Service-Level Differentiation in Many-Server Service Systems via Queue-Ratio Routing

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1 Publshed onlne ahead of prnt October 28, 2009 OPERATIONS RESEARCH Artcles n Advance, pp ssn X essn nforms do /opre INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. Servce-Level Dfferentaton n Many-Server Servce Systems va Queue-Rato Routng Ita Gurvch Kellogg School of Management, Northwestern Unversty, Evanston, Illnos 60208, -gurvch@kellogg.northwestern.edu Ward Whtt Department of Industral Engneerng and Operatons Research, Columba Unversty, New York, New York 10027, ww2040@columba.edu Motvated by telephone call centers, we study large-scale servce systems wth multple customer classes and multple agent pools, each wth many agents. To mnmze staffng costs subect to servce-level constrants, where we delcately balance the servce levels (SLs) of the dfferent classes, we propose a famly of routng rules called fxed-queue-rato (FQR) rules. Wth FQR, a newly avalable agent next serves the customer from the head of the queue of the class (from among those he s elgble to serve) whose queue length most exceeds a specfed proporton of the total queue length. The proportons can be set to acheve desred SL targets. The FQR rule acheves an mportant state-space collapse (SSC) as the total arrval rate ncreases, n whch the ndvdual queue lengths evolve as fxed proportons of the total queue length. In the current paper we consder a varety of servce-level types and explot SSC to construct asymptotcally optmal solutons for the staffng-and-routng problem. The key assumpton n the current paper s that the servce rates depend only on the agent pool. Subect classfcatons: queues, networks: multple classes, server pools; queues, optmzaton: desgn, staffng, routng; queues, lmt theorems: asymptotcoptmalty, many-server heavy-traffclmts. Area of revew: Stochastc Models. Hstory: Receved January 2007; revsons receved October 2007, September 2008, January 2009; accepted Aprl Publshed onlne n Artcles n Advance. 1. Introducton Large call centers usually serve multple classes of customers havng dfferent servce requrements and dfferent perceved value. The servces provded by the call center agents usually requre specal sklls, but t s usually not possble or cost effectve for all agents to have all sklls. Wth current technology, call centers have the capablty of routng calls to approprate agents wth the requred sklls, usng some form of skll-based routng (SBR), but t remans challengng to perform SBR effectvely; see 5 of Gans et al. (2003). Call centers usually specfy ther operatonal obectves n the form of qualty-of-servce (QoS) constrants. Followng common practce, we wll focus on the x-y servcelevel (SL) constrant, whch stpulates that x% of the calls should be answered wthn y seconds. We let the call center have dfferent SL constrants for dfferent customer classes; e.g., wth both regular and VIP customers, we mght am to respond to 80% of regular customers wthn 30 seconds, but 80% of VIP customers wthn 10 seconds. In ths context, the total problem has three components: desgn, staffng, and routng. In the desgn phase, we start by groupng the customers nto classes and the agents nto servce pools. (In dong so, we assume that the customers wthn classes are homogeneous, as are the agents wthn pools.) Then we must decde what sklls each pool should have,.e., whch classes they are allowed to serve. In the staffng phase, we must decde how many agents should be n each servce pool. Fnally, n the routng phase we must decde how the agents should be assgned to customers n real tme. The total problem s typcally large and complex, so that t s unproductve to search for an optmal soluton. Thus we look for a good, smple soluton, that produces nearoptmal performance n a relatvely smple way. In partcular, we hope to turn the large scale nto an advantage nstead of a dsadvantage by fndng relatvely smple procedures that become more effectve as the scale ncreases. Indeed, we want to fnd a relatvely smple approach that s asymptotcally optmal for specfed problems as the scale ncreases. Our goal s to acheve smplcty and asymptotc optmalty A Smple Intutve Routng Rule: FQR When consderng possble controls, we thnk we should seek controls that are ntutve and structurally smple. Controls that lack any evdent structure or nsght are unlkely to be used by call center managers. A good example of a smple and ntutve control that s applcable to very general network structures (but essentally lmted to 1

2 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems 2 Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. sngle-agent servce pools) s the generalzed-c (Gc) rule, frst ntroduced by Van Meghem (1995) for a model wth multple classes and a sngle server (also known as the V model), and generalzed to more complcated networks by Mandelbaum and Stolyar (2004). A parallel to Mandelbaum and Stolyar (2004) n a many-server settng has been provded by Atar (2005), who characterzes a famly of controls that acheve asymptotcally optmal performance n the QED regme. (See Gurvch and Whtt 2009b for more dscusson.) Although the controls n Atar (2005) can be mplemented easly n a computerzed envronment, they are not nearly as smple as the Gc rule. Thus, t seems desrable to seek a famly of controls for manyserver systems that brdge the gap between the smple and ntutve Gc rule n Mandelbaum and Stolyar (2004) and the more complcated controls n Atar (2005). Wth that goal n mnd, we propose fxed-queue-rato (FQR) routng. We assume that there s a queue for each customer class. When an agent becomes free, he chooses the customer from the head of the lne (from one of the classes he can serve) for whch the queue length most exceeds a fxed proporton p of the total queue length (for all classes). The proportons p are n turn chosen to depend on the specfed SL constrants. The FQR rule s a specal case of the queue-and-dleness-rato (QIR) famly of controls that we ntroduce n Gurvch and Whtt (2009a). A consequence of Gurvch and Whtt (2009b) s that FQR makes the separate queue lengths asymptotcally proportonal to the total queue length. In other words, FQR produces a very mportant state-space collapse (SSC), causng the vector-valued queue-length process to evolve, asymptotcally, as a one-dmensonal process. In addton, FQR s a smple balancng rule lke the Gc rule and, lke the Gc rule, t s a hghly decentralzed control. The key assumpton that we make here s that the servce rates are pool dependent,.e, that the servce tme of a customer depends on the type (pool) of the agent that provdes the servce, but not on the customer class. Under ths assumpton, SSC reduces the multdmensonal system dynamcs to a tractable one-dmensonal process. Ths reducton allows us to provde closed-form expressons for the staffng levels and prove that our proposed soluton s not only feasble, but also asymptotcally optmal. Our soluton stands