzed Mrov ch of he rocess deced Fre 3 s show Fre Fre 4

11 Dl rocesses To Solve Sle Server Sysems By rdomzo wh b he rso robbly f re e [ ]! I order o ob he rse robbles for he M/M// sysem s Corollry of seco we eed o clcle For ow ssme re resrced o he follow se of ses { } Cosder he se of smle hs o from se o se -ses Fre 4 These smle hs my be vslzed s lce hs cofed o horzol sr boded by les y - d y s show Fre 5 The hs ever och eher of hese bodres I order o deerme we wsh o L co he mber of hs L Fre 5 x As seco 3 d d d - where s he mber of wrd ses d d s he mber of dowwrd ses Solv for d d ves so d d d L

12 Dl rocesses To Solve Sle Server Sysems where L deoes he mber of hs L To co he mber of smle hs L we se he follow eerlzo of he refleco rcle whch my be fod Mohy 979 d Nry 979 l l l L where l es ll eer vles osve eve d zero d z y z y whe y z d z y whe z < or y < z We recs hs resl s he follow lemm Lemm m m m m 3 3 mod m m where < m m d re osve eers d m ex s he m h roo of y roof Aly he boml heorem y x y x m-mes Frs le x d y he le x d y he le 3 x d y ec l m x d y So we hve d for m Mlly boh sdes of hs eo by d sm over m e m m m

13 Dl rocesses To Solve Sle Server Sysems The resl follows by relz h m m whe m mod becse s he m h roo of y d m whe m mod whch s eece of be he m h roo of y Ths comlees he roof of he lemm Aly Lemm wh m d d l d ex o ob l l m m Smlrly l l m m follows from Lemm whe m d l d ex Therefore L l l l by Lemm by sbso

14 Dl rocesses To Solve Sle Server Sysems by fcor s s by sbs ex d s Eler s relo The rel r of hs ls exresso els L self rel mber herefore L s s For referece we se hs resl s he follow rooso Accord o Mohy 979 form of hs resl lso ers Kemerm 96 rooso Assme d re ses he follow se { } d sose L s he colleco of ll smle hs o from se o se -ses Fre 4 These smle hs corresod o lce hs h move or dow by oe ech se d re cofed o he horzol sr boded by les y - d y ever och eher of hese bodres see Fre 5 The he mber of lce hs L deoed by L s ve by he follow exresso L s s We ow se or h co resl o deerme he -se rso robbly of he rdomzed Mrov ch cred Fre 4 see Lemm Recll hs Mrov ch ores from he dl rocess Trse robbly f for he rse ses of he dl rocess wll he follow s eece of rdomzo Lemm 3 A secl cse

15 Dl rocesses To Solve Sle Server Sysems cocer he rse robbly fco of he dl rocess hv bsorb se - s he reseed Lemm 4 Lemm Sose d The -se rso robbles for he Mrov ch corresod o Fre4 s ve by he follow exresso s s where d roof By he lyss follow Fre 5 d L where d d re he mber of wrd d dowwrd ses e from se o se ses The forml from rooso for L ves he resl Lemm 3 Sose d he he rso robbly f for he dl Mrov rocess corresod o Fre 3 s ve by he follow exresso ex s s ex s s where d roof By Theorem Rdomzo d Lemm [ ] s s! e

16 Dl rocesses To Solve Sle Server Sysems [ ]! s s e ex s s by he Tylor seres for { }! ex x x Lemm 4 Sose he he rso robbly f for he dl Mrov rocess corresod o Fre 3 s ve by he follow exresso ex s s roof Kolmoorov s forwrd eo Q d d mles h d d Lemm 4 follows by sbs o hs eo he exresso ve for from Lemm 3 d er erm by erm wh resec o The of ero s deermed by he l vle whe > - v deermed he rse robbly f of he dl rocess we re ow oso o fd he rse robbly f of he orl M/M// ee sysem Theorem 3 The rse robbly f of he M/M// ee sysem corresod o Fre re s follows Cse d ex s s s ex s s s

17 Dl rocesses To Solve Sle Server Sysems Cse d ex s s s Cse 3 d ex s ex s s Cse 4 d ex s Cse 5 d ex s s ex s s Cse 6 d ex s s

18 Dl rocesses To Solve Sle Server Sysems roof Cses d The resl follows from [ ] [ ] fod Corollry d s Lemms 3 d 4 I Cse he smmo erm s mss Cses 3 d 4 Corollry becomes [ ] becse - s bsorb se The forml he follows s before by sbs exressos from Lemms 3 d 4 I Cse 4 he smmo erm s oe Cses 5 d 6 Corollry s ow [ ] becse s bsorb se d oce Lemms 3 d 4 rodce he reced exresso I Cse 6 he smmo erm s oe Theorem 4 Tcs The rse robbly f of he M/M// ee sysem corresod o Fre re s follows If he ex s s s s If he s s s s ex roof The erm hs heorem s mos esly obed by relz corresods o he well-ow sedy se solo of he M/M// ee sysem see Gross d rrs 985 Ths erm s lso ssoced wh he eevle of hs sysem see Kr d Morese 4 The rem oro of he rse robbly fco my be derved hroh lebrc mlos of he solos fod Theorem 3 I rclr he follow eomerc seres dey

