MANY-SERVER QUEUES WITH CUSTOMER ABANDONMENT: A SURVEY OF DIFFUSION AND FLUID APPROXIMATIONS

Transcription

1 J Sys Sci Sys Eg (Mar 212) 21(1): 1-36 DOI: 1.17/s y ISSN: (Paper) (Olie) CN /N MANY-SERVER QUEUES WITH CUSTOMER ABANDONMENT: A SURVEY OF DIFFUSION AND FLUID APPROXIMATIONS J. G. DAI 1 Shuagchi HE 2 1 H. Milo Sewar School of Idusrial ad Sysems Egieerig, Georgia Isiue of Techology, Alaa, GA 3332, USA. dai@gaech.edu() 2 Deparme of Idusrial ad Sysems Egieerig, Naioal Uiversiy of Sigapore, Sigapore isehes@us.edu.sg Absrac The performace of a call ceer is sesiive o cusomer abadome. I his survey paper, we focus o G/ GI / GI parallel-server queues ha serve as a buildig block o model call ceer operaios. Such a queue has a geeral arrival process (he G ), idepede ad ideically disribued (iid) service imes wih a geeral disribuio (he firs GI ), ad iid paiece imes wih a geeral disribuio (he GI ). Followig he square-roo safey saffig rule, his queue ca be operaed i he qualiy- ad efficiecy-drive (QED) regime, which is characerized by large cusomer volume, he waiig imes beig a fracio of he service imes, oly a small fracio of cusomers abadoig he sysem, ad high server uilizaio. Operaioal efficiecy is he ceral arge i a sysem whose saffig coss domiae oher expeses. If a moderae fracio of cusomer abadome is allowed, such a sysem should be operaed i a overloaded regime kow as he efficiecy-drive (ED) regime. We survey rece resuls o he may-server queues ha are operaed i he QED ad ED regimes. These resuls iclude he performace isesiiviy o paiece ime disribuios ad diffusio ad fluid approximae models as pracical ools for performace aalysis. Keywords: Heavy raffic, square-roo safey saffig, qualiy- ad efficiecy-drive regime, efficiecy-drive regime, piecewise OU process 1. Iroducio Cusomer call ceers have become a impora par of he service ecoomy i a moder sociey. To ake advaage of he ecoomy of scale, call ceers wih hudreds of ages are ubiquious i may idusries. These sysems face a large amou of daily raffic ha is irisically sochasic ad has emporal variaios. I a call ceer, a cusomer waiig for service may hag up he phoe before beig served. This is called cusomer abadome. Such a pheomeo is commo because cusomers usually have limied paiece. Cusomer expecaio demads ha a proper Research suppored i par by NSF gras CMMI ad CMMI Sysems Egieerig Sociey of Chia & Spriger-Verlag Berli Heidelberg 212

2 2 J Sys Sci Sys Eg saffig level be maiaied i he call ceer so ha mos cusomers are served wihou waiig for a log ime ad oly a small fracio of cusomers abado he sysem. As poied by Gare e al. (22), cusomer abadome is a crucial facor for call ceer operaios. I may sigificaly impac he sysem performace ad mus be modeled explicily i order for a operaioal model o be releva for decisio makig. I his paper, we focus o a mahemaical model ha is deoed by a G/ GI / GI queue. I his queue, we model cusomer abadome by assigig each cusomer a paiece ime. Whe a cusomer s waiig ime for service exceeds his paiece ime, he abados he sysem wihou service. I he G/ GI / GI oaio, he G refers o a geeral arrival process, he firs GI refers o idepede ad ideically disribued (iid) service imes wih a geeral disribuio, is he umber of ideical servers, ad GI refers o iid paiece imes wih a geeral disribuio. As we do o assume iid ierarrival imes, he symbol for he arrival process is G, o GI. We call a G/ GI / GI queue wih a large umber of parallel servers a may-server queue. Such a queue serves as a buildig block o model large-scale call ceers. For call ceer operaios, i is reasoable o assume ha he paiece imes are iid, as he queue is usually ivisible o waiig cusomers. As argued by Halfi & Whi (1981), he performace of may-server queues is qualiaively differe from ha of sigle-server queues or queues wih a small umber of servers. Due o he sochasic variabiliy i ier arrival ad service imes, he mea cusomer waiig ime i a sigle-server queue goes o ifiiy as he server approaches 1% uilizaio. A maager has o make a paiful choice bewee qualiy of service (shor waiig imes) ad operaioal efficiecy (high server uilizaio) i a sigle-server service sysem. I coras, a may-server sysem ca be operaed i he qualiy- ad efficiecy-drive (QED) regime ha is characerized by large cusomer volume, he mea waiig ime beig a fracio of he mea service ime, oly a small fracio of cusomers abadoig he queue, ad high server uilizaio. This regime is also called he raioalized regime i Gare e al. (22) because i mos cases, a maager should operae his service sysem i such a regime. To achieve boh qualiy ad efficiecy, he maager ca exploi he poolig effec by operaig a large umber of servers i parallel. More specifically, he maager ca apply he square-roo safey saffig rule o drive he sysem o he QED regime. This rule is a impora saffig priciple ha is boh heoreically jusified ad widely praciced. I cerai service sysems, he saffig coss domiae he coss of cusomer delay ad abadome. I such sysems, a more reasoable operaioal regime is he efficiecy-drive (ED) regime. I he ED regime, he service capaciy is se below he cusomer arrival rae by a moderae fracio. I such a may-server sysem, he mea waiig ime is comparable o he mea service ime, a moderae fracio of cusomers abado he sysem, ad all servers are almos always busy. These wo operaioal regimes are elaboraed i Secio 2. I is empirically repored ha i call ceers, boh he service ime disribuios ad he paiece ime disribuios are far from