on two pllars: () staffng va reducton of the multclass multpool system to a sngle-class multpool system, and () a smple routng rule that smultaneously makes the system perform as effcently as the sngle-class multpool system and takes care of the servcelevel dfferentaton. Our reducton approach to staffng llustrates our emphass on smplcty: We propose frst choosng the total number of agents by aggregatng the servce-level constrants and actng as f all customers have access to all agents. Thus, we reduce the orgnal SBR system nto a sngleclass multpool call center known as the nverted-v model; see Fgure 1. Fgure 1. SBR An SBR model and ts correspondng model = Model In the second step, we am to satsfy the ndvdual class-level QoS constrants by approprately routng the customers n the system. Because the staffng and desgn decsons are hghly nterdependent, our proposed approach may seem nave and nadequate. However, we show that the FQR routng rule that we use n the second step guarantees that the two-step soluton s nearly optmal, thus decomposng the ont optmzaton problem of desgn, staffng, and routng nto more elementary problems that can be addressed sequentally. In our sequel, Feldman et al. (2007), we remove the pooldependence assumpton and provde a general asymptotc feasblty (but not optmalty) result. Based on the feasblty result, we then construct smple smulaton-based optmzaton procedures to solve the desgn, staffng, and routng problem for more-general SBR systems Related Lterature Several recent papers have used the smplfed reducton approach to staffng. Theoretcal support s contaned n Armony (2005) and Gurvch et al. (2008). These papers establshed asymptotcoptmalty of that staffng approach wth approprate routng for specal classes of models as the total arrval rate ncreases. The frst paper consdered models wth a sngle customer class and multple agent types, whereas the second consdered symmetrc models wth multple customer classes, but a sngle agent pool. Ther asymptotcoptmalty follows Borst et al. (2004), whch formulated and establshed asymptotcoptmalty for the sngle-class, sngle-pool M/M/N queue. The asymptotc framework s the now-famlar many-server heavy-traffc lmtng regme ntroduced by Halfn and Whtt (1981), whch s also known as the qualty-and-effcency-drven (QED) regme. In the QED regme the arrval rate and numbers of servers both ncrease, whereas the servce-tme dstrbuton remans unchanged. These two lmts are coordnated so that the probablty of delay approaches a lmt strctly between 0 and 1. Borst et al. (2004) showed that the QED regme arses naturally from economc consderatons. We wll be consderng the QED regme throughout ths paper.

3 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS 3 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. The smplfed reducton approach to staffng was also a central dea n Wallace and Whtt (2005), whch developed a smulaton-based teratve algorthm for staffng an SBR call center that starts by choosng an ntal total number of agents by actng as f the call center were a sngleclass sngle-skll call center. After ntal skll requrements are assgned, smulaton s used teratvely to fnd detaled staffng and skll requrements so that the SL and other QoS constrants are met. The approach n Wallace and Whtt (2005) has two shortcomngs, that we address here. Frst, that approach requres an teratve smulaton algorthm to adust staffng levels and skll assgnments n order to satsfy the class-dependent QoS constrants. Because servce s performed n a relatvely short tme scale compared to staffng, we thnk t should be more effectve to prmarly rely on the routng rather than the staffng n order to acheve desred servce dfferentaton. In ths paper we provde a way to do that. Second, although the approach n Wallace and Whtt (2005) seems to become more effectve as the scale ncreases, t has not yet been shown to be asymptotcally feasble or optmal as the scale ncreases. Here, n contrast, we establsh asymptotc optmalty. In ths paper we assume that the servce rates are pool dependent. A specal case s the system wth common servce rates, e.g., as consdered n Wallace and Whtt (2005). The analyss n ths paper reles heavly on our prevous paper (Gurvch and Whtt 2009a), whch establshes SSC results for the QIR generalzaton of FQR. We establsh asymptotc optmalty for the case of convex holdng costs n Gurvch and Whtt (2009b). An mportant contrbuton here s smultaneously addressng the three problems of desgn, staffng, and routng. Conventonally, these are treated separately and herarchcally. Wallace and Whtt (2005) also addressed these three problems together, but the only prevous work we are aware of that establshes asymptotcfeasblty or optmalty for all three problems s Bassamboo et al. (2006a, b), Bassamboo and Zeev (2008). They establsh asymptotcoptmalty for the problem of mnmzng costs assocated wth watng, abandonments, and customer reectons. In Bassamboo and Zeev (2008) they also consder abandonment constrants, but not tal-probablty SL constrants. Ther analyss s nterestng because t focuses on uncertanty n the arrval rates. As a consequence, they consder a dfferent lmtng regme, the effcency-drven (ED) regme. Ther general settng allowng for uncertanty n the arrval rates comes at the prce of havng to restrct the analyss to a cruder noton of asymptotc optmalty than the one we use here. Our fner analyss, although more lmted n ts scope, allows us to dentfy key system characterstcs and, n turn, to construct ntutve routng schemes. Moreover, t allows us to tackle drectly watng-tme tal-probablty SL constrants that are wdely used n the ndustry. In addton, the routng scheme we propose n ths paper s used n Gurvch et al. (2008) to construct a staffng and routng algorthm for a call center wth uncertan arrval rates operatng under SL constrants. Wthn the context of sngle-server statons, several papers have tackled the problem of SL constrant satsfacton. Notable s Van Meghem (2003), whch embeds the constrant-satsfacton problem nto the convex holdng cost settng of Van Meghem (1995), rather than dealng wth t drectly. Organzaton of the Paper. In 2 we ntroduce the model and ntal problem formulaton. The proposed desgn, staffng, and routng soluton s ntroduced n 3 and the asymptotcoptmalty results are stated n 4. We ntroduce and solve addtonal problem formulatons n 5. We state conclusons n 6. Some proofs and auxlary results appear n the e-companon, whch s avalable as part of onlne verson that can be found at nforms.org/. 