Delay-Based Service Differentiation with Many Servers and Time-Varying Arrival Rates

Transcription

1 hp://pubsolie.iforms.org/joural/ssy/ STOCHASTIC SYSTEMS Vol. 8, No. 3, Sepember 218, pp ISSN pri, ISSN olie Delay-Based Service Differeiaio wih May Servers ad Time-Varyig Arrival Raes Xu Su, a Ward Whi a a Deparme of Idusrial Egieerig ad Operaios Research, Columbia Uiversiy, New York, New York 127 Coac: xs2235@columbia.edu, hp://orcid.org/ XS; ww24@columbia.edu, hp://orcid.org/ WW Received: July 8, 217 Revised: Jauary 22, 218 Acceped: April 9, 218 Published Olie i Aricles i Advace: Sepember 24, 218 hps://doi.org/1.1287/ssy Copyrigh: 218 The Auhors Absrac. We sudy he problem of saffig specifyig a ime-varyig umber of servers ad schedulig assigig ewly idle servers o a waiig cusomer from oe of K classes i he may-server V model wih class-depede ime-varyig arrival raes. I order o sabilize performace a class-depede delay arges, we propose he blid model-free head-of-lie delay-raio HLDR schedulig rule, which exeds a earlier dyamic-prioriy rule ha explois he head-of-lie delay iformaio. We sudy he HLDR rule i he qualiy-ad-efficiecy-drive may-server heavy-raffic MSHT regime. We saff o he MSHT fluid limi plus a corol fucio i he diffusio scale. We esablish a MSHT limi for he Markov model, which has dramaic sae-space collapse, showig ha he argeed raios are aaied asympoically. I he MSHT limi, meeig saffig goals reduces o a oe-dimesioal corol problem for he aggregae queue coe, which may be approximaed by recely developed saffig algorihms for ime-varyig sigle-class models. Simulaio experimes cofirm ha he overall procedure ca be effecive, eve for o-markov models. Ope Access Saeme: This work is licesed uder a Creaive Commos Aribuio 4. Ieraioal Licese. You are free o copy, disribue, rasmi ad adap his work, bu you mus aribue his work as Sochasic Sysems. Copyrigh 218 The Auhors. hps://doi.org/1.1287/ssy , used uder a Creaive Commos Aribuio Licese: hps://creaivecommos.org/liceses/by/4./. Fudig: This research was suppored by he Naioal Sciece Foudaio [Gra CMMI ]. Supplemeal Maerial: The olie appedix is available a hps://doi.org/1.1287/ssy Keywords: service differeiaio may-server heavy-raffic limi ime-varyig arrivals raio corol schedulig of cusomers o eer service sample-pah Lile s law 1. Iroducio ad Summary I his paper, we sudy delay-based service differeiaio via raio corols i a muliclass may-server service sysem wih ime-varyig arrival raes. We aim o keep he raios of he delays of differe classes early cosa over ime a specified arges A Time-Varyig V Model i he Qualiy-ad-Efficiecy-Drive May-Server Heavy-Traffic Regime I paricular, we sudy he ime-varyig TV V model ha is, he muliclass exesio of he M /M/s + M may-server Markovia queueig model wih ulimied waiig space ad abadome from queue. There is a TV umber s of homogeeous servers workig i parallel. Arrivals from K classes come accordig o idepede ohomogeeous Poisso processes NHPPs, wih arrivals of class-i occurrig a a TV rae λ i. If possible, class-i cusomers eer service immediaely upo arrival; oherwise, hey joi he ed of a class-i queue, hereafer o be served i order of arrival. The cusomer service imes ad paiece imes ime o abado from queue afer arrival are muually idepede expoeial radom variables, idepede of he arrival process. The mea service ime ad paiece ime of each class i cusomer are 1/µ i ad 1/θ i, respecively. For his model, we sudy he combied problem of saffig choosig he fucio s ad schedulig assigig a ewly idle server o he head-of-lie HoL cusomer i oe of he K queues. We do o allow a server o be idle whe here is a waiig cusomer. We propose a varia of he square-roo-saffig SRS rule for saffig ad a head-of-lie delay-raio HLDR schedulig rule ad esablish supporig resuls. This approach is aracive because i is raspare ad flexible; for example, i ca be applied o o-markov models; see Secio

2 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors Saffig. I paricular, our SRS saffig fucio is s m+ c m, 1 where m is he offered load ha is, he expeced umber of busy servers i he associaed ifiie-server model obaied by acig as if s, ad c is a corol fucio o mee desired performace arges. Because he classes ca be cosidered separaely i a ifiie-server model, he offered load m is he sum of he correspodig sigle-class offered loads m i, each of which ca be represeed as he iegral m i or as he soluio of he ordiary differeial equaio e µ i s λ i s ds, 2 ṁ i λ i µ i m i. 3 The SRS approach o TV saffig i 1 follows Jeigs e al ad Feldma e al. 28 for he sigle-class case, wih 2 comig from heorems 1 ad 6 of Eick e al. 1993; see Gree e al. 27 ad Whi 217 for reviews Releasig Busy Servers. Wih TV saffig, we eed o specify wha happes a he imes whe saffig is scheduled o decrease bu all servers are busy. Eve for cosa saffig levels, varias of his issue commoly arise i service sysems whe he servers are people, because huma servers work o shifs ad may be busy a he ed of he shif. I applicaios, we may assume ha he server complees he service i progress afer compleig he shif, bu he he saffig is acually higher ha sipulaed a hose imes. Whe he service imes are relaively log, we may wa o allow server swichig upo deparure, which we assume is beig used here; see Igolfsso e al. 27 ad Liu ad Whi 212a. For simpliciy i he mahemaical aalysis, we ry o avoid his issue as much as possible. Thus, we assume ha server swichig is beig used, so ha he server ha has compleed service mos recely is released. The, o maiai work coservaio, we assume ha he cusomer ha was beig served he mos rece cusomer o eer service is pushed back io a queue. Because he service-ime disribuio is expoeial, he remaiig service ime has he same disribuio as a ew service ime. This push-back scheme is he sadard approach for he M /M/s + M model; for example, see Puhalskii 213. However, here is a addiioal complicaio for muliclass queues, because we wa o maiai work coservaio ad class ideiy. Hece, we assume ha he mos rece arrival is pushed ou of service ad placed i a special high-prioriy queue, so ha he order of eerig service is o alered by his feaure; see Secio 3.3. As par of our proof of he may-server heavy-raffic MSHT fucioal ceral limi heorem FCLT, we show ha he impac of his high-prioriy queue is asympoically egligible; see Sep 2 of he proof of Theorem The HLDR Schedulig Rule. HLDR explois he HoL waiig ime U i of class-i a ime. HLDR uses a prespecified TV vecor fucio v v 1,..., v K. The HLDR schedulig rule assigs he ewly available server o he HoL class-i cusomer ha has he maximum value of U i /v i. The HLDR rule is appealig because i is a blid schedulig policy ha is, i does o deped o ay model parameers A New Qualiy-ad-Efficiecy-Drive MSHT FCLT. We esablish a ew may-server heavy-raffic fucioal ceral limi heorem ha suppor he combied SRS saffig ad HLDR schedulig for he TV model. As usual, we cosider a sequece of models idexed by he umber of servers,, ad le. We keep he service ad abadome raes uchaged, bu le he arrival-rae ad saffig fucios i model be λ i λ i, so ha he offered load is m m, ad s m + c m m+ c for c c m, 4 where m correspods o he MSHT fluid limi, obaied from he associaed fucioal weak law of large umbers FWLLN. I is sigifica ha he MSHT fluid limi coicides wih he appropriae scalig by wih he offered load i for he ifiie-server model, as give i Secio 1.1.1; for example, see secio 9 i Massey ad Whi 1993, Madelbaum e al. 1998, ad secio 4 of Liu ad Whi 212a. The secod expressio i 4 is appealig for he simple direc way ha appears. We show ha he scalig i 4 pus he model io he qualiy-ad-efficiecy-drive QED MSHT regime; ha is, we esablish a odegeerae joi MSHT FCLT for he appropriaely scaled umber of class-i cusomers i he