3 J Sys Sci Sys Eg 3 expoeial; see, e.g., Brow e al. (25). Therefore, oe mus use geeral disribuios o model service ad paiece imes. Rece papers, such as Zely & Madelbaum (25), Dai & He (21), ad Madelbaum & Momčilović (212), have demosraed ha he performace of a may-server queue i he QED regime is isesiive o he paiece ime disribuio as log as he paiece ime desiy a he origi is fixed. This pheomeo is discussed i Secio 3. Whe he service ad paiece ime disribuios are geeral, excep by compuer simulaio, o aalyical or umerical mehods are available o evaluae he performace of such a queue. We survey approximae models for may-server queues i Secios 4 ad 5. I Secio 4, we sudy diffusio approximaios for may-server queues i he QED regime. I hese diffusio models, he service ime disribuio is modeled by a phase-ype disribuio ad he paiece ime disribuio is assumed o be geeral. We demosrae ha he diffusio models are accurae i predicig he sysem performace i he QED regime. For a may-server queue i he ED regime, fluid approximaios have bee show o be useful. I Secio 5, we survey a fluid model proposed by Whi (26). This fluid model is show o be adequae i esimaig he performace of a may-server queue i he ED regime. 2. Operaioal Regimes i he Presece of Cusomer Abadome As argued by Whi (26), mos service sysems ca be classified io wo ypes: reveue-geeraig sysems ad service-orieed sysems. The former ype aims o maiai high qualiy of service while he laer ype focuses more o operaioal efficiecy. As he scale of service sysems goes large, he cogesio due o variabiliy i cusomer service demads ca be offse by poolig service faciliies. If he cusomer arrival rae ad he service capaciy are well balaced, i is possible o achieve boh qualiy ad efficiecy i a service sysem wih may servers. I his case, he service sysem is said o be workig i he qualiy- ad efficiecy-drive (QED) regime. We demosrae i Secio 2.1 ha oe ca follow he square-roo safey saffig rule o operae a service sysem i he QED regime. Cusomer abadome may have a sigifica impac o he performace of a service sysem. For a service sysem whose cusomers are huma beigs, oe mus cosider he abadome pheomeo i decisio makig ad operaios. I Secio 2.2, we demosrae ha i he presece of cusomer abadome, he square-roo safey saffig rule ca sill lead he sysem o he QED regime while keepig a small abadome fracio. To achieve he mos operaioal efficiecy i a service-orieed sysem, a cerai perceage (say, 15% o 2%) of cusomer abadome is usually allowed. I his case, he service sysem ca be operaed i a overloaded regime while sill maiaiig cerai service levels. The key isigh here is ha cusomer abadome could compesae for a sligh excess i he arrival rae over he service capaciy. Such a service sysem is highly efficie because all servers are busy almos all he ime. This overloaded regime for a may-server sysem is called he efficiecydrive (ED) regime. We iroduce he ED regime i Secio 2.3. Usually, he arrival rae of

4 4 J Sys Sci Sys Eg a service sysem is ime-varyig. I is worh meioig ha despie he bes effors, here would be cerai ime periods i which he service sysem has o be operaed i he overloaded regime. These wo regimes, QED ad ED, were coied by Madelbaum for may-server queues; see Gas e al. (23) for more deails. 2.1 The QED Regime ad he Square-roo Safey Saffig Rule I a service sysem wih a sigle or a small umber of servers, he maager has o srike a compromise bewee service qualiy ad operaioal efficiecy. I coras, i is possible o achieve boh of hem i a service sysem wih may parallel servers. We use a umerical example o demosrae his disicio. To evaluae he qualiy of service i a service sysem, he fracio of cusomers who have o wai before receivig service (kow as he delay probabiliy) ad he mea cusomer waiig ime are wo impora performace measures. These wo measures should be maiaied uder cerai levels o mee cusomer expecaios. Le us cosider a M / M / queue ha has a Poisso arrival process wih rae, expoeially disribued service imes wih mea 1, ad ideical parallel servers. The raffic iesiy of his queue is defied by. (1) We assume ha 1, which is also equal o he average uilizaio per server. The delay probabiliy is give by he well-kow Erlag-C formula (see, e.g., Gross & Harris (1985)), P w 1 k ( ) ( ) ( ) ( 1 )! k k!! The mea waiig ime is 1. (2) Pw W. (3) (1 ) ad by Lile s law, he mea queue legh is Q W. (4) We evaluae he waiig ime facor f w, which is defied as he raio of he mea waiig ime o he mea service ime, i he followig umerical example. By his defiiio, he waiig ime facor is Pw fw W. (5) (1 ) for he M / M / queue. I Figure 1, we plo he delay probabiliy ad he waiig ime facor as he umber of servers icreases from 1. All hese queues have he same raffic iesiy.95. The figure shows ha he delay probabiliy decreases gradually while he waiig ime facor decreases rapidly. For example, whe 18, he delay probabiliy is 76.7% ad he waiig ime facor is.85. Thus, he average uilizaio per server is 95%, 76.7% of cusomers are delayed before receivig service, ad he mea waiig ime is less ha he mea service ime. This level of qualiy of service is cosidered good for may service sysems. If oe furher icreases he umber of servers o 1 ad he average uilizaio per server is kep a 95%, he delay probabiliy decreases o 5.7% ad he waiig ime facor is.11. I his case, early half of cusomers are served wihou delay ad he mea waiig ime is oly aroud 1% of he mea service ime. This level of service is highly aracive

5 J Sys Sci Sys Eg 5 despie he fac ha he servers are 95% uilized. Such a sysem is operaed i he QED regime. I his regime, he sysem has a large umber of parallel servers, he arrival rae is high, ad he arrival rae ad he service capaciy are approximaely equal so ha he server uilizaio is close o 1. Eve hough he average server uilizaio is close o 1, oly a fracio of cusomers eed o wai i he queue wih may parallel servers. This pheomeo is i sharp coras o he observaio ha almos all cusomers have o wai i a sigle-server queue. To illusrae how he waiig ime facor chages wih he server uilizaio, we also plo he waiig ime facor curves for a sigle-server queue ad for a queue wih 18 servers i Figure 2. Compared wih he curve for he sigle-server queue, he waiig ime facor for he muli-server queue icreases much more slowly as he server uilizaio approaches 1. Figure 1 Delay probabiliy ad waiig ime facor vs umber of servers, for M / M / queues wih.95 Figure 2 Waiig ime facor vs server uilizaio, for a M / M /1 queue ad for a M / M /18 queue

