Distribution of the delay in polling systems in heavy traffic

Mei, R.D. van der

doi:10.1016/s0166-5316(99)00048-6

Cited by 28 publications

(47 citation statements)

References 25 publications

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“…Thus, in this particular case, the server follows a fixed alternating (or cyclic) routing mechanism. The heavytraffic behaviour of cyclic polling models that are of a branching type and consist of an arbitrary number of queues has already been established in e.g., [21,27,28]. Translating this to our setting with two queues, exhaustive service and cyclic routing ( p 1 = p 2 = 0), these results readily imply the following.…”

Section: Initial Study Of the Heavy-traffic Behavioursupporting

confidence: 55%

“…This principle implies that, although the total scaled workload in the system tends to a Bessel-type diffusion in the heavy-traffic regime, the total workload in the system may be considered as a constant during the course of a polling cycle, while the loads of the individual queues fluctuate like a fluid model. In [27], several heavy-traffic limits have been established by taking limits in known expressions for the Laplace-Stieltjes transform (LST) of the waiting-time distribution. Alternatively, [21] provides similar results, by studying the behaviour of the descendant set approach (a numerical computation method, cf.…”

Section: Introductionmentioning

confidence: 99%

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On two-queue Markovian polling systems with exhaustive service

2014

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We consider a class of two-queue polling systems with exhaustive service, where the order in which the server visits the queues is governed by a discrete-time Markov chain. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we obtain explicit expressions for the Laplace-Stieltjes transforms of the waitingtime distributions and the probability generating function of the joint queue length distribution at an arbitrary point in time. We also study the heavy-traffic behaviour of properly scaled versions of these distributions, which results in compact and closedform expressions for the distribution functions themselves. The heavy-traffic behaviour turns out to be similar to that of cyclic polling models, provides insights into the main effects of the model parameters when the system is heavily loaded, and can be used to derive closed-form approximations for the waiting-time distribution or the queue length distribution.

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Section: Initial Study Of the Heavy-traffic Behavioursupporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

On two-queue Markovian polling systems with exhaustive service

2014

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“…The following result, which was shown in [25], characterizes the limiting behavior of the waiting-time distribution in heavy traffic. …”

Section: First-come-first-servedmentioning

confidence: 91%

“…The following result gives a characterization for the limiting behavior of the cycle-time distributions, stating that the (scaled) cycle times (1 − ρ)C i converge to a gamma distribution with known parameters (proven in [25,26]). …”

Section: Model and Notationmentioning

confidence: 99%

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The impact of scheduling policies on the waiting-time distributions in polling systems

Bekker¹,

Vis²,

Dorsman

et al. 2014

Queueing Syst

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We consider polling models consisting of a single server that visits the queues in a cyclic order. In the vast majority of papers that have appeared on polling models, it is assumed that at each of the individual queues the customers are served on a First-Come-First-Served (FCFS) basis. In this paper we study polling models where the local scheduling policy is not FCFS, but instead, is varied as Last-ComeFirst-Served (LCFS), Random Order of Service (ROS), Processor Sharing (PS) and Shortest-Job-First (SJF). The service policies are assumed to be either gated or globally gated. The main result of the paper is the derivation of asymptotic closed-form expressions for the Laplace-Stieltjes transform (LST) of the scaled waiting-time and sojourn-time distributions under heavy-traffic assumptions. For FCFS service the asymptotic sojourn-time distribution is known to be of the form U Γ, where U and Γ are uniformly and gamma distributed with known parameters. In this paper we show that the asymptotic sojourntime distribution (1) for LCFS is also of the form U Γ, (2) for ROS is of the formŨ Γ whereŨ has a trapezoidal distribution, and (3) for PS and SJF is of the formŨ * Γ whereŨ * has a generalized trapezoidal distribution. These results are rather intriguing and lead to new fundamental insight in the impact of the local scheduling policy on the performance of polling models. As a by-product the heavy-traffic results suggest simple closed-form approximations for the complete waiting-time and sojourn-time distributions for stable systems with arbitrary load values. The accuracy of the approximations is evaluated by simulations.

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The Impact of Priority Policy in a Two-Queue Markovian Polling System with Multi-Class Priorities

Chu

Liu

2017

Queueing Theory and Network Applications

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Distribution of the delay in polling systems in heavy traffic

Cited by 28 publications

References 25 publications

On two-queue Markovian polling systems with exhaustive service

On two-queue Markovian polling systems with exhaustive service

The impact of scheduling policies on the waiting-time distributions in polling systems

The Impact of Priority Policy in a Two-Queue Markovian Polling System with Multi-Class Priorities

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