To provide useful practical insight into the performance of service-oriented (non-revenue-generating) call

Transcription

1 MANAGEMENT SCIENCE Vol. 50, No. 10, October 2004, pp in ein inform doi /mnc INFORMS Efficiency-Driven Heavy-Traffic Approximation for Many-Server Queue with Abandonment Ward Whitt Department of Indutrial Engineering and Operation Reearch, Columbia Univerity, Mudd Building, 500 Wet 120th Street, New York, New York , To provide ueful practical inight into the performance of ervice-oriented (non-revenue-generating) call center, which often provide low-to-moderate quality of ervice, thi paper invetigate the efficiency-driven (ED), many-erver heavy-traffic limiting regime for queue with abandonment. Attention i focued on the M/M//r + M model, having a Poion arrival proce, exponential ervice time, erver, r extra waiting pace, exponential abandon time (the final +M), and the firt-come firt-erved ervice dicipline. Both the number of erver and the arrival rate are allowed to increae, while the individual ervice and abandonment rate are held fixed. The key i how the two limit are related: In the now common quality-and-efficiency-driven (QED) or Halfin-Whitt limiting regime, the probability of initially being delayed approache a limit trictly between 0 and 1, while the probability of eventually being erved (not abandoning) approache 1. In contrat, in the ED limiting regime, the probability of eventually being erved approache a limit trictly between 0 and 1, while the probability of initially being delayed approache 1. To obtain the ED regime, it uffice to let the arrival rate and the number of erver increae with the traffic intenity held fixed with >1 (o that the arrival rate exceed the maximum poible ervice rate). The ED regime can be realitic becaue with the abandonment, the delay need not be extraordinarily large. When the ED appropriation are appropriate, they are appealing becaue they are remarkably imple. Key word: call center; contact center; queue; multierver queue; queue with cutomer abandonment; multierver queue with cutomer abandonment; Erlang-A model; heavy-traffic limit; many-erver heavy-traffic limit; efficiency-driven limiting regime Hitory: Accepted by Wallace J. Hopp, tochatic model and imulation; received February 6, Thi paper wa with the author 9 day for 2 reviion. 1. Introduction Recently, there ha been great interet in multierver queue with a large number of erver, motivated by application to telephone call center (ee Mandelbaum 2001, Gan et al. 2003). For the baic Erlang-C model, i.e., for the M/M// model (having a Poion arrival proce, independent and identically ditributed (IID) exponential ervice time, erver, unlimited waiting pace, and the firt-come firterved ervice dicipline) and generalization with more general arrival and ervice procee, ueful inight can be gained by conidering many-erver heavy-traffic limit in which the number of erver increae along with the arrival rate (with the individual ervice rate held fixed) o that the traffic intenity / increae too, with 1 a (1.1) where 0 << (ee Halfin and Whitt 1981; Puhalkii and Reiman 2000; Jelenkovic et al. 2004; Whitt 2005a, b). In thi o-called Halfin-Whitt limiting regime or Quality and Efficiency-Driven (QED) limiting regime, the teady-tate probability of delay approache a limit trictly between 0 and 1. In contrat, if we only increae, keeping fixed, then the teady-tate probability of delay approache 1 a 1; if intead we only increae, keeping fixed with <1, then the teady-tate probability of delay approache 0. Garnett et al. (2002) howed that the ame QED limiting regime i alo ueful for multierver queue with cutomer abandonment, pecifically for the aociated Erlang-A model, i.e., for the purely Markovian M/M//+M model with exponential abandonment time (the final+m) (alo ee Whitt 2005b for a generalization to G/M//+M). Then, becaue the abandonment enure tability, the limit in (1.1) can be negative a well a poitive. Again, in thi QED limiting regime, the teady-tate probability of initially being delayed approache a limit trictly between 0 and 1. However, the probability of abandonment i aymptotically negligible; pecifically, the teady-tate probability of abandonment i aymptotically of order 1/ a. Our purpoe here i to point out that, in the preence of ignificant cutomer abandonment, the 1449

2 1450 Management Science 50(10), pp , 2004 INFORMS QED limiting regime i not the only many-erver heavy-traffic limiting regime worth conidering to generate approximation that are ueful in practice. Here we focu attention on the alternative Efficiency- Driven (ED) limiting regime, in which the probability of abandonment approache a limit trictly between 0 and 1, while the probability of initially being delayed approache 1. A hown by Garnett et al. (2002) for the M/M/ + M model, uch a limit occur when approache a limit trictly greater than 1 a and. More imply, it uffice to keep fixed with >1a and. In practice, of coure, we are uually intereted in a ingle queueing ytem with pecified parameter, not a equence of queueing ytem. To apply one of thee heavy-traffic limit, we think of our given ytem a one term (term ) in a equence of ytem indexed by, where (e.g., ee pp. 58 and 158 of Whitt 2002 or Equation (2.21) and (3.4) here). Thu, if there are two different limit that might be conidered, it i not evident a priori which of thee limit hould be more ueful. Then, the reulting approximation can be judged by their effectivene, i.e., by their accuracy and eae of ue. It i our experience that in typical cenario with a large number of erver (e.g., = 100) and only moderate abandonment that the QED approximation are uually more