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16
Coping with TimeVarying Demand When Setting Staffing Requirements for a Service System
, 2007
"... We review queueingtheory methods for setting staffing requirements in service systems where customer demand varies in a predictable pattern over the day. Analyzing these systems is not straightforward, because standard queueing theory focuses on the longrun steadystate behavior of stationary mode ..."
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Cited by 73 (26 self)
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We review queueingtheory methods for setting staffing requirements in service systems where customer demand varies in a predictable pattern over the day. Analyzing these systems is not straightforward, because standard queueing theory focuses on the longrun steadystate behavior of stationary models. We show how to adapt stationary queueing models for use in nonstationary environments so that timedependent performance is captured and staffing requirements can be set. Relatively little modification of straightforward stationary analysis applies in systems where service times are short and the targeted quality of service is high. When service times are moderate and the targeted quality of service is still high, timelag refinements can improve traditional stationary independent periodbyperiod and peakhour approximations. Timevarying infiniteserver models help develop refinements, because closedform expressions exist for their timedependent behavior. More difficult cases with very long service times and other complicated features, such as endofday effects, can often be treated by a modifiedofferedload approximation, which is based on an associated infiniteserver model. Numerical algorithms and deterministic fluid models are useful when the system is overloaded for an extensive period of time. Our discussion focuses on telephone call centers, but applications to police patrol, banking, and hospital emergency rooms are also mentioned.
Dynamic routing in largescale service systems with heterogeneous servers
, 2005
"... Motivated by modern call centers, we consider largescale service systems with multiple server pools and a single customer class. For such systems, we propose a simple routing rule which asymptotically minimizes the steadystate queue length and virtual waiting time. The proposed routing scheme is ..."
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Cited by 52 (12 self)
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Motivated by modern call centers, we consider largescale service systems with multiple server pools and a single customer class. For such systems, we propose a simple routing rule which asymptotically minimizes the steadystate queue length and virtual waiting time. The proposed routing scheme is FSF which assigns customers to the Fastest Servers First. The asymptotic regime considered is the HalfinWhitt manyserver heavytraffic regime, which we refer to as the Quality and Efficiency Driven (QED) regime; it achieves high levels of both service quality and system efficiency by carefully balancing between the two. Additionally, expressions are provided for system limiting performance measures based on diffusion approximations. Our analysis shows that in the QED regime this heterogeneous server system outperforms its homogeneous server counterpart.
Heavytraffic limits for waiting times in manyserver queues with abandonments
, 2008
"... In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stop ..."
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Cited by 22 (10 self)
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In this online supplement we provide results that we have omitted from the main paper. First, in Appendix A, we give a proof of Lemma 2.1. In Appendix B we give a proof of Theorem 6.1 using the technique described in [2]. Finally, in Appendix C, we give an alternative proof of Theorem 5.2 using stopped arrival processes as in the proof of Theorem 6.3.
Multiclass multiserver queueing system in the halfinwhitt heavy traffic regime. asymptotics of the stationary distribution. Queueing Systems 71
, 2012
"... We consider a heterogeneous queueing system consisting of one large pool of O(r) identical servers, where r → ∞ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfi ..."
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Cited by 13 (1 self)
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We consider a heterogeneous queueing system consisting of one large pool of O(r) identical servers, where r → ∞ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the HalfinWhitt sense, namely the nominal utilization is 1 − a / √ r where a> 0 is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue Q r (∞). While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form O ( √ r) on both Q r (∞) and the number of idle servers. The bounds are uniform w.r.t. parameter r and the service policy. In particular, we show that lim sup r E exp(θr − 1 2 Q r (∞)) < ∞. Therefore, the sequence r − 1 2 Q r (∞) is tight and has a uniform exponential tail bound. We further consider the system with strictly positive abandonment rates, and show that in this case every weak limit ˆ Q(∞) of r − 1 2 Q r (∞) has a subGaussian tail. Namely E[exp(θ ( ˆ Q(∞)) 2)] < ∞, for some θ> 0. 1
Queueing systems with many servers: null controllability in heavy traffic
, 2005
"... A queueing model has J ≥ 2 heterogeneous service stations, each consisting of many independent servers with identical capabilities. Customers of I ≥ 2 classes can be served at these stations at different rates, that depend on both the class and the station. A system administrator dynamically control ..."
