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33
Fluid limits of manyserver queues with reneging.
 Ann. Appl. Prob.
, 2010
"... Abstract. This work considers a manyserver queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service excee ..."
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Cited by 34 (3 self)
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Abstract. This work considers a manyserver queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measurevalued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essentially only requiring that the service and reneging distributions have densities, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a fluid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.
Fair dynamic routing in largescale heterogeneousserver systems
, 2008
"... In a call center, there is a natural tradeoff between minimizing customer wait time and fairly dividing the workload amongst agents of different skill levels. The relevant control is the routing policy; that is, the decision concerning which agent should handle an arriving call when more than one a ..."
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Cited by 24 (5 self)
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In a call center, there is a natural tradeoff between minimizing customer wait time and fairly dividing the workload amongst agents of different skill levels. The relevant control is the routing policy; that is, the decision concerning which agent should handle an arriving call when more than one agent is available. We formulate an optimization problem for a call center with two heterogeneous agent pools, one that handles calls at a faster speed than the other, and a single customer class. The objective is to minimize steadystate expected customer wait time subject to a “fairness” constraint on the workload division. The optimization problem we formulate is difficult to solve exactly. Therefore, we solve the diffusion control problem that arises in the manyserver heavytraffic QED limiting regime. The resulting routing policy is a threshold policy that prioritizes faster agents when the number of customers in the system exceeds some threshold level and otherwise prioritizes slower agents. We prove our proposed threshold routing policy is nearoptimal as the number of agents increases, and the system’s load approaches its maximum processing capacity. We further show simulation results that evidence that our proposed threshold routing policy outperforms a common routing policy used in call centers (that routes to the agent that has been idle the longest) in terms of the steadystate expected customer waiting time for identical desired workload divisions.
Queues with Many Servers: The Virtual WaitingTime Process in the QED Regime
, 2007
"... We consider a multiserver queue (G/GI/N) in the Quality and EfficiencyDriven (QED) regime. In this regime, which was first formalized by Halfin and Whitt, the number of servers N is not small, servers ’ utilization is 1 − O(1/√N) (EfficiencyDriven) while waiting time is O(1/ N) (QualityDriven). ..."
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Cited by 24 (1 self)
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We consider a multiserver queue (G/GI/N) in the Quality and EfficiencyDriven (QED) regime. In this regime, which was first formalized by Halfin and Whitt, the number of servers N is not small, servers ’ utilization is 1 − O(1/√N) (EfficiencyDriven) while waiting time is O(1/ N) (QualityDriven). This is equivalent to having the number of servers N being approximately equal to R + β R, where R is the offered load and β is a positive constant. For the G/GI/N queue in the QED regime, we analyze the virtual waiting time VN (t), as N increases indefinitely. Assuming that the service time distribution has a finite support, it is shown that, in the limit, the scaled virtual waiting time V̂N (t) = NVN (t)/ES is representable as a supremum over a random weighted tree (S denotes a service time). Informally, it is then argued that, for large N,
Steadystate analysis of a multiserver queue in the HalfinWhitt regime
, 2008
"... We examine a multiserver queue in the HalfinWhitt (Quality and EfficiencyDriven) regime: as the number of servers n increases, the utilization approaches 1 from below at the rate Θ(1 / √ n). The arrival process is renewal and service times have a latticevalued distribution with a finite suppor ..."
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Cited by 14 (0 self)
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We examine a multiserver queue in the HalfinWhitt (Quality and EfficiencyDriven) regime: as the number of servers n increases, the utilization approaches 1 from below at the rate Θ(1 / √ n). The arrival process is renewal and service times have a latticevalued distribution with a finite support. We consider the steadystate distribution of the queue length and waiting time in the limit as the number of servers n increases indefinitely. The queue length distribution, in the limit as n → ∞, is characterized in terms of the stationary distribution of an explicitly constructed Markov chain. As a consequence, the steadystate queue length and waiting time scale as Θ ( √ n) and Θ(1 / √ n) as n → ∞, respectively. Moreover, an explicit expression for the critical exponent is derived for the moment generating function of a limiting (scaled) steadystate queue length. This exponent depends on three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a singleserver queue in the conventional heavytraffic regime. The results are derived by analyzing Lyapunov functions.
Workload forecasting for a call center: Methodology and a case study
, 2009
"... Today’s call center managers face multiple operational decisionmaking tasks. One of the most common is determining the weekly staffing levels to ensure customer satisfaction and meeting their needs while minimizing service costs. An initial step for producing the weekly schedule is forecasting the ..."
