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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.
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 34 (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.
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
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 22 (10 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.
Manyserver diffusion limits for G/Ph/n+GI queues
, 2009
"... This paper studies manyserver limits for multiserver queues that have a phasetype service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded G/Ph/n + GI queues, where the patience times are independent, identically distributed following ..."
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Cited by 19 (9 self)
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This paper studies manyserver limits for multiserver queues that have a phasetype service time distribution and allow for customer abandonment. The first set of limit theorems is for critically loaded G/Ph/n + GI queues, where the patience times are independent, identically distributed following a general distribution. The next limit theorem is for overloaded G/Ph/n + M queues, where the patience time distribution is restricted to be exponential. We prove that a pair of diffusionscaled totalcustomercount and serverallocation processes, properly centered, converges in distribution to a continuous Markov process as the number of servers n goes to infinity. In the overloaded case, the limit is a multidimensional diffusion process, and in the critically loaded case, the limit is a simple transformation of a diffusion process. When the queues are critically loaded, our diffusion limit generalizes the result by Puhalskii and Reiman (2000) for GI/Ph/n queues without customer abandonment. When the queues are overloaded, the diffusion limit provides a refinement to a fluid limit and it generalizes a result by Whitt (2004) for M/M/n / + M queues with an exponential service time distribution. The proof techniques employed in this paper are innovative. First, a perturbed system is shown to be equivalent to the original system. Next, two maps are employed in both fluid and diffusion scalings. These maps allow one to
A Network of TimeVarying ManyServer Fluid Queues with Customer Abandonment
"... To describe the congestion in largescale service systems, we introduce and analyze a nonMarkovian open network of manyserver fluid queues with customer abandonment, proportional routing and timevarying model elements. A proportion of the fluid completing service at each queue is routed immediate ..."
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Cited by 17 (14 self)
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To describe the congestion in largescale service systems, we introduce and analyze a nonMarkovian open network of manyserver fluid queues with customer abandonment, proportional routing and timevarying model elements. A proportion of the fluid completing service at each queue is routed immediately to each other queue, while the fluid not routed to other queues leaves the network. The fluid queue network serves as an approximation for the corresponding nonMarkovian open network of manyserver queues with Markovian routing, where all model elements may be time varying. We establish the existence of a unique vector of (net) arrival rate functions at each queue and the associated timevarying performance. In doing so, we provide the basis for an efficient algorithm, even for networks with many queues. Key words: queues with timevarying arrivals; queueing networks; manyserver queues; deterministic fluid model; customer abandonment; nonMarkovian queues. History: Submitted on February 7, 2010 1.
Distributionvalued heavytraffic limits for the G/GI/∞ queue
, 2009
"... We study the G/GI/∞ queue from two different perspectives in the same heavytraffic regime. First, we represent the dynamics of the system using a measurevalued process that keeps track of the age of each customer in the system. Using the continuousmapping approach together with the martingale fun ..."
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Cited by 14 (0 self)
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We study the G/GI/∞ queue from two different perspectives in the same heavytraffic regime. First, we represent the dynamics of the system using a measurevalued process that keeps track of the age of each customer in the system. Using the continuousmapping approach together with the martingale functional central limit theorem, we obtain fluid and diffusion limits for this process in a space of distributionvalued processes. Next, we study a measurevalued process that keeps track of the residual service time of each customer in the system. In this case, using the functional central limit theorem and the random time change theorem together with the continuousmapping approach, we again obtain fluid and diffusion limits in our space of distributionvalued processes. In both cases, we find that our diffusion limits may be characterized as distributionvalued OrnsteinUhlenbeck processes. Further, these diffusion limits can be analyzed using standard results from the theory of Markov processes.
A manyserver fluid limit for the Gt/GI/st + GI queueing model experiencing periods of overload
 OPERATIONS RESEARCH LETTERS
, 2012
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