accurate than the ED approximation. However, when the quality of ervice i omewhat low, the ED approximation become appropriate. The ED approximation become appropriate when queue length and waiting time are relatively large, e.g., in environment uch a the Internal Revenue Service help line. The ED approximation may be relevant when abenteeim i a problem; then agent work cheduling may be aiming to be in the QED regime, but end up in the ED regime unintentionally. The great appeal of the ED approximation, when they are appropriate, i their implicity; they often can be ued for quick back-of-the-envelope calculation. The ED approximation require having >1, o we may not expect them to be very ueful. However, data from ervice-oriented call center how that the arrival rate often exceed the maximum poible ervice rate over meaurement interval, even when target performance level occurring in ervicelevel agreement (SLA) are being met. In addition, computational reult from computer imulation and numerical algorithm ubtantiate that performance target often can be met when the arrival rate exceed the maximum poible ervice rate. Specifically, in Whitt (2005c), we developed a numerical algorithm for calculating approximation for all the tandard performance meaure in the M/GI//r + GI model with large, in which the ervice time and abandon time come from independent equence of IID random variable with general ditribution. While tudying how the approximation perform compared to imulation, we aw that it i often reaonable to have the arrival rate exceed the maximum poible ervice rate when there are many erver and ignificant abandonment. An initial example ha 100 agent (erver) handling call with mean holding (ervice) time and mean abandon time both equal to five minute. The SLA may tipulate that at mot 5% of the cutomer hould abandon and that 80% of the call that eventually are erved hould be anwered within 30 econd. The M/M/ + M model indicate that the 100 agent can handle an arrival rate of 20.4 call per minute, yielding a traffic intenity of = 102 > 1. With that arrival rate, the SLA i jut met: 5% of the arrival abandon and 80% of the anwered call are anwered within 30 econd. Some call center provide even lower quality of ervice; then we may encounter even higher traffic intenitie. That initial example illutrate that, with ubtantial abandonment, we may well have >1 when the SLA i met. However, even though = 102 > 1, the QED approximation perform well, being far uperior to the ED approximation. For example, the QED approximation for the 0.05 abandonment probability i 0.051, wherea the ED approximation i In thi cae, we will how that the ED approximation provide only a rough indication of performance. Neverthele, the ED approximation may be ueful. The ED approximation become more appropriate with a further degradation of ervice. For example, uppoe that the arrival rate increae by 8% from 20.4 call per minute to 22.0 call per minute, increaing the traffic intenity from = 102 to = 110. Now, a hown by Table 1 and 2 in 5, the ED approximation are quite good. For example, the exact abandonment probability i 0.10, while the ED approximation i 0.09; the exact mean queue length i 10.9, while the ED approximation i Moreover, further refined ED approximation have le than 1% error. Without a finite waiting room or cutomer abandonment, teady-tate ditribution do not exit when >1. Then >1 i imply an overloaded regime, becaue the queue length explode a time evolve. Then the overloaded regime i intereting primarily to decribe tranient behavior. Without a finite waiting room or cutomer abandonment, it i poible to define an underloaded ED limiting regime, in which i le than 1 but extremely cloe to 1. Indeed, that i the conventional heavy-traffic limiting regime with a ingle erver. However, with a large number of erver, ha to be o cloe to 1 that the ytem become untable in the ene that a very mall increae in the traffic intenity puhe the ytem into the overloaded regime.

3 Management Science 50(10), pp , 2004 INFORMS 1451 But the ED many-erver heavy-traffic limiting regime become viable with cutomer abandonment. Even a mall amount of abandonment can keep the ytem table when the arrival rate exceed the maximum poible ervice rate. Even though the teadytate probability of initially being delayed approache 1 in the ED limit, thi alternative ED heavy-traffic limit can be realitic becaue the delay experienced need not be extraordinarily large. We believe that the ED regime doe decribe the operation of many exiting call center remarkably well, epecially when a great emphai i placed on efficiency. An emphai on efficiency i more common among call center that are ervice oriented intead of revenue generating. Even though in thi paper we focu on a regime upporting low-to-moderate quality of ervice, we do not advocate providing low-to-moderate quality of ervice in call center. Indeed, a uggeted in Whitt (1999), it may be poible to provide pectacular quality of ervice without requiring a commitment of exceive reource with good planning and good execution. However, to improve the quality of ervice, it i important to undertand the performance of exiting call center. We contend that the ED limiting regime can be helpful to undertand the performance of exiting call center providing low-to-moderate quality of ervice. In thi paper, we etablih limit in the ED heavytraffic regime and develop approximation baed on thoe limit. We only conider Markovian birth-anddeath model. For the Markovian model conidered here, the limit are not difficult to obtain; indeed, we etablih them by the ame argument ued in the eminal heavy-traffic paper on multierver queue