6 6 J Sys Sci Sys Eg I geeral, he variabiliy i cusomer arrival ad service processes coribues o he sysem cogesio, degradig he qualiy of service, paricularly whe he sysem is heavily loaded. Figure2, however, illusraes ha he ifluece of he variabiliy ca be offse by poolig service faciliies. The poolig priciple has bee widely used i resource maageme uder uceraiy. The QED regime ca be achieved by applyig he square-roo safey saffig rule. Le R / be he offered load of he queue. Oe expecs ha a appropriae saffig level (i.e., he umber of servers) should be R, where is he excess service capaciy agais he sysem s sochasic variabiliy. To keep he server uilizaio high, should be much smaller ha R. The square-roo safey saffig rule recommeds a amou of excess service capaciy of R for some. Thus, followig he square-roo safey saffig rule, he saffig level is R R, (6) whe he offered load R is high. Of course, he value of give by (6) should be rouded o a ieger. I urs ou ha wih a fixed, as he offered load icreases, he correspodig saffig level i (6) sabilizes he delay probabiliy ad makes he waiig ime facor o 1/2 he order of O ( ). I was proved by Halfi & Whi (1981) ha whe he offered load R is high i he M / M / seig, 1 Pw ( )/ ( ) 1. (7) Where ad are he probabiliy desiy ad he cumulaive disribuio fucio, respecively, of he sadard ormal disribuio. Formulas (6) ad (7) ca be used for performace aalysis wih a give saffig level, or o deermie he saffig level ha achieves a give delay probabiliy. For a give saffig level ad a give uilizaio level 1, we ca se (1 ) (8) ad use he righ side of (7) o approximae he delay probabiliy P w. Wih he delay probabiliy, he waiig ime facor f w i (5) becomes Pw fw. (9) For example, whe 1 ad.95, oe has.5, he righ side of (7) predics P w o be.55, compared o he exac value.57 from (2). The waiig ime facor compued hrough (5) is.11 based o boh he exac ad approximae values of P w. The secod ad more impora usage of (6) is ha i leads o he followig saff provisio i he M / M / seig. Suppose ha he delay probabiliy is required o be less ha a arge value 1. Oe eeds o se he saffig level so ha he delay probabiliy is approximaely. For his, oe firs solves for from he followig equaio 1, ( )/ ( ) 1 ad he se he saffig level usig (6) for a give offered load R. 2.2 Modelig Cusomer Abadome Cusomer abadome is prese i mos service sysems ha serve huma beigs. For a service sysem wih sigifica cusomer abadome, ay queueig model ha igores

7 J Sys Sci Sys Eg 7 he abadome pheomeo is likely irreleva o operaioal decisios. To demosrae he sigifica ifluece of cusomer abadome o he sysem performace, le us cosider a M / M / M queue. I has ideical servers, he arrival process is Poisso wih rae, ad he service imes are iid followig a expoeial disribuio wih mea 1/. I his queue, each cusomer has a paiece ime ad he paiece imes are iid followig a expoeial disribuio wih mea 1/. This model is also kow as he Erlag-A model. The raffic iesiy of a queue wih cusomer abadome is sill give by (1), bu i should o loger be udersood as he average server uilizaio sice he cusomers who abado he sysem do o receive ay service. The M / M / M queue is also a racable model whose performace measures have explici formulas. For example, he delay probabiliy is 1 1 (, ) E1, Pw, (1) ( (, ) 1) E 1, where xexp( y) y 1 (, ) x x y exp( )d y x for x ad y, ad k 1 ( ) ( ) E1,! k k! deoes he blockig probabiliy i he M / M / / (Erlag-B) model. I addiio, he fracio of cusomers who abado he sysem is 1 1 Pa 1 P 1 1 w, (, ) (11) he mea waiig ime (amog all cusomers icludig hose who have abadoed he sysem) is P W a, (12) ad he mea queue legh is Q W (13) by Lile s law. Usig he formula of he abadome fracio P a, he average uilizaio per server ca be compued by (1 Pa ) (1 Pa ). (14) See Madelbaum & Zely (27) for he complee deails o he Erlag-A model. We assume ha he M / M / M queue has 5 servers, he arrival rae is 55 cusomers per miue, he mea service ime is 1 miue, ad he mea paiece ime is 2 miues. Several performace measures of his queue, obaied via formulas (1) (14), are lised i Table 1. I he same able, we also lis he performace measures for a modified queue. The modified queue is a M / M / queue wih he same mea service ime, he same umber of servers, ad he same hroughpu as he origial queue, bu i has o cusomer abadome. The arrival rae of he modified queue is equal o he hroughpu of he origial queue, i.e., 55 (1.12) The correspodig performace measures ca be obaied by formulas (2) (4) ad he fac ha he average server uilizaio is equal o he raffic iesiy. Table 1 shows ha boh he mea waiig ime ad he mea queue legh i he origial queue are much shorer ha he correspodig quaiies i he modified queue. I oher words, wih he same service capaciy ad hroughpu, some key performace measures i a queue wih abadome are much beer ha i a queue wihou abadome. The performace

8 8 J Sys Sci Sys Eg measures of he origial queue idicae ha he sysem is workig i he QED regime, eve hough i is slighly overloaded (i.e., 1). To mee a cerai service requireme wihou cosiderig cusomer abadome, oe eds o overesimae he saffig level. Of course, cusomer abadome ca be cosly. Oe eeds o fid a rade-off bewee cusomer abadome ad saffig cos usig a correc model. The beer performace o he waiig imes i a queue wih cusomer abadome ca be explaied iuiively as follows. I he origial queue wih abadome, whe he sysem is i a cogesio period, he cusomers who experiece log waiig abado he sysem. Their waiig imes are capped by heir paiece imes. I he modified queue wihou abadome, hese cusomers will experiece exremely log delays, which degrades he overall waiig ime saisics. Wih cusomer abadome, a service sysem ca reach a seady sae eve if he cusomer arrival rae is larger ha is service capaciy, i.e., 1. As more ad more cusomers accumulae i he buffer, he abadome rae keeps icreasig uil arrivals ad deparures (icludig boh service compleios ad abadomes) reach a equilibrium. The square-roo safey saffig rule (6) also applies o a service sysem wih high offered load i he presece of cusomer abadome. I was proved by Gare e al. (22) ha whe he saffig level of he M / M / M queue follows (6), he delay probabiliy ca be approximaed by where P w 1 1 h 1, 1 h( ) ( x) hx ( ) 1 ( x) (15) is he hazard rae fucio of he sadard ormal disribuio. Therefore, o mee a arge delay probabiliy 1, oe ca se he saffig level usig (6), wih he value of deermied by solvig 1 1 h 1. 1 h( ) (16) Table 1 Compariso bewee queues wih ad wihou cusomer abadome M / M /5 M M / M /5 Delay probabiliy (%) Abadome fracio (%) 1.2 N/A Mea waiig ime (i secods) Mea queue legh Server uilizaio (%)

