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The Impact of Dependent Service Times on LargeScale Service Systems
"... This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic ..."
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This paper investigates the impact of dependence among successive service times upon the transient and steadystate performance of a largescale service system. That is done by studying an infiniteserver queueing model with timevarying arrival rate, exploiting a recently established heavytraffic limit, allowing dependence among the service times. That limit shows that the number of customers in the system at any time is approximately Gaussian, where the timevarying mean is unaffected by the dependence, but the timevarying variance is affected by the dependence. As a consequence, required staffing to meet customary qualityofservice targets in a largescale service system with finitely many servers based on a normal approximation is primarily affected by dependence among the service times through this timevarying variance. This paper develops formulas and algorithms to quantify the impact of the dependence among the service times upon that variance. The approximation applies directly to infiniteserver models, but also indirectly to associated finiteserver models, exploiting approximations based on the peakedness (the ratio of the variance to the mean in the infiniteserver model). Comparisons with simulations confirm that the approximations can be useful to assess the impact of the dependence.
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.
A manyserver fluid limit for the Gt/GI/st + GI queueing model experiencing periods of overload
 OPERATIONS RESEARCH LETTERS
, 2012
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Choosing arrival process models for service systems: tests of a nonhomogeneous Poisson process.
 Nav. Res. Logist.
, 2014
"... Abstract: Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to per ..."
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Abstract: Service systems such as call centers and hospital emergency rooms typically have strongly timevarying arrival rates. Thus, a nonhomogeneous Poisson process (NHPP) is a natural model for the arrival process in a queueing model for performance analysis. Nevertheless, it is important to perform statistical tests with service system data to confirm that an NHPP is actually appropriate, as emphasized by Brown et al.
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
Largetime asymptotics for the Gt/Mt/st + GIt manyserver fluid queue with customer abandonment
, 2010
"... We previously introduced and analyzed the Gt/Mt/st +GIt manyserver fluid queue with timevarying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper we establish an asympt ..."
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We previously introduced and analyzed the Gt/Mt/st +GIt manyserver fluid queue with timevarying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper we establish an asymptotic loss of memory (ALOM) property for that fluid model; i.e., we show that there is asymptotic independence from the initial conditions as time t evolves, under regularity conditions. We show that the difference in the performance functions dissipates over time exponentially fast, again under the regularity conditions. We apply ALOM to show that the stationary G/M/s + GI fluid queue converges to steady state and the periodic Gt/Mt/st + GIt fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.
The Gt/GI/st + GI ManyServer Fluid Queue
, 2012
"... This paper introduces a deterministic fluid model that approximates the manyserver Gt/GI/st + GI queueing model, and determines the timedependent performance functions. The fluid model has timevarying arrival rate and service capacity, abandonment from queue, and nonexponential service and patie ..."
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This paper introduces a deterministic fluid model that approximates the manyserver Gt/GI/st + GI queueing model, and determines the timedependent performance functions. The fluid model has timevarying arrival rate and service capacity, abandonment from queue, and nonexponential service and patience distributions. Two key assumptions are that: (i) the system alternates between overloaded and underloaded intervals, and (ii) the functions specifying the fluid model are suitably smooth. An algorithm is developed to calculate all performance functions. It involves the iterative solution of a fixedpoint equation for the timevarying rate that fluid enters service and the solution of an ordinary differential equation for the timevarying headofline waiting time, during each overloaded interval. Simulations are conducted to confirm that the algorithm and the approximation are effective.
The heavily loaded manyserver queue with abandonment and deterministic service times
, 2010
"... Abstract We consider the GI/D/n + GI manyserver queue, with renewal arrival process (the first GI), deterministic (D) service times, a large number n of servers and customer abandonment, with i.i.d. abandonment times having a general distribution (the +GI), when the arrival rate exceeds the maximum ..."
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Abstract We consider the GI/D/n + GI manyserver queue, with renewal arrival process (the first GI), deterministic (D) service times, a large number n of servers and customer abandonment, with i.i.d. abandonment times having a general distribution (the +GI), when the arrival rate exceeds the maximum total service rate. The customer abandonment keeps the model stable: In great generality, the number of customers in the system converges to a unique stationary distribution. However, when n is large, if the system does not start with that steady state distribution, e.g., if it starts empty, then the timedependent performance tends to rapidly become periodic with a period equal to the service time. This anomalous behavior is explained through a manyserver heavytraffic fluid limit in which n → ∞. The limiting deterministic fluid model also has a unique stationary point, but that stationary point is not approached from any other initial state. Instead, the fluid model performance approaches one of its uncountably many periodic steady states. Keywords manyserver queues · deterministic fluid model · customer abandonment · deterministic service times · periodic steady state · transient behavior · multiple equilibria · nearly deterministic queues · asymptotic stability
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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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.
Infiniteserver queues with batch arrivals and dependent service times
, 2011
"... Motivated by largescale service systems, we consider an infiniteserver queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be timevarying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d ..."
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Motivated by largescale service systems, we consider an infiniteserver queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be timevarying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavytraffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavytraffic approximations for the stationary model and the model with a timevarying arrival rate. In the simulation experiments, we use the MarshallOlkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of MarshallOlkin multivariate hyperexponential distributions to model dependent hyperexponential service times within a batch.