STAFFING OF TIME-VARYING QUEUES TO ACHIEVE TIME-STABLE PERFORMANCE

Transcription

1 STAFFING OF TIME-VARYING QUEUES TO ACHIEVE TIME-STABLE PERFORMANCE by Z. Feldman A. Mandelbaum Technion Insiue Technion Insiue Haifa Haifa ISRAEL ISRAEL W.A. Massey W. Whi Princeon Universiy Columbia Universiy Princeon, NJ New York, NY U.S.A. U.S.A. November 2004, Revision: November 25, 2005

2 Absrac This paper develops mehods o deermine appropriae saffing levels in call ceners and oher many-server queueing sysems wih ime-varying arrival raes. The goal is o achieve argeed ime-sable performance, even in he presence of significan ime-variaion in he arrival raes. The main conribuion is a flexible simulaion-based ieraive-saffing algorihm (ISA) for he M /G/s + G model - wih nonhomogeneous Poisson arrival process (he M ) and cusomer abandonmen (he +G). For Markovian M /M/s + M special cases, he ISA is shown o converge. For ha M /M/s +M model, simulaion experimens show ha he ISA yields imesable delay probabiliies across a wide range of arge delay probabiliies. Wih ISA, oher performance measures - such as agen uilizaions, abandonmen probabiliies and average waiing imes - are sable as well. The ISA saffing and performance agree closely wih he modified-offered-load (MOL) approximaion, which was previously shown o be an effecive saffing algorihm wihou cusomer abandonmen. While he ISA algorihm so far has only been exensively esed for M /M/s + M models, i can be applied much more generally, o M /G/s + G models and beyond. Keywords: Conac ceners; call ceners; saffing; non-saionary queues; queues wih imedependen arrival raes; many-server queues; capaciy planning; queues wih abandonmen; ime-varying Erlang models.

3 1. Inroducion In his paper we develop mehods o deermine appropriae saffing levels in call ceners and oher many-server queueing sysems wih ime-varying arrival raes. For background on call ceners, see Gans e al. (2003). In seing saffing levels, we are faced wih wo sources of variabiliy: predicable variabiliy ime-variaions of he expeced load and sochasic variabiliy random flucuaions around his ime-dependen average. (There may also be model uncerainy, bu we do no consider i.) Mos available saffing algorihms are designed o cope only wih sochasic variabiliy, avoiding he predicable variabiliy in various ways. For example, when he service imes are relaively shor, as in many call ceners when service is provided by a elephone call, i is cusomary o use a poinwise saionary approximaion (PSA), i.e., o ac as if he sysem a ime were in seady-sae wih he arrival rae occurring a ha insan (or during ha half hour); see Green and Kolesar (1991) and Whi (1991). In call ceners, saffing ypical is held consan over saffing inervals of minues. The effec of saffing inervals can be imporan, see Green e al. (2001), bu here we do no consider saffing inervals. However, service imes are no always shor, even in call ceners. If relaively lenghy ineracions are no uncommon or if arrival raes change quie rapidly, hen PSA can produce poor performance. As a consequence, some pars of he day may be oversaffed, while ohers are undersaffed. For a review of saffing mehods o cope wih ime-varying arrivals, see Green e al. (2005). In his paper we address he saffing problem wih boh predicable and sochasic variabiliy: Given a daily performance goal, and faced wih boh predicable and sochasic variabiliy, we seek o find he minimal saffing levels ha mee his performance goal sably over he day. We aim o undersand when PSA is appropriae and o do significanly beer han PSA when i is no appropriae. We emphasize he imporance of achieving ime-sable performance. Wih ime-sable performance, he nearly-consan qualiy of service is easily adjused up or down, as desired. Moreover, our experience suggess ha cusomers end o prefer consisen performance even a he expense of some service level. Our main conribuion in his paper is a flexible simulaion-based ieraive-saffing algorihm (ISA). We develop he ISA for he many-server M /G/s + G queueing model, which has a nonhomogeneous Poisson arrival process (he M ) wih ime-varying arrival-rae funcion λ(), independen and idenically disribued (i.i.d.) random service imes wih a general cu- 1

4 mulaive disribuion funcion or cdf (he firs G), a ime-varying number of servers s, which is for us o se, and i.i.d. random imes o abandon (before saring service) wih a general cdf (he final +G). Allowing non-exponenial service-ime and ime-o-abandon disribuions is imporan, because hey have been found o occur in pracice; see Boloin (1994) and Brown e al. (2005). We show ha he ISA saffing funcion s ISA yields ime-sable delay probabiliies across a wide range of delay-probabiliy arges for he Markovian M /M/s +M special case, where he service-ime and ime-o-abandon cdf s are exponenial wih means µ 1 and θ 1, respecively. Even hough we only repor resuls for ISA applied o Markovian M /M/s + M models, he mehod is developed for more general M /G/s + G models. (Indeed, we obained similar resuls for log-normal and deerminisic service-ime disribuions.) Moreover, he ISA applies much more generally, so ha i has he poenial of far-reaching applicaions. Indeed, by being based on simulaion, ISA has wo imporan advanages: Firs, by using simulaion, we achieve generaliy: We can apply he approach o a large class of models; we are no limied o models ha are analyically racable. We are able o include realisic feaures, no ordinarily considered in analyical models. For example, we can carefully consider wha happens o agens who are in he middle of a call when heir scheduled shif ends. Second, by using simulaion, we achieve auomaic validaion: In he process of performing he algorihm, we direcly confirm ha ISA achieves is goal; we direcly observe he performance of he sysem under he final saffing funcion {s ISA sill needs o be verified. : 0 T }. Of course, in oher seings he effeciveness of he ISA Here is how he res of his paper is organized: In 2 we specify he ISA. Then in 3 we review he infinie-server and modified-offered-load (MOL) approximaions from Jennings e al. (1996). We will show ha he ISA saffing levels and performance agree closely wih MOL and ha boh perform well. In 4 and 5 we illusrae he performance of ISA by considering M /M/s + M examples, firs wih a sylized sinusoidal arrival-rae funcion and long service imes, and hen wih a realisic arrival-rae funcion from a medium-sized financial-services call cener, aken from Green e al. (2001) and shorer (cusomary) service imes. In 6 we presen some supporing heory for he case θ = µ. In 7, we discuss he dynamics of he ieraive algorihm, esablishing convergence of he ISA in he M /M/s + M special case (for all µ and θ). Finally, in 8 we draw conclusions and indicae some direcions for furher research. We presen addiional maerial in a longer unabridged version available on line as an Inerne Supplemen. There we consider he M /M/s model (wihou abandonmen) wih he same 2