19 Dl rocesses To Solve Sle Server Sysems s ϖ s ϖ s ϖ ϖ s ϖ s sefl redc he doble sms er Theorem 3 o sle sm exresso The rem dels my be fod Ksfy 4 I s eres o comre Theorem 3 d s dervo o he rdol mehod of solo see for exmle Theorem o es 3- of Tcs 96 where s deermed s ler lebrc eevle roch I cors or roch here ses dl rocesses rdomzo lce h comborcs rooso d he Tylor seres exso of he exoel fco 5 M/M// wh Csrohes I hs seco he clsscl M/M// ee sysem s modfed o clde csrohe re whch oes o se Csrohes re ssmed ossble y o-zero se b wh re d whe hey occr hey redce he ee sysem o zero csomers The rsos re show he follow se re rso drm Fre Fre 6 Alhoh he sl rso mrx of hs sysem hs dmeso by we rerese rsos meed by mrx hv res s show below The reso for sch elred Q-mrx s co of fd s dl rocess

20 Dl rocesses To Solve Sle Server Sysems Q As before he obecve s o deerme he e rse robbly f corresod o Q Forely he heory of dl rocesses hs bee develoed s brhdeh rocesses d here s exeso of Corollry Secfclly by resl of Krse see Aderso 99 Theorem 34 es s sochsclly moooe for he M/M// wh csrohe rocess d herefore by heorem of Semd see Aderso 99 he rme wh he roof of rooso 4 d he remrs follow he roof o es 5-5 follows h [ ] where s he e rso robbly fco corresod o he dl rocess hv fesmls res Q ve by Q

21 Dl rocesses To Solve Sle Server Sysems I hs seco we wll refer o [ ] s Semd s eo where s he ew dded se sed of - s seco I hs wy Semd s eo whch coecs he rso robbly f of he orl d dl rocesses s see o be he sme s Corollry shfed o he rh by The dl rocess hs rso re drm s ve Fre 7 The se sce of he dl rocess s { } - Fre 7 Rdomz hs sysem ves he follow rdomzed Mrov Ch of he dl rocess r r r r r - Fre 8 where r Noc he smlry bewee Fres 4 d 8 he follow resls loy wh he heorems of seco 4 my be dedced

22 Dl rocesses To Solve Sle Server Sysems Lemm 5 Sose d The -se rso robbles for he Mrov ch corresod o Fre 8 s ve by he follow exresso s s wh roof The mber of lce hs from se o se resrced o he horzol bd srcly bewee d els he mber of lce hs from se - o se - cofed o horzol bd srcly bewee - d owever by rooso here re s s sch hs The roof ow roceeds s Lemm Lemm 6 Sose d he he rso robbly f for he dl Mrov rocess corresod o Fre 7 s ve by he follow exresso ex s s where roof By Theorem Rdomzo d Lemm 5 [ ] s s! e [ ]! s s e ex s s

23 Dl rocesses To Solve Sle Server Sysems by he Tylor seres for { }! ex x x Lemm 7 Sose he he rso robbly f for he dl Mrov rocess corresod o Fre 7 s ve by he follow exresso ex s s roof Kolmoorov s forwrd eo Q d d mles h d d s ow from Lemm 6 d my be deermed by er wh resec o s Lemm 4 The of ero s oce deermed by he l vle whch holds for > A hv deermed he rse robbly f of he dl rocess we my ow sbse o Semd s eo [ ] o fd he rse robbly f of he M/M// wh csrohe ee sysem Theorem 5 The rse robbly f of he M/M// sysem wh csrohes corresod o Fre 6 re s follows where Cse d ex s s ex s s s s

24 Dl rocesses To Solve Sle Server Sysems Cse d ex s s s Cse 3 d s s ex ex s Cse 4 d ex s Cse 5 d ex s s ex s s Cse 6 d ex s s

25 Dl rocesses To Solve Sle Server Sysems roof Cses d The resl follows from [ ] [ ] d s Lemms 6 d 7 I Cse he smmo erm s oe Cses 3 d 4 Semd s eo becomes [ ] becse s bsorb se The forml he follows s before by sbs exressos from Lemms 6 d 7 I Cse 4 he smmo erm s mss Cses 5 d 6 Now [ ] becse s bsorb se d oce Lemms 6 d 7 rodce he reced exresso I Cse 6 he smmo erm s oe Ths comlees he roof of Theorem 5 The close coeco bewee he rse solos of M/M// d M/M// wh csrohes becomes cler whe loo Fres 4 d 8 The h co s he sme oly he robbles re re-ormlzed Or h co resls wold hve ered excly he sme for ech sysem hd Semd s eo bee sed hroho hs rcle lce of Corollry Ths leves he rl rem sse of wheher Tc s Theorem 4 eerlzes srhforwrd wy o M/M// wh csrohes? The swer s yes d s ve Theorem 6 whch fes he dffere cses of rse robbly f lsed wh Theorem 5 Theorem 6 Sose > he he rse robbly f for d of he M/M// wh csrohes sysem show Fre 6 re s follows Ar Br ex s s s s

