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105
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.
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.
Exact and asymptotic ntuple laws at first and last
, 2010
"... Understanding the space–time features of how a Lévy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory ..."
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Cited by 30 (6 self)
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Understanding the space–time features of how a Lévy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes, to name but a few. In Doney and Kyprianou [Ann. Appl. Probab. 16 (2006) 91–106] a new quintuple law was established for a general Lévy process at first passage below a fixed level. In this article we use the quintuple law to establish a family of related joint laws, which we call ntuple laws, for Lévy processes, Lévy processes conditioned to stay positive and positive selfsimilar Markov processes at both first and last passage over a fixed level. Here the integer n typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive selfsimilar Markov processes issued from the origin. Although the relation between the ntuple laws for Lévy processes and positive selfsimilar Markov processes are straightforward thanks to the Lamperti transformation, by interplaying the role of a (conditioned) stable processes as both a (conditioned) Lévy processes and a positive selfsimilar Markov processes, we obtain a suite of completely explicit first and last passage identities for socalled Lampertistable Lévy processes. This leads further to the introduction of a more general family of Lévy processes which we call hypergeometric Lévy processes, for which similar explicit identities may be considered.
The Skorokhod problem in a timedependent interval
"... Abstract: We consider the Skorokhod problem in a timevarying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on th ..."
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Cited by 23 (4 self)
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Abstract: We consider the Skorokhod problem in a timevarying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of twodimensional reflected Brownian motions.
Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies
 ANN. APPL. PROBAB
, 2005
"... We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks ..."
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Cited by 23 (0 self)
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We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks
Service Level Differentiation in Call Centers with Fully Flexible Servers
, 2004
"... We study largescale service systems with multiple customer classes and many statistically identical servers. The following question is addressed: How many servers are required (staffing) and how does one match them with customers (control) in order to minimize staffing cost, subject to class level ..."
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Cited by 18 (8 self)
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We study largescale service systems with multiple customer classes and many statistically identical servers. The following question is addressed: How many servers are required (staffing) and how does one match them with customers (control) in order to minimize staffing cost, subject to class level quality of service constraints? We tackle this question by characterizing scheduling and staffing schemes that are asymptotically optimal in the limit, as system load grows to infinity. The asymptotic regimes considered are consistent with the Efficiency Driven (ED), Quality Driven (QD) and Quality and Efficiency Driven (QED) regimes, first introduced in the context of a single class service system. Our main findings are: a) Decoupling of staffing and control, namely (i) Staffing disregards the multiclass nature of the system and is analogous to the staffing of a single class system with the same aggregate demand and a single global quality of service constraint, and (ii) Class level service differentiation is obtained by using a simple Idle server based ThresholdPriority (ITP) control (with stateindependent thresholds), b) Robustness of the staffing and control rules: Our proposed SingleClass Staffing (SCS) rule and ITP control are approximately optimal
On the number of collisions in Λcoalescents
 ELECTRON. J. PROBAB
, 2007
"... We examine the total number of collisions Cn in the Λcoalescent process which starts with n particles. A linear growth and a stable limit law for Cn are shown under the assumption of a powerlike behaviour of the measure Λ near 0 with exponent 0 < α < 1. ..."
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Cited by 14 (1 self)
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We examine the total number of collisions Cn in the Λcoalescent process which starts with n particles. A linear growth and a stable limit law for Cn are shown under the assumption of a powerlike behaviour of the measure Λ near 0 with exponent 0 < α < 1.
Asymptotics of the allele frequency spectrum associated with the BolthausenSznitman coalescent
, 2007
"... We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it a ..."
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Cited by 14 (0 self)
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We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it and group together individuals whose most recent mutations are the same. The number of blocks of each of the different possible sizes in this partition is the allele frequency spectrum. The celebrated Ewens sampling formula gives precise probabilities for the allele frequency spectrum associated with Kingman’s coalescent. This (and the degenerate starshaped coalescent) are the only Λcoalescents for which explicit probabilities are known, although they are known to satisfy a recursion due to Möhle. Recently, Berestycki, Berestycki and Schweinsberg have proved asymptotic results for the allele frequency spectra of the Beta(2 − α,α) coalescents with α ∈ (1,2). In this paper, we prove full asymptotics for the case of the BolthausenSznitman coalescent.