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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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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
Stabilizing Performance in a SingleServer Queue with TimeVarying Arrival Rate”. Working paper, Columbia University, available at: www.columbia.edu.
, 2014
"... Abstract We consider a general G t /G t /1 singleserver queue with unlimited waiting space and a timevarying arrival rate, where the the service rate at each time is subject to control. We first study the ratematching control, where the the service rate is made proportional to the arrival rate. W ..."
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Abstract We consider a general G t /G t /1 singleserver queue with unlimited waiting space and a timevarying arrival rate, where the the service rate at each time is subject to control. We first study the ratematching control, where the the service rate is made proportional to the arrival rate. We show that the model with the ratematching control can be regarded as a deterministic time transformation of a stationary G/G/1 model, so that the queue length distribution is stabilized as time evolves. However, the timevarying virtual waiting time is not stabilized. We show that the timevarying expected virtual waiting time with the ratematching servicerate control becomes inversely proportional to the arrival rate in a heavytraffic limit. We also show that no control that stabilizes the queue length asymptotically in heavytraffic can also stabilize the virtual waiting time. Then we consider a squareroot servicerate control, where the service rate exceeds the arrival rate by a constant multiple of the square root of the arrival rate. We show that this alternative servicerate control stabilizes the waiting time, but not the queue length, when the arrival rate changes very slowly relative to the average service time. This behavior is supported by a limit theorem supporting the pointwisestationary approximation.
Stabilizing Performance in a Service System with TimeVarying Arrivals and Customer Feedback
, 2015
"... Analytical approximations are developed to determine the timedependent offered load (effective demand) and appropriate staffing levels that stabilize performance at designated targets in a manyserver queueing model with timevarying arrival rates, customer abandonment from queue and random feedba ..."
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Analytical approximations are developed to determine the timedependent offered load (effective demand) and appropriate staffing levels that stabilize performance at designated targets in a manyserver queueing model with timevarying arrival rates, customer abandonment from queue and random feedback with additional delay after completing service. To provide a flexible model that can be readily fit to system data, the model has historydependent Bernoulli routing, where the feedback probabilities, servicetime and patience distributions all may depend on the visit number. Before returning to receive a new service, the fedback customers experience delays in an infiniteserver or finitecapacity queue, where the parameters may again depend on the visit number. A new refined modifiedofferedload approximation is developed to obtain good results with low waitingtime targets. Simulation experiments confirm that the approximations are effective. A manyserver heavytraffic FWLLN shows that the performance targets are achieved asymptotically as the scale increases.