9 J Sys Sci Sys Eg 9 Whe solvig (16) for, i is possible o have a egaive soluio ha resuls i a saffig level below he offered load. Because he small fracio of cusomer abadome reduces he excess service demads, he service sysem ca sill achieve a saisfacory qualiy of service whe he sysem is slighly overloaded (as idicaed by he performace of he M / M / M queue i Table 1). Zely & Madelbaum (25) proved ha whe he saffig level follows (6) wih a give (which is o ecessarily posiive), he fracio of cusomers who abado he M / M / M queue is approximaely P a h( ) 1 1 h( ) 1. ` h( ) (17) 1/2 We ca see ha P a is o he order of O ( ) as. I follows from (12) ha he mea waiig ime is o he same order, ad i follows from (14) ha he average server uilizaio is close o oe. Therefore, i he presece of cusomer abadome, he square-roo safey saffig rule sill leads he sysem o he QED regime ad yields high server uilizaio, shor waiig imes, ad a very small abadome fracio. Diffusio models have bee demosraed o be accurae i esimaig he performace of may-server queues i he QED regime. We will survey hese diffusio models i Secio The ED Regime I cerai service sysems, he saffig coss domiae he expeses of cusomer delay ad abadome. For hese sysems, he raioal operaioal regime is he ED regime ha emphasizes server uilizaio over qualiy of service. I his regime, he arrival rae exceeds he service capaciy by a moderae fracio (e.g., 2%). More precisely, he ED regime requires ha he raffic iesiy be greaer ha 1 ad ha he order of 1 be O(1) as. Sice he fracio of cusomers who cao be served is a leas 1, he fracio of cusomer abadome i he ED regime mus also be o he order of O (1) as. Noe ha if he square-roo safey saffig rule (6) is applied wih, he resulig service capaciy is also below he arrival rae. I his case, however, boh 1 ad he fracio of cusomer 1/2 abadome are o he order of O ( ). The sysem is i he QED regime, o i he ED regime. Because he sysem is overloaded, almos all cusomers are delayed i he buffer ad all servers are busy early 1% of he ime i he ED regime. Alhough i migh be coueriuiive, a service sysem operaed i he ED regime ca sill resul i reasoable performace as measured by he mea waiig ime ad he fracio of cusomer abadome. This is because he los service demads of he abadoed cusomers compesae for he excess i he arrival rae over he service capaciy. A fluid model proposed by Whi (26) has bee show o be useful i esimaig he performace of a may-server queue i he ED regime. We will survey his model ad discuss he performace of queues i he ED regime i Secio 5. A fluid model was sudied by Bassamboo & Radhawa (21) o solve he saffig problem

10 1 J Sys Sci Sys Eg of a M / M / GI queue. The goal is o balace he saffig coss ad he coss from cusomer delay ad abadome. Sice he exac opimizaio is o possible, hey employed he fluid model o approximae he queue. They proved ha if he paiece ime disribuio has a o-decreasig hazard rae, i is asympoically opimal for he sysem o operae i he QED regime wih he service capaciy approximaely equal o he arrival rae. However, if he paiece ime disribuio has a decreasig hazard rae, he operaio coss are reduced whe he sysem is overloaded. Hece, he opimized saffig level drives he queue o he ED regime. I he same paper, he auhors also proved ha i he seady sae, he accuracy gaps of he fluid approximaios for he mea queue legh ad he rae of cusomer abadome do o icrease wih he arrival rae. This implies ha he fluid approximaios could be paricularly accurae whe he uderlyig sysem is operaed i he ED regime. 3. Performace Isesiiviy o Paiece Time Disribuios i he QED Regime As we have demosraed i he previous secios, service sysems operaed i he QED regime are characerized by shor cusomer waiig imes. For a M / M / M queue i he QED regime wih (1 ) beig fixed, i ca be see from (12) ad (17) ha he mea waiig ime decreases o zero a rae 1/2 as he saffig level goes o ifiiy. If a service sysem has hudreds of parallel servers ad he service imes are ypically several miues, he i he QED regime, he waiig imes should be o he order of secods. The above observaio implies ha whe is large, he paiece ime disribuio, ouside a small eighborhood of zero, has lile ifluece o he sysem dyamics. Such a resul ca be cofirmed by he umerical example below. Cosider a M / M / GI queue. Le F be he cumulaive disribuio fucio of he paiece imes ha saisfies x 1 F( ) ad lim x F( x), (18) where is he desiy of F a he origi. I paricular, is ideical o he abadome rae whe he paiece ime disribuio is expoeial. If he waiig imes are shor, he abadome process should deped o he paiece ime disribuio mosly hrough is desiy a he origi. Suppose ha he queue has 1 servers, he Poisso arrival process has rae 15, ad he service imes are expoeially disribued wih mea 1. This sysem is slighly overloaded bu sill i he QED regime. A small fracio of raffic, a leas ( 1) / 4.8% of he arrivals, has o abado he sysem. We cosider hree paiece ime disribuios wih he same desiy a he origi: a expoeial disribuio (Exp) wih rae, a uiform disribuio (Uiform) o he ierval [,1/ ], ad a wo-phase hyperexpoeial disribuio ( H 2). A wo-phase hyperexpoeial disribuio ca be deermied by is iiial disribuio p ( p1, p2) wih p1 p2 1 ad is rae vecor ( 1, 2). For such a hyperexpoeially disribued radom variable, wih probabiliy p 1 i is expoeially disribued wih mea 1/ 1, ad wih probabiliy p2 i is expoeially disribued wih mea 1/ 2. I our example, he hyperexpoeial paiece ime disribuio has

11 J Sys Sci Sys Eg 11 p (.21,.79) ad (.3,79 / 3). Thus, 21% of cusomers have log paiece imes wih mea 1 / (3 ) ad79% of cusomers have shor paiece imes wih mea 3/ (79 ). Equivalely, he desiy fucio of he hyperexpoeial paiece ime disribuio is give by f ( x).21exp(.3 x).79exp( 79 x/ 3), x. (19) The squared coefficie of variaio of his disribuio is All hree disribuios have desiy a he origi. The exac formulas for several performace measures of he M / M / GI model are summarized i Secio 9 of Zely & Madelbaum (25). We follow hese formulas o obai he abadome fracio ad he mea queue legh. Table 2 displays he resuls for differe values ad differe paiece ime disribuios. For each row wih a fixed, he performace is very close for differe paiece ime disribuios. (The Diffusio colum i Table 2 will be explaied i Secio 4.2.) This example idicaes ha i he QED regime, he sysem performace is geerally ivaria wih he paiece ime disribuio as log as is desiy a he origi is fixed ad posiive. This ivariace also suggess ha o obai performace measures for a may-server queue wih a geeral paiece ime disribuio, i is geerally accurae o replace he paiece ime disribuio by a expoeial disribuio wih he same desiy a he origi. A expoeial paiece ime disribuio is aracive i may aspecs. For example, whe he service ime disribuio is phase-ype, someimes he marix-aalyic mehod ca be effecive o compue he performace of a queue wih a expoeial paiece ime disribuio. The compued performace is i ur used o approximae he origial queue wih a geeral paiece ime disribuio. Secio 4 will have more discussio o phase-ype disribuios ad he marix-aalyic mehod. Table 2 suppors he replaceme of a M / M /1 GI queue by a M / M /1 M queue. However, i is impora ha he wo sysems mach he paiece ime desiy a he origi, o ay oher saisics such as he mea paiece ime. To highligh his poi, suppose ha a maager uses a M / M /1 M sysem o replace a M / M /1 GI sysem. Bu his ime, he maager maches he mea paiece ime, a pracice ha is ofe used i idusry. I Table 3, for a fixed mea paiece ime m, he mea queue leghs are give for differe paiece ime disribuios, icludig a expoeial disribuio wih rae 1/ m, a uiform disribuio o [, 2 m ] wih 1/(2 m), ad a hyperexpoeial disribuio give by (19) wih / m. Table 3 shows ha for each fixed m, he performace is drasically differe as he paiece ime disribuio chages. This example illusraes ha he mea paiece ime is a wrog saisic o focus o ad oe should ever use i o calibrae a cusomer abadome model. The pheomeo of performace isesiiviy o paiece ime disribuios was firs sudied by Zely & Madelbaum (25) for he seady-sae aalysis of M / M / GI queues ad was laer elaboraed by Dai & He (21) for he process level aalysis uder he G/ G/ GI seig. I Dai & He (21), a