3 232 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors sysem a ime for all i, ogeher wih associaed delay processes, where he arge HoL delay raios hold almos surely for all i he limi process; see Theorem 1. Our MSHT FCLT is cosise wih previous QED MSHT limis for boh saioary models i Halfi ad Whi 1981 ad Gare e al. 22 ad for osaioary models i Madelbaum e al. 1998, heorem 2 i Puhalskii 213, ad secio 2.6 i Whi ad Zhao 217. Jus like c i 1, he fucio c i 4 is a corol ha we use o achieve performace objecives, for example, sabilize performace of he K classes over ime a desigaed arges The Accumulaig-Prioriy Disciplie for Healhcare Applicaios The saioary versio of HLDR, where he vecor fucio v above is idepede of coicides wih he accumulaig-prioriy AP schedulig rule sudied by Saford e al. 214, Sharif e al. 214, Li ad Saford 216, ad Li e al. 217, which i ur coicides wih a dyamic-prioriy rule proposed by Kleirock If v i 1 for all i ad ; ha is, all classes accumulae prioriy a a equal cosa rae, he he HLDR reduces o global firs-come-firs-serve, as i Talreja ad Whi 28. As discussed i Sharif e al. 214, here is srog moivaio for his schedulig policy i healhcare. I paricular, Caadia emergecy deparmes EDs classify paies io five acuiy levels. Accordig o he Caadia Triage ad Acuiy Scale CTAS guidelie Bullard e al. 214, p. 2, CTAS level i paies eed o be reaed wihi w i miues wih w 1, w 2, w 3, w 4, w 5, 15, 3, 6, 12. I his coex, we esablish addiioal isigh for AP by 1 sudyig saffig as well as schedulig; ad 2 esablishig MSHT limi ad exedig o a TV seig. There is also moivaio for he TV exesio here from healhcare, because he arrival raes i EDs are srogly ime-depede ad he service imes are relaively log, as ca be see from Armoy e al. 215 ad Whi ad Zhag 217. Oe of he grea appeals of he AP ad HLDR schedulig rules is ha hey also apply wihou chage i a TV evirome, bu we coribue by exposig how hese rules perform i a TV evirome. Our framework also allows TV arges Exedig Previous Raio Corols o a Time-Varyig Seig The HLDR schedulig rule is also closely relaed o raio rules cosidered by Gurvich ad Whi 29a, b; 21; also see Dai ad Tezca 28, 211. These papers cosidered more geeral saioary models wih muliple pools of servers, ad he associaed rouig as well as schedulig, bu we oly cosider a sigle service pool i his paper. The papers Gurvich ad Whi 29a, b; 21 esablish MSHT limis for hese raio corols, showig ha hey iduce a simplifyig sae-space collapse, ha permi achievig performace goals asympoically. Here, we exed hose resuls for a sigle service pool o a TV seig. The echical complexiy is sigificaly less here, because by resricig aeio o he sigle-pool case, we do o eed o cosider he hydrodyamic limis i Gurvich ad Whi 29a Fixed-Queue-Raio Schedulig i a TV Seig. The paper by Gurvich ad Whi 21 showed ha aalogs of HLDR based o queue leghs isead of HoL delays, called fixed-queue-raio FQR corols, are effecive for achievig delay-based service-differeiaio. Gurvich ad Whi 29b also cosiders varias of HLDR i secios 3.3 ad 3.4 ad is iere suppleme. Jus as wih HLDR ad AP, FQR exeds direcly o a TV evirome. We sared his sudy by coducig simulaio experimes o ivesigae how FQR ad HLDR perform i a TV evirome. We prese some of he resuls here i Secio 2. These wo schedulig rules ofe boh work well i a TV evirome, bu o always: If he raios of he arrival raes of differe classes are ime-varyig, he FQR ca seriously fail o sabilize delays. However, we also iroduce a modified TVQR ha achieves he same performace as HLDR asympoically; see Theorem 2. I coras o HLDR, TVQR is o a blid corol, because i requires he arrival raes, alhough hose ca be esimaed, as suggesed i defiiio 3.4 of Gurvich ad Whi 29b Sae Space Collapse ad he Sample-Pah TV MSHT Lile s Law. The successes ad failures of FQR i he TV seig ca be explaied by a sample-pah SP MSHT Lile s law LL ha is a cosequece of he TV MSHT limis i Theorems 1 ad 2, which geeralize he SP-MSHT-LL for he saioary model ha is a cosequece of heorem 4.3 i Gurvich ad Whi 29a ad is discussed afer equaio 13 i secio 3 of Gurvich ad Whi 21. I paricular, for large-scale sysems ha are approximaely i he QED MSHT regime, Q i λ i V i, T < for all i, 5