5 sinusoidal arrival-rae funcion used for he M /M/s +M model in 4, and show ha ISA also works well for i. We also revisi he challenging example in Jennings e al. (1996), again showing ha ISA performs well, jus like MOL. We expand he analysis of he M /M/s + M example in 4 by considering differen abandonmen raes, in paricular, θ = 0.2 and θ = 5.0 wih µ = 1, represening relaively paien and impaien cusomers, respecively. We presen addiional maerial for he realisic example discussed in 5. We also provide addiional heoreical perspecive for he square-roo-saffing algorihm from a uniform-acceleraion perspecive, as in Mandelbaum e al. (1998) and Massey and Whi (1998) and references herein. 2. The Simulaion-Based Ieraive-Saffing Algorihm (ISA) In his secion we specify he ISA. For our implemenaion of he algorihm, we assume ha we have an M /G/s +G model, bu i will be eviden ha he mehod applies much more generally. To sar, we specify a ime-horizon [0, T ], an arrival-rae funcion {λ(); 0 T }, a service-ime cdf and a ime-o-abandon cdf. The algorihm is ieraive, coninuing unil he observed error is negligible. Le s (n) be he saffing level a ime in ieraion n and le N (n) be he oal number of cusomers in he sysem a ime under his saffing funcion. The final ieraion yields he ISA saffing s ISA and he sochasic process N ISA number of cusomers in he sysem wih ha saffing funcion. represening he Alhough our algorihm is ime-coninuous, we make saffing changes only a discree imes. Tha is achieved by dividing he ime-horizon ino small inervals of lengh. In all experimens presened in his paper, we use = 0.1/µ, where 1/µ is he mean service ime. We hen le he number of servers be consan wihin each of hese inervals. For any specified saffing funcion, he sysem simulaion can be performed in a convenional manner. We generae a coninuous-ime sample pah for he number in sysem by successively advancing he nex generaed even. The candidae nex evens are of course arrivals, service compleions, abandonmens and ends of shifs (he imes a which he saffing funcion is allowed o change). For non-saionary Poisson arrival process, we generaed arrival imes by hinning a single Poisson process wih a consan rae λ exceeding he maximum of he arrival-rae funcion λ() for all, 0 T. Then an even in he Poisson process a ime (a poenial arrival ime) is in an acual arrival in he sysem wih probabiliy λ()/λ, independen of he hisory up o ha ime; see Secion 5.5 of Ross (1990). We esimae he disribuion of N (n) for each n and by performing muliple (5000) independen replicaions. We hink of saring off wih infiniely many servers. Since his is a simulaion, we choose (afer experimening) a large finie number, 3

6 ensuring ha he probabiliy of delay (i.e., of having all servers busy upon arrival) is negligible for all. The algorihm ieraively performs he following seps, unil convergence is obained. Convergence means ha he saffing levels do no change more han some hreshold τ afer an ieraion, which we ake o be Given he i h saffing funcion {s (i) all using simulaion. : 0 T }, evaluae he disribuion of N (i) for 2. For each, 0 T, le s (i+1) be he leas number of servers so ha he delayprobabiliy consrain is me a ime ; i.e., le s (i+1) = arg min {k N : P {N (i) k} α}. 3. If here is negligible change in he saffing from ieraion i o ieraion i + 1, hen sop; i.e., if max { s (i+1) s (i) : 0 T } τ, hen sop and le s (i+1) be he proposed saffing funcion, denoed by s ISA. Oherwise, advance o he nex ieraion, i.e., replace i by i + 1 and go back o sep 1. As indicaed before, s ISA denoes he final saffing level a ime and N ISA number in sysem a ime wih ha saffing funcion. necessarily P (N ISA s ISA ) α, 0 T. denoes he If he algorihm converges, hen Our implemenaion of ISA was wrien in C++. For he special case of he Markovian M /M/s + M model wih individual service rae µ = 1/E[S] and individual abandonmen rae θ, we rigorously esablish convergence of he algorihm in 7. Experience indicaes ha he algorihm consisenly converges relaively rapidly. Experience also indicaes ha he final ime-dependen delay probabiliies, and oher performance measures, are remarkably sable. The number of ieraions required depends on he parameers, especially he raio r θ/µ. If r = 1, corresponding o an infinie-server queue - see 6, hen no more han wo ieraions are needed, since he disribuion of he number in sysem does no depend upon he number of servers in ha special case. As r depars from 1, he number of required ieraions ypically increases. For example, when r = 10, he number of ieraions can ge as high as When r is very small and he raffic inensiy is very high, so ha we are a he edge of sabiliy, he number of ieraions can be very large. For more discussion, see 7. 4

7 3. Infinie-Server and Modified-Offered-Load Approximaions In his secion we review saffing algorihms based on infinie-server (IS) and modifiedoffered-load (MOL) approximaions from our (wih Ois B. Jennings) previous paper Jennings e al. (1996). These approximaions were developed for he M /G/s model wihou cusomer abandonmen, bu he mehods exend direcly o he corresponding model wih cusomer abandonmen. The effeciveness of hese mehods wih abandonmens was no demonsraed previously, hough. Our simulaion experimens here will show ha ISA produces essenially he same resuls as MOL, wih and wihou cusomer abandonmen, and ha boh are effecive. (Our repored experimens are limied o Markovian M /M/s + M models, bu limied experimenaion for oher M /G/s + G models indicae ha excellen resuls hold here oo.) To describe our goal in saffing, le N be he number of cusomers in he M /G/s + G sysem a ime, eiher waiing or being served. We focus on he probabiliy of delay (of a poenial arrival, i.e., P (N s )), aiming o choose he ime-dependen saffing level s such ha P (N s ) α < P (N s 1) for all, (3.1) where α is he arge delay probabiliy. The Infinie-Server Approximaion. We discuss he MOL and infinie-server approximaions ogeher, because he MOL approximaion builds on he infinie-server approximaion. We sar by considering he infinie-server approximaion. Why would anyone consider an infinie-server approximaion? From a mahemaical perspecive, he reason is ha he finieserver M /G/s + G model of ineres is analyically inracable, whereas he corresponding infinie-server M /G/ model is remarkably racable. From an engineering perspecive, he reason is ha he infinie-server model can be used o show he amoun of capaciy ha would acually be used (and is hus needed) if here were no capaciy consrains (i.e., a limied number of servers). For he Markovian M /M/s + M model, where θ = µ, here is even a sronger connecion: In ha special case, he disribuion of he number of cusomers in he infinieserver M /M/ model acually coincides wih he disribuion of he number of cusomers in he M /M/s + M model, as we explain in 6, so here is addiional srong moivaion for considering he infinie-server approximaion. So wha does he infinie-server approximaion do? The infinie-server approximaion for he M /G/s + G model approximaes he random variable N by he number N of busy 5