12 12 J Sys Sci Sys Eg Table 2 Performace isesiiviy o paiece ime disribuios Abadome fracios (%) Mea queue legh Exp Uiform H 2 Diffusio Exp Uiform H 2 Diffusio Table 3 Mea paiece ime is a wrog saisic Abadome fracios (%) Mea queue legh Exp Uiform H 2 Exp Uiform H 2 m m m m m deermiisic relaioship is esablished bewee he abadome processes ad he queue legh processes for may-server queues. This relaioship says ha for may-server queues i he QED regime, he cumulaive umber of cusomers who have abadoed he sysem is approximaely equal o a cosa muliple of he cumulaive amou of waiig ime amog all cusomers. Clearly his cosa should be ierpreed as he abadome rae per ui of waiig ime. I was proved by Dai & He (21) ha his cosa is equal o he paiece ime desiy a he origi whe i is sricly posiive. More specifically, if A() is he umber of abadomes by ime ad Q () is he queue legh (i.e., he umber of waiig cusomers) a ime, he Qs ()d s is he cumulaive waiig ime by ime amog all cusomers. The relaioship says ha he scaled differece 1 A () Qs ( )ds is close o zero for ay ime whe is large. Hece, oe may use A() Q()d s s (2) o approximae he abadome process for a may-server queue i he QED regime. 4. Diffusio Models for May-Server Queues i he QED Regime The exac aalysis of a may-server queue wih cusomer abadome has largely bee limied o he M / M / M model, which has a Poisso arrival process ad expoeial service ad paiece ime disribuios. However, as poied ou by Brow e al. (25), he service ime disribuio i a call ceer appears o

13 J Sys Sci Sys Eg 13 follow a log-ormal disribuio. I Zely & Madelbaum (25), he paiece ime disribuio i a call ceer has also bee observed o be far from expoeial. Wih geeral service ad paiece ime disribuios, here is o fiie-dimesioal Markovia represeaio of he queue. Excep compuer simulaios, o mehods are available o aalyze such a queue eiher aalyically or umerically. Much aeio has bee devoed o he approximae aalysis of such a queue. I our approximae aalysis, we approximae a geeral service ime disribuio wih a phase-ype disribuio. A phase-ype radom variable is defied o be he ime uil absorpio of a rasie, fiie-sae Markov chai. Ay posiive-valued disribuio ca be approximaed by phase-ype disribuios. See Neus (1981) for more discussio o phase-ype disribuios. For a GI / Ph / GI queue wih a phase-ype service ime disribuio, wo mulidimesioal diffusio processes were proposed by He & Dai (211) o approximae he dyamics of he queue. I Secio 4.1, we iroduce Browia moio ad illusrae how a arrival process such as a Poisso process ca be approximaed by a Browia moio model. I Secio 4.2, we illusrae he diffusio approximaio for M / M / GI queues. Because he service ime disribuio is expoeial, we are able o spell ou he deails of every sep i derivig he diffusio approximaio. The resulig diffusio process is a oe-dimesioal piecewise Orsei Uhlebeck (OU) process, whose saioary disribuio has a explici formula. I Secio 4.3, he diffusio model for M / H2 / GI queues is preseed. The resulig diffusio process is wo-dimesioal, whose saioary disribuio ca be compued umerically usig he algorihm developed i He & Dai (211). Diffusio approximaios are rooed i he limi heorems for may-server queues i heavy raffic. These heorems require ha he umber of servers go o ifiiy. Secio 4.4 shows ha he diffusio approximaio is accurae, someimes eve for queues wih as few as 2 servers. The paiece ime disribuio is buil io he above diffusio models oly hrough is desiy a he origi. Whe he paiece ime desiy is zero a he origi or chages rapidly ear he origi, we prese i Secio 4.5 a aleraive diffusio model ha uses he hazard rae fucio of he paiece ime disribuio. The hazard rae diffusio model is show o be accurae whe he previous diffusio model works poorly or fails. 4.1 Browia Approximaio Le E{ E( ): } be a Poisso process wih rae 1 arrivals per miue. I Figure 3a, we plo a sample pah of he Poisso process i he firs 1 miues. Oe ca see ha E () evolves aroud he sraigh lie give by is expecaio. To focus o he sochasic variabiliy, we plo he sample pah of he ceered process { E( ) : } i Figure 3b. The ceered process records he flucuaio of he Poisso process aroud is mea. I he plo, he x -axis represes he ime, i a spa of 1 miues. The flucuaio represeed by he y -axis is scaled auomaically by he ploig sofware. To furher examie he effec of he scalig, i Figure 4 we plo he ceered process whe 1,. I urs ou ha he magiude of he ceered process is o he order of as becomes large.