2. The SBR System and the Problem Formulaton We consder a system wth a set I = 1I of customer classes and a set = 1J of agent types. The number of agents of type (whch wll be a decson varable) s denoted by N ; let N= N 1 N J. Class- customers arrve accordng to a Posson process wth rate and = I s the aggregate arrval rate. If a type- agent can serve a class- customer, we let be the correspondng servce rate. Alternatvely, the mean handlng tme of a class- customer by a type- agent s 1/. Akey assumpton n ths paper s that the servce rates are pool dependent. That s, for each, wehave = for all I that can be served by pool. Throughout, we wll assume, wthout loss of generalty, that the servce rates are labeled n decreasng order: 1 2 J At ths pont, we do not consder customer abandonment; see 5.2 for extensons to that case. The possble-routng graph for ths SBR system has a natural representaton as a bpartte graph (see Fgure 1) wth vertces V = I;.e., V s the unon of the set of customer classes and the set of agent pools. Then, the only edges we consder connect customer classes to agent pools: E = I can serve. An edge s present n the routng graph f class- customers can be served by type- agents. We let I be the set of customer classes that can be served by pool,.e, I= I E, and, symmetrcally, we defne J= I E to be the set of pools that are qualfed to serve class- customers. In addton, we assume that agents of type- ncur a cost of c per unt of tme. We also wll allow the system to mpose addtonal constrants on the staffng vector to reflect unon contracts, hrng and

4 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems 4 Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. tranng constrants, or other manageral consderatons; see 2.1 for the formal modelng of these constrants. For the asymptotc analyss, we wll construct a sequence of SBR systems ndexed by the aggregate arrval rate. The servce rates and the routng graph are held fxed. We wll make the dependence on the ndex explct by addng the superscrpt to all the relevant parameters and processes. We assume that the ratos a = / reman constant for all. Also, we let the SL target T scale wth to put the system nto the QED regme. Assumpton 2.1 (QED Scalng for SL Targets). The SL targets T, I are scaled so that T = T / for some strctly postve constants T, I. The proposed staffng and routng soluton wll be completely defned n terms of the orgnal targets T ; there wll be no use of the constants T. The scalng s only used for the proof of asymptotcoptmalty The Problem Formulaton To formulate our optmzaton problem, let A t be the number of class- customers to arrve by tme t. Let W T be the average watng tme of all customers that arrved up to tme T ; let F T be the emprcal dstrbuton of the watng tme of class- customers up to tme T ; and let be ts complement;.e., F T I A T W T =1 k=1 wk = and A T A T F T k=1 1wk y = >y A T where A T = I A T, w k s the realzed watng tme of the kth class- customer to arrve to the system after tme 0, and 1B s the ndcator of the event B, whch s equal to 1 f B occurs, and 0 otherwse. An ntal formulaton, representng common call-center goals, can then be stated as follows: mnmze c N subect to lm sup T N F T T 1 I where s a subset of J + that s generated by lnear constrants. Specfcally, we allow constrants of the form N = b, where b = N J + : A N b, for some matrx A d J, d +, and b = ˆb for some ˆb d. We also let = b be the set obtaned from by relaxng the ntegralty assumptons. That s, = N J + : A N b. If the set of possble staffng vectors s unconstraned, we set +. We observe that to consder optmalty, (2) s not well formulated, because we have not yet suffcently constraned (1) (2) the polces. So far, the formulaton permts gvng some customers satsfactory performance at the expense of gvng other customers (n the proporton 1 ) arbtrarly poor performance. Ths problem s dscussed extensvely at the end of 2 of Gurvch et al. (2008), so we wll be bref here. To llustrate the dffcultes, note that we could elect not to serve class- customers who have wated longer than T. Even f we requred frst-come frst-served (FCFS) servce wthn each class, we could satsfy all the constrants wth relatvely lmted staffng by dsallowng any watng,.e., by usng a pure-loss model. Clearly, n a loss model all the customers that do enter the system do not experence any wat, and we may choose the number of agents so that the blockng probablty s less than 1. That s clearly an undesrable outcome because many customers are blocked and do not receve servce at all. Even when requrng that all customers be served, hghly undesrable polces are possble, such as the alternatng-prorty control dscussed by Gurvch et al. (2008). As a consequence, we modfy (2) by addng an addtonal constrant; n partcular, we ntally consder the followng best-effort optmzaton problem: mnmze c N subect to lm sup T lm sup T N W T T I F T T 1 I 1 We emphasze that t s not suffcent to add the global average-watng-tme constrant. It s also mportant to remove the ndvdual SL constrant of class I. Otherwse, the problems wth formulaton (2) reman unresolved. Indeed, f the global average-watng-tme constrant s very loose, one can show that t s possble to construct alternatng prorty controls as llustrated n Gurvch et al. (2008) to construct polces under whch, part of the tme, each class experences extremely low servce levels. The dffcultes n formulaton (2) can be avoded n other ways, e.g., consderng only average-watng-tme constrants. Such a formulaton and ts correspondng soluton are consdered n 5. We frst focus on the formulaton (3). We now defne the set of admssble polces. To ths end, we say that all customers are served f t s not allowed to block or overflow customers;.e., we requre that for all t 0, Q t = A t D t Z t, where D t and Z t are, respectvely, the number of class- departures from the system up to tme t and the number of class- customers n servce at tme t. We say that a routng polcy s nonantcpatve f a decson at any tme s based on the hstory up to that tme and not upon future events. We say that a routng polcy s nonpreemptve f customers stay n servce wth the agent frst assgned to them untl ther servce s complete once an agent has been assgned. (3)

5 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS 5 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. Defnton 2.1 (Admssble Routng Polces). We say that a routng polcy s admssble f: 1 t s nonantcpatve, 2 t s nonpreemptve, and 3 all customers are served. Let be the set of all admssble routng polces. We conclude ths secton wth the defnton of asymptotcfeasblty. Defnton 2.2 (Asymptotc Feasblty). A sequence of staffng vectors and routng polces N s asymptotcally feasble for (3) f (a), for all, and (b) for every >0, there exsts T such that, for all T T, { W T lm sup P TI and lm sup } 1 + (4) PF T T + 1 I 1 (5) Asymptotcfeasblty holds for (2) nstead of (3) f (4) and (5) above are replaced by lm sup PF T T + I (6) Gven two postve real-valued functons f and g, we say that fx s ogx (as x ) f f x/gx 0 as x ; we say that fx s Ogx f f x/gx s bounded as x. The defnton of asymptotcoptmalty wll be the same for all the formulatons that we consder n ths paper. Hence, we do not specfy a specfc formulaton wthn the defnton. For the rest of the paper, asymptotc optmalty wll always be n the sense of Defnton 2.3 below, where asymptotcfeasblty wll depend on the context. The o n the