The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue

Transcription

1 Te Veraly o MMAP[K] ad e MMAP[K]/G[K]/ Queue Q-Mg E eparme o dural Egeerg altec aloue Uvery ala ova Scoa Caada B3J X4 E-mal Q-Mge@dalca To appear Queueg Syem T paper ude a gle erver queueg yem mulple ype o cuomer Te r par o e paper dcue ome modelg ue aocaed e Marov arrval procee mared arrval MMAP[K] ere K a eger repreeg e umber o ype o cuomer Te ueule o MMAP[K] modelg po procee o by a umber o ereg eample Te ecod par o e paper ude a gle erver queueg yem a MMAP[K] a pu proce Te buy perod vrual ag me ad acual ag me are uded Te ocu o e acual ag me o dvdual ype o cuomer Eplc ormula are obaed or e aplace Sele raorm o ee acual ag me Key ord Queueg yem ag me mulple ype o cuomer po proce mar aalyc meod AMS 99 Subec Clacao Prmary 6K5 roduco Te obecve o paper o o o o model po procee mulple ype o cuomer by ug Marov arrval procee mared arrval MMAP[K] ad o udy a gle erver queueg yem mulple ype o cuomer T udy a movaed by e poeal applcao o e reul obaed paper elecommucao mauacurg ad ervce yem ere mulple ype o cuomer are pree elecommucao yem uc a c ceer are requred o adle dere ype o daa eg voce vdeo ad acmle mulaeouly Te aure o ee daa procee uually dramacally dere Some o em are bury ome are g volume ome cao aord ay lo ad ome cao ave log delay Tee daa compee or yem reource ad u ave muual luece e ey ravel a eor Tereore o do yem perormace aaly ueul o uderad o eac ype o daa beave e eor mauacurg a cuomer order may co o everal uborder or dere compoe Tu e demad procee o dvdual compoe are depede ad o ould be er repleme or produco procee To ga g o uc produco yem a aaly a e compoe level a mpora a a aaly a e aggregae level e ervce dury cuomer ca be dgued o dere group eac requrg a parcular ype o ervce mpora o uderad o a yem erve dvdual ype o cuomer e ey compee or e ame reource

2 For all e above cae a ormal ad covee ormalm o e pu proce val yem modelg A dealed aaly o e correpodg queueg yem coderg e beavor or perormace o dvdual ype o cuomer ueul pracce or pog ou a eve or queueg yem oly oe ype o cuomer e e pu proce bury e ag me o dere cuomer ca be gcaly dere Tu mae ee o dgu cuomer o ubgroup ad aalyze e queueg procee o dvdual group o cuomer repecvely ee Eample E [8] T paper ugge e ue o MMAP[K] o model po procee mulple ype o cuomer MMAP[K] a geeralzao o Marov arrval procee MAP c ave bee uded ad ued eevely queueg eory MAP a roduced eu [8] o model o-marova po procee MAP or ome o pecal cae are dely ued by reearcer ad pracoer elecommucao ad mauacurg ee Ce ad Reder [4] ucao [4 5] ucao e al [6 7] eu [ ] ad Ramaam [6 7] le MAP a ueul ool o model po procee oe ype o cuomer MMAP[K] ueul e mulple ype o cuomer are pree MMAP[K] a roduced by eu ee E ad eu [] Amue ad Koole [3] roduced MMAP[K] depedely Cloely relaed or ca alo be oud Paceco [4] ad Prabu [5] ere e erm MMAP a roduced or pecal cae o mulvarae Marov addve procee paper ll be o a MMAP[K] provde lebly modelg correlaed po procee pecal arrval paer ll alo be o a queueg yem MMAP[K] a pu procee are aalycally ad compuaoally racable Queueg yem mulple ype o cuomer ave bee uded eevely e e pu procee are depede Poo procee ad prore are aged o dere ype o cuomer ee Saord ad Fcer [9] Taag [3 3] ad reerece ere Ueul reul ave bee obaed or yem ably codo queue leg ag me ad buy perod drbuo To eed applcao o ee reul queueg yem more geeral pu procee ere roduced ad uded Tae Segupa ad Yeug [34] ad Yeug ad Segupa [36] codered Marov modulaed Poo procee ad uperpoo procee o Marov arrval procee E [9] ad E ad Ala [] uded queueg yem MMAP[K] P-drbued ervce me ad a la-come-r-erved ervce dcple E [8] ad Tae ad aegaa [33] a queueg yem depede arrval procee o mulple ype o cuomer ad a r-come-r-erved FCFS ervce dcple a uded Some reul ere obaed or e udameal perod queue leg ad ag me T paper geeralze mo o e reul obaed E [8] Te or o Tae ad aegaa [33] alo ee Tae [3] cloely relaed o paper ac e aaly o e buy perod buy cycle dle probable vrual ag me o e queueg yem uded paper ca be carred ou by ug reul obaed [33] Te derece beee paper ad [33] are paper dcue ome modelg ue relaed o MMAP[K]; paper coder e acual ag me o ay parcular ype o cuomer ad derve e correpodg aplace Sele raorm