by Iglehart (1965), drawing on Stone (1963). The tochatic-proce limit here alo can be viewed a conequence of more general reult for tatedependent Markovian queue in Mandelbaum and Pat (1995; ee Theorem 4.1 and 4.2 and 5.3 there), but the relatively complicated full framework of Mandelbaum and Pat (1995) i not needed to treat the pecial cae conidered here. The main contribution here, we believe, i communicating that thi ED many-erver heavy-traffic limiting regime can indeed yield ueful approximation. When thi ED regime i appropriate, the heavy-traffic limit i very helpful becaue it generate remarkably imple approximation (e.g., ee (3.1) (3.5)). In particular, the ED approximation are imple even in comparion to the QED approximation. The ED approximation can be ueful even if they are le accurate than the QED approximation. Indeed, becaue there already are effective exact numerical algorithm to calculate all deired performance meaure in the M/M//r+M model, a in Whitt (2005c), only truly imple approximation can provide ignificant value added. The ret of thi paper i organized a follow. We begin in 2 by etablihing the tochatic-proce limit for the number in ytem in the M/M//r + M model in the ED regime. We alo etablih a determinitic fluid limit and a limit for the teady-tate ditribution in the ED regime. In 3, we develop heuritic approximation for teady-tate performance meaure baed on the limit. Interetingly, the approximation for the teady-tate queue length in the ED regime (ee (3.4) and (3.6)) depend on the number of erver,, and the individual abandonment rate,, only through the ratio /. It i thu natural to wonder if we can obtain a related limit a 0 and, indeed, uch a limit wa etablihed by Ward and Glynn (2003) (auming fixed ; ee Part 4 of Theorem 1 and Remark 5 there). In 4, we how that the ame ED limit hold more generally when i held fixed with >1 and /, becaue either the individual abandonment rate become mall or the number of erver become large or both. The main approximation developed by Ward and Glynn (2003) for fixed tem from a double limit in which 1 and 0, o that 1 / (1.2) in the pirit of (1.1), which define the QED limiting regime. When 0, we alo emphaize the value of the alternative ED regime in which i fixed with >1. In 5, we compare the ED approximation to exact numerical olution for the M/M//r + M model, which we obtain uing the algorithm in Whitt (2005c). (That algorithm producing approximate performance meaure for the M/GI//r +GI model produce exact numerical reult for the M/M//r + M pecial cae.) We how that the performance of the ED approximation range from poor to pectacular. We provide way to judge when the ED approximation will be effective. The paper end in 6 with our concluion. We conclude thi introduction by mentioning two companion paper: In Whitt (2005a), we develop determinitic fluid approximation for the general G/GI//r + GI model in the ED many-erver heavy-traffic regime. Unlike here, the emphai there i on trying to account for the impact on performance of a nonexponential ervice-time ditribution and a nonexponential abandon-time ditribution. In Whitt (2005e), we extend the ED many-erver heavy-traffic limit here to Mn/Mn//r + Mn model with tatedependent rate. Our motivation there i to gain additional inight into the performance of the M/M//r + Mn approximation of the M/GI//r + GI model developed in Whitt (2005c).

4 1452 Management Science 50(10), pp , 2004 INFORMS For additional dicuion about cutomer abandonment in queue, ee Brandt and Brandt (1999, 2002), Zohar et al. (2002), Ward and Glynn (2003), Mandelbaum and Zeltyn (2004), and the reference therein. 2. Limit for the Erlang-A Model in the ED Regime In thi ection, we etablih ED many-erver heavytraffic limit for the M/M/+M model, alo known a the Erlang-A model. We actually treat the M/M//r+ M model, allowing finite waiting room r a well a infinite waiting room (r ). When r<, we require that r be ufficiently large that it i not a factor, e.g., ee (2.2) below. Arrival finding all erver buy and all waiting pace full are blocked and lot, without affecting future arrival. Entering cutomer are erved in order of arrival by the firt available erver, but waiting cutomer may elect to abandon before they tart ervice. Cutomer do not abandon after they tart ervice. We chooe meauring unit for time o that the individual mean ervice time i 1/ = 1. The model i thu characterized by four parameter: (1) the arrival rate,, (2) the number of erver,, (3) the number of extra waiting pace, r, and (4) the individual abandonment rate,. The aumption of exponential abandonment rate i equivalent to the cutomer having IID time to abandon before beginning ervice, with a common exponential ditribution having mean 1/. We conider a equence of thee M/M//r + M model indexed by. Let, r, and be the remaining parameter, a a function of. We increae and r with, but we leave the individual ervice rate = 1 and the individual abandonment rate = fixed, independent of. We let the traffic intenity remain fixed. (It uffice to let >1.) In particular, we aume that and with = where >1 (2.1) r = where >q (2.2) q 1 (2.3) for all. Condition (2.1) determine the ED regime. Condition (2.2) enure that, aymptotically, no cutomer are blocked, a we will how below. Let N t be the number of cutomer in the ytem at time t when there are erver. The ED regime i relatively tractable becaue, in the ED regime, N t tend to concentrate about a fixed value; i.e., for large, we will how that N t 1 + q (2.4) for q defined in (2.3). Heuritically, we obtain q in (2.3) by finding the point, ay x, where the input rate equal the output rate. Clearly, that can occur only with all erver