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Martingale proofs of manyserver heavytraffic limits for Markovian queues
 PROBABILITY SURVEYS
, 2007
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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.
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
Staffing Call Centers with Uncertain Demand Forecasts: A ChanceConstrained Optimization Approach
, 2010
"... We consider the problem of staffing call centers with multiple customer classes and agent types operating under qualityofservice (QoS) constraints and demand rate uncertainty. We introduce a formulation of the staffing problem that requires that the QoS constraints are met with high probability wi ..."
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Cited by 8 (1 self)
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We consider the problem of staffing call centers with multiple customer classes and agent types operating under qualityofservice (QoS) constraints and demand rate uncertainty. We introduce a formulation of the staffing problem that requires that the QoS constraints are met with high probability with respect to the uncertainty in the demand rate. We contrast this chanceconstrained formulation with the averageperformance constraints that have been used so far in the literature. We then propose a twostep solution for the staffing problem under chance constraints. In the first step, we introduce a random static planning problem (RSPP) and discuss how it can be solved using two different methods. The RSPP provides us with a firstorder (or fluid) approximation for the true optimal staffing levels and a staffing frontier. In the second step, we solve a finite number of staffing problems with known arrival rates—the arrival rates on the optimal staffing frontier. Hence, our formulation and solution approach has the important property that it translates the problem with uncertain demand rates to one with known arrival rates. The output of our procedure is a solution that is feasible with respect to the chance constraint and nearly optimal for large call centers.
Routing and staffing in largescale service systems: The case of homogeneous impatient customers and heterogeneous servers
, 2011
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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.
State Space Collapse in ManyServer Diffusion Limits of Parallel Server Systems
, 2006
"... We consider a class of queueing systems that consist of server pools in parallel and multiple customer classes. Customer service times are assumed to be exponentially distributed. We study the asymptotic behavior of these queueing systems in a heavy traffic regime that is known as the Halfin and Wh ..."
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Cited by 6 (3 self)
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We consider a class of queueing systems that consist of server pools in parallel and multiple customer classes. Customer service times are assumed to be exponentially distributed. We study the asymptotic behavior of these queueing systems in a heavy traffic regime that is known as the Halfin and Whitt manyserver asymptotic regime. Our main contribution is a general framework for establishing state space collapse results in this regime for parallel server systems. In our work, state space collapse refers to a decrease in the dimension of the processes tracking the number of customers in each class waiting for service and the number of customers in each class being served by various server pools. We define and introduce a “state space collapse ” function, which governs the exact details of the state space collapse. We show that a state space collapse result holds in manyserver heavy traffic if a corresponding deterministic hydrodynamic model satisfies a similar state space collapse condition. Our methodology is similar in spirit to that in Bramson [10], which focuses on the conventional heavy traffic regime. We illustrate the applications of our results by establishing state space collapse results in manyserver diffusion limits of staticbufferpriority Vparallel server systems, Nmodel parallel server systems, and minimumexpecteddelay–fasterserverfirst distributed server pools systems. We show for these systems that the condition on the hydrodynamic model can easily be checked using the standard tools for fluid models.
Critical care in hospitals: When to introduce a Step Down Unit?
, 2014
"... medicalsurgical wards. Because SDUs are less richly staffed than ICUs, they are less costly to operate; however, they also are unable to provide the level of care required by the sickest patients. There is an ongoing debate in the medical community as to whether and how SDUs should be used. On one ..."
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Cited by 2 (1 self)
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medicalsurgical wards. Because SDUs are less richly staffed than ICUs, they are less costly to operate; however, they also are unable to provide the level of care required by the sickest patients. There is an ongoing debate in the medical community as to whether and how SDUs should be used. On one hand, an SDU alleviates ICU congestion by providing a safe environment for postICU patients before they are stable enough to be transferred to the general wards. On the other hand, an SDU can take capacity away from the already overcongested ICU. In this work, we propose a queueing model to capture the dynamics of patient flows through the ICU and SDU in order to determine how to size the ICU and SDU. We account for the fact that patients may abandon if they have to wait too long for a bed, while others may get bumped out of a bed if a new patient is more critical. Using fluid and diffusion analysis, we examine the tradeoff between reserving capacity in the ICU for the most critical patients versus gaining additional capacity achieved by allocating nurses to the SDUs due to the lower staffing requirement. Despite the complex patient flow dynamics, we leverage a statespace collapse result in our diffusion analysis to establish the optimal allocation of nurses to units. We find that under some circumstances the optimal size of the SDU is zero, while in other cases, having a sizable SDU may be beneficial. The insights from our work will be useful for hospital managers determining how to allocate nurses to the hospital units, which subsequently determines the size of each unit.
Staffing CallCenters With Uncertain Demand Forecasts: A ChanceConstraints Approach. working paper
, 2009
"... We consider the problem of staffing callcenters with multiple customer classes and agent types operating under qualityofservice (QoS) constraints and demand rate uncertainty. We introduce a formulation of the staffing problem that requires that the QoS constraints are met with high probability wi ..."
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Cited by 2 (0 self)
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We consider the problem of staffing callcenters with multiple customer classes and agent types operating under qualityofservice (QoS) constraints and demand rate uncertainty. We introduce a formulation of the staffing problem that requires that the QoS constraints are met with high probability with respect to the uncertainty in the demand rate. We contrast this chanceconstrained formulation with the averageperformance constraints that have been used so far in the literature. We then propose a twostep solution for the staffing problem under chance constraints. In the first step, we introduce a Random Static Planning Problem (RSPP) and discuss how it can be solved using two different methods. The RSPP provides us with a firstorder (or fluid) approximation for the true optimal staffing levels and a staffing frontier. In the second step, we solve a finite number of staffing problems with known arrival rates–the arrival rates on the optimal staffing frontier. Hence, our formulation and solution approach has the important property that it translates the problem with uncertain demand rates to one with known arrival rates. The output of our procedure is a solution that is feasible with respect to the chance constraint and nearly optimal for large call centers. 1