3 Compared o E [8] paper gve a more dealed dcuo o e modelg ue o MMAP[K] Queueg yem geeral bac arrval are codered paper le [8] ocued o e cae ere eac bac a oly oe cuomer T paper alo coder e ervce equece o cuomer eac bac a ue a rarely dcued e eg leraure addo paper pree ome o e proo gve [8] a ormal ay e g o y ome o e meod or are gaed ad eplaed Te udy o e ag me a gle erver queue dae bac o e Pollacze- Kc ormula or e M/G/ queue ee Coe [6] ad Kleroc [] For e M/G/ queue ad relaed Poo arrval queueg yem PASTA ol [35] guaraee a e acual ag me ad e vrual ag me ave e ame drbuo oever or queueg yem more geeral pu procee eg BMAP PASTA doe o old e e vrual ag me ad e acual ag me ave dere probably drbuo For e BMAP/G/ queue Pollacze-Kc ype ormula or vrual ad acual ag me ave bee oud ee ucao [4 5] ucao e al [6 7] eu [9 ] ad Ramaam [6] e mulple ype o cuomer are pree o oly are e vrual ag me ad e acual ag me dere bu alo e acual ag me o dvdual ype o cuomer are dere ee E [8] Tereore a aaly o e ag procee o dvdual ype o cuomer eceary o ga g o uc queueg yem Suc a aaly coduced Seco 5 ad 6 o paper Te re o e paper orgazed a ollo Seco gve e deo o MMAP[K] ad pree ome lmg propere aocaed MMAP[K] Seco 3 o e ueule o MMAP[K] by preeg a umber o ereg eample ad roduce e MMAP[K]/G[K]/ queue a FCFS ervce dcple Seco 4 pree ome reul abou e buy perod ad e aplace Sele raorm o e vrual ag me Seco 5 ad 6 udy e acual ag me o dvdual ype o cuomer Seco 5 e aplace Sele raorm o acual ag me o varou ype o cuomer are preeed uve proo Formal proo o ee reul are gve Seco 6 Fally Seco 7 reul obaed paper are ummarzed ad ome dcuo gve o uure reearc eo o MMAP[K] A Marov arrval proce o mared arrval MMAP[K] a ocac po proce mulple arrval bace occurrg couou me or dcree me Eac arrval bac repree e arrval o a umber o cuomer o e yem o ere Tere are K dere ype o cuomer Trougou o paper e ord arrval ad bac are equvale Te ord arrval ued moly e r par o e paper le bac ued maly e ecod par o e paper For laer ue paper a deo o MMAP[K] gve o e couou me cae T deo a gve by eu E ad eu [] e ℵ K < 3

4 ere K a pove eger a repree e umber o cuomer ype Eac ℵ repree e ype o cuomer a arrval ad er relave order Te leg o ℵ deoed by e umber o eger cuomer Coder a m-ae Marov reeal proce a rreducble embedded Marov ca rao probably mar P p ad epoeal oour me drbuo ae gve by ep-σ or m T alo a Marov ca couou me e be e emal geeraor o Marov ca Te mar ad e parameer P ad σ m o e Marov reeal proce are relaed by p σ or m ad p σ or m e J deoe e ae o Marov reeal proce a me J called e uderlyg Marov proce emal geeraor ee a Marov reeal proce J τ o e ae pace [ m ℵ] [ e rao probably mar or m ℵ P J τ J ep u du ere e marg varable Te marce ℵ are oegave Te mar a egave dagoal eleme ad oegave o-dagoal eleme aumed o be ogular Te relaop beee e emal geeraor ad ℵ 3 ℵ A arrval called a ype arrval bac e arrval mared by ℵ Te marg rae mar o ype arrval A ype arrval clude cuomer oe ype are ad equeced e arrval accordgly e e r cuomer e arrval a ype cuomer e ecod a ype cuomer ad e la a ype cuomer e be e oal umber o ype cuomer o arrved Te e K co o e coug procee o dvdual ype o cuomer eoe by p P J J m K K K 4 ad P K a m m mar eleme p K e P z z K be e mome geerag uco o P K ca be proved a K P z zk Ez zk ep z 5 ℵ ere repree deo ad E repree maemacal epecao 4