buy (x>1). Before caling by dividing by, we have the equation = + x (2.5) after dividing by and letting, we obtain the equation = 1 + x (2.6) The olution to thee equation i x = q for q in (2.3). The diffuion approximation i a refinement of the determinitic approximation in (2.4) and (2.3). To etablih convergence to a diffuion proce, we form the normalized tochatic proce N t N t 1 + q t 0 (2.7) for q in (2.3). (Throughout thi paper, we ue boldface to denote normalized procee and their limit.) Let the initial tate N 0 be pecified independently, o that the tochatic proce N t t 0 i Markov. To etablih a tochatic-proce limit for the procee N, let D D0 denote the pace of all right-continuou real-valued function on the poitive half line 0 with left limit everywhere in 0, endowed with the uual Skorohod J 1 topology (ee Billingley 1999 or Whitt 2002). Let denote convergence in ditribution (weak convergence), both for equence of tochatic procee in D or for equence of random variable in. Let Norm 2 denote a random variable that i normally ditributed with mean m and variance 2. Theorem 2.1 (Stochatic-Proce Limit for the Erlang-A Model in the ED Regime). Conider the equence of M/M//r + M model pecified above, atifying (2.1) (2.3). If N 0 N0 a, then N N in D a (2.8) where N i the caled proce in (2.7) and N i an Orntein-Uhlenbeck (OU) diffuion proce with infiniteimal mean tate-dependent drift) and infiniteimal variance which ha teady-tate ditribution mx = x (2.9) 2 x = 2 (2.10) N d = Nor0 / (2.11)

5 Management Science 50(10), pp , 2004 INFORMS 1453 Proof. Becaue N i a birth-and-death proce and the limiting OU diffuion proce ha no boundarie, we can apply the weak convergence theory in Stone (1963), jut a Iglehart (1965) did in hi eminal paper. Given Stone (1963), with the caling in (2.7), it uffice to how that the infiniteimal mean and variance converge to the infiniteimal mean and variance of the limit proce. Becaue N t i nonnegative integer valued, the poible value of N t are k 1 + q/ for k 0. Hence, for arbitrary real number x, we conider a equence x 1, where x i an allowed value of N t for each and x x a. For example, for all ufficiently large, we can contruct an allowed value by letting x 1 + q+ x 1 + q where t i the floor function, i.e., the greatet integer le than or equal to t. When x<0, we need to be ufficiently large to guarantee that 1+q+x 0. For any real number x and equence of allowed value x 1, the infiniteimal mean are m x lim EN t + h N t/h N t = x h 0 [ N t + h N = lim E t h 0 h ] N t = + 1/ + x = 1/ + x x mx a and the infiniteimal variance are 2 x lim EN t + h N t 2 /h N t = x h 0 [ N t + h N = lim E t 2 h 0 h ] N t = + 1/ + x = + + 1/ + x 2 2 x a It i well known that the OU diffuion ha a normal teady-tate ditribution with variance equal to the infiniteimal variance divided by twice the tatedependent drift rate (e.g., ee p. 218 of Karlin and Taylor 1981). The tochatic-proce limit in Theorem 2.1 i often called a functional central limit theorem (FCLT) (e.g., ee Whitt 2002). A imple conequence of the FCLT i a functional weak law of large number (FWLLN), which formalize the heuritic dicuion in (2.4) (2.6). It i obtained imply by dividing by before letting in the etting of Theorem 2.1. To tate the FWLLN, let N t N t t 0 (2.12) Corollary2.1 (FWLLN for the Erlang-A Model in the ED Regime). Under the condition of Theorem 2.1, where for q in (2.3). N N in D a (2.13) Nt = 1 + q t 0 (2.14) Proof. When we divide the caled proce in (2.7) by and let, we obtain convergence in probability to the zero function by an application of a verion of the continuou mapping theorem Theorem in Whitt (2002) implying the reult. It i alo poible to etablih a more general determinitic fluid approximation by jut changing the initial condition in Corollary 2.1. When we cale by dividing by throughout, we obtain an ordinary differential equation (ODE) for the limit, which i ueful for decribing the tranient behavior of the Erlang-A model. For a real number x, let x + max x 0 and x min x 0. Theorem 2.2 (ED Fluid Limit for the Erlang-A Model). Conider the equence of M/M//r + M model pecified above, atifying (2.1) (2.3), and let ˆN t be the caled number in ytem in (2.12). If ˆN 0 n0 a, where n0 i a real number (determinitic), then ˆN n in D a (2.15) where n i a degenerate diffuion proce with continuou piecewie-linear infiniteimal mean (tate-dependent drift) mx = 1 x x 1 (2.16) and infiniteimal variance 2 x = 0; i.e., n i the ODE ṅt dn dt t = 1 nt nt 1 (2.17) with initial value n0. Proof. We firt extend the proce N to the entire real line by letting the birth rate be and the death rate be 0 for negative integer, and by letting the

6 1454 Management Science 50(10), pp , 2004 INFORMS birth rate be 0 and the death rate be + for poitive integer greater than. With that contruction the caled proce ˆN will never viit tate outide the interval 0, but at the ame time will have no boundarie. The proof now i eentially the ame a for Theorem 2.1. Now we need to calculate the infiniteimal mean and variance when we cale by dividing by intead of. Now we let x be a poible value of ˆN t for each, uch that x x a. Now the limit for the infiniteimal mean i m x lim E [ ] ˆN t + h ˆN t/h ˆN t = x h 0 [ N t + h N t = lim E h 0 h { 1 x 1 x = 1 x x 1 { 1 x 1 x = x x 1 1 x x 1 ] N t = x The limit for the infiniteimal variance i 2 x lim E [ ] ˆN t + h ˆN t 2 /h ˆN t = x h 0 [ N t + h N = lim E t 2 ] N t = x h 0 h x 1 x = 1 + x x x a It i well known that the degenerate OU diffuion (with 0 infiniteimal variance) i the ODE in (2.17). Remark 2.1 (Steady-State of the Fluid Limit). It i eay to ee that the determinitic fluid limit function in Theorem 