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Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

Part of: Special processes Computer system organization Operations research and management science

Published online by Cambridge University Press:  21 March 2016

P. Vis ,
R. Bekker  and
R. D. van der Mei
P. Vis*
Affiliation:
VU University Amsterdam and Centre for Mathematics and Computer Science
R. Bekker*
Affiliation:
VU University Amsterdam
R. D. van der Mei*
Affiliation:
VU University Amsterdam and Centre for Mathematics and Computer Science
*
∗ Postal address: VU Amsterdam, Department of Mathematics, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands.
∗ Postal address: VU Amsterdam, Department of Mathematics, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands.
∗ Postal address: VU Amsterdam, Department of Mathematics, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands.
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Abstract

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We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS (first-come-first-served) local service orders, namely last-come-first-served with and without preemption, random-order-of-service, processor sharing, the multi-class priority scheduling with and without preemption, shortest-job-first, and the shortest remaining processing time policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e. when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queue i is fully characterized and of the form Γ Θi, with Γ and Θi independent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θi which explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight into the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values.

Keywords

Polling systemservice disciplinewaiting-time distributionheavy traffic

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service 68M20: Performance evaluation; queueing; scheduling
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 4 , December 2015 , pp. 989 - 1014
Copyright
Copyright © Applied Probability Trust 2015 

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