5 eoe by θ e aoary probably vecor o e mar e θ θe ere e e colum vecor all compoe oe ee Gamacer [7] or more abou mar eory Ug equao 5 eay o very a e aoary arrval rae o ype arrval gve by θ e or ℵ ee Apped eu e al [3] uvely e umber o ype arrval durg e perod δ gve by δ a bac ac a ll be ued repeaedly Furermore e arrval rae o dvdual ype o cuomer a group o cuomer ad all e cuomer ca be deed ad obaed a mlar maer For ace e ollog aoary arrval rae ca be obaed ealy For ad le θ e e aoary arrval rae o ype cuomer c come rom ype arrval ad are e cuomer er correpodg arrval; θ e e aoary arrval rae o ype cuomer c come rom ype arrval; θ e e aoary arrval rae o ype cuomer; ℵ ℵ b θ e e aoary arrval rae o arrval bace; ℵ ℵ θ e e aoary arrval rae o all cuomer regardle o ℵ er ype ℵ ere e dcao uco Some codoal probable ad lmg reul o MMAP[K] c all be ued laer eco ca be obaed mmedaely e p be e probably a e ae o e uderlyg Marov proce J rg aer a ype arrval gve a e arrval o ype ad e uderlyg Marov proce a ae u beore e arrval a a arbrary me Coder e eve a ere a ype arrval δ eady ae e probably a ere a ype arrval δ θ eδ δ Te probably a e ae cage rom o δ Te δ o δ p lm δ δ o δ 6 e p be e probably a e ae o e uderlyg Marov proce J rg aer e arrval o a ype cuomer c e cuomer a ype arrval gve a e arrval o ype ad e uderlyg Marov proce a ae u beore e arrval Te p 7 5

6 Smlarly le p be e probably a e ae o e uderlyg Marov proce J rg aer e arrval o a ype cuomer gve a a ype cuomer u came ad e uderlyg Marov proce a ae u beore e cuomer arrve e p a arrval be e probably a e ae o e uderlyg Marov proce J rg aer a arrval gve a ere a arrval ad e uderlyg Marov proce a ae u beore e arrval a a arbrary me Te p ℵ ad p a arrval ℵ 8 Propery For e MMAP[K] deed above e ave e ollog ueul cocluo a Te probably a a arbrary arrval bac o ype / b b Te probably a a arbrary ype cuomer rom a ype arrval / c Te probably a a arbrary ype cuomer rom e poo o a ype arrval / d Te probably a a arbrary cuomer o ype / Fally eco e rao o e umber o oal cuomer arrved ad e umber o a parcular ype o bac arrved obaed e ξ be e umber o ype bace a arrved beore ad e e cuomer arrve Te lm /ξ e average umber o cuomer o arrved beee o coecuve ype bace e be e oal umber o cuomer o arrved beee o coecuve ype bace cludg oe ype bac Te mome geerag uco o ca be obaed a è Ez z e z ℵ 9 ereag epreo 9 repec o z ad eg z yeld E / Te eay o prove a ξ lm E More aympoc ad lmg reul ca be derved or MMAP[K] oeele oly ee o be ued laer paper are preeed eco Src proo o all e cocluo ca be obaed ollog e approace ued or e pecal cae Marov arrval procee See eu e al [3] or more deal 3 Modelg MMAP[K] ad e MMAP[K]/G[K]/ queue 6