4 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors 233 where Q i is he queue legh, λ i is he arrival rae ad V i is he class-i poeial delay a ime he delay of a hypoheical class-i cusomer if i were o arrive a ime ad had ifiie paiece. Formally, his ca be expressed via he followig corollary o our Theorem 1: Corollary 1 SP TV MSHT Lile s Law. Uder he codiios of Theorem 1, he relaios i 5 are valid asympoically as. The SP-MSHT-LL i 5 holds because of he dramaic sae-space collapse SSC associaed wih he QED MSHT limis uder he raio rules. Because of he SSC, he queue ad delay raios are relaed by Q i Q j λ i V i, T < for all i. 6 λ j V j As a cosequece, we also obai he followig corollary o our Theorem 1: Corollary 2 Queue Raios ad Delay Raios. Uder he codiios of Theorem 1, asympoically as, a queue raio ca be sable over ime ogeher wih a delay raio if ad oly if he TV raio of he arrival raes is sable. I is sigifica ha he raio of class-depede arrival raes may be TV i applicaios. For isace, secio 3.5 of Whi ad Zhag 217 shows ha he proporio of arrivals o he Israeli emergecy deparme ha are admied o a ieral ward of he hospial varied srogly over ime. Because he admied paies ed o be amog he more criical paies, we ifer ha here is likely o be a differece i he TV arrival raes of paies classified by acuiy Scalig he Tail-Probabiliy Delay Targes i he QED MSHT Limi. Oe-way o achieve delay-based service differeiaio is o have class-depede arges for he delay ail probabiliies. I paricular, for he sequece of models idexed by, he goal may be expressed as PVi w i α, T for all i, 7 where he class-i arges w i are chose o produce he desired service-level differeiaio. The arges w i could be TV as well, bu we leave ha ou because we are usually ieresed i sable performace over ime i he TV seig. A key compoe of he QED MSHT FCLT supporig 7 is he QED scalig of he delay probabiliy arges, which follows assumpio 2.1 of Gurvich ad Whi 21. Because he QED MSHT scalig makes queue leghs be of order O, while waiig imes are of order O1/, waiig imes ad queue leghs are scaled very differely i he QED MSHT scalig. I order o ge a odegeerae QED MSHT limi for PVi w i,we assume ha w i w i as for < w i <, for all i. 8 As a cosequece, we ge PVi w i P V i w i P ˆV i w i, for all i, T, 9 where ˆV ˆV 1,..., ˆV K is he limi process we shall esablish wih he seleced schedulig policy. The scalig of he delay arges here ad i Gurvich ad Whi 21 makes he ail probabiliies similar o he delay probabiliies PVi ha are kow o be well sabilized i he QED MSHT regime for he sigle-class model; for example, see Feldma e al. 28. I coras, if we do o scale he delay probabiliy arges, he we are forced io he efficiecy-drive ED MSHT regime. Liu 218 has show ha his ED scalig ca also be effecive for sabilizig ail probabiliies. The approach i Liu 218 evidely should become relaively more effecive as he delay arges icrease. Such large arges ofe occur i healhcare; for example, as i he six-hour boardig ime limi i he Sigapore hospial discussed i secio of Shi e al Reducio o a Oe-Dimesioal Sochasic Corol Problem. Give he oe-dimesioal MSHT limi associaed wih HLDR, i suffices o sabilize he ail probabiliy of he limi of he oal aggregae queue legh,

5 234 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors ˆQ a a specified arge. Parallelig he big display bewee equaios 13 ad 14 of Gurvich ad Whi 21, we have lim PV i w i P ˆV i w i P ˆQ i λ i w i P K ˆQ i K K λ i w i P ˆQ λ i w i. i 1 i 1 i 1 1 Hece, give λ i ad w i for all i, we ca sabilize all processes a he arge levels ha is, we ca achieve PVi w i α for all i, if we ca fid a corol fucio c ha achieves P ˆQ K λ i w i α. 11 i The Beefis of Addiioal Srucure: Four Cases. Give ha we are akig advaage of he SSC provided by HLDR, he form of he limi reveals how difficul is he overall corol problem. The difficuly depeds criically upo he model parameers µ i ad θ i. I his paper, we ideify four cases. Case 1 is he geeral model wih parameers µ i ad θ i depedig o he class i, for which Theorem 1 shows ha he limi i reducio above is ˆQ [ˆX c] +, where ˆX is a sum of he compoes of a K-dimesioal diffusio process ad so o iself a diffusio process. We obai he oher hree cases by imposig addiioal codiios o he service ad abadome raes. Case 2 has θ i µ i for all i; he, he limi process has he srucure of a TV K-dimesioal Orsei Uhlebeck diffusio process, complicaed by a ime-varyig variace. The K-dimesioal srucure of he limi process i cases 1 ad 2 reveals ihere challeges i aalyzig he muliclass model. The sroges posiive coclusios are for cases 3 ad 4. Case 3 has θ i θ ad µ i µ for all i; he, he limi process is a 1-dimesioal diffusio process. I Case 3, we ca esablish asympoic opimaliy for he proposed soluios o he combied saffig ad schedulig problem ad effecively reduce he saffig compoe o he saffig problem for he associaed sigle-class M /M/s + M model. I remais o solve he 1-dimesioal diffusio corol problem o fid he saffig fucio. For pracical applicaios, his resul srogly suppors applyig HLDR ogeher wih heurisic saffig algorihms for he sigle-class M /M/s + M model, such as he modifiedoffered-load approximaio or he ieraive-saffig-algorihm i Feldma e al. 28; hese are surveyed i Whi 217 ad Whi ad Zhao 217. Case 4 combies cases 2 ad 3, havig θ i µ i µ for all i. Case 4 is he ideal siuaio where we ca provide a explici soluio for he saffig fucio. The simplificaio provided by havig he abadome rae equal o he service rae ca be explaied by he coecio o ifiie-server models; see secio 6 of Feldma e al. 28. We verify he effeciveess of our HLDR policy wih a simulaio example i Secio Saffig for he Aggregae Queueig Model. I Secio 1.3.4, we observed ha we ca apply he limi process from he MSHT limi o obai a sochasic corol problem for he saffig. A aleraive is o use a saffig algorihm for he aggregaed queueig model associaed wih he give model. Wihi he QED MSHT framework, we ca obai a appropriae model by cosrucig a associaed sequece of sigle-class models for which he aggregae queue legh process has he same QED MSHT limi as obaied for he TV muliclass model. For example, i Case 3 i Secio 1.3.5, he aggregae model is direcly a M /M/s + M model, which has bee sudied i Feldma e al. 28 ad Liu ad Whi 212b ad subsequely. Ideed, as log as he service ad paiece disribuios are he same for all classes, he aggregae model is a G /GI/s + GI model, for which saffig algorihms have bee developed i He e al. 216, Liu ad Whi 212b, ad Whi ad Zhao 217. We illusrae for he case of a muliclass M /GI/s /M model wih a logormal service disribuio i Secio 5.3. However, here are sigifica difficulies i he geeral case 1, because he service imes ad paiece imes lose he idepedece propery. Aalogous difficulies i coveioal heavy-raffic limis for muliclass sigle-server queues were exposed ad sudied i Fedick e al. 1989, 1991 ad Fedick ad Whi Opimizig ad Saisficig by Focusig o Raio Rules The sadard approach o he saffig-ad-schedulig problem for he Markovia queueig model is o formulae a Markov decisio process, as i Puerma 1994, sarig by specifyig releva coss for example, for

