Multiclass Queueing Systems in Heavy Traffic: An Asymptotic Approach Based on Distributional and Conservation Laws

Bertsimas, Dimitris; Mourtzinou, Georgia

doi:10.1287/opre.45.3.470

Cited by 18 publications

(28 citation statements)

References 29 publications

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“…Based on the heavy-traffic results for the M/G/1 queue (see [13]), the distribution of the scaled total workload converges to an exponential distribution with mean ρE [R], where R is a residual service time and…”

Section: Resultsmentioning

confidence: 99%

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Heavy-traffic asymptotics of a priority polling system with threshold service policy

Liu

Chu

2016

Computers & Operations Research

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In this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system consisting of three M/M/1 queues with threshold policy. It turns out that the scaled queue-length of the critically loaded queue is exponentially distributed, independent of that of the stable queues. In addition, the queue lengths of stable queues possess the same distributions as a priority polling system with N -policy vacation. Based on this fact, we provide the exact tail asymptotics of the vacation polling system to approximate the tail distribution of the queue lengths of the stable queues, which shows that it has the same prefactors and decay rates as the classical M/M/1 preemptive priority queues. Finally, a stochastic simulation is taken to test the results aforementioned.

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Section: Resultsmentioning

confidence: 99%

“…By applying the multiclass distributional law of Bertsimas and Mourtzinou [13] it directly follows that the scaled waiting time at Q 3 follows an exponential distribution with parameter µ 3 η.…”

Section: Equating the Second-order Termsmentioning

confidence: 99%

Heavy-traffic asymptotics of a priority polling system with threshold service policy

Liu

Chu

2016

Computers & Operations Research

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“…Our analysis relies on results of Bertsimas and Mourtzinou [1], who state that the equations describing the physics of the system with renewal arrivals in HT are (almost) identical to the ones for the system with Poisson arrivals under any traffic intensity. More specifically, they show that the mean delay E[W i ] in case of renewal arrivals in the limit of ρ tending to 1 is given by, for…”

Section: Renewal Arrivalsmentioning

confidence: 99%

“…Bertsimas and Mourtzinou [1] prove that, in case of HT, the unknown variables V ar[C i ] again satisfy the set of equations formed by (7), i.e., as ρ ↑ 1 for…”

Section: Renewal Arrivalsmentioning

confidence: 99%

“…More specifically, we consider a gated polling model with a general number of queues, general service-time distributions, zero switch-over times (see also Remark 3.2) under the assumption of general renewal arrivals. For this model, we use a result by Bertsimas and Mourtzinou [1], who derive a set of linear equations for the variance of the cycle times for gated polling models with renewal arrivals in HT. Taking the proper HT limits of this set, in combination with the conservation law for the GI/G/1 queue in HT, we derive explicit closed-form expressions for the expected delay in HT for the model under consideration.…”

Section: Introductionmentioning

confidence: 99%

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Polling models with renewal arrivals: A new method to derive heavy-traffic asymptotics

Mei

Winands

2007

Performance Evaluation

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We consider asymmetric cyclic polling systems with an arbitrary number of queues, general service-time distributions, zero switch-over times, gated service at each queue, and with general renewal arrival processes at each of the queues. For this classical model, we propose a new method to derive closed-form expressions for the expected delay at each of the queues when the load tends to 1, under proper heavy-traffic (HT) scalings. In the literature on polling models, rigorous proofs of HT limits have only been obtained for polling models with Poisson-type arrival processes, whereas for renewal arrivals HT limits are based on conjectures [6,7,15]. Therefore, the main contribution of this paper lies in the fact that we propose a new method to rigorously prove HT limits for a class of non-Poisson-type arrivals. The results are remarkably simple and provide new fundamental insight and reveal explicitly how the expected delay at each of the queues depends on the system parameters, and in particular on the interarrival-time distributions at each of the queues. The results also suggest simple approximations for the expected delay in stable polling systems. Numerical results show that the approximations are highly accurate when the system load is roughly 90% or more.

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Scheduling policies for an antiterrorist surveillance system

Lin

Kress

Szechtman

2009

Naval Research Logistics

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This paper concerns scheduling policies in a surveillance system aimed at detecting a terrorist attack in time. Terrorist suspects arriving at a public area are subject to continuous monitoring, while a surveillance team takes their biometric signatures and compares them with records stored in a terrorist database. Because the surveillance team can screen only one terrorist suspect at a time, the team faces a dynamic scheduling problem among the suspects. We build a model consisting of an M/G/1 queue with two types of customers-red and white-to study this problem. Both types of customers are impatient, but the reneging time distributions are different. The server only receives a reward by serving a red customer, and can use the time a customer has spent in the queue to deduce its likely type. In a few special cases, a simple service rule-such as first-come-first-serve-is optimal. We explain why the problem is in general difficult, and develop a heuristic policy motivated by the fact that terrorist attacks tend to be rare events.

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Multiclass Queueing Systems in Heavy Traffic: An Asymptotic Approach Based on Distributional and Conservation Laws

Abstract: We

Cited by 18 publications

References 29 publications

Heavy-traffic asymptotics of a priority polling system with threshold service policy

Heavy-traffic asymptotics of a priority polling system with threshold service policy

Polling models with renewal arrivals: A new method to derive heavy-traffic asymptotics

Scheduling policies for an antiterrorist surveillance system

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