2.2, nt, converge monotonically a t to it teady-tate limit n = q 1/. For cutomary application, we are primarily intereted in approximation for the teady-tate performance meaure in the M/M//r + M model. Such approximation can be generated heuritically from Theorem 2.1, but limit for the teady-tate performance meaure do not follow directly from Theorem 2.1. They do with additional argument, however. They can alo be etablihed directly, tarting from the teady-tate ditribution in the M/M//r + M model. Here we apply Theorem 2.1. Here are the performance meaure we conider: N, the teady-tate number of cutomer in the ytem; Q, the teady-tate number of cutomer waiting in queue; W, the teady-tate waiting time (before beginning ervice or abandoning) of a typical cutomer (which ha the ame ditribution a the virtual waiting time of an arrival at an arbitrary time, becaue of the Poion arrival proce); and P ab, the teady-tate abandonment probability. Let S denote the event that a cutomer eventually i erved; necearily PS = 0 = 1 PS = 1 = P ab. Let W S denote a random variable with the conditional ditribution of the waiting time given that the cutomer eventually will be erved, i.e., PW S x PW x S. For the Erlang-A model, it i well known that thee teadytate quantitie are well defined. Theorem 2.3 (ED Heavy-Traffic Limit for Steady- State Quantitie in the Erlang-A Model). Conider the equence of M/M//r + M model pecified above, atifying (2.1) (2.3). Then, a, N N ( 1 + q N = d Nor 0 ) (2.18) N N N = 1 + q (2.19) PN 0 (2.20) Q N ( + q Q = d Nor 0 ) (2.21) Q Q Q = q 1 (2.22) P ab Pab 1 (2.23) W S w (2.24) W W (2.25) where w i the determinitic quantity w 1 log = 1 e log e1 Pab > 0 (2.26) and W i the random variable with PW >x= e x 0 x w and PW >w= 0 (2.27) for w in (2.26), which ha expected value EW = Pab = q (2.28) Proof. For the firt limit in (2.18), mot of the work ha been done by Theorem 2.1. To make ue of Theorem 2.1, we can follow the argument in the proof of Theorem 4 in Halfin and Whitt (1981). We can deduce that the equence of normalized teadytate random variable N 1 i tight by contructing upper and lower bounding procee

7 Management Science 50(10), pp , 2004 INFORMS 1455 that have proper limit a (ee Halfin and Whitt 1981 for detail). The tightne implie relative compactne by Prohorov Theorem, Theorem of Whitt (2002), thu every ubequence ha a convergent ub-ubequence. We how convergence by howing that all convergent ubequence mut have the ame limit. Conider any convergent ubequence. That convergent ubequence can erve a the equence of initial ditribution in the condition of Theorem 2.1. But becaue thee particular initial ditribution are tationary ditribution, the limiting ditribution mut be a tationary ditribution for the limiting OU diffuion proce. The OU diffuion proce ha a unique tationary ditribution, however. Thu, all convergent ubequence mut have that normal tationary ditribution a their limiting ditribution. With that additional argument, Theorem 2.1 implie (2.18). The next limit (2.19) follow by dividing by and letting, jut a in Corollary 2.1. Then (2.20) i an immediate conequence. The limit for the caled queue-length procee in (2.21) and (2.22) follow from (2.18) by continuou mapping theorem. To etablih (2.23), note that in teady tate, the erver are all buy aymptotically, by (2.20). Hence, after dividing by, the ervice rate i aymptotically 1. Becaue the arrival rate i aymptotically after dividing by, the total abandonment rate after dividing by necearily i aymptotically 1 and P ab Pab 1/. We now turn to the waiting-time reult. The waiting time for a cutomer that eventually will be erved W S i the firt paage time to the zero tate, tarting with Q cutomer, if we turn off the arrival proce directly after that arrival. With the arrival proce turned off, 1 Q t qt by the law of large number, where the limit qt atifie the ODE qt dq = 1 qt (2.29) dt with initial condition q0 = q. Aymptotically a, at time t, the caled queue-length proce i being depleted by ervice completion at rate 1 and by abandonment at rate qt. Arguing more carefully, for any >0, the caled number of departure in the interval t t +, given Q t, where 1 Q t qt, i aymptotically 1 + qt + o a. Hence we indeed have (2.29). Next, it i eay to ee that the unique olution to the ODE in (2.29) i qt = ( q + 1 ) e t 1 t 0 (2.30) Thu, the waiting time of a cutomer that will eventually be erved approache the value w uch that qw = 0. Solving qw = 0, we get w = 1 log e ( q ) = 1 log e (2.31) a given in (2.26). Given that erved cutomer wait exactly w in the limit a, we immediately obtain (2.25) and (2.27). We oberve that (2.28) i conitent with two exact relation for the M/M/ + M model. Firt, by Little Law or L = W, we have EQ = EW (2.32) even without the M/M/ + M aumption, e.g., ee Whitt (1991). Second, for the M/M/ + M model, we can expre the total abandonment rate in two way, yielding P ab = EQ (2.33) or P ab = EQ = EQ (2.34) Combining (2.32) and (2.33), we obtain P ab = EW (2.35) Thu, when we know any one of the three performance meaure P ab, EQ, or EW, we know all three. Formula (2.28) how that the relation remain valid in the limit. 3. ED Approximation The main ED approximation for the M/M//r + M model are the three imple approximation that follow directly from (2.22) (2.26): P ab Pab EQ q 1 1 EW S w 1 log e (3.1) It i important to recognize, however, that only the firt imple approximation for the abandonment probability P ab i generally valid beyond the Markovian M/M//r + M model, a can be een from Whitt (2005d). In particular, the approximation for the mean teady-tate queue length and waiting time depend critically on the exponential ditribution aumption. Thee M/M//r + M ED approximation can be very ueful more generally, however, a rough approximation and to tet whether an M/M//r + M model i appropriate. We chooe to focu on the conditional waiting time given that the arrival i eventually erved, W S, but we alo have reult for the unconditional waiting time W, from which we can obtain reult for the conditional waiting time given that the arrival eventually abandon, W S c, where Sc i the

