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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 56 (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.
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.
TwoParameter HeavyTraffic Limits for InfiniteServer Queues
"... Abstract: In order to obtain Markov heavytraffic approximations for infiniteserver queues with general nonexponential servicetime distributions and general arrival processes, possibly with timevarying arrival rates, we establish heavytraffic limits for twoparameter stochastic processes. We ..."
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Cited by 26 (13 self)
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Abstract: In order to obtain Markov heavytraffic approximations for infiniteserver queues with general nonexponential servicetime distributions and general arrival processes, possibly with timevarying arrival rates, we establish heavytraffic limits for twoparameter stochastic processes. We
A manyserver fluid limit for the Gt/GI/st + GI queueing model experiencing periods of overload
 OPERATIONS RESEARCH LETTERS
, 2012
"... ..."
Manyserver heavytraffic limits for queues with timevarying parameters. Annals of Applied Probability 24: 378–421
, 2014
"... A manyserver heavytraffic FCLT is proved for the Gt/M/st+GI queueing model, having timevarying arrival rate and staffing, a general arrival process satisfying a FCLT, exponential service times and customer abandonment according to a general probability distribution. The FCLT provides theoretical ..."
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Cited by 10 (7 self)
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A manyserver heavytraffic FCLT is proved for the Gt/M/st+GI queueing model, having timevarying arrival rate and staffing, a general arrival process satisfying a FCLT, exponential service times and customer abandonment according to a general probability distribution. The FCLT provides theoretical support for the approximating deterministic fluid model the authors analyzed in a previous paper and a refined Gaussian process approximation, using variance formulas given here. The model is assumed to alternate between underloaded and overloaded intervals, with critical loading only at the isolated switching points. The proof is based on a recursive analysis of the system over these successive intervals, drawing heavily on previous results for infiniteserver models. The FCLT requires careful treatment of the initial conditions for each interval. 1. Introduction. This paper is a sequel to
HEAVYTRAFFIC LIMITS FOR MANYSERVER QUEUES WITH SERVICE INTERRUPTIONS
, 2008
"... We establish manyserver heavytraffic limits for G/M/n + M queueing models, allowing customer abandonment (the +M), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queuelength process where the limit is an ordinary diffe ..."
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Cited by 8 (0 self)
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We establish manyserver heavytraffic limits for G/M/n + M queueing models, allowing customer abandonment (the +M), subject to exogenous regenerative service interruptions. With unscaled service interruption times, we obtain a FWLLN for the queuelength process where the limit is an ordinary differential equation in a twostate random environment. With asymptotically negligible service interruptions, we obtain a FCLT for the queuelength process, where the limit is characterized as the pathwise unique solution to a stochastic integral equation with jumps. When the arrivals are renewal and the interruption cycle time is exponential, the limit is a Markov process, being a jumpdiffusion process in the QED regime and an OU process driven by a Levy process in the ED regime (and for infiniteserver queues). A stochasticdecompostion property of the steadystate distribution of the limit process in the ED regime (and for infiniteserver queues) is obtained.
Manyserver queues with customer abandonment: numerical analysis of their diffusion models
, 2011
"... The performance of a call center is sensitive to customer abandonment. In this survey paper, we focus on / /G GI n GI parallelserver queues that serve as a building block to model call center operations. Such a queue has a general arrival process (the G), independent and identically distributed ..."
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Cited by 7 (1 self)
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The performance of a call center is sensitive to customer abandonment. In this survey paper, we focus on / /G GI n GI parallelserver queues that serve as a building block to model call center operations. Such a queue has a general arrival process (the G), independent and identically distributed (iid) service times with a general distribution (the first GI), and iid patience times with a general distribution (the GI ). Following the squareroot safety staffing rule, this queue can be operated in the quality and efficiencydriven (QED) regime, which is characterized by large customer volume, the waiting times being a fraction of the service times, only a small fraction of customers abandoning the system, and high server utilization. Operational efficiency is the central target in a system whose staffing costs dominate other expenses. If a moderate fraction of customer abandonment is allowed, such a system should be operated in an overloaded regime known as the efficiencydriven (ED) regime. We survey recent results on the manyserver queues that are operated in the QED and ED regimes. These results include the performance insensitivity to patience time distributions and diffusion and fluid approximate models as practical tools for performance analysis.
A fluid approximation for the Gt/GI/st + GI queue
, 2010
"... We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general timevarying arrival process (the Gt), a general servicetime distribution (the first GI), a timedependent number of servers (the st) and allows ..."
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Cited by 3 (3 self)
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We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general timevarying arrival process (the Gt), a general servicetime distribution (the first GI), a timedependent number of servers (the st) and allows abandonment from queue according to a general abandonmenttime distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We characterize performance in the fluid model over alternating intervals in which the system is overloaded and underloaded (including critically loaded). For each t ≥ 0 and y ≥ 0, we determine the amount of fluid that is in service (in queue) at time t and has been so for time at most y. We obtain the service content density by applying the Banach contraction fixed point theorem. We also determine the timevarying potential waiting time, i.e., the virtual waiting time of a quantum of fluid arriving at a specified time, assuming that it will not abandon. The potential waiting time is determined by an ordinary differential equation. We show that a timevarying service capacity can be chosen to stabilize delays at any fixed target. Key words: queues with timevarying arrivals; nonstationary queues; manyserver queues; deterministic fluid model; fluid approximation; queues with abandonment; nonMarkovian queues.