8 servers in he associaed M /G/ model, having infiniely many servers bu he same arrival process and service imes. The infinie-server saffing funcion s is obained by applying (3.1) wih N insead of N. As we now explain, ha approximaion provides grea simplificaion because (i) he ail probabiliy P (N s ) a ime depends on he saffing funcion {s : 0} only hrough is value a he single ime and (ii) he exac ime-dependen disribuion of N is known. The firs simplificaion follows from he fac ha he disribuion of he sochasic process {N : 0} is oally independen of he saffing funcion {s : 0}. When we calculae P (N s ), he saffing level s jus serves as he argumen of he ail-probabiliy funcion. The second simplificaion sems from basic properies of M /G/ queues. In paricular, as reviewed in Eick e al. (1993a), for each, N has a Poisson disribuion whenever he number in he sysem a he iniial ime has a Poisson disribuion. (Being empy is a degenerae case of a Poisson disribuion.) Tha Poisson disribuion is fully characerized by is mean m. As in previous work, such as Eick e al. (1993a,b) and Jennings e al. (1996), our work repored here shows ha he ime-dependen mean m exac ime-dependen mean m for he M /G/s + G model. is he crucial quaniy. We regard his in he M /G/ model as he (ime-dependen) offered load We now observe ha convenien formulas exis for he offered load m. Eick e al. (1993a) showed ha he offered load has he racable represenaion [ ] m E [N ] = G c ( u)λ(u) du = E λ(u) du = E [λ( S e )] E[S], (3.2) where λ() is he arrival-rae funcion, S is a generic service ime wih cdf G, G c () 1 G() P (S > ), and S e is a random variable wih he associaed saionary-excess cdf (or equilibriumresidual-lifeime cdf) G e associaed wih he service-ime cdf G, defined by G e () P (S e ) 1 E[S] S 0 G c (u) du, 0, (3.3) wih k h momen E[S k e ] = E[S k+1 ]/((k + 1)E[S]); see Theorem 1 of Eick e al. (1993a) and references herein. The differen expressions in (3.2) provide useful insigh; see Eick e al. (1993a, b) and Secion 4.2 of Green e al. (2005). For he special case in which λ() is consan, m m = λe[s]. Accordingly, he PSA approximaion for m λ()e[s]. We call m P SA in he M /G/ model is m P SA he PSA (ime-dependen) offered load for he M /G/s + G model. In addiion, here are convenien explici formulas for m in special cases as well as useful approximaions. We will use he explici formula for sinusoidal arrival-rae funcions in 4. 6

9 Based on a second-order Taylor-series approximaion for λ abou, he offered load can be approximaed by m λ( E[S e ])E[S] + λ(2) () V ar(s e )E[S], (3.4) 2 where λ (2) () is he second derivaive of he funcion λ evaluaed a ime ; see Theorem 9 of Eick e al. (1993a). Approximaion (3.4) shows ha he approximae offered load in (3.4) coincides wih he PSA offered load m P SA λ()e[s] excep for a ime shif by E[S e ] and a space shif by λ (2) ()V ar(s e )E[S]/2. The ime shif is especially imporan. A simple refinemn of PSA based on (3.4) suggesed by Eick e al. (1993a) is lagged PSA, where we ignore he space shif and approximae m by λ( E[S e ])E[S]. We now coninue, exploiing he esablished Poisson disribuion wih a known imedependen mean m. Assuming ha m is no exremely small, we can apply a normal approximaion for he Poisson disribuion, obaining firs P (N s ) P (N hen P (N s ) P (N(m, m ) s ) = P s ) and ( ) ( ) N(0, 1) s m s m = 1 Φ m m, (3.5) where N(m, σ 2 ) denoes a normally disribued random variable wih mean m and variance σ 2, and Φ(x) P (N(0, 1) x) is he sandard normal cdf. From (3.5), we see ha we can obain a sable approximae delay probabiliy if we can choose he saffing funcion s o make (s m )/ m sable in he final erm of (3.5). Accordingly, we obain he square-roo-saffing formula: s = m + β m, 0 T, (3.6) where x is he leas ineger greaer han or equal o x and he consan β is a measure of he qualiy of service. Combining he arge in (3.1) and he normal approximaion in (3.5), we see ha he qualiy-of-service parameer β in (3.6) should be chosen so ha 1 Φ(β) = α. The normal approximaion and he square-roo-saffing formula for saionary many-server queues are classic resuls, see Whi (1992) and references herein. Wha is less well undersood is he role of he offered load m wih ime-varying arrivals. The noaion s means ha we saff according o he infinie-server approximaion. In doing so, we no only apply he normal approximaion and he square-roo-saffing formula, bu we also use he infinie-server mean m as he offered load. 7