8 1456 Management Science 50(10), pp , 2004 INFORMS complement of the event S. Theorem 2.3 implie that W S c W S in the limit a,o contraint on the ditribution of W S will tend to be omewhat more tringent than contraint on the ditribution of W, but thee will differ relatively little when the abandonment probability i not great. Combining the firt approximation in (3.1) with the exact relation in (2.33), we obtain the approximation EW = P ab Pab 1 = (3.2) We alo obtain eentially the ame approximation for EW S, baed on an approximation for w, EW S w = 1 log e1 Pab Pab 1 = (3.3) For approximation (3.3), we ue the approximation log1 x x for mall x, which i aymptotically correct (the ratio approache 1) a x 0. From (2.21), we alo obtain a normal approximation for the entire teady-tate queue-length ditribution, in particular, ( 1 Q Nor ) (3.4) A conequence i an approximation for the variance VarQ (3.5) An immediate practical inight to glean from the approximation for the teady-tate queue length in (3.1), (3.4), and (3.5) i that the coefficient of variation (tandard deviation divided by the mean) approache 0a for all parameter >1 and >0. Thu the crude determinitic analyi begin to tell more and more of the tory a increae. For example, the 4 5 = 20 cae in Table 1 in 5 have queue-length coefficient of variation ranging from 0.33 to (And thee value would be maller if we looked at the conditional queue length given that it i poitive.) We next dicu refined heuritic approximation that do not follow directly from Theorem 2.3. An obviou imple refinement to (3.4) i ( 1 Q Nor ) + (3.6) where x + max0x. Let and be the cumulative ditribution function (cdf) and probability denity function (pdf) of a tandard normal random variable, repectively, i.e., y PNor0 1 x y x dx where x = 1 2 e x2 /2 (3.7) Let c be the aociated complementary cdf (ccdf), defined by c x 1 x, and let h be the aociated hazard function hx x (3.8) c x To implify expreion, we ue the following notation: q 1 v and qv q (3.9) v We then obtain the aociated approximation and PQ >0 PNorqv>0= c (3.10) EQ Q >0 ENorqvNorqv>0 =q+ vh (3.11) EQ 2 Q >0 ENorq v 2 Norq v > 0 = q 2 + v + 2q vh + vh (3.12) for q, v, and in (3.9). The conditional normal moment in (3.11) and (3.12) are tandard (e.g., ee Propoition 18.3 of Browne and Whitt 1995). Clearly, we can combine the lat three formula to obtain approximation for the mean EQ and variance VarQ. Becaue the three performance meaure P ab, EQ, and EW are all related by the exact relation in (2.32) (2.34), we can ue a refined approximation for any one to obtain refined approximation for all three. Thu, we obtain the refined approximation for the abandonment probability P ab Pab EQ (3.13) 1/ Of coure, approximation for EQ tranlate immediately into approximation for EW by virtue of Little Law, (2.32). We now conider refined approximation for PW x S, the cdf of the conditional teadytate waiting time until beginning ervice, given that

9 Management Science 50(10), pp , 2004 INFORMS 1457 the cutomer i erved. Becaue the waiting time tend to be the um of a relatively large number of ervice time, we apply an approximation baed on the law of large number together with the fluid approximation in (2.26), aying that PW x S Q = q 1 if 1 ( ) 1 log e x (3.14) 1 + q and 0 otherwie. However, 1 ( ) 1 log e x 1 + q if and only if q 1 ex 1 (3.15) Thu, we obtain the approximation ( PW x S P Q ) ex 1 for v in (3.9). ( P Norq v ) ex 1 ( ) /e x 1 q = v (3.16) 4. Slow Abandonment in the ED Regime: The Limit a / From Equation (3.4) and (3.6), we ee that the normal approximation for the teady-tate queue length Q depend on the parameter and only through the ratio /. Thu, it i natural to conider limit in which we let 0 intead of and, indeed, that ha already been done by Ward and Glynn (2003). However, they did not emphaize the ED regime, where i fixed with >1. Intead, they emphaize the limiting regime in (1.2). We go further for the ED limiting regime here by etablihing limit a /, allowing either or 0 or both. For the pecial cae in which only 0, we recover Part 4 of Theorem 1 in Ward and Glynn (2003); for the pecial cae in which only, we recover Theorem 2.1. We tart by defining caled procee, indexed by both and, Y t [ N t/ 1 ] for t 0 (4.1) We now tate the reult, omitting the proof becaue it i jut like the proof of Theorem 2.1. Theorem 4.1 (The ED Limit a /. Conider the M/M//r + M model defined in 2, atifying (2.1) and (2.3). If Y 0 Y0 in a /, where Y i the caled proce in (4.1), then Y Y in D a / (4.2) where Y i an OU diffuion proce with infiniteimal mean mx = x and infiniteimal variance 2 x = 2. From Theorem 4.1, we obtain the ame approximation for the teady-tate queue length Q a before, i.e., a in (3.4) and (3.6). The fact that the ratio / play uch a critical role invite an explanation. We would like to identify the fundamental role it play, in the pirit of the traffic intenity, = /. In fact, the role of the ratio / i brought out by the limit. Indeed, we ee the role of / from the firt heuritic approximation developed in (2.4) (2.6), baed on finding the point where the input rate matche the output rate. That informal reaoning tend to make ene if either i large or i mall or both. We alo ee that, in