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Fluid models for multiserver queues with abandonments
 Operations Research
"... Deterministic fluid models are developed to provide simple firstorder performance descriptions for multiserver queues with abandonment under heavy loads. Motivated by telephone call centers, the focus is on multiserver queues with a large number of servers and nonexponential servicetime and ti ..."
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Cited by 72 (33 self)
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Deterministic fluid models are developed to provide simple firstorder performance descriptions for multiserver queues with abandonment under heavy loads. Motivated by telephone call centers, the focus is on multiserver queues with a large number of servers and nonexponential servicetime and timetoabandon distributions. The first fluid model serves as an approximation for the G/GI/s + GI queueing model, which has a general stationary arrival process with arrival rate λ, independent and identically distributed (IID) service times with a general distribution, s servers and IID abandon times with a general distribution. The fluid model is useful in the overloaded regime, where λ> s, which is often realistic because only a small amount of abandonment can keep the system stable. Numerical experiments, using simulation for M/GI/s + GI models and exact numerical algorithms for M/M/s + M models, show that the fluid model provides useful approximations for steadystate performance measures when the system is heavily loaded. The fluid model accurately shows that steadystate performance depends strongly upon the timetoabandon distribution beyond its mean, but not upon the servicetime distribution beyond its mean. The second fluid model is a discretetime fluid model, which serves as an approximation for the Gt(n)/GI/s + GI queueing model, having a statedependent and timedependent arrival process. The discretetime framework is exploited to prove that properly scaled queueing processes in the queueing model converge to fluid functions as s → ∞. The discretetime framework is also convenient for calculating the timedependent fluid performance descriptions. Subject classifications: Queues, approximations: multiserver queues with abandonment. Queues, multichannel: approximation of nonMarkovian multichannel queues with customer abandonment.
Martingale proofs of manyserver heavytraffic limits for Markovian queues
 PROBABILITY SURVEYS
, 2007
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Staffing of timevarying queues to achieve timestable performance
, 2005
"... Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probab ..."
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Cited by 70 (35 self)
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Continuing research by Jennings, Mandelbaum, Massey and Whitt (1996), we investigate methods to perform timedependent staffing for manyserver queues. Our aim is to achieve timestable performance in face of general timevarying arrival rates. It turns out that it suffices to target a stable probability of delay. That procedure tends to produce timestable performance for several other operational measures. Motivated by telephone call centers, we focus on manyserver models with customer abandonment, especially the Markovian Mt/M/st + M model, having an exponential timetoabandon distribution (the +M), an exponential servicetime distribution and a nonhomogeneous Poisson arrival process. We develop three different methods for staffing, with decreasing generality and decreasing complexity: First, we develop a simulationbased iterativestaffing algorithm (ISA), and conduct experiments showing that it is effective. The ISA is appealing because it applies to very general models and is automatically validating: we directly see how well it works. Second, we extend the squarerootstaffing rule, proposed by Jennings et al., which is based on the associated infiniteserver model. The rule dictates that the staff level at time t be st = mt + β √ mt, where mt is the offered load (mean number of busy servers in the infiniteserver model) and the constant β reflects the service grade. We show that the service grade β in the staffing formula can be represented as a function of the target delay probability α by
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 70 (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.
Staffing a Call Center with Uncertain Arrival Rate and Absenteeism
 Production and Operations Management
"... This paper proposes simple methods for staffing a singleclass call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The arrival rate and the proportion of servers present are treated as random variables. The basic model is a multiserver queue with customer aba ..."
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Cited by 46 (5 self)
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This paper proposes simple methods for staffing a singleclass call center with uncertain arrival rate and uncertain staffing due to employee absenteeism. The arrival rate and the proportion of servers present are treated as random variables. The basic model is a multiserver queue with customer abandonment, allowing nonexponential servicetime and timetoabandon distributions. The goal is to maximize the expected net return, given throughput benefit and server, customerabandonment and customerwaiting costs, but attention is also given to the standard deviation of the return. The approach is to approximate the performance and the net return, conditional on the random modelparameter vector, and then uncondition to get the desired results. Two recentlydeveloped approximations are used for the conditional performance measures: first, a deterministic fluid approximation and, second, a numerical algorithm based on a purely Markovian birthanddeath model, having statedependent death rates. Key words: modelparameter uncertainty; contact centers; employee absenteeism; customer abandonment; fluid models
Engineering solution of a basic callcenter model
 Management Science
, 2005
"... An algorithm is developed to rapidly compute approximations for all the standard steadystate performance measures in the basic callcenter queueing modelM/GI/s/r+GI, which has a Poisson arrival process, IID service times with a general distribution, s servers, r extra waiting spaces and IID custom ..."