7 eco a umber o pecal cae ad eample o MMAP[K] are preeed ad e MMAP[K]/G[K]/ queue roduced a ell Te obecve o eco o demorae e ueule o MMAP[K] e modelg o po procee ad ece e poeal applcao o MMAP[K] o elecommucao mauacurg ad ervce dure Specal cae 3 Te uperpoo proce o K depede Poo procee a MMAP[K] Suppoe a e arrval rae o e K Poo procee are K Te mar repreeao o correpodg MMAP[K] K ad K K Specal cae 3 A bac Marov arrval proce BMAP mar repreeao a MMAP[K] K e arrval rae mar o bace o e ze or See ucao [4 5] ucao Meer-eller ad eu [7] eu [8] or more deal abou BMAP Specal cae 33 A MMAP[K] mar repreeao K decrbe a po proce K ype o cuomer Eac arrval bac co o a gle cuomer T ype o MMAP[K] a bee uded E ad eu [] ad ued a pu procee o queueg yem E [8 9] ad E ad Ala [] e a e ereg eample are preeed o o o o ue MMAP[K] o model ocac po procee a pecal arrval paer Eample 34 Coder a po proce ype o cuomer A arrval may co o a gle ype cuomer a gle ype cuomer or a ype cuomer ad a ype cuomer Suc po procee ca be modeled by MMAP[] a careully coe uderlyg Marov proce J ad marce e e order o e o cuomer a arrval bac mu be codered or ome reao ca e be pl o o marce o dgu a bac rom a bac Eample 35 e e arrval o a po proce ype o cuomer poe a cyclc paer e proce ca be modeled by e ollog MMAP[] d d 3 d d ere d d d ad d are marce dmeo coe properly Marce d ad d ave egave dagoal eleme ad oegave o-dagoal eleme Marce d ad d are oegave MMAP[] a ype cuomer ollo a ype cuomer ad a ype cuomer ollo a ype cuomer ormulao o oly e equece o cuomer modeled bu alo e erarrval me beee arrval o dere ype ca be peced 7

8 Eample 36 Coder a po proce o ype o cuomer oberved a ay ype cuomer olloed by a lea oe ype cuomer Suc a po proce ca be modeled by a MMAP[] a mar repreeao d d d d d 3 ere d d d d ad d are ozero marce dmeo coe properly Marce d ad d ave egave dagoal eleme ad oegave o-dagoal eleme Marce d d ad d are oegave Eample 37 Coder a po proce ree ype o cuomer Prore e amog dere ype o cuomer eac bac oever uc prore may cage rom bac o bac MMAP[K] ca be ued o model uc po procee For ace a MMAP[K] decrbe a po proce c ype 3 cuomer ave ger prore over ype cuomer e a ype cuomer pree; oere ype cuomer ave ger prore over ype 3 cuomer a bac geeral po procee mulple ype o arrval ad more complcaed arrval paer ca be modeled by ug MMAP[K] Te MMAP[K]/G[K]/ queue Te MMAP[K]/G[K]/ queue a gle erver queueg yem a MMAP[K] pu proce Te pu proce repreeed by marce ℵ Tere are K ype o cuomer Te ervce me o ype cuomer ave a commo drbuo uco F aplace Sele raorm ad mea ervce me / K e aume a all ervce me are depede o eac oer ad depede o e pu proce All cuomer o a gle queue ad are erved baed o a r-come-r-erved FCFS ba oever ervce prore may be aged o cuomer e ame bac Ta eac bac cuomer are erved a ey are equeced For eample or a bac o e ype 3 e r ype cuomer erved r e e ype cuomer e e ype 3 cuomer ad ally e oer ype cuomer Becaue o e lebly o MMAP[K] modelg e pu proce a varey o ervce prore bace ca be cluded ee e rac ey o e queueg yem a ρ K / 33 aumed a e rac ey ρ< o a e queueg yem ca reac eady ae 8

9 9 To ed eco e pree a eample o o a e coderao o e correlao or paer e arrval proce doe mae a derece e perormace aaly o queueg yem mulple ype o cuomer Eample 38 Coder a queueg yem a MMAP[] pu proce Te ervce me o e o ype o cuomer ave epoeal drbuo parameer ad repecvely e / ad / For e pu proce e coder o cae / b ; a > > 34 For e o pu procee e average arrval rae o ype ad ype cuomer are e ame Ug e reul obaed Seco 4 ad 5 e aplace Sele raorm o e ag me o ype cuomer eady ae or e o cae ca be obaed a b ; ] [ ] [ a y y y e 35 ere vecor y y y ad y are e vecor probable a e queueg yem empy a a arbrary me e e pu procee are gve by par a ad par b equao 34 repecvely eay o ee a y e y / / More deal abou y ca be oud Seco 4 Te derece beee epreo a ad b equao 35 o a e modelg o e pu proce ueul ad may be eceary o oba good appromao o perormace meaure 4 Te buy perod ad e vrual ag me Te buy perod o e MMAP[K]/G[K]/ queue oly relaed o e oal ervce me o eac bac Tu e dcuo ca be coed a e bac level o a e dvdual cuomer level e covee o coder dvdual bace a uper cuomer Te ervce me o a uper cuomer o e ype ℵ e ummao o e ervce me o all cuomer e bac Te drbuo uco ad e correpodg aplace Sele raorm o e ervce me o ℵ are gve by ad F F F F 4 For e buy perod o e MMAP[K]/G[K]/ queue may reul obaed E [8] ad Tae ad aegaa [33] ca be appled mmedaely mor cage For ace e