firt order, the queue length hould be proportional to /. The important role of the ratio / i further brought out by the tochatic-proce limit in Theorem 2.1 and 4.1. Even from the baic limit in Theorem 2.1, we can ee the role of and. Firt, the role of i clear from the caling in (2.7). We ee that the mean and variance of N t hould both be approximately proportional to. Next, the role of i firt een by it contribution to q in (2.3) and the way it affect the centering term in (2.7). We then ee that the tate-dependent drift of the limiting OU diffuion proce i proportional to, which implie that the variance of the teady-tate ditribution i inverely proportional to. Combining thoe inight, we ee that both the mean and variance of the teady-tate queue length hould be approximately proportional to /. The proof of Theorem 4.1 how that it uffice to let /. We can alo provide a more informal direct argument. If we conider a waiting cutomer, that cutomer tend to abandon at rate and tend to move toward ervice at rate =. Thu / i the ratio of two rate: the rate a cutomer move toward entering ervice, and the rate the cutomer tend to abandon. When the ratio / i large, the tendency to abandon i le, o that the queue ize i likely to be larger. In other word, the key ratio / can be thought of a the tendency to be erved intead of abandon. For given proportion of abandonment, etimated by 1, a higher ratio / caue larger queue, in particular, a larger value of q, puhing u more into the ED regime. 5. Numerical Comparion In thi ection, we evaluate the ED approximation in 3, baed on the limit in 2 and 4, by compar-

10 1458 Management Science 50(10), pp , 2004 INFORMS Table 1 A Comparion of Approximation with Exact Numerical Value for Several Performance Meaure in the M/M/+ M Model Performance meaure a a function of with / fixed Approximation Exact with erver Perf. mea. Simple Refined ,000 10,000 / = 1000 and 1 = 010 P ab EQ SDQ EW S / = 1000 and 1 = 002 P ab EQ SDQ EW S / = 100 and 1 = 010 P ab EQ SDQ EW S / = 100 and 1 = 002 P ab EQ SDQ EW S ing them to exact numerical reult for the Erlang-A model, uing the algorithm decribed in Whitt (2005c). We vary with both the ratio / and the limiting abandonment rate / = 1 held fixed. We conider two value for the ratio /, 1,000 and 100; we conider two value for the caled total abandonment rate 1, 0.10 and The two cae with / = 100 are the two cae dicued in the introduction. We conider four different performance meaure: the probability of abandonment, P ab; the mean teady-tate queue length, EQ ; the tandard deviation of the teady-tate queue length, SDQ ; and the conditional mean teady-tate waiting time given that the cutomer eventually will be erved, EW S. It hould be noted that the firt two, P ab and EQ, are connected by the exact relation (2.34) o that they have the ame relative error. It i intereting to ee the actual value, however. In Table 1 we diplay two different approximation: firt, the imple approximation, and, econd, a refined approximation. The imple approximation are given in (3.1) and (3.5). The refined approximation for the mean queue length i obtained by combining (3.10) and (3.11). The refined approximation for the tandard deviation of the queue length i obtained by combining (3.10) (3.12). The refined approximation for the probability of abandonment, P ab, i obtained from (3.13), uing the refined approximation for the mean queue length. From Table 1 we ee that, a expected, the quality of the reult improve a the ratio / increae and a the caled total abandonment rate 1 increae. Conitent with intuition, that evidently i a general property. We have found that a good way to etimate, in advance, the overall quality of the ED approximation i to look at the product of thee quantitie / and 1, which i jut the approximate teady-tate queue length, i.e., EQ q = 1 = 1 (5.1) In the four cae here, q i 100, 20, 10, and 2. Accordingly, the approximation are accurate when / = 1000, 1 = 010, and q = 100, but the approximation are crude when / = 100, 1 = 002, and q = 2. Importantly, the approximation appear ueful in the middle two cae in which q = 20 and q = 10. Very roughly, the percentage error in the approximation can be etimated by the reciprocal of q, i.e., the percentage error hould be of order 1/q. For example, the approximation error hould be about 10% when q = 10. Except poibly in the lat cae with / = 100 and 1 = 002, Table 1 how that the four performance meaure do not vary greatly with, over a very wide range, when / and 1 are held fixed. The mean teady-tate queue length i approximately proportional to /, but independent of for fixed /. On the other hand, by Little Law, the mean teady-tate

11 Management Science 50(10), pp , 2004 INFORMS 1459 waiting time i approximately inverely proportional to for fixed /. Except in the bet cae with / = 1000 and 1 = 010, the refinement are much better than the imple approximation. Neverthele, we feel the imple approximation are epecially ueful for making quick rough etimate. The difference between the refined approximation and the imple approximation give a good idea of the accuracy of the imple approximation. The weaket approximation in Table 1 i clearly the imple approximation for the mean conditional waiting time, EW S. The reult ugget that it would be better to jut focu on the unconditional expected waiting time, EW, and ue Little Law with the approximation for the mean queue length. Then we obtain the ame accuracy a the mean queue length. In ummary, we regard the numerical reult in Table 1 a trong evidence that the approximation can be very ueful. To evaluate call center performance, there i great interet in ervice level. It i tandard