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Cited by 45 (27 self)
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An algorithm is developed to rapidly compute approximations for all the standard steadystate performance measures in the basic callcenter queueing modelM/GI/s/r+GI, which has a Poisson arrival process, IID service times with a general distribution, s servers, r extra waiting spaces and IID customer abandonment times with a general distribution. Empirical studies indicate that the servicetime and abandontime distributions often are not nearly exponential, so that it is important to go beyond the MarkovianM/M/s/r+M special case, but the general servicetime and abandontime distributions make the realistic model very difficult to analyze directly. The proposed algorithm is based on an approximation by an appropriate Markovian M/M/s/r+M(n) queueing model, where M(n) denotes statedependent abandonment rates. After making an additional approximation, steadystate waitingtime distributions are characterized via their Laplace transforms. Then the approximate distributions are computed by numerically inverting the transforms. Simulation experiments show that the approximation is quite accurate. The overall algorithm can be applied to determine desired staffing levels, e.g., the minimum number of servers needed to guarantee that, first, the abandonment rate is below any specified target value and, second, that the conditional probability that an arriving customer will be served within a specified deadline, given that the customer eventually will be served, is at least a specified target value.
The impact of delay announcements in manyserver queues with abandonments: supplementary material
, 2006
"... This is a supplement to the main paper, having the same title. In this work we develop methods to study the impact upon steadystate performance of delay announcements made to arriving customers in a manyserver queue with customer abandonment. We assume that the queue is not visible to waiting cust ..."
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Cited by 31 (6 self)
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This is a supplement to the main paper, having the same title. In this work we develop methods to study the impact upon steadystate performance of delay announcements made to arriving customers in a manyserver queue with customer abandonment. We assume that the queue is not visible to waiting customers, as in most customer contact centers, when contact is made by telephone, email or instant messaging. We propose simple robust announcement schemes: (i) the delay of the last served (DLS) customer and (ii) a fixed delay announcement (FDA) based on an appropriate longrun average delay. For any singlenumber delay announcement made immediately upon arrival, customers may balk or have new abandonment behavior as a function of the announced delay. To perform a roughcut performance analysis, prior to detailed simulation, we introduce a fluid model, which provides an approximate and highly simplified description for large systems in an overloaded regime. In the fluid model, all customers are faced with the same delay and consequently can be given the same delay announcement. That property motivates considering a second approximation scheme: an equilibrium fixed delay announcement (FDA) in the stochastic model. We show that these two approximate descriptions of aggregate performance are effective by comparing to simulations of systems with statedependent DLS announcements. Here we present additional material, supplementing the main paper.
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 23 (9 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.
A Fluid Approximation for Service Systems Responding to Unexpected Overloads
"... In a recent paper we considered two networked service systems, each having its own customers and designated service pool with many agents, where all agents are able to serve the other customers, although they may do so inefficiently. Usually we want the agents to serve only their own customers, but ..."
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Cited by 22 (15 self)
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In a recent paper we considered two networked service systems, each having its own customers and designated service pool with many agents, where all agents are able to serve the other customers, although they may do so inefficiently. Usually we want the agents to serve only their own customers, but we want an automatic control that activates serving some of the other customers when an unexpected overload occurs. Assuming that we will not know the timing or the extent of the overload, we proposed a queueratio control with thresholds: When a prespecified threshold is crossed, serving the other customers is activated, so that a certain queue ratio, which is determined by only observing the queue lengths, is maintained. We then developed a simple fluid approximation in which this control was shown to be optimal, and we showed how to calculate the control parameters. In this paper, we focus on the fluid approximation itself, and develop its transient equations, which depend on a heavytraffic averaging principle. Although deterministic, the new fluid model includes stochastic balance equations. The full fluid model here enables us to analyze scenarios we could not analyze previously, e.g., when the thresholds are small, and are being crossed over frequently. We also use this new fluid approximation to refine our previous results. Simulation experiments show that the refined fluid model greatly increases the accuracy of our performance predictions. Key words: service systems; unexpected overloads; fluid approximation; averaging principle; efficiencydriven regime 1.