10 epoeal equao gve Teorem 6 E [8] or e o raorm o e leg o a buy perod ad e umber o dere ype o cuomer erved e buy perod ll old Ug ee equao epreo o e mome o e leg o a buy perod ad umber o cuomer erved a buy perod ca be derved Sce ee reul are ragorard geeralzao ad are o ued laer eco deal are o preeed For laer ue e drbuo o e leg o a arbrary buy perod regardle o e ype o e r arrval o e buy perod dcued Te reul are mlar o oe obaed Tae ad aegaa [33] ad proo are omed e ψ y be e probably a e leg o a buy perod y or le e ae o e uderlyg Marov proce J e e buy perod ed gve a e al ae ad e al orload Ψ y a m m mar eleme ψ y Ψ e aplace Sele raorm o Ψ y repec o y ee Q ℵ Ψ F d 4 Te ca be o a mar Q ae Tae ad aegaa [33] Q ℵ F d ep Q 43 e Q lm Q ca be proved a mar Q e emal geeraor o e uderlyg Marov proce a obaed by ecg e buy perod Mar Q ae Q F d ep Q 44 ℵ ealed erpreao o Q ca be oud Amue [] or Tae ad aegaa [33] A compuao meod o Q ca be oud Tae ad aegaa [33] Syem empe probable are alo mpora uco o e queueg yem o ere epecally e ey ll be eeded or e drbuo uco o e vrual ad acual ag me Tu ome bac reul abou dle probable are preeed e y be e probably a e queueg yem dle a a arbrary me ad e ae o e uderlyg Marov proce J m e y be a m-dmeo vecor compoe y Smlar o Seco 4 [33] ca be proved a y ae y Q 45 Sce Q a rreducble emal geeraor e oluo o y uque up o a coa Te coa all be deermed a ρ Teorem 4 uvely equao 45

11 old ce Q e emal geeraor o e uderlyg Marov proce durg e dle perod ad y e aoary drbuo up o a coa durg e dle perod Equao 44 or mar Q ad equao 45 or vecor y are epecally mpora compug e drbuo o vrual ad acual ag me Te vrual ag me e oal orload e yem a a arbrary me Apparely e vrual ag me deped oly o e ervce me o bace ad doe o requre ormao abou o dvdual cuomer a bac are erved Tu a relavely mple aaly ca be coduced o e vrual ag me e v be e aplace Sele raorm o e orload e queueg yem a a arbrary me e e ae o e uderlyg Marov proce J e v v v v m Teorem 4 e e queueg yem o ere ca reac eady ae a or > v y ℵ 46 ere deed equao 4 Proo Equao 46 ca be proved ug e reul obaed Seco 5 Tae ad aegaa [33] eal are omed Equao 46 gve a complee aer o e aplace Sele raorm o e drbuo o e vrual ag me ecep a e vecor y eed o be deermed o a y e le vara vecor up o a coa o mar Q Tereore y deermed compleely e y e o Teorem 4 e e queueg yem o ere ca reac eady ae y e ρ Proo T proo mlar o e oe gve or Teorem 4 E [8] eal are omed oe e reul obaed eco or e buy perod ad vrual ag me old o oly or e FCFS cae bu alo or may or coervg ervce dcple 5 Te acual ag me reul ad a ormal proo eco e FCFS ervce dcple mpoed Te acual ag me o varou ype o arrval ad cuomer are vegaed To mae eay or reader o ollo e reul abou e acual ag me o bace or e ag me o e r cuomer o be erved a bac are gve r

