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30
On Customer Contact Centers with a CallBack Option: Customer Decisions, Routing Rules, and System Design
, 2004
"... Organizations worldwide use contact centers as an important channel of communication and transaction with their customers. This paper describes a contact center with two channels, one for realtime telephone service, and another for a postponed callback service offered with a guarantee on the maxim ..."
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Cited by 78 (13 self)
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Organizations worldwide use contact centers as an important channel of communication and transaction with their customers. This paper describes a contact center with two channels, one for realtime telephone service, and another for a postponed callback service offered with a guarantee on the maximum delay until a reply is received. Customers are sensitive to both realtime and callback delay and their behavior is captured through a probabilistic choice model. The dynamics of the system are modeled as an M/M/N multiclass system. We rigorously justify that as the number of agents increases, the system’s load approaches its maximum processing capacity. Based on this observation, we perform an asymptotic analysis in the manyserver, heavy traffic regime to find an asymptotically optimal routing rule, characterize the unique equilibrium regime of the system, approximate the system performance, and finally, propose a staffing rule that picks the minimum number of agents that satisfies a set of operational constraints on the performance of the system.
Martingale proofs of manyserver heavytraffic limits for Markovian queues
 PROBABILITY SURVEYS
, 2007
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Servicelevel differentiation in manyserver service systems: A solution based on fixedqueueratio routing
 OPERATIONS RESEARCH
, 2007
"... Motivated by telephone call centers, we study largescale service systems with multiple customer classes and multiple agent pools, each with many agents. For the purpose of delicately balancing service levels of the different customer classes, we propose a family of routing controls called FixedQue ..."
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Cited by 55 (27 self)
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Motivated by telephone call centers, we study largescale service systems with multiple customer classes and multiple agent pools, each with many agents. For the purpose of delicately balancing service levels of the different customer classes, we propose a family of routing controls called FixedQueueRatio (FQR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified proportion of the total queue length. We show that the proportions can be set to achieve desired servicelevel targets for all classes; these targets are achieved asymptotically as the total arrival rate increases. The FQR rule is a special case of the QueueandIdlenessRatio (QIR) family of controls which in a previous paper where shown to produce an important statespace collapse (SSC) as the total arrival rate increases. This SSC facilitates establishing asymptotic results. In simplified settings, SSC allows us to solve a combined designstaffingandrouting problem in a nearly optimal way. Our analysis also establishes a diminishingreturns property of flexibility: Under FQR, very moderate crosstraining is sufficient to make the call center as efficient as a singlepool system, again in the limit as the total arrival rate increases.
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 51 (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 the G/H ∗ 2 /n/m Queue
, 2005
"... We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit the ..."
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Cited by 49 (12 self)
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We establish heavytraffic stochasticprocess limits for queuelength, waitingtime and overflow stochastic processes in a class of G/GI/n/m queueing models with n servers and m extra waiting spaces. We let the arrival process be general, only requiring that it satisfy a functional central limit theorem. To capture the impact of the servicetime distribution beyond its mean within a Markovian framework, we consider a special class of servicetime distributions, denoted by H ∗ 2, which are mixtures of an exponential distribution with probability p and a unit point mass at 0 with probability 1 − p. These servicetime distributions exhibit relatively high variability, having squared coefficients of variation greater than or equal to one. As in Halfin and Whitt (1981, Heavytraffic limits for queues with many exponential servers, Oper. Res. 29 567–588), Puhalskii and Reiman (2000, The multiclass GI/PH/N queue in the HalfinWhitt regime. Adv. Appl. Probab. 32 564–595), and Garnett, Mandelbaum, and Reiman (2002. Designing a call center with impatient customers. Manufacturing Service Oper. Management, 4 208–227), we consider a sequence of queueing models indexed by the number of servers, n, and let n tend to infinity along with the traffic intensities �n so that √ n�1 − �n � → � for − � <�<�. To treat finite waiting rooms, we let mn / √ n → � for 0 <�≤�. With the special H ∗ 2 servicetime distribution, the limit processes are
Scheduling flexible servers with convex delay costs in manyserver service systems
 MANUFACTURING AND SERVICE OPERATIONS MANAGEMENT. FORTHCOMING
, 2007
"... In a recent paper we introduced the queueandidlenessratio (QIR) family of routing rules for manyserver service systems with multiple customer classes and server pools. A newly available server next serves the customer from the head of the queue of the class (from among those he is eligible to se ..."