6 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors 235 waiig ad for abadome ad rewards for compleed service, for example, hroughpu. For queueig problems such as hese, a direc applicaio is difficul, so ha i is aural o seek asympoic opimaliy i he presece of heavy-raffic scalig. Followig grea success for queueig models wih coveioal high-raffic HT scalig, for example, as for he cµ rule Va Mieghem 1995, Madelbaum ad Solyar 24, his approach was applied o schedulig i may-server queues by Aar e al. 24, Harriso ad Zeevi 24, ad Aar 25 ad coiues o be a major direcio of research, as ca be see from Araposahis e al. 215adAraposahis ad Pag 216. The subsaial body of relaed work ca be raced from hese refereces. The MSHT limis are used o produce a limiig diffusio corol problem. Uforuaely, he resulig Hamilo Jacobi Bellma equaios for he limiig diffusio corol problems ed o be difficul o solve, so ha i is hard o exrac useful applied resuls. Tha impasse led Gurvich ad Whi 29a, b; 21 o focus o raio schedulig ad rouig policies. Isead of opimal policies, hey sough good policies. I he laguage of Herber Simo Simo 1947, 1979, hey suggesed saisficig isead of opimizig. I par, his was because he implicaios of a seemigly aural opimizaio framework are o so evide; see Miler ad Olse 28 ad secio 2 of Gurvich e al. 28. For example, a ailprobabiliy cosrai ca permi he scheduler simply o o serve ay class-i cusomer who has waied loger ha he performace arge. I coras, if fixed raios over ime are maiaied, he we direcly udersad he implicaios of he schedulig rule. To pu his i a formal opimizaio framework, we would say ha obaiig fixed or early fixed raios is o a meas o aoher ed, bu is i fac par of he goal he objecive. From ha perspecive, he SSC associaed wih he MSHT limi shows ha he raio rules are asympoically opimal. Neverheless, Gurvich ad Whi 29a, b; 21 devoed cosiderable effor o esablishig asympoic opimaliy of raio rules for coveioal cos models, where i exiss, for example, for he geeralized cµ rule i secio 3.2 of Gurvich ad Whi 29b. Where he raio rules fell shor, hey focused o he weaker oio of asympoic feasibiliy. We will do he same here Orgaizaio I Secio 2, we prese resuls of iiial simulaio experimes o show he value of HLDR ad TVQR schedulig rules wih TV arrival raes. I Secio 3, wedefie he model ad iroduce he saffig miimizaio problem. I Secio 4, we sae our mai aalyical resuls ad describe he proposed soluios o he joi saffig ad schedulig problems. I Secio 5, we prese he simulaio resuls implemeig he full algorihm for examples from Case 4 ad showig ha i performs well. We iclude a example wih a logormal service-ime disribuio. I Secios 6 ad 7, we provide he proofs of he MSHT limis ad for asympoic feasibiliy ad opimaliy. I Secio 8, we coclude wih discussig direcios for fuure research. We provide backgroud o he simulaio mehodology ad more umerical resuls i he olie appedix. 2. Iiial Simulaio Experimes We illusrae he FQR, HLDR, ad TVQR schedulig rules wih a wo-class M /M/s + M model havig siusoidal arrival-rae fucios ad saffig chose o sabilize he aggregae performace. TVQR is defied i Secio The Experimeal Seig Le he wo arrival-rae fucios be Le he TV saffig fucios be as i 4. λ i a i + b i sid i for T, i 1, Saioary Arrivals We sar wih he saioary case wihou cusomer abadome from queue, leig a 1, b 1 6, ad a 2, b 2 9, i 12 so ha he ime-scalig facors d i play o role wih µ 1 µ 2 µ 1adθ 1 θ 2. Suppose ha he objecive is oachieveadelayraiov 1/2. From he SP MSHT Lile s lawi5, we ifer ha he queue raio should be approximaely equal o 1/26/9 1/3.Hece,oewouldwaouseheFQRrule wih arge queue raio r 1/3. Wih his value, we udersad ha he raio Q 1 /Q 2 is expeced o be aroud he arge 1/3, while he delay raio should be abou 1/2. We se he fixed saffig level usig he SRS saffig rule wih c.25, yieldig he cosa saffig level s 17 o mee he cosa offered load of 15. We obai

7 236 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors Figure 1. Queue ad Delay Raios for a Two-Class Saioary M/M/s Queue wih Arrival Rae Fucios λ 1 6, λ 2 9, Commo Service Rae µ 1, Wihou Abadome θ 1 θ 2, ad c.25 our simulaio esimaes by performig 2, idepede replicaios; see he olie appedix for furher explaaio. Figure 1 shows he queue raio ad wo delay raios over he ime ierval [5, 7] for he FQR rule lef ad he HLDR rule righ. We plo boh he poeial delay ad he HoL delay. Because he HoL delay a ime is he elapsed delay of he cusomer i queue ha is ex o eer service, he HoL cusomer will experiece addiioal delay before eerig service, we expec i o be somewha less ha he HoL poeial delay. Figure 1 shows ha boh FQR ad HLDR sabilize he queue raio a he arge r 1/3 ad he delay raio a he associaed level v 1/2. For FQR, his is as prediced by heorem 4.3 of Gurvich ad Whi 29a TV Arrivals Wihou Abadome Now cosider TV arrival-rae fucios by choosig a 1, b 1, d 1 6, 2, 1/2 ad a 2, b 2, d 2 9, 3, 1/2 i 12, so ha he overall arrival-rae fucio is λ λ 1 +λ si/2. Agai, le µ 1 µ 2 µ 1 ad θ 1 θ 2. Wih d 1 d 2 1/2, he cycle legh is 4π 12.57, which is abou oe half day if we measure ime i hours. Figure 2 shows he resuls. Figure 2, a ad b, plos he same se of performace measures for FQR ad HLDR show i Figure 1. Figure 2a shows ha FQR is agai effecive a Figure 2. Queue ad Delay Raios for a Two-Class M /M/s Queue wih Arrival Rae Fucios λ si/2, λ si/2, Commo Service Rae µ 1, Wihou Abadome θ 1 θ 2, ad c.25

8 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors 237 Figure 3. Queue ad Delay Raios for a Two-Class M /M/s Queue wih Arrival-Rae Fucios λ si/2, λ si/2, Commo Service Rae µ 1, Wihou Abadome θ 1 θ 2, ad c.25 sabilizig he queue leghs, bu is ow highly ieffecive a idirecly sabilizig delays. Similarly, Figure 2b shows ha HLDR is remarkably effecive a direcly sabilizig he raio of he delays, bu i does o idirecly sabilize he queue leghs. Figure 2c shows ha he specially desiged TV modificaio of FQR performs much like HLDR. Wha we see i Figure 2 ca be explaied by 5: The raio of he arrival raes varies from 6 2/ /3 o 6 + 2/9 3 4/3, a facor of 4. To see ha, we ecouer o such difficuly if he aggregae arrival rae is highly TV, while he raio AR is cosa. To illusrae, Figure 3 shows he correspodig resuls whe we simply chage he sig of b 1 from o +, which makes AR 2/3 for all TV Arrivals wih Abadome We ow cosider hese same schedulig rules i he wo-class model whe here is cusomer abadome. For simpliciy, le he abadome raes be class-ivaria wih rae θ.5. The mea ime o abado is wice he mea service ime. From our experimes, we see ha abadome affecs our abiliy o sabilize he raios, bu ha i has less ad less impac as he scale icreases ad has oe a all i he MSHT limi. To demosrae he impac of scale, we plo he queue ad delay raios as a fucio of sysem size for he wo-class example i Figure 4. Here, we use safey saffig fucio c, which is cosise wih he heurisic of simply saffig o he offered load, as discussed i paragraph 3 of secio 6 of Feldma e al. 28, p Figure 4 shows he queue ad delay raios as a fucio of sysem size for he same wo-class M /M/s + M queue, bu wih abadome raes θ 1 θ 2.5. Figure 4 shows ha hese schedulig corols become more effecive as he scale icreases, cosise wih ou laer MSHT limi. Remark 1 Class-Depede Service. The olie appedix shows he correspodig resuls for he wo-class M /M/s + M queue wih class-depede service imes. 3. Formulaio We specify our oaio ad coveios i Secio 3.1 ad lay ou he prelimiaries of he ime-varyig muliclass queueig model i Secio 3.2. We formalize he high-prioriy queue for cusomers pushed ou of service because of saffig decrease i Secio 3.3. We he defie he poeial delay i Secio 3.4 ad iroduce problem formulaios wih differe SL ypes i Secio 3.5. Wedefie he HLDR ad TVQR rules i Secios 3.6 ad 3.7, respecively Noaio ad Coveios We deoe by R, R +, ad N, respecively, he ses of all real umbers, oegaive reals ad oegaive iegers. For real umbers a ad b, a b mia, b, a b maxa, b ad [a] + a. We use a o deoe he leas ieger ha is greaer ha or equal o a. 1A deoes he idicaor fucio of eve se A. The space of righ-coiuous R-valued fucios o R + wih lef-had limi is deoed by $ $R +, R ad is edowed wih Skorokhod s J 1 -opology ad he Borel σ-algebra. For a fucio {x; R + } i $, le x represe he lef-had limi a for > ad Δx x x. All sochasic processes are assumed o be radom elemes of $. Covergece i disribuio weak covergece i $ has he sadard meaig ad is deoed by. The quadraic variaio process of a locally square iegrable marigale {M; R + } is deoed by { M ; R + }.