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Cited by 13 (1 self)
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Today’s call center managers face multiple operational decisionmaking tasks. One of the most common is determining the weekly staffing levels to ensure customer satisfaction and meeting their needs while minimizing service costs. An initial step for producing the weekly schedule is forecasting the future system loads which involves predicting both arrival counts and average service times. We introduce an arrival count model which is based on a mixed Poisson process approach. The model is applied to data from an Israeli Telecom company call center. In our model, we also consider the effect of events such as billing on the arrival process and we demonstrate how to incorporate them as exogenous variables in the model. After obtaining the forecasted system load, in large call centers, a manager can choose to apply the QED (QualityEfficiency Driven) regime’s “squareroot staffing” rule in order to balance the offeredload per server with the quality of service. Implementing this staffing rule requires that the forecasted values of the arrival counts and average service times maintain certain levels of precision. We develop different goodness of fit criteria that help determine our model’s practical performance under the QED regime. These show that during most hours of the day the model can reach desired precision levels.
CrossSelling in a Call Center with a Heterogeneous Customer Population
, 2006
"... This is the technical appendix accompanying the paper, “CrossSelling in a Call Center with a Heterogeneous Customer Population, ” [3]. The organization of this appendix is as follows: we begin in §B with the completion of the proof of Proposition 1, whose sketch was given in §A of [3]. We continue ..."
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Cited by 11 (3 self)
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This is the technical appendix accompanying the paper, “CrossSelling in a Call Center with a Heterogeneous Customer Population, ” [3]. The organization of this appendix is as follows: we begin in §B with the completion of the proof of Proposition 1, whose sketch was given in §A of [3]. We continue in §C with some preliminaries required for the performance analysis of (S)(C).
Routing and staffing in largescale service systems: The case of homogeneous impatient customers and heterogeneous servers
, 2011
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Queueing Models for FullFlexible Multiclass Call Centers with RealTime Anticipated Delays
 International Journal of Production Economics
, 2007
"... In this paper, we consider two basic multiclass call center models, with and without reneging. Customer classes have different priorities. The content of different types of calls is assumed to be similar allowing their service times to be identical. We study the problem of announcing delays to cus ..."
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Cited by 6 (2 self)
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In this paper, we consider two basic multiclass call center models, with and without reneging. Customer classes have different priorities. The content of different types of calls is assumed to be similar allowing their service times to be identical. We study the problem of announcing delays to customers upon their arrival. For the simplest model without reneging, we give a method to estimate virtual delays that is used within the announcement step. For the second model, we first build the call center model incorporating reneging. The model takes into account the change in customer behavior that may occur when delay information is communicated to them. In particular, it is assumed that customer reneging is replaced by balking that depends on the state of the system in this case. We develop a method based on Markov chains in order to estimate virtual delays of new arrivals for this model. Finally, some practical issues concerning delay announcement are discussed.
When promotions meet operations: Crossselling and its effect on callcenter performance
, 2006
"... We study crossselling operations in call centers. The following question is addressed: How many customer service representatives are required (staffing) and when should crossselling opportunities be exercised (control) in a way that will maximize the expected profit of the firm while maintaining a ..."
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Cited by 6 (3 self)
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We study crossselling operations in call centers. The following question is addressed: How many customer service representatives are required (staffing) and when should crossselling opportunities be exercised (control) in a way that will maximize the expected profit of the firm while maintaining a prespecified service level target. We tackle these questions by characterizing scheduling and staffing schemes that are asymptotically optimal in the limit, as the system load grows to infinity. Our main finding is that a threshold priority (TP) control, in which crossselling is exercised only if the number of callers in the system is below a certain threshold, is asymptotically optimal in great generality. The asymptotic optimality of TP reduces the staffing problem to the solution of a simple deterministic problem, in some cases, and to a simple search procedure in others. Our asymptotic approach establishes that our staffing and control scheme is nearoptimal for large systems. In addition, we numerically demonstrate that TP performs extremely well even for relatively small systems.
Call centers with delay information: Models and insights
 Manufacturing Service Oper. Management
, 2011
"... In this paper, we analyze a call center with impatient customers. We study how informing customers about their anticipated delays affects performance. Customers react by balking upon hearing the delay announcement, and may subsequently renege, particularly if the realized waiting time exceeds the de ..."
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Cited by 6 (2 self)
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In this paper, we analyze a call center with impatient customers. We study how informing customers about their anticipated delays affects performance. Customers react by balking upon hearing the delay announcement, and may subsequently renege, particularly if the realized waiting time exceeds the delay that has originally been announced to them. The balking and reneging from such a system are a function of the delay announcement. Modeling the call center as an M/M/s+M queue with endogenized customer reactions to announcements, we analytically characterize performance measures for this model. The analysis allows us to explore the role announcing different percentiles of the waiting time distribution, i.e., announcement coverage, plays on subsequent performance in terms of balking and reneging. Through a numerical study we explore when informing customers about delays is beneficial, and what the optimal coverage should be in these announcements. It is shown how managers of a call center with delay announcements can control the tradeoff between balking and reneging, through their choice of announcements to be made.