10 The MOL Approximaion. Secion 4 of Jennings e al. (1996) also inroduced a refinemen of he infinie-server approximaion for he ime-dependen delay probabiliies, which is anamoun o a modified-offered-load (MOL) approximaion, as in Jagerman (1975) and Massey and Whi (1994, 1997). The MOL approximaion for N in he M /G/s + G model a ime, denoed by N MOL, is he limiing seady-sae number of cusomers in he sysem in he corresponding saionary M/G/s + G model (wih he same service-ime and ime-oabandon disribuions and he same number of servers s a ime ), bu using m as he saionary offered load operaing a ime. Since he saionary offered load is λe[s], ha means leing he homogeneous Poisson arrival process in he saionary M/G/s + G model have ime-dependen arrival rae λ MOL The MOL saffing funcion s MOL m E[S] = m µ a ime. (3.7) is obained by applying (3.1) wih N MOL insead of N. The imporan insigh is ha he righ ime-dependen offered load in he M /G/s + G model should be he ime-dependen mean number of busy servers in he associaed infinieserver model - m. Since he righ offered load for he saionary model is λe[s], he obvious direc ime-dependen generalizaion is he PSA offered load m P SA λ()e[s]. However, λe[s] is also he mean number of busy servers in he associaed saionary infinie-server model. I urns ou ha he mean number of busy servers in he infinie-server model is a beer generalizaion of offered load han he PSA ime-dependen offered load for mos ime-varying many-server models. Indeed, i may be considered exacly he righ definiion for he infinie-server model iself. The MOL approximaion in 4 of Jennings e al. (1996) was no applied direcly. Insead of calculaing he seady-sae delay probabiliy for he saionary M/M/s model, we exploied an approximaion for he delay probabiliy based on a many-server heavy-raffic limi in Halfin and Whi (1981). Tha produces a simple formula relaing he delay probabiliy α and he service qualiy β. Moreover, he heavy-raffic limi provides an alernaive derivaion of he square-roo saffing formula in (3.6), wihou relying on an infinie-server approximaion or a normal approximaion. We will do he same hing here wih cusomer abandonmens, relying on he heavy-raffic limis for he M/M/s + M model esablished by Garne e al. (2002). Jennings e al. (1996) showed ha he mehod for seing saffing requiremens in he M /G/s model oulined above is remarkably effecive. This was demonsraed by doing numerical comparisons for he M /M/s special case. For any given saffing funcion, he ime- 8

11 dependen disribuion of N in ha Markovian model can be derived by solving a sysem of ime-dependen ordinary differenial equaions (ODE s). We oo could have exploied ODE s for he M /M/s + M model, bu we waned o develop a mehod ha applies o much more general models. The mos imporan conclusion from hose previous experimens in Jennings e al. (1996) is ha i is indeed possible o achieve ime-sable performance for he M /M/s model by an appropriae choice of a saffing funcion s, even in he face of a srongly ime-varying arrivalrae funcion. Here we show he same is rue for he M /M/s + M model. And we provide a means o go far beyond hese Markovian models. 4. An M /M/s + M Example wih a Sinusoidal Arrival-Rae Funcion We demonsrae he performance of ISA by considering M /M/s + M examples. We sar in his secion wih a sinusoidal arrival-rae funcion λ() = a + b sin(c), 0 T, (4.1) leing a = 100, b = 20 and c = 1. Here we le he individual service rae µ and he individual abandonmen rae θ boh be 1. Leing µ = 1 is wihou loss of generaliy, because we are free o choose he ime unis. For he special case θ = µ ha we consider, we have srong supporing heory in 6, bu we also found ha ISA is effecive wih oher abandonmen raes. We show corresponding resuls for θ = 0.2 and θ = 5.0 in he Inerne Supplemen. cener. Since m P SA λ()e[s] = λ(), his example capures he many-server spiri of a call However, he sinusoidal form of he arrival-rae funcion is clearly a mahemaical absracion, which has he essenial propery of producing significan flucuaions over ime, i.e., significan predicable variabiliy. This paricular arrival-rae funcion is by no means criical for our analysis; our mehods apply o an arbirary arrival-rae funcion. An imporan issue, however, is he rae of flucuaion in he arrival-rae funcion compared o he expeced service ime. To be concree, we will measure ime in hours, and focus on a 24-hour day, so ha T = 24. A cycle of he sinusoidal arrival-rae funcion in (4.1) is 2π/c; since we have se c = 1, a cycle is 2π 6.3 hours. Thus here will be abou 4 cycles during he day. Since we le he mean service ime be 1 and have chosen o measure ime in hours, he mean service ime in his example is 1 hour. Tha clearly is relaively long for mos call ceners, where he ineracions are shor elephone calls. If we were o change he ime unis in order 9

12 Figure 1: The offered load m for he sinusoidal arrival-rae funcion in (4.1) wih parameers a = 100, b = 20 and four possible values of c: 0.5, 1, 2 and offered load c=0.5 c=1.0 c=2.0 c= ime o recify ha, making he expeced service ime 10 minues, hen a cycle of he arrival-rae funcion would become abou 1 hour, making for more rapid flucuaions in he arrival rae han are normally encounered in call ceners. Thus our example is more challenging han usually encounered in call ceners, bu may be approached in evolving conac ceners if many ineracions do indeed ake an hour or more. We consider a more realisic example in 5. Since we have a sinusoidal arrival-rae funcion, we can apply formula (15) of Eick e al. (1993b) o obain m = a + b [sin(c) c cos(c)]. (4.2) 1 + c2 For he specific parameers a = 100, b = 20 and c = 1, we ge m = [sin() cos()]. In order o pu our model ino perspecive, in Figure 1 we plo he ime-dependen offered load m in (4.2) for he sinusoidal arrival-rae funcion in (4.1) for he parameers a = 100 and b = 20, as in our example, bu wih four differen values of he ime-scaling parameer c: 0.5, 1, 2 and 8. Noe ha he ime-dependen offered load m is also a periodic funcion wih he same period 2π/c as he arrival-rae funcion λ(), bu he number of cycles increases and he ampliude (size of he flucuaions) decrease as c increases. As c increases, m he average value a = 100. approaches In Figure 2 we presen wo graphs, showing he ISA saffing funcions for wo values of α: 0.1 and 0.9. In each graph, we plo hree curves: he arrival rae λ() m P SA (doed), he 10

13 Figure 2: Saffing - number of servers as a funcion of ime - for he sinusoidal example: (1) α = 0.1 (QD), (2) α = 0.9 (ED). saffing level ime Arrival Rae Offered Load Saffing offered load m saffing level ime Arrival Rae Offered Load Saffing (dashed) and he ISA saffing funcion s ISA (solid). Noe ha we sar our sysem empy. This allows us o observe he behavior of he ransien sage. In paricular, here is a rampup a he lef side of he plo. Our mehods respond appropriaely o ha rampup. The wo values of α used in Figure 2 plus α = 0.5 characerize hree differen regimes of operaion, as discussed by Garne e al. (2002): Qualiy-Driven (QD) - arge α = 0.1, Efficiency-Driven (ED) - arge α = 0.9, and Qualiy-and-Efficiency-Driven (QED) - arge α = 0.5. In he QD regime, he ISA saffing funcion is well above he ime-dependen offered load, while in he ED regime he ISA saffing funcion is well below he ime-dependen offered load. However, in he QED regime, he ISA saffing funcion falls righ on op of he imedependen offered load. (For ha reason, we omi he plo, since i is unnecessary.) In ha 11