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Cited by 33 (19 self)
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In a recent paper we introduced the queueandidlenessratio (QIR) family of routing rules for manyserver service systems with multiple customer classes and server pools. A newly available server next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified proportion of the total queue length. Under fairly general conditions, QIR produces an important statespace collapse as the total arrival rate and the numbers of servers increase in a coordinated way. That statespace collapse was previously used to delicately balance service levels for the different customer classes. In this sequel, we show that a special version of QIR stochastically minimizes convex holding costs in a finitehorizon setting when the service rates are restricted to be pooldependent. Under additional regularity conditions, the special version of QIR reduces to a simple policy: Linear costs produce a prioritytype rule, in which the leastcost customers are given low priority. Strictly convex costs (plus other regularity conditions) produce a manyserver analogue of the generalizedcµ (Gcµ) rule, under which a newly available server selects a customer from the class experiencing the greatest marginal cost at that time.
Queueandidlenessratio controls in manyserver service systems
, 2007
"... Motivated by call centers, we study largescale service systems with multiple customer classes and multiple agent pools, each with many agents. We propose a family of routing rules called QueueandIdlenessRatio (QIR) rules. A newly available agent next serves the customer from the head of the queu ..."
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Cited by 32 (10 self)
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Motivated by call centers, we study largescale service systems with multiple customer classes and multiple agent pools, each with many agents. We propose a family of routing rules called QueueandIdlenessRatio (QIR) rules. A newly available agent next serves the customer from the head of the queue of the class (from among those he is eligible to serve) whose queue length most exceeds a specified statedependent proportion of the total queue length. An arriving customer is routed to the agent pool whose idleness most exceeds a specified statedependent proportion of the total idleness. We identify regularity conditions on the network structure and system parameters under which QIR produces an important statespace collapse (SSC) result in the QualityandEfficiencyDriven (QED) manyserver heavytraffic limiting regime. The SSC result is applied in two subsequent papers to solve important staffing and control problems for largescale service systems.
Queue lengths and waiting times for multiserver queues with abandonment and retrials
 Working Paper, Technion
, 2000
"... Abstract. We consider a Markovian multiserver queueing model with time dependent parameters where waiting customers may abandon and subsequently retry. We provide simple fluid and diffusion approximations to estimate the mean, variance, and density for both the queue length and virtual waiting time ..."
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Cited by 29 (7 self)
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Abstract. We consider a Markovian multiserver queueing model with time dependent parameters where waiting customers may abandon and subsequently retry. We provide simple fluid and diffusion approximations to estimate the mean, variance, and density for both the queue length and virtual waiting time processes arising in this model. These approximations, which are generated by numerically integrating only 7 ordinary differential equations, are justified by limit theorems where the arrival rate and number of servers grow large. We compare our approximations to simulations, and they perform extremely well.
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 22 (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,
Waiting time asymptotics for time varying multiserver queues with abandonment and retrials
 IN: PROCEEDINGS OF THE 37 ANNUAL ALLERTON CONFERENCE ON COMMUNICATION, CONTROL AND COMPUTING, ALLERTON, IL
, 1999
"... We consider a nonstationary Markov multiserver queueing model where waiting customers may abandon and subsequently retry. In this paper we derive fluid and diffusion approximations for the associated waiting time process. The fluid and diffusion approximations for the corresponding queue length proc ..."
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Cited by 21 (4 self)
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We consider a nonstationary Markov multiserver queueing model where waiting customers may abandon and subsequently retry. In this paper we derive fluid and diffusion approximations for the associated waiting time process. The fluid and diffusion approximations for the corresponding queue length process were obtained in [4] (see also [5]).