9 238 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors Figure 4. Queue ad Delay Raios as a Fucio of Sysem Size for a Two-Class M /M/s + M Queue wih Arrival Rae Fucios λ 1 β 6 2 si/2, λ 2 β si/2, Service Rae µ 1, Abadome Raes θ 1 θ 2.5, ad Safey Saffig Fucio c : The Cases β 1 ad β 8 We refer he reader o Jacod ad Shiryaev 213, Pag e al. 27, ad Whi 22 for backgroud i weak-covergece ad marigale heory. All radom eiies iroduced i his paper are suppored by a complee probabiliy space Ω, ^, P Prelimiaries There is a se {1,..., K} of cusomer classes. As idicaed i Secio 1.1.4, for he MSHT FCLT, we cosider a sequece of models idexed by he umber of servers. I model, he arrival processes A i are idepede NHPPs wih raes λ i. For i, le Λ i λ i udu,  i 1/2 A i Λ i. 13 The sequece of processes { i } saisfies a FCLT; ha is,  i W i Λ i  i i $ as, 14 where W i represes a sadard Browia moio for each i. Deoe by A i A i he aggregae arrival process. By he assumed idepedece, A is a NHPP saisfyig a FCLT as well wih arrival rae fucio λ i λ i ad associaed cumulaive rae fucio Λ λudu. As i Secio 1, he service imes ad paiece imes are muually idepede, idepede of he arrival processes, ad expoeially disribued, bu hese ca be class-depede. Le µ i ad θ i deoe he service rae ad abadome rae of class-i cusomers, respecively. Remark 2 More Geeral Arrival Processes. We could geeralize he arrival processes from M o G, ad he aalysis would sill go hrough, provided ha we follow he composiio cosrucio as by equaio 2.2 i Whi 215 ad assume a FCLT for he base process; see secio 7.3 of Pag e al. 27. As i Secio 1.1.4, we saff accordig o 4, which maches he iflow ad ouflow o he fluid scale; ha is, boh he queue ad he idleess are zero o he fluid scale. As idicaed i Secio 1.1.2, wih ime-varyig saffig s,

10 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors 239 we eed o specify how we maage he sysem whe all servers are busy ad he saffig is scheduled o decrease. Wha we do is o immediaely eforce ha saffig chage, so ha we force a cusomer ou of service. I he sigleclass case, we ca le oe cusomer reur o he head of he queue, as i Puhalskii 213. I he muliple-class case, he ideiy of he class ha is moved ou of service has a effec o he sysem sae. Our remedy is o creae a highprioriy queue HPQ ad le ay cusomer ha was forced ou of service joi he back of he HPQ. To be specific, we assume ha he mos rece cusomer o eer service is forced back io he HPQ, so ha eerig service i order of arrival is maiaied. We sipulae ha cusomers i he HPQ have he highes service prioriy; ha is, he ex available server always chooses o serve he HoL cusomer i he HPQ firs. I addiio, we require ha o cusomers abado he HPQ. Heceforh, we use Q,i o deoe he umber of class-i cusomers i he HPQ. We will show ha he high-prioriy queue has o impac o he asympoic behavior, regardless of he class ideiies of pushed-back cusomers; ha is, he coe of his high-prioriy queue is asympoically egligible i he MSHT scalig, ad hus does o affec he limi. We assume a work-coservig policy ha is, o cusomers wai i queue if here is a available server. Le Qi represe he umber of cusomers i he ih queue, le Ψi represe he umber of cusomers ha have eered service icludig ay pushed back io he high-prioriy queue, if ay, ad le Ri represe he umber of abadomes of class-i cusomers, respecively, all up o ime. Byflow coservaio Q i Q i +A i Ψ i R i Q i +Πa i Λ i Ψ i Πab i θ i Qi udu, 15 where Πi a ad Πi ab are idepede ui-rae Poisso processes. Le Bi be he umber of busy servers servig a class-i cusomer a ime ad Di he cumulaive umber of class-i cusomer ha have depared due o service compleio up o ime. Agai by flow coservaio, we ge Q,i +B i Q,i +B i +Ψ i D i Bi +Ψ i Πd i µ i Bi udu, 16 where Πi d are ui-rae Poisso processes idepede of Πi a ad Πi ab give i 15. Le Xi umber of class-i cusomers i sysem a ime. Addig up 15 ad 16 yields deoe he oal Xi Q i +Q,i +B i X i +A i D i R i. 17 Aleraively, oe ca derive 17 direcly from flow coservaio. Fially, le Q i Q,i, Q i Qi, ad X i Xi be he oal umber of high- ad lowprioriy cusomers i queues ad he aggregae umber of cusomers i sysem respecively. Addig up 17 over i yields X Q +Q +B X +A D R i, 18 i where we have defied B i B i ad D i D i The High-Prioriy Queue To formally describe he dyamics of he HPQ, we use 6a {u [, ] : Δs u 1} 6 d {u [, ] : Δs u 1} o represe he collecio of ime isaces a which he saffig decreases icreases. The cusomers eer he HPQ accordig o he process A 1B u s u. 19 u 6 a Le D deoe he umber of deparures from he HPQ umber of cusomers ha reeer he service faciliy from he HPQ up o ime. The, i holds ha D 1Q u > + u 6 d 1Q u > dd u. 2