14 Figure 3: Delay probabiliies for he sinusoidal example wih nine delay-probabiliy arges α, ranging from 0.1 o 0.9. delay probabiliy ime QED case (α = 0.5), i would have sufficed o simply le s = m. This phenomenon held in all our experimens. Tha iself is quie sunning. (Noe ha saffing o he offered load is much easier han he full MOL approximaion. Clearly, cusomer abandonmen plays a crucial role in saffing o he offered load.) We now show ha ISA achieves he desired ime-sable performance. In Figure 3 we show he ISA delay probabiliies obained wih arge α for α = 0.1, 0.2,..., 0.9. These delay probabiliies are esimaed by performing muliple (5000) independen replicaions wih he final saffing funcion deermined by our algorihm. (We verified ha his was sufficien by repeaing he experimen wih independen random numbers. We saw negligible change in he plos. The observed flucuaions are largely due o he inheren discreeness: The saffing levels mus be inegers.) Under he ISA saffing levels, he delay probabiliies are remarkably accurae and sable; he observed delay probabiliies flucuae around he arge in each case. In addiion o sabilizing he delay probabiliies, oher performance measures (e.g. uilizaion, ail probabiliies abandonmen probabiliies, ec.) are found o be quie sable as well. However, as he arge delay probabiliy increases oward heavy loading, he abandonmen probabiliies become much less ime-sable, as shown in Figure 4. (Like he delay probabiliy, we le he abandonmen probabiliy be for a poenial arrival a ime ; a precise definiion is given afer (6.1).) We discuss his phenomenon furher in he Inerne Supplemen. Oher measures of congesion such as average waiing ime and average queue lengh were also found o 12

15 Figure 4: Abandonmen probabiliies for he same sinusoidal example wih he same nine delay-probabiliy arges abandonmen probabiliy ime be relaively sable, bu like he abandonmen probabiliies, hese oo become less ime-sable under heavy loads. Deails are given in he Inerne Supplemen. We now validae he square-roo-saffing rule in (3.6). implied empirical qualiy of service {β ISA β ISA sisa : 0 T } by seing For ha purpose, we define an m, 0 T, (4.3) m where m funcion β ISA is again he offered load in (3.2) and (4.2). Since s ISA is obained from ISA, he is iself obained from ISA. I hus becomes ineresing o see if he implied service grade is approximaely consan as a funcion of ime. Tha would empirically jusify he square-roo-saffing formula in (3.6). And, indeed, i is. Again we consider nine values of α ranging from 0.1 o 0.9 in seps of 0.1. As α increases, he qualiy of service refleced by β ISA 1.3, hiing 0 for α = 0.5. Bu he main poin is ha β ISA decreases, from abou +1.3 o is approximaely consan as a funcion of for each α over he full range from 0.1 o 0.9. The oscillaions in he plos are essenially he same as in Figure 3 (see he Inerne Supplemen). The ime-sabiliy of β ISA is exremely imporan because i validaes he square-roosaffing formula in (3.6) for his example. Firs, Figure 3 shows ha ISA is able o produce he arge delay probabiliy α for a wide range of α. When his is done, he square-roo-saffing formula holds empirically. In oher words, we have shown ha we could have saffed direcly by he infinie-server approximaion and he square-roo-saffing formula insead of by he ISA. The single criical non-rivial elemen is he offered load m. 13

16 However, one issues remains: In order o saff direcly by he square-roo saffing formula, we need o be able o relae he qualiy of service β o he arge delay probabiliy α. Indeed, we wan a funcion mapping α ino β. We propose a simple answer: MOL. For he M /M/s +M model wih ime-varying arrival-rae funcion λ(), saffing funcion s and parameers µ and θ, we use he associaed saionary M/M/s + M model, wih he same service and abandonmen raes µ and θ, and wih s = s, λ = λ MOL = m µ (as in (3.7)) for he approximaion a ime. We used exac M/M/s + M formulas from Garne e al. (2002). Moreover, paralleling wha was done in 4 of Jennings e al. (1996), we sugges using simple formulas obained from he many-server heavy-raffic limi for he M/M/s + M model in Garne e al. (2002). The Garne funcion mapping β ino α is [ θ α = 1 + µ h( ˆβ) ] 1, < β <, (4.4) h( β) where ˆβ = β θ/µ, wih µ he individual service rae and θ he individual abandonmen rae (boh here se equal o 1 now) and h(x) = φ(x)/(1 Φ(x)) is he hazard rae of he sandard normal disribuion, wih φ being he probabiliy densiy funcion (pdf) and Φ he cdf. To obain he desired funcion mapping α ino β, we can use he inverse of he Garne funcion, which is well defined. For his example, he Garne funcion yields essenially he same formula as he exac values for he M/M/s + M model. We also looked a addiional simulaion oupu, aimed a esablishing he validiy of he ISA and MOL approximaions. Firs, we compared he empirical disribuion of he cusomer waiing imes, wih ISA, o he heoreical disribuion of hose waiing imes in he saionary M/M/s+M model. To illusrae, in Figure 5 we plo he empirical condiional waiing ime pdf given wai, i.e. he disribuion of he waiing ime for hose who were in fac delayed, during he enire ime-horizon, for he case α = 0.1. We plo he proporions experiencing delays in inervals of lengh In doing so, we are looking a all he waiing imes experienced across he day. As before, we obain saisically precise esimaes by averaging over a large number of independen replicaions (here again 5000). In his case, he empirical condiional disribuion is based on saisics gahered from he ime of reaching seady sae unil he end of he horizon. We compared he empirical condiional waiing-ime disribuion o many-server heavy-raffic approximaions for he condiional waiing-ime disribuion in he saionary M/M/s + M queue, drawing on Garne e al. (2002). Figure 5 shows ha he approximaion for he condiional waiing-ime disribuion in he saionary queues maches he performance of our ime-varying model remarkably well. Plos for α = 0.5 and α = 0.9 in he Inerne 14