14 14 J Sys Sci Sys Eg (a) A sample pah (b) The sample pah of he ceered process Figure 3 Poisso process wih rae 1 (a) A sample pah (b) The sample pah of he ceered process Figure 4 Poisso process wih rae 1 Le ad 2 be give. A sochasic process B{ B( ): } is called a 2 (, ) -Browia moio if (i) B() ad almos every sample pah is coiuous, (ii) B has saioary, idepede icremes, ad (iii) B () is ormally disribued wih mea variace ad 2 ad for every. The parameers 2 are called he drif ad he variace, respecively, of he Browia moio. The process B is called a sadard Browia moio if ad 2 1. By he well-kow Dosker s heorem, (see, e.g., Billigsley (1999)), E { E ( ): } coverges i disribuio o a sadard Browia moio as, where he scaled, ceered process is defied by E E () E (). (21) For a Poisso process, Dosker s heorem suggess ha oe may replace is scaled

15 J Sys Sci Sys Eg 15 flucuaio i (21) by he sadard Browia moio whe is large. Dosker s heorem is a fucioal ceral limi heorem. Dosker s heorem holds for much more geeral processes icludig reewal processes. For a geeral reewal process E associaed wih a sequece of iid radom variables ha has mea 1/ ad squared coefficie of variaio c 2 a, is scaled flucuaio process E i (21) coverges o a Browia moio wih 2 2 drif ad variace c a as. The ceral idea of diffusio approximaios is o replace a scaled flucuaio process such as he oe i (21) by a appropriae Browia moio. For a may-server queue wih a reewal arrival process (or a arrival process ha saisfies he codiios of a fucioal ceral limi heorem) ad a cerai service ime disribuio, we may use Browia moios o approximae he radom variaios i arrival ad service. A diffusio model is obaied by replacig cerai scaled processes i sysem equaios by Browia moios. 4.2 Diffusio Model for M / M / GI Queues To illusrae he diffusio approximaio of a queue, le us cosider a M / M / GI queue ha has arrival rae, service rae, ad a paiece ime disribuio saisfyig (18). We use X () o deoe he umber of cusomers i he sysem a ime, icludig hose i service ad hose waiig. Le 1 X () ( X() ). We call X { X ( ): } he scaled cusomer-cou process. Whe he arrival rae is high ad he square-roo safey saffig rule is adoped so ha (1 ) is a moderae umber, we ca use a diffusio process Y o approximae X. The diffusio process may be described as follows. Le be he space of fucios u : ha are righ coiuous o [, ) ad has lef limis o (, ). For each u, oe ca fid a uique fucio y ha saisfies y () u () ys () d s ys () d, s, where is he paiece ime desiy a he origi i (18), x max{ x,}, ad x max{ x,} for x. Thus, u y defies a map from a arbirary fucio u o aoher fucio y. Le U () X () B (), 2 where B is a (, ) -Browia moio wih variace 2 ( ) 1. (22) Each sample pah of U is a fucio i. Thus, Y ( U) is a well-defied fucio o each sample pah. Noe ha Y saisfies he sochasic differeial equaio Y () X () B () Ys () ds Ys () d. s (23) The sochasic differeial equaio (23) is he diffusio model for he M / M / GI queue. Is soluio Y ( U) is he diffusio process ha we use o approximae he scaled cusomer-cou process X. The drif coefficie of Y is piecewise

16 16 J Sys Sci Sys Eg liear, give by x whe x, bx ( ) x whe x. Suppose ha. A ay ime, he drif is egaive if Y () / ad is posiive if Y (). Whe Y () is eiher well above or well below zero, his drif will pull i back o a equilibrium level. The process eds o evolve aroud is log-erm mea over ime. A Orsei Uhlebeck (OU) process ha has a liear drif has he similar mea-reverig propery. Because of is piecewise liear drif, Y is called a piecewise Orsei Uhlebeck (OU) process. The piecewise OU process is aalyically racable. I admis a piecewise ormal saioary disribuio, whose desiy is 1 2 ( x ) a1 2 exp whe x, g( x) 2 ( x ) a2 exp whe x, 2 (24) where a 1 ad a 2 are ormalizig cosas ha make g() x coiuous a zero; see Browe & Whi (1995). Oe may derive formula (15) for he delay probabiliy i a M / M / M queue by usig (24) as well as he approximaio Pw gx ( )d x. Because of he performace isesiiviy o paiece ime disribuios, formula (15) applies o he M / M / GI model so log as is ake o be he paiece ime desiy a he origi. Recall ha Q () is he umber of cusomers waiig i he buffer a ime. Le Z() be he umber of cusomers i service a ime. Clearly, Q () X() ad Z () X (). Oe ca compue performace esimaes such as he mea queue legh Q ad he fracio of cusomer abadome P a usig he diffusio model. For ha, le Y( ) be a radom variable ha has he saioary disribuio of Y. Usig he saioary desiy i (24), he mea queue legh Q ca be compued by Q [ Y( ) ] xg( x)dx (25) ad he mea umber of idle servers I ca be compued by [ ] I Y( ) xg( x)d x. Sice I is he mea umber of busy servers, he abadome fracio P a ca be compued via ( I ) Pa 1. (26) We show he performace esimaes compued by (25) ad (26) from he diffusio model i Table 2 uder he Diffusio colums. The diffusio esimaes agree well wih he exac resuls. I he res of his secio, we give a deailed derivaio of he diffusio model (23). Le E () be he umber of cusomer arrivals by ime, ad le S { S( ): } be a Poisso process wih rae 1. We assume ha X (), E{ E( ): }, ad S are muually idepede. Le T () Zs ()d, s which is he cumulaive service ime received by all cusomers up o ime. Sice is he service rae, ST ( ( )) mus be equal i disribuio o he umber of service compleios. Recall ha A() is he cumulaive

17 J Sys Sci Sys Eg 17 umber of abadoed cusomers by ime. Oe mus have X () X() E () S( T ()) A (). (27) To derive Browia approximaios, we defie several scaled processes by 1 E () ( E () ), 1 S () ( S ( ) ), 1 1 Q () Q (), Z () ( Z () ), 1 A () A (). Correspodigly, he dyamical equaio (27) has a scaled versio 1 X () X () E () S ( T ()) Z ()d s s A (), (28) wih give i (8). I he diffusio model, we replace he scaled primiive processes i (28) by cerai Browia moios. These approximaios ca be jusified by Dosker s heorem. Whe he umber of servers is large, he correspodig diffusio process ca be proved close o X. Please refer o Dai e al. (21) for relaed covergece resuls. Sice E is a Poisso process wih rae, he scaled process E { E ( ): } is close o a Browia moio. Noe ha E () has mea zero ad variace. We use a Browia moio BE { BE( ): } wih variace o replace E i (28). Because S is a Poisso process wih rae 1, he scaled process S ca be replaced by a sadard Browia moio B S. We assume ha X (), B E, ad B S are muually idepede. Sice T () is he cumulaive service ime for all cusomers up o, T ()/( ) should be close o he average uilizaio per server, i.e., 1 T () ( 1). Because Z () is he scaled umber of idle servers ad Q () is he scaled queue legh, we have Q () X () ad Z () X (). Because of (2), we may approximae he scaled abadome process by A () X () s d. s (29) I follows from (28) ha X() X () B () B ( 1) E S ()d ( ) X s s X ()d. s s Le B () BE() BS( ( 1)). The B is a drifless Browia moio wih variace ( 1), he same oe as i (22). Thus, X is approximaely a soluio o he sochasic differeial equaio (23). I he proposed diffusio approximaio, we use he soluio Y o he sochasic differeial equaio (23) o replace X. 4.3 Diffusio Model for M / H2 / GI Queues Via a similar Browia replaceme procedure as i Secio 4.2, a diffusio model has bee derived by He & Dai (211) for GI / Ph / GI queues i he QED regime. A wo-phase hyperexpoeial disribuio ( H 2 ), which has bee discussed i Secio 3, is a special case of phase-ype disribuios. I his secio, we resric our discussio o H 2 service ime disribuios, ad illusrae he diffusio approximaio proposed by He & Dai