11 24 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors From 19 ad 2, i follows ha Q A D 1B u s u 1Q u > 21 u 6a u 6 d 1Q u > dd u. We ow develop a more racable upper-boud process for he coes of he HPQ. For ha purpose, we cosider a e-ipu process ha allows addiioal arrivals, bu has he same deparure rules. For ha purpose, le he ew e-ipu process be defied by ad apply he oe-dimesioal reflecio mappig ψ o Z o ge Z s s D,, 22 Υ ψz Z if u {Z u}; 23 for example, see secio 13.5 i Whi 22. The followig lemma shows ha Υ serves as a upper boud for Q. Lemma 1. Le Q ad Υ be as give i 21 ad 23, respecively. The Q Υ for all w.p.1. Proof of Lemma 1. By 23 ad 22, i is o hard o see ha Υ 1 Υ u > u 6 a 1 u 6 d 1 Υ u > dd u. 24 Combiig 21 ad 24 gives he desired resul. We ca apply mahemaical iducio over successive eve imes. We see ha he upper boud sysem ca have exra arrivals, bu mus have he same deparures wheever he wo processes are equal. I Secio 6, we will show ha Υ is asympoically egligible i he MSHT scalig, ad so Q has o impac o he MSHT limi Poeial Delays Wihou cusomer abadome, he poeial delay i queue i a ime ca be represeed as he followig firspassage ime: V i if { s : Ψ i + s Q i +A i }. Oe may aemp o icorporae he abadome process R i io he expressio ad wrie V i if { s : Ψ i + s+r i + s Q i +A i }, 25 bu he represeaio 25 isicorrec, because he erm Ri + s may iclude class-i cusomers ha arrived afer ime ad he abadoed; see secio 1 i Talreja ad Whi 29. To formally defie he poeial delay of class i a some ime, we exclude he abadome of cusomers who arrived afer ime ; see secio 4 of Talreja ad Whi 29. Followig he oaio of ha paper, we defie R, i s o be he umber of class-i cusomers who arrived before ime bu have abadoed over he ime ierval [, s. The, he poeial delay i queue i a ime ca be represeed as he followig firs-passage ime Vi if { s : Ψi + s+r, i + s > Qi +A i } The Opimizaio Formulaio We ow iroduce several formulaios, each aimig o miimize he oal cos over a fiie ierval [, T], subjec o he service-level cosrais.

12 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors A Mea-Waiig-Time Formulaio. We sar wih mea-waiig-ime formulaio miimize T s udu, 27 subjec o: E[Vi u] w i u for u T, i, where Vi, asi26, represes he waiig ime of a virual cusomer of class i ha arrives a ime. These SL cosrais sipulae ha he expeced delay i queue i a ime shall o exceed he arge w i. Here, we allow he SL arges w i be fucios i ime. As i Secio 1.3.3, we scale w i wih o pu our sysem io he QED MSHT regime. Assumpio 1 QED Scalig for SL Targes. The SL arge fucios w i are scaled so ha w i w i / for some prespecified fucios w i, i. We ow defie he se of admissible policies. To his ed, we say ha a schedulig policy is oaicipaive if a decisio a ay ime is based o he hisory up o ha ime ad o upo fuure eves. Defiiio 1 Admissible Policies. We say ha a joi-saffig-ad-schedulig policy s, π is admissible if 1 he saffig compoe s follows he SRS rule 4; ad 2 he schedulig compoe π is oaicipaive. We le Π be he se of all admissible policies. Defiiio 2 Asympoic Feasibiliy for he Mea-Waiig-Time Formulaio. A sequece of saffig fucios ad schedulig policies {s, π } is said o be asympoically feasible for 27 ifs, π Π ad lim sup E[Vi /w i ] 1 for all, i. 28 Defiiio 3 Asympoic Opimaliy for he Mea-Waiig-Time Formulaio. A sequece of saffig fucios ad schedulig policies {s, π } is said o be asympoically opimal for 27, if i is asympoically feasible ad for ay oher sequece {s, π } ha is asympoically feasible. [s s ] + o 1/2 as, for all A Tail-Probabiliy Formulaio. We ex cosider a aleraive formulaio represeig he goal of commo call ceers. This formulaio aims o corol he ail probabiliy of he waiig ime of each class. The opimizaio problem is miimize T s udu 3 subjec o: P Vi u > w i u α for u T, i. The se of cosrais requires ha he probabiliy ha a class i cusomer who arrives a ime wais loger ha w i ime uis is o greaer ha α. As meioed i Secio 1.4, his seemigly reasoable formulaio ca be problemaic; for example, because oe ca simply choose o o serve ay class-i cusomer who has waied loger ha he performace arge, wihou violaig ay of he SL cosrais. The difficuly ca be circumveed by addig a global SL cosrai as was doe i secio of Gurvich ad Whi 21. Such a formulaio ad is correspodig soluio will be cosidered shorly. A he mome, we will discuss he asympoical feasibiliy for problem 3 despie he fac ha his formulaio is somewha problemaic. Defiiio 4 Asympoic Feasibiliy for he Tail-Probabiliy Formulaio. A sequece of saffig fucios ad schedulig policies {s, π } is said o be asympoically feasible for 3 if, s, π Π, ad for every ɛ >, lim sup P Vi /w i 1 + ɛ α for all, i. 31

13 242 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors A Mixed Formulaio. As idicaed above, a global SL cosrai is someimes required for he ailprobabiliy formulaio o be well-posed, which aurally leads o our hird formulaio which we call he mixed formulaio: miimize T s udu subjec o: E [Q u] q u for u T, 32 P Vi u w i α for u T, i 1,..., K 1. We recall ha Q represes he oal umber of waiig cusomers i sysem a ime. Agai, we le he arge fucio q scale wih so as o force he sysem o operae i he QED regime. I paricular, we make he followig assumpio by which he uderlyig saffig rule has o be i he form of 4. Assumpio 2 QED Scalig for SL Targes. The SL arge fucio q is scaled so ha q q for some prespecified fucio q. Defiiio 5 Asympoic Feasibiliy for he Mixed Formulaio. A sequece of saffig fucios ad schedulig policies {s, π } is said o be asympoically feasible for 32, if s, π Π, ad for every ɛ >, lim sup E [Q /q ] 1 for all, ad lim sup P Vi /w i 1 + ɛ α for all, i 1,..., K Defiiio 6 Asympoic Opimaliy for he Mixed Formulaio. A sequece of saffig fucios ad schedulig policies {s, π } is said o be asympoically opimal for 32, if i is asympoically feasible ad for ay oher sequece {s, π } ha is asympoically feasible, [s s ] + o 1/2 as, for all The HLDR Corol We ow formalize he HLDR schedulig rule ha uiquely deermies he assigme processes Ψ i. Le Ui be he HoL delay of cusomer i. The, he HoL cusomer i queue i arrived a ime Hi U i. Now, iroduce a se of weigh/corol fucios v v 1,..., v K ad defie a weighed HoL delay Ũ i U i /v i for each i. 35 I addiio, use Ũ o represe he maximum of hose weighed HoL delays ha is, } Ũ max {Ũ1,..., Ũ K max {U 1 /v i,..., UK /v K}. 36 i i Le τ deoe he cusomer class ha has he maximum weighed HoL delay; ha is, { } τ i : Ũi Ũ. 37 We ca he spell ou he assigme processes Ψ i : Ψ i u 7 1τu i, 38 where 7 is he collecio of ime isaces up o ime a which a schedulig decisio is o be made ad τ is give by 37. Here, ies are broke arbirarily. For isace, if Ũ i Ũ i Ũ for i i, he he ex-available server chooses o serve eiher queue i or queue i wih equal probabiliies The TVQR Corol As idicaed earlier, our HLDR corol is iimaely relaed o TV versio of he QR rule sudied i Gurvich ad Whi 29a. We briefly review he FQR corol, which is a special case of he more geeral QR corol iroduced by Gurvich ad Whi 29a, i he coex of muliclass queue wih a sigle pool of idepede ad ideically disribued i.i.d. servers. Agai, le Q i be he queue legh of class i ad Q be he correspodig aggregae