asymptotcoptmalty condton (7) below corresponds to asymptotc optmalty n the dffuson scale, whch s more refned than asymptotc optmalty n the flud scale, whch would nvolve a larger bound on the error of o as. (The staffng levels wll be O.) Defnton 2.3 (Asymptotc Optmalty). A sequence of staffng vectors and routng polces N s asymptotcally optmal f t s asymptotcally feasble, and c N c N + = o as (7) for any other sequence N of asymptotcally feasble staffng vectors and routng polces. We end ths secton wth a bref dscusson of the relaton between our notons of asymptotcfeasblty and optmalty, and the more tradtonal steady-state feasblty and optmalty. Remark 2.1 (Steady-State Constrants). Although actual call-center operatons nvolve fnte-horzon decsons and constrants, the tradtonal way of call-center modelng would be to wrte both (2) and (3) wth steady-state constrants nstead of the fnte-horzon ones. To obtan the steady-state formulaton, one would replace the ndvdual tal constrants wth PW >T, where W s the class- steady-state watng tme. The global average delay constrant s replaced wth the constrant EW TI where W s the steady-state global watng tme,.e, W s equal n dstrbuton to I /W. Even though these alternatve formulatons dffer lttle from a practcal perspectve, they are mathematcally dfferent. They are asymptotcally equvalent only f one can establsh a certan lmt-nterchange result. Such lmt-nterchange arguments are elementary for some models, such as the nverted-v model, but t s a complex task for the general SBR settng. In ths paper we restrct the attenton to the long-run average formulaton n (2) and (3). What we do parallels what s done wth smulaton A Lower Bound Our soluton wll be based on a reducton of the SBR system to a more elementary model n whch multple agent types serve a sngle customer class, also known as the nverted-v (or ) model. Gven an SBR system, the assocated model has the same set of agent-pools, the same staffng levels N, and the same servce rates. In addton, the arrval rate of the sngle customer class s the sum of the arrval rates n the SBR system. An example of an SBR system and ts correspondng model s gven n Fgure 1. Clearly, the model s not as smple as the M/M/N queue. However, when t s optmally operated, ts asymptotcperformance leads to smple expressons for staffng, as has been shown by Armony (2005). We wll explot the results n Armony (2005) here. In partcular, we wll explot a result for the model, statng that EW T I only f where s the unque soluton to P 1 P 1 = = T I = T I and N + + o (8) [ 1 + / 1 / ] 1 1 / 1 wth and beng, respectvely, the standard normal pdf and cdf. Here, P 1 s the asymptotcdelay probablty n the model operated under the fastest-serverfrst (FSF) polcy as ntroduced by Armony (2005). That s, assumng that N = + + o and that FSF s used (plus addtonal techncal condtons), Armony (2005) shows that PW >0 P 1, wth W beng the steady-state watng tme n the model. We note that the necessary condton (8) s gven n Armony (2005) for steady-state asymptotcfeasblty, (9)

6 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems 6 Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. whereas we focus here on the somewhat weaker noton that appears n Defnton 2.2. However, we fnd that the same necessary condton holds f one consders the same model, but wth long-run average constrants (as n (3)) and wth our noton of asymptotcfeasblty. Ths s proved n Lemma EC.2.2 of the onlne appendx whch, n turn, s a key step n the proof of Theorem 2.1 below. We now use ths necessary condton to construct a lower bound for the staffng of the SBR model. In dong ths constructon we wll use two facts: () In contrast to the SBR system, customers n the model have access to all agent pools. It s ntutvely clear, then, that f a gven staffng vector s not suffcent for the gven aggregate watng-tme target n the model, t wll also not be suffcent n the less effcent SBR system. Consequently, to meet the global watng-tme constrant t s necessary that N + + o. () Because we have no abandonments n the system, the capacty should suffce to serve all customers (at least at a flud scale). In partcular, any feasble staffng vector must satsfy that N y I J for some vector y such that Iy 1 and y, E are postve. Together, these two nformal arguments suggest that an asymptotclower bound for the optmzaton problem (3) should be gven by Mnmze c N Subect to: N + J J N y y 1 I N I y 0 E (10) We call a staffng vector determned through the soluton of (10) a -based staffng. A standard argument shows that the optmzaton problem (10) can be solved by solvng an assocated LP that yelds the same set of optmal solutons. Because we are not concerned wth the way n whch (10) s solved, we wll use (10) drectly. The next theorem provdes the formal lower-bound result. Its proof s gven n the onlne appendx. Theorem 2.1 (Lower-Bound Capacty). Consder the sequence of SBR systems and let N be a sequence of asymptotcally feasble staffng and routng rules such that lm nf N > 0 Then, N 0 satsfes that N + + o (11) where s the -model parameter n (9). Theorem 2.1 only provdes a lower bound. We wll next propose a soluton that we wll prove acheves ths lower bound. The soluton wll be based on the -based staffng and the FQR routng rule. 3. The Proposed Soluton Our soluton conssts of a staffng component and a routng component. The staffng that we use s the -based staffng determned by an optmal soluton to (10). For the routng component, we use FQR wth ratos that wll be explctly determned as functons of the servce-level targets T. Let Q t be the number of class- customers n queue. Let Z t be the number of type- servers busy gvng servce to class- customers, so that X t = Q t+ Z s the overall number of class- customers present n the system at tme t, and I t = N I =1 Z t be the number of dle agents n pool at tme t n the th system. Accordngly, I t = J =1 I t s the total number of dle agents n the system. Let X t be the overall number of customers n the system (n servce and n queue),.e., ( I I J X t = X t = Q t + Z ) t =1 =1 and let N t = N be the aggregate number of agents. Below we use arg max and let t have the standard defnton;.e., gven a functon f A, wth A a fnte set, let arg max f= y A f y = max x A fx. Defnton 3.1 (FQR for the SBR Model). Gven two probablty vectors v= v and p= p I, FQR for the SBR model s defned as follows: Upon arrval of a class- customer at tme t, the customer wll be routed to an avalable agent n pool, where =1 t arg max I t v X t N JI t>0.e., the customer wll be routed to an agent pool wth the greatest dleness mbalance. If there are no such agents, the customer wats n queue, to be served n order of arrval. Upon servce completon by a type- agent at tme t, the agent wll admt to servce the customer from the head of queue where t arg max Q t p X t N + IQ t>0.e., the agent wll admt a customer from the queue wth the greatest queue mbalance. If there are no such customers, the agent wll reman dle.