17 Figure 5: The empirical condiional waiing ime disribuion, given posiive wai, for he M /M/s + M example wih delay-probabiliy arge α = 0.1 (QD) proporion ime Wai Wai>0 Garne Funcion Supplemen show an excellen mach across he full range of delay-probabiliy arges. We nex relaed he empirical (α, β) pairs o he Garne funcion in (4.4). We define he empirical values ᾱ and β as simply he ime-averages of he observed (ime-sable) ISA values (for α, displayed in he plo in Figure 3). In Figure 6, we plo he pairs of (ᾱ i, β i ) alongside he Garne funcion. Needless o say, he agreemen is phenomenal! We close his secion by observing ha, jus as in Jennings e al. (1996), oher common approximaions, such as he PSA or he SSA (he simple saionary approximaion, using he overall ime-average arrival rae) perform poorly for his example; again see he Inerne Supplemen. 5. A Realisic M /M/s + M Example In his secion we consider a more realisic example: a medium-sized financial-services call cener aken from Green e al. (2001). The hourly call volumes are shown in Figure 7. The mean service ime is E[S] = 6 minues. Tha is achieved wih our hourly ime scale by leing µ = 10. Corresponding o ha, we le θ = 10, so ha we have θ = µ as in Secion 4. (Green e al. (2001) did no consider cusomer abandonmen.) Once again, ISA is very effecive. To show ha, we plo he ISA delay probabiliies as a funcion of he delay-probabiliy arge α for hree values of α in Figure 8. Wih such shor 15

18 Figure 6: A comparison of he empirical relaion beween α and β wih he Garne funcion for he sinusoidal example. Theoreical & Empirical Probabiliy Of Delay vs Alpha Bea Garne Empirical service imes, we migh hink ha ha his should be an easy problem, for which simple PSA would also work well. Indeed, when we look a he saffing for hree values of α in Figure 9, we do no see much difference, bu here acually is a difference. Even hough he service imes are indeed shor here, he arrival-rae funcion is changing rapidly a some imes, especially in hours 4 6. For his example, Figure 8 shows ha simple PSA performs significanly worse han ISA. As before, we find ha ISA produces essenially he same resuls as MOL. Moreover, he dominan effec in MOL is capured by he ime lag in (3.4); i.e., here i suffices o use lagged PSA, wih approximae offered load λ( E[S e ])E[S]. Since he service-ime disribuion is exponenial, S e and S have a common exponenial disribuion, and he lagged-psa offered load is jus λ( E[S])E[S]. The good performance of lagged PSA is consisen wih he various 16

19 Figure 7: Hourly call volumes o a medium-size financial-services call cener Calls per Hour Hour of Day Figure 8: A comparison of ISA, PSA and lagged PSA for he same hree delayprobabiliy arges. delay probabiliy ime Targe Alpha=0.1(ISA) Targe Alpha=0.1(LPSA) Targe Alpha=0.1(PSA) Targe Alpha=0.5(ISA) Targe Alpha=0.5(LPSA) Targe Alpha=0.5(PSA) Targe Alpha=0.9(ISA) Targe Alpha=0.9(LPSA) Targe Alpha=0.9(PSA) refinemens proposed by Green e al. (2001). We show ha simple PSA performs worse han ISA and lagged PSA by ploing he delay probabiliies for hese hree saffing rules in Figure 8. The performance of simple PSA here is nowhere near as bad as i was in he challenging M /M/s example in Jennings e al. (1996), and as i is for he example here in 4 (see he Inerne Supplemen), bu here are clear deparures from he performance arges in Figure 8. The PSA delay probabiliies are significanly below he arges during he hours 4 6 wih 17

20 Figure 9: A comparison of saffing levels based on ISA, PSA and lagged PSA for he realisic example, for hree delay-probabiliy arges: 0.1, 0.5 and saffing levels ime Targe Alpha=0.1(ISA) Targe Alpha=0.1(LPSA) Targe Alpha=0.1(PSA) Targe Alpha=0.5(ISA) Targe Alpha=0.5(LPSA) Targe Alpha=0.5(PSA) Targe Alpha=0.9(ISA) Targe Alpha=0.9(LPSA) Targe Alpha=0.9(PSA) rapidly increasing arrival raes. The differences among he corresponding saffing funcions in Figure 9 look small, bu hose small differences can have a significan impac, because he arrival-rae funcion changes rapidly. We also observe ha ISA is no as successful as before, because he arge delay probabiliy is no achieved accuraely a he beginning and a he end of he day. This phenomenon is even more eviden for oher performance measures; see he Inerne Supplemen. However, his weak performance is due o he exremely low arrival raes ha prevail a he beginning and he end of he day. When he load is small, he addiion or removal of a single server will grealy affec he delay probabiliy. On he posiive side, here is a clear ime-inerval - from hours 5 o 18, in which all performance measures are sable. Finally, we remark ha here is excellen maching beween he Garne funcion and he empirical resuls, jus as in Figure 6; see he Inerne Supplemen. 6. Theoreical Suppor in he Case θ = µ Relaion o oher models. In one special case, we can analyze he M /M/s +M model in considerable deail. Tha is he case we considered in 4 and 5, in which θ = µ. (As in 4, we le hose boh be 1.) Wih he condiion θ = µ, i is easy o relae he M /M/s +M model o, firs, he corresponding M /M/ model wih he same arrival-rae funcion and service rae 18