18 18 J Sys Sci Sys Eg (211). Whe he service imes i a queue follow a wo-phase hyperexpoeial disribuio wih iiial disribuio p ( p1, p2) ad rae ( 1, 2), oe ca evisio wo ypes of cusomers. Wih probabiliy p 1, a cusomer belogs o he firs ype ad his service ime is expoeially disribued wih mea 1/ 1 ad wih probabiliy p 2, he is of ype wo ad he service ime is expoeially disribued wih mea1/ 2. The, he service rae is give by 1. (3) p1 / 1 p2 / 2 I he seady sae, oe expecs ha he cusomers i service are disribued bewee he wo ypes followig a disribuio ( 1, 2), give by p1 / 1 p2 / 2 1 ad 2. p1 / 1 p2 / 2 p1 / 1 p2 / 2 (31) Le X 1 () ad X2 () be he respecive umbers of cusomers of hese wo ypes a ime. Sice he cusomers i service are disribued followig disribuio, we defie is scaled versio afer ceerig by 1 X j() ( X j() j), j 1,2. I he diffusio model, we use a wo-dimesioal diffusio process ( Y 1, Y 2 ) o approximae ( X 1, X 2), where ( Y, Y ) saisfies 1 2 he followig sochasic differeial equaio Y () Y () p p B () j j j j E j1 M j j j j ( Y () ( j pj Y1() Y2()) )ds pj Y1 s Y2 s s ( 1) B ( ) B (( 1) ) ( ( ) ( )) d (32) for j 1, 2. I (32), B E is he same Browia moio as i Secio 4.2, B 1 ad B 2 are wo idepede sadard Browia moios, ad B M is a Browia moio wih drif zero ad variace pp 1 2. I has bee proved by Dieker & Gao (211) ha Y has a uique saioary disribuio. The algorihm proposed by He & Dai (211) ca be used o compue he saioary disribuio umerically. Secio 4.4 preses he performace esimaes obaied from his diffusio approximaio. I he res of his secio, we derive he diffusio approximaio ha uses ( Y1, Y 2) o replace ( X 1, X 2). Le Ci () ( C 1 (), i C 2 ()) i be a wo-dimesioal radom vecor idicaig he i h cusomer s ype. The radom vecor akes (1, ) wih probabiliy p 1 ad akes (,1) wih probabiliy p 2. We assume ha C(1), C(2), are iid. The, k M ( k) C ( i), j 1,2, j i1 j is he umber of ype j cusomers amog he firs k arrivals. Le M j { Mj( k): k 1,2, } ad Sj { Sj( ): } be a Poisso process wih rae 1. We assume ha ( X1(), X 2()), ( M1, M 2), S 1, S 2, ad E are muually idepede. Le Z j () deoe he umber of ype j cusomers i service a ime.the, T () Z ()d s s (33) j j is he cumulaive service ime received by ype j cusomers. Le Lj () be he cumulaive umber of ype j cusomers who have abadoed he sysem by ime. The, he umber of ype j cusomers i he sysem mus follow

19 J Sys Sci Sys Eg 19 X () X () M ( E()) j j j S ( T ( )) L ( ). j j j j We defie he scaled processes by 1 S j() ( Sj() ), 1 Z j() ( Z() j), 1 L j() Lj(), 1 M () ( C () i p ). j i 1 j j (34) The usig (3), (31), (33), ad (34), we have he followig scaled sysem equaio X () X () p p E () j j j j M ( E( )) S ( T ( )) 1 1 j j j j j j j Z ()d s s L () for j 1, 2. I he diffusio model for M / H2 / GI queues, we replace E wih he Browia moio B E as i Secio 4.2. The processes S 1 ad S 2 are replaced by B1 ad B 2, wo idepede sadard Browia moios. Noe ha we always have M 1() M 2(). Hece, he process M 1 is replaced by a Browia moio B M wih variace pp 1 2 ad M 2 is replaced by B M. Whe he umber of servers is large, boh he abadoed cusomers ad he waiig cusomers i he queue are approximaely disribued bewee he wo ypes accordig o disribuio p. Hece, L () p A (), j j where A () is he scaled umber of abadoed cusomers by ime as defied i Secio 4.2. Recall ha Q () is he queue legh a ime. The, Z () X () p Q(). j j j Sice Q () ( X1() X2( ) ), his approximaio has a scaled versio Z () X () p ( X () X ()). j j j 1 2 We also exploi he approximaios as well as E () Tj (), ( 1) j, A () Q ()d s s ( X () s X ()) s d. s 1 2 These replacemes lead o he diffusio model (32) for M / H2 / GI queues. I our diffusio model, a wo-dimesioal diffusio process is used o approximae he scaled umber of cusomers of each ype. Whe his procedure applies o a geeral phase-ype service ime disribuio wih d phases, he correspodig diffusio model is a d -dimesioal piecewise OU process. 4.4 Performace Esimaio Usig he Diffusio Model To obai he performace esimaes of a queue usig he diffusio model, oe eeds o kow he saioary disribuio of he mulidimesioal diffusio process. Excep for he oe-dimesioal case, he saioary disribuio of a mulidimesioal piecewise OU process does o have a explici formula. I He & Dai (211), he auhors also developed a fiie eleme algorihm compuig he saioary disribuio of a mulidimesioal diffusio process. Usig he umerical resuls obaied by his algorihm, hey demosraed ha he diffusio model is a good approximaio of a