7 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS 7 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. Tes are broken n an arbtrary but consstent manner, so that the vector-valued stochastc process Q Z s a contnuous-tme Markov chan (CTMC) wth statonary transton probabltes. To explctly express the dependence on the vectors p and v, we wll use the notaton FQRpv. We pont out that f p > 0 for all I, then FQR s equvalently gven by havng each newly avalable agent choose the customer from the head of queue, where { Q t arg max t IQ t>0 p } (12) whch makes the use of X t N + unnecessary. Choosng p and v. For the routng component of our soluton, we wll be usng FQR wth the rato vectors p v, where v = 001 and p s the unque soluton to P 1 e I 1/p I 1 TI 1 = and p = T (13) p I 1 I 1 TI 1 for P 1 defned n (9). Because TI = T (see Assumpton 2.1) and / I 1 = a /a I 1, the value of pi 1 s ndependent of. Consequently, so are the values p for I 1. The choce of the rato vector p wll be ustfed by () a sample-path verson of Lttle s law that holds for the many-server servce system, () the SSC that s nduced by FQR, and () the fact that t performs asymptotcally as effcently as the model. Informally, SSC ustfes the followng sequence of approxmatons PW I 1 >T I 1 PQ I 1 > I 1 T I 1 Pp I 1 Q > I 1 T I 1 P {Q > } I 1TI 1 p I 1 PQ > 0e N / I 1 TI 1 /pi 1 where Q s the steady-state queue length n the model that s constructed from the SBR system, as n the begnnng of 2.2. The last step follows from smple expressons for the dstrbuton of the queue length for the model. By the analyss of model n Armony (2005), the probablty PQ > 0 converges to P 1 f we use the -based staffng. Also, N = + + o. Hence, PW I 1 >T I 1 P {Q > } I 1TI 1 p I 1 P 1 e I 1/p I 1 TI 1 (14) Smlar nformal arguments can be repeated for each of the customer classes. Fnally, because was chosen so that the rght-hand sde n (14) equals, the -based staffng and FQR should provde an asymptotcally feasble soluton to (3). Because the -based staffng s a lower bound, ths soluton s also asymptotcally optmal. Ths nformal argument s formalzed n the next secton. 4. Asymptotc Feasblty and Optmalty We begn to consder the lmtng behavor as. We wll show that the -based staffng and FQR, wth approprately chosen ratos, yeld an asymptotcally feasble and optmal soluton for (3). Frst, however, we consder the desgn of the system. In order to dentfy the desgn, t suffces to look at (10) wth one constrant removed,.e, consder the followng nonlnear optmzaton problem: Mnmze c N Subect to: J N y y 1 I N I y 0 E (15) The soluton to the mathematcal program (15) can be regarded as a frst-order determnstcflud approxmaton for the SBR system, as n Whtt (2006). From that pont of vew, gven a selected soluton Nȳ, we would then use N to provde an ntal estmate of the staffng and ȳ to provde an ntal estmate of the approprate routng. We pont out that the soluton to (15) s ndependent of. Indeed, by the defnton of and the assumpton that = a, (15) s equvalent to the mathematcal program: Mnmze c Subect to: J x a x 1 I A b 0 I x 0 E (16) Both mathematcal programs (15) and (16) can be replaced wth lnear programs (LP) that yeld the same optmal soluton. Henceforth, we only refer to optmal solutons of (15), wthout consderng how they are obtaned. We denote an optmal soluton to (15) by N ȳ. Our frst assumpton requres a weak form of unqueness of optmal solutons to (15). Assumpton 4.1 (Unqueness of Staffng). Fx and let N ȳ and N ỹ be two optmal solutons to (15). Then N = N. We note that due to the equvalence between (15) and (16), f Assumpton 4.1 holds for a gven, then t holds for all. Smlarly, the equvalence between (15) and (16) mples that f N > 0 for one, then the same holds for all values of. Informally, Assumpton 4.1 s requred because we wll want to use the optmal solutons of (15) as a frstorder (flud) approxmaton for the staffng (and not only

8 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems 8 Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. the staffng cost) that we get from the -based staffng as defned by the soluton to (10). In some smplfed settngs such as the one consdered n 5.4, ths assumpton can be removed. The second assumpton s a crtcal loadng assumpton, needed to put the system n heavy traffc. Assumpton 4.2 (Crtcal Loadng). For any 0, N = for any optmal soluton N ȳ to (15). Fnally, we make the followng structural assumpton. Below, E and V are as defned n the begnnng of 2. Also, we say that a graph s connected f there exsts a path between every two nodes n the graph. Assumpton 4.3 (Connected Routng Graph). For any 0, there exsts an optmal soluton N ȳ for (15) such that the graph E N ȳ = I ȳ > 0 s a connected subgraph of GV E. As before, the equvalence between (15) and (16) guarantees that f Assumpton 4.3 holds for a gven, then t holds for all. Ths connected-graph assumpton s crucal for the ablty to nstantaneously balance the system by redrectng capacty from one customer class to the other; see 2.7 of Atar (2005) for elaboraton. Assumptons are assumed to hold throughout the rest of the paper. Wth the above defntons, we can state our asymptotc optmalty result. Theorem 4.1 (Asymptotc Optmalty for the SBR Model wth Pool-Dependent Rates). Suppose that any optmal soluton for (15) has N > 0 for all. Let N be determned through the -based staffng n (10) wth as n (9). Set to FQRp v wth p as n (13) and v = Then, the sequence N s asymptotcally optmal for (3). Remark 4.1 (Choosng the Rato Vector v ). In lght of our SSC result n Theorem 3.1 of Gurvch and Whtt (2009a), the choce v = 0 01 wll cause all the dleness to be concentrated n pool J, whch s the slowest agent-pool. Ths choce guarantees that all the faster servers wll be constantly busy, thus maxmzng the depleton rate of customers from the system. Informally, then, ths choce