12 e b be e aplace Sele raorm o e acual ag me b o e r cuomer o be erved a arbrary ype bac e e ae o e uderlyg Marov proce J rg aer e arrval o e bac or m ad ℵ e b b b bm e b be e aplace Sele raorm o e acual ag me o e r cuomer o be erved a arbrary bac e e ae o e uderlyg Marov proce J rg aer e arrval o e bac e b b b bm Teorem 5 e e queueg yem o ere ca reac eady ae e acual ag me o a arbrary ype ℵ bac gve by b y 5 ℵ Te acual ag o a arbrary bac regardle o e ype o e bac gve by b y ℵ ℵ b 5 Proo By Teorem 4 e orload a a arbrary me gve by v e e uderlyg Marov proce J ae Accordg o reul Seco e probably a a ype bac arrve e e uderlyg Marov proce J ae rg aer e arrval gve a e uderlyg Marov proce a ae u beore e arrval / Te eay o ee a m b v / m 53 T lead o equao 5 Equao 5 ca be proved mlarly T complee e proo For e acual ag me o dvdual cuomer ueul o ee a e ag me o a cuomer e ag me o bac plu e ervce me o e cuomer e ame bac bu equeced aead o For ace coder e ag me o a ype cuomer o arrve a ype bac ad e cuomer o e bac eoe by ag me aume a Te ere v v v 54 b v e ervce me o e cuomer e bac T equao combg Teorem 5 lead o e ollog elemeary reul

13 3 Teorem 5 e be e aplace Sele raorm o e acual ag me o a arbrary ype cuomer o come rom e poo o a ype bac e e ae o e uderlyg Marov proce rg aer e cuomer or bac arrved e m e e queueg yem o ere ca reac eady ae e ave y ℵ 55 Proo oe a e T complee e proo Teorem 5 gve e acual ag me drbuo o a parcular ype o cuomer rom a parcular poo o a parcular bac T reul ueul dg e ag me drbuo or may pecal group o cuomer For ace le be e aplace Sele raorm o e acual ag me o a arbrary ype cuomer c come rom a ype ℵ bac; e aplace Sele raorm o e acual ag me o a arbrary ype cuomer; ad e aplace Sele raorm o e acual ag me o a arbrary cuomer Teorem 53 e e queueg yem o ere ca reac eady ae e ave ; ; ℵ ℵ ℵ ℵ ℵ y y y 56 Proo Te aplace Sele raorm equao 56 ca be obaed by codog o e ype o e arbrary cuomer uder coderao Accordg o Propery e probably a a ype cuomer come rom e poo o a ype bac gve a a ype cuomer / Te 57 c lead o e r epreo equao 55 Smlarly e ollog relaop old

14 ℵ K ℵ ; 58 Tee o relaop lead o e ecod ad rd epreo equao 56 T complee e proo 6 A ormal proo o e acual ag me drbuo A a o e ey provg all e ormula obaed Seco 5 o prove Teorem 5 Te obecve o eco o provde a rgorou proo or Teorem 5 e be e probably a e queueg yem empy aer e deparure o a arbrary cuomer regardle o ype e e ae o e uderlyg Marov proce J rg aer e deparure m eoe by e m-dmeo vecor compoe Fr e o a e e queueg yem o ere ca reac eady ae e acual ag me o a arbrary ype ℵ bac gve by b 6 ℵ e all e eabl a relaop beee e vecor ad y o complee e proo o Teorem 5 Suppoe a e queueg yem o ere eady ae e be e ag me o a arbrary ype ℵ bac Beee ad e e ype bace ere mg be ome oer bace deoed by e ere are oer bace o arrved beee e o coecuve ype bace oe a a radom varable Te e ag me o e e ype bac gve by v U v U v U 6 ere v v e e oal ervce me o all cuomer bac ad U e erarrval me e e e arrval o ype ; ad ma oce a v U ad v U are dere or eve e To derve e aoary drbuo uco o rom equao 6 coder e ollog mple cae r e 4

15 v U 63 eoe by e drbuo o e ag me o a ype bac e e ae o e uderlyg Marov proce J rg aer e ype bac arrved eoe by e drbuo o e ag me o a ype bac ollog a ype bac e e ae o e uderle Marov proce J rg aer e bac arrved Te codog o e erarrval me U ad e ervce me o e bac yeld m u ep F du d 64 Equao 64 ca be re o vecor orm a ollo ere - u ep u du F du d u ep < < F u d u Te e aplace Sele raorm o u u e du ep d e u F du 67 Eed e uco o < o oba uco ˆ Te or < ˆ y u u ep u ep y ep F F du d y u F du d du dy 68 5

16 6 ep ep ep ep ep ep ep C du F u y dy du F u dy y y For e eeded uco ˆ << aplace Sele raorm gve a ˆ du u d e - 69 O e oer ad e aplace Sele raorm o e eeded uco ca be oud by ep ˆ ] [ d e d e C C C 6 Combg equao 69 ad 6 yeld C 6 Coder e bac < aer e r ype bac ad beore e e ype bac Te ag me e ype bac gve a ollo U v U v U v U v 6 ducvely e aplace Sele raorm o e drbuo uco o gve a C C 63