14 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors 243 quaiy. The FQR corol uses a vecor fucio r r 1,..., r K. Upo service compleio, he available server admis o service he cusomer from he head of he queue i where i i arg max i {Q i r iq }; ha is, he ex-available-server always chooses o serve he queue wih he greaes queue imbalace. Here, isead of usig fixed raios, we iroduce a ime-varyig vecor fucio r r 1,..., r K ad he ex-available-server choose o serve a class i cusomer where i i arg max i {Q i r iq }. 4. Mai Resuls I Secio 4.1, we sae our mai resul ad he discuss impora isighs ha i provides i Secio 4.2. We esablish corollaries for impora special cases i Secio 4.3. ISecio4.4, we esablish he associaed resul for he TVQR rule, ad i Secio 4.5, we discuss he asympoic equivalece. I Secio 4.6,weobservehahe resuls i Gurvich ad Whi 29a hemselves ca be exeded o a large class of TV arrival-rae fucios. Fially, i Secio 4.7, we propose soluios o he joi-saffig-ad schedulig problems formulaed i Secio The MSHT FCLT for HLDR i he QED Regime We firs iroduce he diffusio-scaled processes ˆX i 1/2 X i m i ad ˆX 1/2 X m, 39 where Xi represes he umber of class-i cusomers i sysem a ime. Le ˆQ i 1/2 Qi ad ˆQ,i 1/2 Q,i, 4 be he diffusio-scaled queue-legh processes ad ˆQ 1/2 Q ad ˆQ 1/2 Q be he aggregae quaiies. The same scalig was used by Feldma e al. 28, Puhalskii 213, ad Whi ad Zhao 217. As usual, we scale he delay processes by muliplyig by isead of dividig by as i 4: ˆV i 1/2 Vi ad Û i 1/2 Ui for i. 41 We impose he followig regulariy codiios: Assumpio 3. A1 For each i, he arrival-rae fucio λ i is differeiable wih bouded firs derivaive; ha is, here exiss a cosa M 1 > such ha λ i < M 1 for all i ad. The fucios λ i are bouded away from zero; ha is, here exiss λ > such ha λ mi i if λ i > for all. A2 The safey-saffig fucio c is coiuous. A3 All corol fucios v i are coiuous ad bouded from above ad away from zero; ha is, v mi i if v i > ad v max i sup v i <. Our mai resuls esablishes a MSHT FCLT for HLDR i he QED regime. The limi is a se of ieracig diffusio processes. Theorem 1 QED MSHT FCLT for HLDR. Suppose ha he sysem is saffed accordig o 4, operaes uder he HLDR schedulig rule, ad Assumpios A1 A3 hold. If, i addiio, here is covergece of he iiial disribuio a ime, ha is, if ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K, i R 2K as, he we have he joi covergece ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K, ˆV 1,..., ˆV K, Û 1,..., Û K ˆX1,..., ˆX K, ˆQ 1,..., ˆQ K, ˆV 1,..., ˆV K, Û1,..., ÛK i $ 4K, 42

15 244 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors as, where he diffusio limis ˆX i saisfy ˆX i ˆX i µ i ˆXi udu θ i µ i γu 1 v i uλ i u [ ] +du ˆXu cu + λ i u+µ i m i u dw i u, wih γ i v i λ i, ˆX i ˆX i ad W i i.i.d. sadard Browia moios. For each i, [ ] +, ˆQ i γ 1 v i λ i ˆX c [ ] ˆV i Ûi v i γ 1 ˆX c Impora Isighs We ca draw several impora isighs from Theorem The Role of he SRS Safey Fucios c. Give ha he saffig is doe by 4, he behavior o he fluid scale is deermied by he offered load m m m K, where he idividual per-class offered loads m i deped o he specified λ i ad µ i for i. The remaiig compoe of he saffig i 4 is specified by he SRS safey fucio c, which appears explicily i he diffusio limi. Hece, i he limi, he remaiig flexibiliy i he saffig depeds eirely o he sigle fucio c, which remais o be specified. The limiig performace impac of he saffig fucio c ca be see direcly i he limi Sae-Space Collapse. While he sochasic limi process ˆX 1,..., ˆX K for he K-dimesioal scaled umberi-sysem process ˆX 1,..., ˆX K is a K-dimesioal diffusio, depedig o he K i.i.d. sadard Browia moios W i, he limis for he oher processes are all a fucioal of he oe-dimesioal limi process ˆX, i paricular of [ ] +, ˆX c so ha here is grea sae-space collapse. I paricular, he limi processes ˆQi, ˆV i ad Ûi are deermiisic fucioals of each oher, as show by 44. Alhough he poeial ad HoL delays are o he same, heir limis are he same The Role of Cusomer Abadome. Alhough cusomer abadome does ifluece he queue-legh ad waiig-ime limi processes of ieres hrough he oe-dimesioal limi process ˆX, cusomer abadome plays o roles i deermiig hese limiig raios. I is wiped ou i he heavy-raffic diffusio limi. For he -h model, boh arrivals ad deparures occur a a ime scale of 1. Bu because he queue-leghs live o he order of 1/2 i he QED, abadomes occur a a ime scale of 1/2, idicaig a much slower rae. This observaio is cosise wih Whi 26 for he basic M/M/s + M Erlag-A model The Sample-Pah MSHT Lile s Law. We obai he SP MSHT LL direcly from he coclusio of Theorem 1. I paricular, for each i, we see ha, almos surely, For he -h sysem, we have ˆQ i λ i ˆV i for all. 45 or ˆQ i λ i ˆV i +o1 as 46 Qi λ i V i +o as. 47 Tha is, he limi ells us ha Q 1 is O, whereas he error i he SPLL is of a smaller order. Figure 5 depics he idividual sample pahs of Q i ad λ i V i o he same plo for i 1, 2 wih he HLDR policy for he base case. Figure 5, a ad b, shows ha, wih he HLDR rule, he sample pahs chage over ime, bu he wo curves agree closely wih error of small order, which srogly suppors he SP-MSHT-LL.

16 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors 245 Figure 5. Sample Pahs of he Queue-Legh Process Q i ad he Scaled Delay Process v i V i for i 1, 2 wih he HLDR Schedulig Policy Impac of he Arrival Rae ad he Weigh Fucios. Give he limi for he queue-legh processes i 44, we see ha he proporio of class k queue legh of he oal queue legh is icreasig i is isaaeous arrival rae λ k bu decreasig i he isaaeous rae 1/v k Impora Special Cases Theorem 1 applies o he saioary model as a impora special case. Corollary 3 The Saioary Case. Le λ i λ i, v i v i ad c c for. If, i addiio, ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K, i R 2K as, he we have he joi covergece ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K, ˆV 1,..., ˆV K, Û 1,..., Û K ˆX1,..., ˆX K, ˆQ 1,..., ˆQ K, ˆV 1,..., ˆV K, Û1,..., ÛK i $ 4K, as where he diffusio limis ˆX i saisfy ˆX i ˆX i µ i ˆX i udu [ ] +du θ i µ i γ 1 v i λ i ˆXu c + 2λ iwi, i which γ i v i λ i ad ˆX i ˆX i ; for each i ˆQ i v i λ i γ 1 [ ˆX c ] + ad ˆVi Ûi v i γ 1[ ˆX c ] Corollary 3 is i agreeme wih heorem 4.3 i Gurvich ad Whi 29a if oe replaces he sae-depede raio fucio p i here by a fixed raio parameer γ 1 v i λ i.thissuggesssomeformofasympoicequivalece bewee he HLDR corol ad he TVQR corol. I fac, we will show i Secio 4.5 ha a asympoic equivalece exiss o oly for ime-saioary models bu also i ime-varyig seigs. Theorem 4.3 i Gurvich ad Whi 29a has[ ˆX] + ad [ ˆX] i equaio 6, whereas 43 i he prese paper uses [ ˆX c] + ad [ ˆX c]. The discrepacies are due o differe ceerig compoe beig used. I Gurvich ad Whi 29a, he umber of cusomers i sysem is ceered by he umber of servers, whereas we use m o be he ceerig erm.