of v mnmzes the aggregate queue length n the system by maxmzng the depleton rate. Because ths observaton holds for any staffng level, ths choce of v s essental for the mnmzaton of the number of agents requred to acheve the aggregate watng-tme constrants. Once the aggregate queue length s mnmzed, t only remans to dstrbute t n a proper way to ensure that the SL constrants are met. The queue-rato vector, p, takes care of ths task. Theorem 4.1 llustrates one of the key benefts of FQR. Although the model s a more effcent system, FQR allows the SBR system to work as effcently, asymptotcally, makng the staffng of the model suffcent also for the SBR system. Asymptotc Feasblty for (2). We now dscuss an asymptotcally feasble soluton for the SBR problem (2). Although ths formulaton s somewhat problematc, as dscussed n 2, t s very common n ndustry. Hence, t s of nterest to dscuss the constructon of feasble solutons for ths problem. We now defne p to be p = T a = T (17) k I k Tk k I a k T k and redefne to be the unque soluton of P 1 e I /T = (18) where P 1 defned n (9). As before, we observe that the vector p s ndependent of, because / = a and Assumpton 2.1 holds. Theorem 4.2 (Asymptotc Feasblty for the SBR Model wth Pool-Dependent Rates). Suppose that any optmal soluton for (15) has N > 0 for all. Let N be determned through the -based staffng n (10) wth as n (18). Set to FQRp v wth p as n (17) and v = Then, the sequence N s asymptotcally feasble for (2). Both Theorems 4.1 and 4.2 rely on the fact that FQR admts an mportant SSC result by whch, asymptotcally, the queues of class are equal to the proporton p of the aggregate queue length, and the number of dle servers n pool s equal to a proporton v of the aggregate number of dle servers. Somewhat nformally, the SSC results guarantee that wth the staffng and FQR we wll have that, for all I and Q t v Q t 0 as and I t p I t 0 as where Q t and I t are, respectvely, the aggregate queue length and aggregate number of dle agents at tme t. The SSC result for ths settng s a corollary of our more general result n Theorem 3.1 of Gurvch and Whtt (2009a). 5. Other Formulatons Ths secton s dedcated to alternatve formulaton of the call-center optmzaton problem Constrants on Average Delay Here, we consder the formulaton. mnmze c N subect to lm sup T N W T T I (19)

9 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS 9 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. where W T = A T k=1 wk A T Defnton 5.1 (Asymptotc Feasblty for (19)). A sequence of staffng vectors and routng polces N s asymptotcally feasble for (19) f: (a), for all, and (b) for every >0, there exsts T such that, for all T T, { W T lm sup P T } 1 + I (20) The proposed soluton n ths case s as follows: Staffng: We use the optmal soluton to (10) wth now gven by the unque soluton to P 1 = a T = a T (21) I I where agan P 1 s defned n (9). Routng: FQR wth rato vectors v = 0 01 and p that s gven by p = T a = T (22) k I k Tk k I a k T k Theorem 5.1 (Asymptotc Optmalty for the Average-Watng-Tme Formulaton). Suppose that any optmal soluton for (15) has N > 0 for all. Let N be determned through the -based staffng n (10) wth as n (21). Set to FQRp v, where p s as n (22) and v = 001. Then, the sequence N s asymptotcally optmal for (19) Addng Customer Abandonment In ths secton we augment our model by ntroducng customer abandonment. Specfcally, we assume that a class- customer has an exponental patence wth rate. If hs patence expres before he s admtted to servce, the customer wll abandon. Patence tmes of dfferent customers are mutually ndependent. We wll formulate an optmzaton problem for the abandonment model and provde the correspondng asymptotc feasblty and optmalty results. Our asymptotcoptmalty result for ths augmented model s lmted to the case n whch all customer classes have the same abandonment rate,.e, when. It s of practcal nterest, however, that we can provde asymptotcally feasble solutons for the case n whch these rates are dfferent. To prove the asymptotcresults for the abandonment model, we explot Armony and Mandelbaum (2008), whch extends the results of Armony (2005) to the model wth customer abandonments. Usng Armony and Mandelbaum (2008), we are able to provde proofs that parallel those for the nonabandonment case, repeatng the analyses of (2), (3), and (19). Asymptotcfeasblty results can be obtaned for general patence rates, whereas asymptotcoptmalty can be obtaned only for the homogeneous case. Rather than repeatng the analyss for formulatons (2) and (3), we ntroduce and solve a dfferent formulaton wth constrant on the fracton of abandonng customers. To formulate ths problem, let L t be the number of class- customers that abandoned before beng served, up to tme t. The fracton of customers that abandoned among those that were ntally n the queue (at tme t = 0) and those that arrved after tme 0, s then gven by Ab T = L T (23) Q 0 + A T We then consder the formulaton mnmze c N subect to lm sup Ab T I T N (24) where we assume that = / for some strctly postve constants I, n order to place the system n the QED regme. Defnton 5.2 (Asymptotc Feasblty for (24)). A sequence of staffng vectors and routng polces N s asymptotcally feasble for (24) f: (a) for all ; and (b) for every >0, there exsts T such that for all T T, } 1 + I (25) { Ab T lm sup P We propose the followng staffng and routng soluton: Staffng: Use the optmal soluton to (10) wth now gven by the unque soluton to [ ) = P 1 h( ] and [ P 1 = 1 + where h = /1. h/ 1 1 h / (26) 1 ] = p and = a (27) I I Routng: FQR wth rato vectors v = 0 01 and p gven by p = / a k I k k / = / (28) k k I a k k / k In (26), P 1 s the asymptotcdelay probablty n the correspondng model under the FSF polcy, wth the patence rate and havng N = + +o ; see Propostons 4.5 and 4.6 n Armony and Mandelbaum (2008).