26 Dl rocesses To Solve Sle Server Sysems where 4 r d 4 r A r r r r r r r r r r r d r A B r roof The Ar Br erm hs heorem corresods o he sedy se dsrbo of he M/M// wh csrohes sysem Ths dsrbo s deermed by o h he sedy se dsrbo ssfes ler coeffce secod order recrrece relo The rem oro of he reced rse robbly fco s derved hroh lebrc mlos of he solos fod Theorem 5 follow he sme rmes h rodced Theorem 4 from Theorem 3; see Ksfy 4 for comlee dels of hs clclo Remrs I s r o hve wo ee sysems M/M// d M/M// wh csrohes where we exlcly ow he eevles of Q by forml rher h hv o come hem merclly Frher reserch o deerme whch Q mrces hve eevles follow sch rey ers wold be worh whle Dl rocesses re rov o be sefl ool for fd rse robbly f vrey of ses see: Ch e l4 Gree e l3 Kr e l Kr d Morese 4 Kr e l4 The m dve of wor wh dl rocesses s h s ofe eser o lebrclly or comborlly lyze bsorb Mrov rocesses h recrre Mrov rocesses Acowledeme The hors wsh o cowlede he sedy ecoreme of Sr Gol Mohy who hroho hs dsshed creer hs corbed d romoed he lyss of rse behvor ee heory s comborl mehods

27 Dl rocesses To Solve Sle Server Sysems Refereces Aderso W J 99 Coos-Tme Mrov Chs A Alcos-Oreed Aroch Srer-Verl New Yor Bhchry R N d E C Wymre 99 Sochsc rocesses Wh Alcos Joh Wley & Sos New Yor Bohm W A Kr d S G Mohy 997 The Comborcs of Brh-Deh rocesses d Alcos o Qees Qee Sysems Ch I A Kr R J Swf 4 Brh-Mlle Csrohe rocesses rer Ecoomo A d D Fos A Coos-me Mrov Ch Uder he Iflece of Rel o rocess d Alcos Sochsc Models wh Csrohes rer Gree M L A Kr C Morese G Rbo R J Swf 3 Trse robbly F: A Smle h Aroch Dscr ee Mhemcs d Theorecl Comer Scece rs Gross D d CM rrs 985 Fdmels of Qee Theory Secod Edo Joh Wley d Sos New Yor Ksfy 4 Dl rocesses o Deerme Trse robbly F Msers Thess Clfor Se olyechc Uversy omo Clfor Kemerm J B 96 The sse roblem for Sory Mrov Ch Sscl Reserch Moorhs Volme The Uversy of Chco ress Illos Kr A C Morese G Rbo 4 Coe Bewee Brh-Deh rocesses I: A C Kr Rdll J Swf Eds Sochsc rocesses d Fcol Alyss A Volme of Rece Advces oor of MM Ro Volme 38 he Lecre Noes re d Aled Mhemcl Seres Mrcel Deer New Yor 9-4 Kr A d C Morese 4 Trse robbly F of Fe Brh-Deh rocesses wh Csrohes rer Kr A D Mrcs D Klm d T Che 997 Trse Solo of he M/M/ Qee Sysem v Rdomzo I: J Goldse N Gresy d JJ Uhl Eds Sochsc rocesses d Fcol Alyss Volme 86 he Lecre Noes re d Aled Mhem cl Seres Mrcel Deer New Yor Kr A G Rbo R J Swf M L Gree Lm Dl rocesses o Solve Qee Sysems Wh Csrohesrer

28 Dl rocesses To Solve Sle Server Sysems Kmr B K d D Arvdmb Trse Solo of M/M/ Qee wh Csrohes Comer d Mhemcs wh Alcos Lm 4 Alyz Sle d Two Server Qee Sysems Us Dl rocesses Msers Thess Clfor Se olyechc Uversy omo Clfor Leesdro J ellml G Rbo B Sercol 993 Trse Solo of he M/M/ Qee Adv Al robbly 5 Mohy S G 979 Lce h Co d Alcos Acdemc ress New Yor Nry T V 979 Lce h Comborcs Wh Sscl Alcos Uversy of Toroo ress Toroo Swf R J A Smle Immro-Csrohe rocess Mh Sces Tcs L 96 Irodco o he Theory of Qees Oxford Uversy ress Oxford