20 2 J Sys Sci Sys Eg may-server queue. Cosider a M / H2 / M queue wih 5 servers. We se he arrival rae o be cusomers per miue ad he rae of he expoeial paiece ime disribuio o be.5. The hyperexpoeial service ime disribuio has parameers p (.9351,.649) ad (9.354,.72). So he mea service ime of he secod-ype cusomers is more ha 1 imes loger ha ha of he firs ype. Alhough over 93% of cusomers are of he firs ype, he fracio of is workload is merely 1%. Such a disribuio has 2 a large squared coefficie of variaio cs 24. Oe ca check ha he mea service ime is 1 miue. Hece, he queue is a bi overloaded wih Recall ha X () is he umber of cusomers i he sysem a ime. For his M / H2 / M queue, he process X is a quasi-birh-deah process. Oe ca use he marix-aalyic mehod o solve he saioary disribuio of X. See Neus (1981) for deails o he marix-aalyic mehod. To evaluae he accuracy of he diffusio model, i Figure 5a we plo boh he (approximae) saioary disribuio of X obaied by he diffusio model ad he saioary disribuio produced by he marix-aalyic mehod. We see very good agreeme bewee he wo resuls. Whe he umber of servers is moderae, he diffusio model ca sill capure he dyamics of he queue. Nex, we cosider a M / H2 / M queue wih 2 servers. Le he paiece ad service ime disribuios be he same as i he previous sceario, ad he arrival rae be Thus, As illusraed by Figure 5b, he diffusio model ca sill capure he exac saioary disribuio for a queue wih as few as2 servers. Wih a appropriae algorihm, performace esimaio usig he diffusio model ca be much more compuaioally efficie ha he marix-aalyic mehod. The compuaioal complexiy of he algorihm proposed by He & Dai (211), wheher i compuaio ime or i memory space, does o chage wih he umber of servers. I coras, he marix-aalyic mehod becomes compuaioally expesive whe is large. I paricular, he memory (a) 1.45 ad 5 (b) ad 2 Figure 5 Saioary disribuio of he cusomer umber i he M / H2 / M queue

21 J Sys Sci Sys Eg 21 usage becomes a serious cosrai whe a huge umber of ieraios are required i he marix-aalyic mehod. For he 5 sceario i his example, i ook aroud 2 hours o fiish he marix-aalyic compuaio ad he peak memory usage was early 5GB. Usig he diffusio model ad he proposed algorihm, i ook less ha 1 miue ad he peak memory usage was less ha 2MB o he same compuer. 4.5 A Refied Diffusio Model Usig he Hazard Rae of Paiece Times I he above diffusio model, he paiece ime desiy a he origi is he key parameer for modelig he abadome process. This diffusio model, however, has is ow limiaios. Firs, oe eeds o esimae he paiece ime desiy a he origi usig he daa colleced from he service sysem. Esimaig he desiy a he origi is saisically ureliable. The paiece imes are heavily cesored daa, i.e., a cusomer s paiece ime ca be observed oly if he has abadoed he sysem. For a queue i he QED regime, oly a small fracio of cusomers abado he sysem. Alhough sadard survival aalysis ools, such as he Kapla Meier esimaor (see, e.g., Cox & Oakes (1984)), ca be used o esimae his parameer, oe sill has o record each cusomer's waiig or paiece ime ad a good esimae requires a large amou of daa. Secod, for a queue i he QED regime, o maer how shor he waiig imes are, he abadome process sill depeds o he behavior of he paiece ime disribuio i a eighborhood of he origi, o jus a he origi. Whe he paiece ime desiy ear he origi chages rapidly, usig he desiy a he origi solely may o yield a adequae approximaio for he abadome process. Third, whe, he iegral erm correspodig o he abadome process i he diffusio model, eiher (23) or (32), becomes zero. I his case, he diffusio model approximaes a queue as if here is o cusomer abadome. Bu i a queue wih a zero paiece ime desiy a he origi, cusomer abadome sill occurs ad may affec he sysem performace sigificaly. For example, if such a queue is slighly overloaded (i.e., 1), i sill has a saioary disribuio haks o he cusomer abadome ha reduces service demads. However, he diffusio model, wih ad 1, does o have a saioary disribuio ad fails o provide ay performace esimaes for his queue. Now we prese a refied diffusio model usig he eire paiece ime disribuio. This model was proposed by He & Dai (211). I explois he idea of scalig he paiece ime hazard rae fucio, which was firs proposed by Reed & Ward (28) for sigle-server queues ad was exeded o may-server queues by Reed & Tezca (29). This refied diffusio model provides a more accurae approximaio for may-server queues. I his model, we assume ha F, he cumulaive disribuio fucio of he paiece imes, saisfies F() ad has a bouded hazard rae fucio h F, give by ff () hf (),, 1 F ( ) (35) where f F is he desiy of F. Wih he hazard rae fucio, F ca be wrie by

22 22 J Sys Sci Sys Eg F () 1exp h ()d s s,. I he refied diffusio model, he scaled abadome process A is approximaed by X( s) v A () h d d,. F v s (36) The eire paiece ime disribuio is buil io he approximaio hrough is hazard rae fucio. The iuiio of he hazard rae scalig approximaio was explaied by Reed & Ward (28): Cosider he Qs () waiig cusomers i he buffer a ime s. I geeral, oly a small fracio of cusomers ca abado he sysem whe he queue is workig i he QED regime. The by ime s, he i h cusomer from he back of he queue has bee waiig aroud i / ime uis. Approximaely, his cusomer will abado he queue durig he ex ime uis wih probabiliy hf (/ i ). I follows ha for he sysem, he isaaeous abadome rae Qs ( ) a ime s is aroud h (/ i ). Hece, he F i1 scaled abadome rae ca be approximaed by Qs ( ) 1 i Qs ( ) v hf h d F v i1 X( s) v h d, F v (37) from which (36) follows. Noe ha he arrival rae is o he order of O ( ) ad Qs () is o 1/2 he order of O ( ). The paiece ime disribuio i a small eighborhood of zero, o jus is desiy a zero, is cosidered i he isaaeous abadome rae i (37). Hece, he hazard rae scalig approximaio i (36) is more accurae ha ha i (29). Le k be a oegaive ieger. Suppose ha he hazard rae fucio h F is k imes coiuously differeiable i a eighborhood of F zero. By Taylor s heorem, k ( ) F F F 1 x h ( x) h () h ()! () for x small eough, where h F is he h order derivaive of h F. I his case, he approximaio i (36) urs ou o be A () h () Q ()d s s F k 1 /2 ( ) h F () ( 1)! 1 Qs () d. s Because h F () is ideical o he paiece ime desiy a zero, he approximaio i (29) ca be regarded as he zeroh degree Taylor s approximaio of (36). Whe he paiece imes are expoeially disribued, he hazard rae fucio is cosa ad he wo approximaios i (29) ad (36) are ideical. Usig he hazard rae scalig approximaio ad he Browia moio replacemes, we obai aoher diffusio model for he M / M / GI queue Y () X () B () Ys ( ) ds Y( s) v h d d, F v s (38) i which B is he drifless Browia moio wih variace give by (22). I (38), he diffusio process Y has he same diffusio coefficie as i (23), bu is drif coefficie is x v h d whe, ( ) F v x bx x whe x. Comparig (23) ad (38), oe ca see ha he wo models differ oly i how he paiece ime disribuio is icorporaed. Because a more accurae approximaio is used for he