17 7 ere ep ep ; ep du F u y dy du F u C C 64 For e acual ag me o bac codog o e umber o bace beee o coecuve bace ug equao 6 ad 64 yeld Par - Par C 65 Par ad Par o equao 65 are evaluaed e Par evaluaed a 66 ℵ ℵ Par evaluaed a oce a

18 8 C C 67 C C Equao 66 ad 67 lead o C 68 T lead o C ℵ 69 Te e ep o prove a C / uvely Te vecor C e probably a e yem become dle aer e compleo o e ervce o all cuomer e bac o e ype aer e r bu beore e e ype bac Te e vecor C e probably a e queueg yem become dle a e compleo epoc o bace beee o coecuve ype bace Sce e probably a e yem become dle a e compleo epoc o a arbrary bac a C C 6 Equao 69 ad 6 lead o equao 6 A rgorou proo o equao 6 gve a ollo Suppoe a e queueg yem eady ae Tere a ype bac a arrved a me zero bac zero e be e eve a e cuomer leave a empy yem ad e uderlyg Marov proce J ae rg aer e deparure

19 9 e η be e equeal umber o e r cuomer e ype bac e e r cuomer o e ype bac e l cuomer l η o arrve o e queueg yem eay o ee a η η Te e ave ξ ma η e e umber o ype bace a arrved e e cuomer arrve c a roduced Seco e a ype bac arrve a zero called bac zero e ave η Te accordg o reeal eory ee Clar [5] e e lm lm l l m l m ξ ξ η η ξ 6 Accordg o equao ξ / coverge o / Te oer par coverge o C c proved a ollo Coder e embedded ocac proce a e r deparure epoc aer ype bace arrve clear a e embedded ocac proce a regeerave proce ee Ro [8] Tu lm E η η η ξ ξ m l l m l 6 e ζ be e umber o bace a arrved beee o coecuve ype bace ζ e be e eve a e bac uper cuomer leave a empy yem bed ad e uderlyg Marov proce J ae rg aer e deparure Te ˆ ˆ E E ς η m m 63 ce oly e la cuomer a bac e poo o leave a empy yem bed Te > > ˆ ˆ ˆ ˆ m m E E ς ς ς 64 ] ˆ ˆ [ C C E ς ς > > m

20 c lead o equao 6 oce a order o complee e proo o Teorem 5 e eed o d e relaop beee y ad ac e e queueg yem o ere ca reac eady ae e ollog relaop beee y ad old y ad e ρ 65 To prove equao 65 mlar o Teorem 4 e r o a - e ρ By equao 45 vecor y ae equao y Q Coder e embedded Marov ca a e la deparure o buy perod Te oe ep rao mar o embedded Marov ca gve by Ψ F d Q Q ℵ 66 eay o ee a mu ay equao [- - Q ] c mple a - - Q Tereore - c y Sce y e ρ ad - e ρ e coa c / c lead o equao 65 Combg equao 6 ad 65 e oba Teorem 5 T complee e proo T ormal proo o Teorem 5 log bu become aural e o ey ac are oberved Te r oe equao 6 Oce e relaop equao 6 obaed a proo ollog dley approac ee dley [3] c e ecod ey become poble All oer reul gve Seco 5 ca be proved a mlar ay ac e proo o equao 5 mpler eal o oe proo are omed 7 Summary ad uure reearc Te corbuo o paper o-old Fr o o o ue MMAP[K] o model a varey o po procee ome pecal arrval paer uc a cyclc med bac ad prore Secod pree a dealed aaly o e acual ag me procee o dvdual ype o cuomer queueg yem a MMAP[K] pu proce Combg e o par paper o a queueg yem a complcaed pu proce ad mulple ype o cuomer are ll aalycally racable e e pu proce modeled appropraely