21 and, second, a corresponding family of seady-sae disribuions of saionary M/M/s + M models, indexed by, wih he same service and abandonmen raes, bu wih special arrival rae ha depends on ime. Le {s : 0} be an arbirary saffing funcion. For simpliciy, assume ha all sysems sar empy in he disan pas (a ime ). By having λ() = 0 for 0, we can sar arrivals a any ime 0. The firs observaion is ha, for any arrival-rae funcion {λ() : 0} and any saffing funcion {s : 0}, he sochasic process {N : 0} in he M /M/s + M model wih θ = µ has he same disribuion (finie-dimensional disribuions) as he corresponding process {N deah raes are he same. : 0} in he M /M/ model, because he birh and and N The second observaion is ha, for boh hese models, he individual random variables N have he same Poisson disribuion as he seady-sae number in sysem N () corresponding saionary model wih arrival rae m. in he Waiing imes and abandonmen probabiliies. Le W be he virual waiing ime a ime (unil service, i.e., he waiing ime in queue ha would be spen by an infiniely paien cusomer arriving a ime ), and le P ab be he virual abandonmen probabiliy a ime (i.e., he probabiliy of abandonmen for an arrival ha would occur a ime ), boh in he M /M/s + M model. These quaniies are considerably more complicaed han N. Even hough i is difficul o evaluae he full disribuion of W, we can immediaely evaluae he virual delay probabiliy, because i clearly depends only on wha he cusomer encouners upon arrival a ime. Hence, we have P (W > 0) = P (N s ) = P (N s ) = P (P oisson(m ) s ), (6.1) where m is he offered load in (3.2), jus as in (3.5), only here he infinie-server approximaion is exac. Nex we observe ha P ab = E[F (W )], where F is he ime-o-abandon cdf, so ha i suffices o deermine he waiing-ime disribuion. Here is an imporan iniial observaion: Condiional on he even ha W > 0, whose probabiliy we have characerized above, W is disribued (exacly) as he firs passage ime of he (Markovian) sochasic process {N u : u } from he iniial value N encounered a ime down o he saffing funcion {s u : u }, provided ha we ignore all fuure arrivals afer ime. In oher words, W is disribued as he firs passage ime of he pure-deah sochasic process wih sae-dependen deah rae N u 19

22 for u down from he iniial value N o he curve {s u : u }. As a consequence, he disribuion of W and he value of P ab depend on only N and he fuure saffing levels, i.e., {s u : u }. The ime-dependen arrival-rae funcion conribues nohing furher. I is easy o see ha we can esablish sochasic bounds on he disribuion of W if he saffing level is monoone afer ime : hen seing s u = s for all u will yield a bound. We can go furher based on his observaion if we make approximaions. If he number of servers is large, hen W will end o be small, so ha i is ofen reasonable o make he approximaion s u s for all u >. We make his approximaion, no because he saffing level should be nearly consan for all u afer, bu because we hink we only need o consider imes u slighly greaer han. If he fuure-saffing-level approximaion held as an equaliy, hen we would obain he following approximaions as equaliies: W W and P ab P ab, where he consan saffing level in he saionary M/M/s + M model on he righhand sides is chosen o be s and he consan arrival rae is chosen o be λ MOL in (3.7). Given hese approximaions, we can use esablished resuls for he saionary M/M/s + M model, e.g., as in Garne e al. (2002) and Whi (2005). Algorihms o compue he (exac) disribuion of W are given here, including he corresponding condiional disribuions obained when we condiion on wheher or no he cusomer evenually is served. 7. Algorihm Dynamics In his secion we esablish he convergence of ISA for he M /M/s + M model. In doing so, we disregard saisical error caused by having o esimae he delay probabiliies associaed wih each saffing funcion in he simulaion. To prove convergence, we use sample-pah sochasic order, as in Whi (1981). We say ha one sochasic process {N (1) {N (2) : 0 T }, in sample-pah sochasic order and wrie : 0 T } is sochasically less han or equal o anoher, {N (1) : 0 T } s {N (2) : 0 T }, (7.1) if [ ( )] [ ( )] E f {N (1) : 0 T } s E f {N (2) : 0 T } (7.2) for all nondecreasing real-valued funcions f on he space of sample pahs. We have ordinary sochasic order for he individual random variables N (1) and N (2) and wrie N (1) s N (2) if E[f(N (1) )] E[f(N (2) )] for all nondecreasing real-valued funcions on he real line; see 20

23 Chaper 9 of Ross (1996) and Müller and Soyan (2002). Clearly, sample-pah sochasic order as in (7.1) implies ordinary sochasic order for he individual random variables for all. For he convergence, we only need ordinary sochasic-order for each ime, bu in order o ge ha, we need o properly address wha happens before ime as well. Here is he key sochasic-comparison propery for he M /M/s + M model: Theorem 7.1. (sochasic comparison) Consider he M /M/s +M model on he ime inerval [0, T ], saring empy a ime 0. If r 1 and s (1) s (2) s (1) s (2) for all, 0 T, hen for all, 0 T, or if r 1 and {N (1) : 0 T } s {N (2) : 0 T }. (7.3) Proof. Here is he key fac: The deah raes depend sysemaically on he number of servers s. When r > 1 (r < 1), he deah raes a ime decrease (increase) as s increases. The ordering of he deah raes in he wo birh-and-deah processes makes i possible o achieve he sample-pah ordering. Indeed, we jusify he relaion (7.3) by consrucing special versions of he wo sochasic processes on he same underlying probabiliy space so ha he sample pahs are ordered wih probabiliy 1. As discussed in Whi (1981), and proved by Kamae e al. (1978), ha special consrucion is acually equivalen o he sample-pah sochasic ordering in (7.3). The sample-pah ordering obained ensures ha a deparure occurs in he lower process whenever i occurs in he upper process and he wo sample pahs are equal. To sar he consrucion, we le he wo processes be given idenical arrival sreams. Then we consruc all deparures (service compleions or abandonmens) from hose of he lower process a epochs when he wo sample pahs are equal. Suppose ha a ime he sample pahs are equal: N (1) = N (2) = k. Then, a ha, he deah raes in he wo birh and deah processes are necessarily ordered by δ 1 (k) δ 2 (k). We only le deparures occur in process 2 when hey occur in process 1, so he wo sample pahs can never cross over. When a deparure occurs in process 1 wih boh sample pahs in sae k, we le a deparure also occur in process 2 wih probabiliy δ 2 (k)/δ 1 (k), wih no deparure occurring in process 2 oherwise. This keeps he sample pahs ordered w.p. 1 for all. A he same ime, he wo sochasic processes individually have he correc finie-dimensional disribuions. The simulaion experimens show ha he way he saffing funcions converge o he limi depends on he raio r θ/µ: Whenever r > 1, we encouner monoone dynamics. Whenever r < 1, we encouner oscillaing dynamics; and whenever r = 1, we encouner insananeous 21