to require that y% of all call be anwered within x econd for value uch a y = 80% and x = 30 econd. Thu, we are epecially intereted in having an approximation for the conditional cdf PW x S for appropriate value of x. In our numerical example, the mean waiting time vary from one example to the next, o natural value of x change from one example to the next. Thu, we will tandardize by expreing our ervice-level cut-off a a contant multiple of the mean waiting time. Becaue the mean waiting time can be roughly approximated by (3.2), it i natural to look at argument of the form x = Pab/ for variou value of, centered around 1. We thu examine how the approximation for PW x S in (3.16) perform in Table 2 for argument x = Pab/ for variou value of. Here, we vary both and, keeping the quantitie / and 1 fixed. We conider the ame four cae of / and 1 a in Table 1. We make comparion with exact reult for all power of 10 ranging from = 1to = The approximation for the waiting-time cdf in Table 2 i not a accurate a the approximation in Table 1. The approximation are pretty good for higher value of, e.g., for 1, but not for very mall value, epecially when the argument i very mall, e.g., when = 0. It i ignificant that the conditional waiting-time cdf approximation doe work well for Table 2 A Comparion of the Normal Approximation for the Conditional Waiting-Time cdf Given That a Cutomer I Served, PW x S, with Exact Numerical Value in the M/M/+ M Model PW P ab/ S a a function of and Number of erver Cae ,000 Approx. 10, ,000 / = 1000 and 1 = / = 1000 and 1 = / = 100 and 1 = / = 100 and 1 = Note. The argument x conidered are x = P ab/ a a function of and for fixed ratio / and fixed 1.

12 1460 Management Science 50(10), pp , 2004 INFORMS Table 3 The Probability of Initially Being Delayed, 1 PW = 0S, in Each of the Cae Conidered Previouly Cae 1 / q P S 1 PW = 0S the higher argument needed to determine taffing to meet SLA. When conidering whether to ue QED or ED approximation, it i intructive to look at the abandonment probability and the probability of initially being delayed. In the QED (ED) regime, the firt hould be relatively mall (large), while the econd hould be relatively large (mall). We expect the ED approximation to be reaonably accurate when the probability of initially being delayed i greater than or equal to To illutrate, we plot the probability of not being erved immediately, 1 PW = 0S, for the cae conidered previouly in Table 3. For the two example in the introduction, thi probability of initially being delayed i and , repectively. We would thu expect the ED approximation to be very crude in the firt cae, but relatively good in the econd cae, a we oberved. 6. Concluion In thi paper, we etablihed ED many-erver heavytraffic limit for Markovian queue with cutomer abandonment, pecifically for the M/M//r + M model. Many-erver limiting regime involve a equence of queueing ytem indexed by the number of erver,, in which both and the arrival rate are allowed to increae without bound, while the exponential ervice-time and abandon-time ditribution are held fixed. Within the context of the manyerver heavy-traffic limiting regime for queue with abandonment, the ED regime can be characterized in two equivalent way: (1) aume in addition that the probability of abandonment, P ab, converge to a limit trictly between 0 and 1, or (2) aume in addition that the traffic intenity,, approache a limit with >1. We alo developed direct and refined approximation baed on the ED many-erver heavy-traffic limit. We conclude that the ED many-erver heavytraffic approximation can be very ueful to decribe the performance of many-erver queue with ubtantial abandonment. The firt approximation for key performance meaure in the M/M//r + M model (Erlang-A model) in (3.1) (3.5), obtained directly from the limit, are remarkably imple. The heuritic refinement in (3.6) (3.16) are alo not too complicated. Table 1 and 2 how that the approximation are quite accurate when the probability of initially being delayed i not too mall, e.g., when 1 PW = 0S 080 or the approximate mean queue length q 1/ i not too mall, e.g., when q 10. In extreme cae, a when 1 PW = 0S 099 or when q 100, the ED approximation will be extremely accurate, and clearly better than the QED approximation. However, for le extreme cae when >1, we can expect the QED approximation to perform better than the ED approximation. For uch cae, the great appeal of the ED approximation i not their accuracy but their implicity. They permit back-of-the-envelope calculation. They can greatly help undertand the performance of call center providing low-to-moderate quality of ervice. To etimate the quality of the ED approximation in advance, we ugget looking at q = 1/. Both the theory (Theorem 4.1) and the numerical comparion (Table 1) how that key parameter, determining both (i) the performance of the M/M//r + M model in the ED regime, and (ii) the accuracy of the ED approximation, are the ratio / and the caled abandonment rate / = 1. Indeed, in 4, we how that the ED many-erver heavy-traffic limit hold when /, which can occur if either 0or or both. Acknowledgment The author i grateful to Avihai Mandelbaum of the Technion, Andrew Ro of Lehigh Univerity, and anonymou referee for helpful uggetion. The author wa upported by National Science Foundation Grant DMS Reference Billingley, P Convergence of Probability Meaure, 2nd ed. John Wiley and Son, New York. Brandt, A., M. Brandt On the Mn/Mn/ queue with impatient call. Performance Eval Brandt, A., M. Brandt Aymptotic reult and a Markovian approximation for the Mn/Mn/ +GI ytem. Queueing Sytem

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