21 ceraly mpora o develop algorm or mome o e buy perod vrual ag me ad acual ag me Sce mome ca be obaed by roue calculao ee Abae ad [] E [8] ucao [4] ad eu [8 ] deal are o omed T paper uded e queueg yem a FCFS ervce dcple ereg o vegae queueg yem prore aged o dere ype o cuomer ee Taag [3 3] Toug ome o e meodologe ued paper may o be org or oer cae e ormalm adoped paper ll ceraly be ueul a reearc Some reul alog dreco ca be oud Tae ad aegaa [33] T paper doe o coder e queue leg due o ome eccal dculy For a pecal cae ere eac bac a oly oe cuomer ome reul abou e queue leg ave obaed E [8] E [9] ad E ad Ala [] e eady ae queue leg drbuo obaed e e ervce me ave P-drbuo ad e ervce dcple la-come-r-erved CFS Te udy o e queue leg o e queueg yem o ere le a a ope problem or uure reearc Acoledgeme Te auor ould le o a a reeree or /er valuable comme ad uggeo a mproved e paper T reearc uppored by a SERC reearc gra Reerece [] Abae J ad Te ourer-ere meod or verg raorm o probably drbuo Queueg Syem [] Amue S adder eg ad e Marov-modulaed M/G/ queue Socac Procee ad er Applcao [3] Amue S ad Koole G Mared po procee a lm o Marova arrval ream J Appl Prob [4] Ce T ad Reder M Cyclc Marov modulaed Poo procee rac caracerzao Socac Model [5] Clar E Marov reeal eory Adv Appl Prob [6] Coe J Te Sgle Server Queue or-ollad Amerdam 98 [7] Gamacer FR Te Teory o Marce e Yor Celea 959 [8] E Q-M Queue mared cuomer Adv Appl Prob [9] E Q-M Qua-br-ad-dea Marov procee a ree rucure ad e MMAP[K]/P[K]/ queue Europea Joural o Operaoal Reearc [] E Q-Mg ad AS Ala Te MMAP[K]/P[K]/ queue a la-come-r-erved preempve ervce dcple Queueg yem Vol [] E Q-M ad eu MF Marov arrval procee mared rao Socac Procee ad er Applcao [] Kleroc Queueg Syem Vol Teory lley e Yor 975 [3] dley V Te eory o queue a gle erver Proc Camb Pl Soc [4] ucao M e reul o e gle erver queue a bac Marova arrval proce Socac Model

22 [5] ucao M Te MMAP/G/ queue A uoral Model ad Tecque or Perormace Evaluao o Compuer ad Commucao Syem Edor oaello ad R elo Sprger Verlag [6] ucao M Coudury G ad Te rae BMAP/G/ queue Socac Model [7] ucao M Meer-eller KS ad eu MF A gle erver queue erver vacao ad a cla o o-reeal arrval procee Adv Appl Prob [8] eu MF A verale Marova po proce J Appl Prob [9] eu MF Mar-Geomerc Soluo Socac Model A algormc Approac Te Jo op Uvery Pre Balmore 98 [] eu MF Geeralzao o e Pollacze-Kc egral equao e eory o queue Adv Appl Prob [] eu MF Te udameal perod o e queue Marov-modulaed arrval Probably Sac ad Maemac Paper oor o Samuel Karl pp 87- Academc Pre e Yor 989 [] eu MF Srucured Socac Marce o M/G/ Type ad Ter Applcao Marcel eer e Yor 989 [3] eu MF u ad Surya ocal poocao o e Marova arrval proce Socac Model [4] Paceco Aoo Marov-Addve Procee Arg Sorage Model or Commucao Syem Teccal Repor o 96 Scool o Operao Reearc ad dural Egeerg Corell Uvery 994 [5] Prabu U Marov reeal ad Marov-addve procee a reve ad ome e reul B Co ad J Ym edor Proceedg o KAST Maemc orop Volume 6 page Korea Advaced ue o Scece ad Tecology 99 [6] Ramaam V Te /G/ queue ad dealed aaly Adv Appl Prob 98-6 [7] Ramaam V From e mar-geomerc o e mar-epoeal Queueg yem [8] Ro SM Socac Procee Jo lley & So e Yor 983 [9] Saord A ad Fcer Te erdeparure me drbuo or eac cla e M / G / queue Queueg Syem [3] Taag Queueg Aaly A Foudao o Perormace Evaluao Vol Vacao ad Prory Syem Par Elever Scece Publer BV Amerdam 99 [3] Taag Queueg aaly o pollg model progre Froer Queueg Eded by alalo J [3] Tae T A couou vero o mar-aalyc meod e p-ree o e le propery Socac Model [33] Tae T ad aegaa T Te orload e MAP/G/ queue ae-depede ervce applcao o a queue preempve reume prory Socac Model [34] Tae T Segupa B ad Yeug R A geeralzao o e mar M/G/ paradgm or Marov ca a ree rucure Socac Mdoel

23 [35] ol R Poo arrval ee me average Oper Re [36] Yeug R ad Segupa B Mar produc-orm oluo or Marov ca a ree rucure Adv Appl Prob