24 convergence. As shown in 6, when r = 1, he number in sysem is independen of he saffing funcion, so we obain convergence in one sep. An example of he oscillaing dynamics is shown in Figure 10, where saffing levels are shown for he firs wo and final ieraions for he model in 4 wih µ = 1 and r = θ = 0 (no abandonmen). Figure 10: Oscillaing algorihm dynamics for he model in 4 when r = θ = 0: saffing levels in he 1 s, 2 nd and final ieraions. Targe Alpha= saffing levels ime Ieraion 1 Ieraion 2 Las Ieraion Theorem 7.2. (convergence) Consider he M /M/s + M model on he ime inerval [0, T ], saring empy a ime 0. Suppose ha we consider piecewise-consan saffing funcions ha only can change a muliples of > 0. Suppose ha in each ieraion n we can obain he acual sochasic process {N (n) : 0 T } associaed wih he saffing funcion {s (n) : 0 T } (wihou saisical error). Suppose ha s (0) = for all, 0 T. ha (a) If r > 1, hen s (n) s (m) for all n > m 0 and here exiss a posiive ineger n 0 such s ISA = s (n 0) = s (n) for all and n n 0. (7.4) (b) If r < 1, hen here exis 2 subsequences {s (2n) and s (2n+1) s (odd). s (0) s (2n) s (2n+2) s (2n+3) } and {s (2n+1) }, such ha s (2n) s (even) s (2n+1) s (1) (7.5) for all, 0 T, and for all n n 0. Moreover, here exiss a posiive ineger n 0 such ha s (2n) = s (2n 0) = s even s odd = s (2n 0+1) = s (2n+1) (7.6) 22

25 for all, 0 T, and for all n n 0. Proof. Given ha s (0) =, we necessarily have s ( 0) > s (1) for all, 0 T. Hence we have he ordering of he iniial ordering of he saffing funcions ha les us apply he sochasic order. We hen proceed recursively. As a consequence of he sample-pah sochasic order, we ge ordinary sochasic order in (7.3), we ge ordinary sochasic order N (1) N (2) for all. Ordinary sochasic order is equivalen o he ail probabiliies being ordered: P (N (1) > x) P (N (2) > x) for all x, which implies he ordering for he saffing funcions a ime. In paricular, suppose ha Since P P ( ) ( ) N (2) s (2) α < P N (2) s (2) 1 ( ) ( ) N (1) s (2) P N (2) s (2) α, necessarily s (1) s (2). Case 1: r > 1. For s (0) =, we necessarily sar wih s (0) > s (1) for all, which produces firs N (1) s N (0) and hen s (2) s (1) for all. Coninuing, we ge N (n) decreasing in n and s (n). s sochasically decreasing in n, again for all. Since he saffing levels are inegers, if we use only finiely many values of, as in our implemenaion, hen we necessarily ge convergence in finiely many seps. Case 2: r < 1. For s (0) =, we again necessarily sar wih s (0) > s (1) for all. Tha produces firs N (1) s N (0) and hen s (0) s (2) s (1) for all. Aferwards, we ge N (1) s N (2) s N (0) and s (0) s (2) s (3) s (1) for all. Coninuing, we ge N (2n) sochasically increasing in n, while N (2n+1) s (2n) decreases in n, while s (2n+1) sochasically decreases in n, for all. Similarly, increases in n for all. We hus have convergence, o possibly differen limis. Since he saffing levels are inegers, if we use only finiely many values of, as in our implemenaion, hen we necessarily ge convergence in finiely many seps. We remark ha we also obain he convergence in Theorem 7.2 wih oher iniial condiions. In paricular, i suffices o le s (0) s (0) be sufficienly large for all. For r > 1, i suffices o have s ISA for all. For r < 1, i suffices o have s (0) s even for all. We conclude his secion by making some empirical observaions, for which we have ye o develop supporing heory. We also observed ha he arge delay probabiliy α srongly influenced he dynamics. In paricular, higher values of α cause larger oscillaions in he oscillaing case, and slower convergence o he limi in all cases. Finally, we also observed a ime-dependen behavior in he convergence of s (n). We observed a greaer gap as ime increased. For example, le I inf {j : s (i) = s (j) for all i j}. We observed ha I 2 I 1 23

26 for all 2 > 1. An illusraion can be viewed in Figure 10. This ime-dependen behavior is undersandable, because he gap beween wo differen saffing levels persiss across ime, so ha here is a gap in he deah raes a each. Hence, as ges larger, he wo processes can ge furher apar. Thus he gap can firs decrease more a he iniial imes. When i reaches he limi a earlier imes, he gap will sill have o decrease more a laer imes. 8. Conclusions We have developed a simulaion-based algorihm - ISA - ha generaes saffing funcions for which performance has been shown o be sable in he face of ime-varying arrival raes for he M /M/s + M model. The resuls have been found o be remarkably robus, applying o all forms of ime variaion in he arrival-rae funcion, wih or wihou abandonmen, covering he ED, QD and QED operaional regimes. All experimens were done wih nine arge delay probabiliies, ranging from α = 0.1 (QD) o α = 0.9 (ED). In 7 we proved ha he ISA converges for he M /M/s + M model. In our simulaion experimens, we found ha ISA performs essenially he same as he modified-offered-load (MOL) approximaion (reviewed in 3) wih and wihou cusomer abandonmen. Thus we provided addiional suppor for MOL and he square-roo-saffing formula in (3.6) based on i (using arrival rae λ MOL in (3.7)). As we saw in 5, in many applicaions he MOL approximaion is well approximaed iself by lagged PSA and, in easy cases, by PSA iself. To implemen he MOL approximaion wih abandonmens, we applied many-server heavy-raffic limis from Garne e al. (2002), which yield he Garne funcion in (4.4); jus as Jennings e al. (1996) applied applied many-server heavy-raffic limis from Halfin and Whi (1981) wihou cusomer abandonmen. Finally, we found ha he simple approach of saffing o he offered load is remarkably effecive in he QED regime (when α = 0.5). having he ISA saffing funcion s ISA Tha was subsaniaed ime and again by fall on op of he offered load m, as in case 3 in Figure 2. Of course, abandonmen plays an imporan role; he saffing is always above he offered load wihou abandonmen. When he service imes are shor, he offered load m closely wih he PSA offered load m P SA may agree λ()e[s]; hen saffing o he offered load reduces o he naive deerminisic approximaion: saffing o he PSA offered load m P SA. However, i is good o be careful, because even for he realisic example in 5, PSA performed significanly worse han ISA, MOL and lagged PSA. There is much ye o be done. Here are some naural nex-seps: 24