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

Published online by Cambridge University Press:  21 March 2016

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
Corresponding
E-mail address:
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Abstract

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 Θ i , with Γ and Θ i 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 Θ i 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.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2015 

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