10 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems 10 Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. Theorem 5.2 (Asymptotc Feasblty and Optmalty for the Abandonment Formulaton). Suppose that any optmal soluton for (15) has N > 0 for all. Let N be determned through the -based staffng n (10) wth n (26). Set to FQRp v, where p s as n (28) and v = 001. Then, the sequence N s asymptotcally feasble for (24). If, n addton, for all, then the sequence N s also asymptotcally optmal. We prove Theorem 5.2 n the e-companon. The requred argument s smlar to the prevous case wthout abandonments. The key step s to show that wth FQR, the SBR model wth abandonment s asymptotcally equvalent (n terms of the aggregate number of customers n system) to a model n whch the customers patence s exponental wth a rate that s averaged usng the rato vector p of FQR n Equaton (27). For homogeneous patence rates, namely when, we show that a -based staffng (modfed for the abandonment case) provdes a lower bound on the staffng costs. Asymptotc optmalty then follows from the asymptotcfeasblty. Example 1 (A Two-Class Two-Pool System). We apply the proposed staffng-and-routng soluton to a two-class two-pool system. We assume that pool-1 servers can serve only class-1 customers, whereas pool-2 servers are crosstraned. The resultng N-model s depcted n Fgure 2. The customer arrval and patence parameters are, respectvely, 1 2 = and 1 2 = 2 1. Because we are consderng a fxed system, we omt the superscrpt from all notaton. Because we are consderng a settng wth nonhomogeneous patence rates, we am only to show the feasblty of our soluton. To complete the model descrpton, let the (pooldependent) servce rates be 1 2 = 15 1; let c 1 = c 2 = c; and let = N 2 + N In partcular, the Fgure 2. A two-class two-pool N model and the smulaton results. 1 1, 1 1 (t) 2 (t) 1, 2 2, 2 2 Fracton of customers number of agents n the frst pool can be at most 50. Fnally, we assume that the abandonment constrants are 3% for class 1 and 5% for class 2,.e, 1 2 = Wth these parameters, we construct our (asymptotcally feasble) -based staffng and FQR routng soluton. The parameters for FQR are p1 = 0375, p 2 = 0625, = 1375, and = From (26), for the -based staffng we have = , so that we need 1 N N 2 = Accordngly, we set N 1 = 50 and N 2 = / 2 =76. We smulate ths N model wth the specfed staffng and FQRp v. We run 3,000 replcatons of the system, each up to T = 500. The graph n Fgure 2 dsplays the average proporton of abandonng customers for each customer class and for each tme unt (as a functon of tme). Evdently, the proporton of abandonments s below the target (for class 1) or only slghtly above (for class 2). Also, we fnd that P { Ab T } = 1 2 (29) where P should be nterpreted here as the emprcal probablty dstrbuton over the 3,000 replcatons, whereas (29) corresponds to the asymptotc feasblty condton (25) n Defnton 5.2 wth there taken to be Desgnated-Servce Constrants In practce, t s natural to requre that most customers receve ther desgnated servce,.e., servce by the type of agent that the system desgnates for them. A good example s a multlngual call center, where one would prefer that Spansh-speakng customers be served by agents whose domnant language s Spansh, French-speakng customers be served by agents whose domnant language s French, and so forth. In other settngs, the nterpretaton of Fracton of customers abandoned Class 1 Class 2 N 1 N Tme

11 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS 11 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. desgnated servce mght be dfferent. To treat desgnatedservce constrants, we now assume that for each customer class there s one agent type desgnated to provde servce to that class; the agent type desgnated for class s ; that pool of agents can only serve class. To mpose a lower-bound constrant on the proporton of customers that receve ther desgnated servce, we let D T be the proporton of the arrvng class- customers that are routed to ther desgnated agents (the agents n pool ) by tme T. Lettng > 0 be the nondesgnated-servce proporton upper bound, the constrants are formulated as lm nf T D T 1 I The optmzaton problem that we consder s then gven by mnmze c N subect to lm sup T W T T I lm sup F T T 1 I 1 (30) T lm nf T D T 1 I N We now show that the desgnated servce constrants can be ncorporated nto the framework of 4 by approprately redefnng the set. To ths end, we extend the defnton of asymptotcfeasblty as follows: Defnton 5.3 (Asymptotc Feasblty wth Desgnated-Servce Constrants). A sequence of N s asymptotcally feasble for (30) f (a), for all, and (b) for every >0, there exsts T such that, for all T T, Equatons (4) and (5) hold, as well as lm sup PD T 1 I (31) The follow lemma provdes a property that all asymptotcally feasble solutons must satsfy. Lemma 5.1. If N s a sequence of asymptotcally feasble staffng and routng rules n the sense of Defnton 5.3, then lm nf N 1 I (32) Lemma 5.1 suggests that one may ncorporate the settng wth desgnated-servce constrants wthn the framework of 2 by replacng the optmzaton problem (30) wth mnmze c N subect to lm sup T lm sup T N W T T I F T T 1 I 1 (33) where = B and { } B = N J + N 1 I (34) Note that the set fts n the framework of 2.1 as t s defned by lnear constrants. In partcular, (33) s a specal case of (3) obtaned by lettng the set A there be equal to. The formal asymptotcfeasblty result for ths secton appears n the next proposton. Theorem 5.3. Suppose that the condtons of Theorem 4.1 hold wth respect to formulaton (33). Let N be the asymptotcally optmal -based staffng based on (10), wth replaced by n (34) and set to FQRp v wth p as n (13) and v = 001. Then, N s asymptotcally feasble for (30) n the sense of Defnton 5.3. It s asymptotcally optmal for (30) f one of the followng