17 246 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors Remark 3 Cosise wih Previous AP Resuls. The resul i 48 is i aligme wih previous work o AP by Kleirock 1964 ad Saford e al. 214, where he objecive is o achieve desired raios of saioary mea waiig imes experieced by cusomers from he differe classes. By focusig o he QED MSHT regime, we are able o obai a much sroger samplepah resul. If µ i µ ad θ i θ, u, he he limi of he aggregae coe process ˆX is a oe-dimesioal diffusio. Hece, he limi is esseially he same as ha for he sigle-class M /M/s + M model as cosidered by Whi ad Zhao 217 where he aalysis draws upo Puhalskii 213. Corollary 4 Class-Idepede Services ad Abadomes. Suppose ha he codiios i Theorem 1 are saisfied ad µ i µ, ad θ i θ, i. The ˆX, ˆQ 1,..., ˆQ K, ˆV 1,..., ˆV K, Û 1 K,..., Û ˆX, ˆQ1,..., ˆQ K, ˆV 1,..., ˆV K, Û1,..., ÛK where ˆX ˆX µ [ ] + ˆXu cu du θ µ ˆXu cu du + λu+µmu dwu; 49 for each i, [ ] +, ˆQ i γ 1 v i λ i ˆX c ˆV [ ] +. 5 i Ûi v i γ 1 ˆX c If we assume furher ha θ µ i Corollary 4, he he aggregae model is kow o behave like a M /M/ model. Le θ µ 1i49. From 49 i holds ha ˆX ˆX µ ˆXudu + λu+µmu dwu. Hece, he diffusio limi of he aggregae coe process ˆX is a Orsei Uhlebeck OU process wih imevaryig variace The MSHT FCLT for TVQR i he QED Regime We ow ur o he TVQR corol as described by Secio 3.7. Mimickig he aalysis of Gurvich ad Whi 29a, oe ca esablish he MSHT limis, regardig he TVQR rule, via hydrodyamic limis. However, he proof i Gurvich ad Whi 29a is quie ivolved ad i ur relies o addiioal geeral sae-space collapse resuls from Dai ad Tezca 211. Owig o he simpler srucure of he V sysem, we are able o avoid usig he hydrodyamic fucios ad develop a much shorer ad elemeary proof. The proof, which is deferred o Secio 6, adops a similar soppig-ime argume as used by Aar e al. 211 i he aalysis of a ivered-v sysem uder he Loges-Idle-Pool-Firs rouig rule. Theorem 2 QED MSHT FCLT for TVQR. Suppose ha he sysem is saffed accordig o 4, operaes uder he TVQR schedulig rule ad Assumpios A1 ad A2 hold. If, i addiio, ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K i R 2K as, he we have he joi covergece ˆX 1,..., ˆX K, ˆQ 1,..., ˆQ K, ˆV 1,..., ˆV K, Û 1 K,..., Û ˆX1,..., ˆX K, ˆQ 1,..., ˆQ K, ˆV 1,..., ˆV K, Û1,..., ÛK 51 i $ 4K where he diffusio limis ˆX i saisfy ˆX i ˆX i µ i ˆXi udu θ i µ i [ ] +du r i u ˆXu cu + λ i u+µ i m i u dw i u, 52

18 Sochasic Sysems, 218, vol. 8, o. 3, pp , 218 The Auhors 247 where W i are sadard Browia moios. For each i [ ] +, ˆQ i r i ˆX c ad ˆVi Ûi r i [ ] +. λ i ˆX c 53 We gai several isighs from he heorem above: 1 Wih he TVQR, he desired queue raio is achieved i he limi despie he fac ha arrival raes are chagig; 2 from 53, i follows ha boh he poeial ad he HoL delays are iversely proporioal o he arrival rae ad proporioal o he ime-varyig queue raio Asympoic Equivalece of HLDR ad TVQR We firs observe ha for a specific se of corol fucios v v 1,..., v K used i he HLDR rule, oe ca always cosruc a se of ime-varyig queue-raio fucios r r 1,..., r K such ha he resulig TVQR corol ad he HLDR corol are asympoically equivale. Fix he se of corol fucios v v 1,..., v K. Le r k v k λ k i v i λ i for each k. Oe ca easily verify ha he sochasic Equaio 43 becomes Equaio 52. We he observe ha for a specific se of queue-raio fucios r r 1,..., r K, oe ca always fid a se of corol fucios v v 1,..., v K used i he HLDR rule such ha he resulig HLDR corol ad he TVQR corol are asympoically equivale. I fac, he cosrucio is also sraighforward. Le v k r k λ k for each k. Direc calculaio allows us o raslae Equaio 52 io Exedig he QIR Limis o TV Arrivals Eve hough Gurvich ad Whi 29a esablishes MSHT resuls for saioary models, we ow observe ha hese resuls exed immediaely o a large class of models wih TV arrival raes. I paricular, we ow observe ha he heorems 3.1, 4.1, ad 4.3 i Gurvich ad Whi 29a direcly exed o TV arrival-rae fucios ha are piecewise-cosa, wih all chages i he arrival raes occurrig o a fiie subse of he give bouded ierval [, T]. The give proof he applies recursively over he successive subiervals, usig he covergece of he ermial values o each ierval as he covergece of he iiial values required for he ex ierval. Because ay fucio i $[, ], R o a bouded ierval ca be approximaed by a piecewise-cosa fucio over [, T], his resul is quie geeral. However, o rea he case of smooh arrival rae fucios, as cosidered here, a furher limiierchage argume is required. Alhough he remaiig argume may be complex, here should be lile doub ha he exesio holds The Proposed Soluio For each formulaio iroduced above, we propose a soluio ha cosiss of a saffig compoe ad a schedulig compoe. Recall ha v ad r are he raio fucios i he HLDR ad TVQR rule, respecively, ad c is he TV safey saffig fucio Mea-Waiig-Time Formulaio. We sar wih he mea-waiig-ime formulaio as give by Saffig: Choose c ha saisfies E [ ˆX c ] + ϑ wih ϑ λ i w i. 54 i 8 Schedulig: 1 Apply HLDR wih raio fucios v v 1,..., v K w 1,..., w K, 55 or 2 use TVQR wih raio fucios r r 1,..., r K λ 1w 1,..., λ K w K /ϑ. 56