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Cited by 11 (5 self)
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A queueing model has J ≥ 2 heterogeneous service stations, each consisting of many independent servers with identical capabilities. Customers of I ≥ 2 classes can be served at these stations at different rates, that depend on both the class and the station. A system administrator dynamically controls scheduling and routing. We study this model in the Central Limit Theorem (or heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model on R I with a singular control term, that describes the scaling limit of the queueing model. The singular term may be used to constrain the diffusion to lie in certain subsets of R I at all times t> 0. We say that the diffusion is nullcontrollable if it can be constrained to X−, the minimal closed subset of R I containing all states of the prelimit queueing model for which all queues are empty. We give sufficient conditions for null controllability of the diffusion. Under these conditions we also show that an analogous, asymptotic result holds for the queueing model, by constructing control policies under which, for any given 0 < ε < T < ∞, all queues in the system are kept empty on the time interval [ε, T], with probability approaching one. This introduces a new, unusual heavy traffic ‘behavior’: On one hand the system is critically loaded, in the sense that an increase in any of the external arrival rates at the ‘fluid level ’ results with an overloaded system. On the other hand, as far as queue lengths are concerned, the system behaves as if it is underloaded.
Central limit theorem for a manyserver queue with random service rates
 Ann. Appl. Probab
, 2008
"... Given a random variable N with values in N, and N i.i.d. positive random variables {µk}, we consider a queue with renewal arrivals and N exponential servers, where server k serves at rate µk, under two work conserving routing schemes. In the first, the service rates {µk} need not be known to the rou ..."
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Cited by 11 (3 self)
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Given a random variable N with values in N, and N i.i.d. positive random variables {µk}, we consider a queue with renewal arrivals and N exponential servers, where server k serves at rate µk, under two work conserving routing schemes. In the first, the service rates {µk} need not be known to the router, and each customer to arrive at a time when some servers are idle is routed to the server that has been idle for the longest time (or otherwise it is queued). In the second, the service rates are known to the router, and a customer that arrives to find idle servers is routed to the one whose service rate is greatest. In the manyserver heavy traffic regime of Halfin and Whitt, the process that represents the number of customers in the system is shown to converge to a onedimensional diffusion with a random drift coefficient, where the law of the drift depends on the routing scheme. A related result is also provided for nonrandom environments. 1. Introduction. Manyserver
Blind fair routing in largescale service systems
, 2011
"... In a call center, arriving customers must be routed to available servers, and servers that have just become available must be scheduled to help waiting customers. These dynamic routing and scheduling decisions are very difficult, because customers have different needs and servers have different skil ..."
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Cited by 9 (1 self)
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In a call center, arriving customers must be routed to available servers, and servers that have just become available must be scheduled to help waiting customers. These dynamic routing and scheduling decisions are very difficult, because customers have different needs and servers have different skill levels. A further complication is that it is preferable that these decisions are made blindly; that is, they depend only on the system state and not on system parameter information such as call arrival rates and service speeds. This is because this information is generally not known with certainty. Ideally, a dynamic control policy for making routing and scheduling decisions balances customer and server needs, by keeping customer delays low, but still fairly dividing the workload amongst the various servers. In this paper, we propose two blind dynamic control policies for parallel server systems with multiple customer classes and server pools, one that is based on the number of customers waiting and the number of agents idling, and one that is based on customer delay times and server idling times. We show that, in the HalfinWhitt manyserver heavy traffic limiting regime, our proposed blind policies perform extremely well when the objective is to minimize customer holding or delay costs subject to “server fairness”, as defined by how the system idleness is divided among servers. To do this, we formulate an approximating diffusion control problem (DCP), and compare the performance of the nonblind DCP solution to a feasible policy for the DCP that is blind. We establish that the increase in the DCP objective function value is small over a wide range of parameter values. We then use simulation to validate that a small increase in the DCP objective function value is indicative of our proposed blind policies performing very well. Acknowledgement: We thank Itay Gurvich and Avi Mandelbaum for many valuable discussions.
Fluid models of manyserver queues with abandonment
, 2011
"... We study manyserver queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measurevalued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are establ ..."
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Cited by 8 (2 self)
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We study manyserver queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measurevalued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide firstorder approximation for this model. The fluid model solution, which is proved to uniquely exists, serves as the fluid limit of the manyserver queue, as the number of servers becomes large. Based on the fluid model solution, firstorder approximations for various performance quantities are proposed. Key words and phrases: manyserver queue, abandonment, measure valued process, quality driven, efficiency driven, quality and efficiency driven. 1