holds: () A B,or() c c and. We emphasze that the asymptotcally optmal soluton to (33), although asymptotcally feasble for (30), need not be, n general, asymptotcally optmal. The nequaltes mposed by the set B mght be too restrctve. Actually, as the proof of Proposton 5.3 reveals, we could have B defned through { B = N J + N 1 K } I for some K>0. Ths would be less restrctve, but would suffce to generate an asymptotcally feasble soluton for (30). In the next secton we consder a settng n whch c c and. There, as n the second part of Proposton 5.3, we wll be able to replace (30) wth (33) wthout compromsng asymptotcoptmalty Common Servce Rates Ths secton s devoted to a smple settng: a common servce rate, no abandonments, a common cost c for all agents, and = +, as consdered n Wallace and Whtt (2005). We have three purposes: frst, to contrast the FQR-based soluton wth the smulaton-based approach of Wallace and Whtt (2005), second, to llustrate an explct constructon of a system desgn when costs and system constrants do not pose sgnfcant restrctons; and thrd, to llustrate the dmnshng-return property of flexblty. The optmzaton problem that we consder n ths secton s (30), but wth c c,, and J +. Because we have a common servce rate, we can staff usng (nonasymptotc) formulas for the M/M/N queue nstead of the asymptotcexpressons assocated wth the -based staffng n 4. (In ths settng, these staffng methods are asymptotcally equvalent.) To specfy the staffng method here, let N = mnn +EW FCFS T (35) I

12 Gurvch and Whtt: Servce-Level Dfferentaton n Servce Systems 12 Operatons Research, Artcles n Advance, pp. 1 13, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to permssons@nforms.org. where W FCFS s the steady-state watng tme n an M/M/N queue wth arrval rate and servce rate. For routng, we also can use nonasymptotc expressons. Specfcally, we can use FQR wth rato vector p gven by PW FCFS > 0e N / I 1 TI 1 /pi 1 = p = T p I 1 I 1 TI 1 and (36) Note that the rato vector p here does depend on. Rather than assumng that the desgn s gven, we allow ourselves n ths secton to choose the desgn. In dong ths, we wll take nto account the desgnated-servce constrant. We wll ncorporate ths constrant from 5.3. The specfc desgn we suggest here s based on a concatenaton of M systems, whch we call the generalzed M GM model. An example of a GM model wth three customer classes s depcted n Fgure 3. The GM model has a routng graph constructed by allowng only edges of form, I, and + 1, = I\I (I excludng the element I). The GM model s relatvely nexpensve n terms of cross-tranng, because t uses agents wth at most two sklls, and only a lmted number wth two sklls. The soluton we propose s as follows: Desgn: generalzed M model (GM). Use a GM model. Staffng: Sngle-Class Staffng (SCS). Determne the overall number of agents, N usng (35). Then allocate agents to the pools by N = 1 /2 / N, 1I, N = 1 / N for all = 2I 1, and N +1 = + +1 /2 N for all = 1I 1, where = mn 1 I>0. Control: Fxed-Queue-Rato (FQR). Use FQR wth p as defned n (36) and v defned by v +1 = 1 I 1 Fgure 3. for all I\I (37) The generalzed M model for three classes The common servce rate allows us to use any vector v n the control step above. Ths stands n contrast to the case of dfferent servce rates, where we needed to use v = 001. The specfc vector n (37) s desgned to ncrease the amount of desgnated servce by forcng the system to route customers that fnd agents dle n both pools and + 1 to the desgnated agents n pool. One could also modfy FQR so that all customers that fnd agents dle n more than one agent pool that can serve them wll go to the desgnated agent pool. Ths modfcaton s guaranteed to acheve, asymptotcally, the same performance as FQR. Usng the results n 4, the above combned desgn-staffng-and-control soluton can be shown to be asymptotcally optmal as the arrval rate grows: Theorem 5.4 (Asymptotc Optmalty for the SBR Model wth Common Servce Rates). Consder the smple SBR model specfed above and assume that the GM desgn s used. Let N be determned by SCS staffng and set to FQRp v wth p as n (36) and v as n (37). Then, the sequence N s asymptotcally optmal for (30) wth c c and J Conclusons In ths paper we have proposed the fxed-queue-rato (FQR) routng scheme for the real-tme routng of customers n call centers wth multple customer classes and multple agent types operatng under QoS constrants. FQR routng facltates the constructon of combned staffng-desgnand-routng solutons for some settngs of the complcated skll-based-routng (SBR) problem, wth precsely specfed goals. In ths paper, we used FQR to to construct an asymptotcally optmal soluton for the staffng-and-routng problem subect to QoS constrants. The key assumpton that we made s that the servce rates are pool dependent. However, as dscussed n 2, ths s not enough, and we need to be careful about the formulaton; to get asymptotc optmalty, we need to replace the ntal formulaton (2) wth the best-effort formulaton n (3); Theorem 4.1 shows that FQR s asymptotcally optmal n ths settng. FQR also produces asymptotc optmalty for other mportant formulatons, specfed n 5. Some modfcaton s needed for each new formulaton, but a verson of FQR apples n each case. A key component of the proof was showng that our SBR problem s asymptotcally equvalent to the model prevously analyzed by Armony (2005). It s especally nstructve to see what can be done n the specal case of a common servce rate and a common agent cost c consdered n 5.4, whch was prevously consdered by Wallace and Whtt (2005) usng an teratve smulaton-based staffng algorthm. In that case, we need nether the -based staffng n (10), nor the optmzaton problem (15). Consequently, for ths specal case we do not need Assumptons Ths smple model llustrates how FQR smplfes tremendously the constructon of the