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On the stability of a batch clearing system with Poisson arrivals and subadditive service times

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

David Aldous ,
Masakiyo Miyazawa  and
Tomasz Rolski
David Aldous*
Affiliation:
University of California, Berkeley
Masakiyo Miyazawa*
Affiliation:
Science University of Tokyo
Tomasz Rolski*
Affiliation:
Wrocław University
*
∗ Postal address: Department of Statistics, University of California, Berkeley, CA 94720-3860, USA.
∗∗ Postal address: Department of Information Sciences, Science University of Tokyo, Noda, Chiba 278-8510, Japan.
∗∗∗ Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland. Email address: rolski@math.uni.wroc.pl
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Abstract

We study a service system in which, in each service period, the server performs the current set B of tasks as a batch, taking time s(B), where the function s(·) is subadditive. A natural definition of ‘traffic intensity under congestion’ in this setting is ρ := limt→∞t-1Es (all tasks arriving during time [0,t]). We show that ρ > 1 and a finite mean of individual service times are necessary and sufficient to imply stability of the system. A key observation is that the numbers of arrivals during successive service periods form a Markov chain {An}, enabling us to apply classical regenerative techniques and to express the stationary distribution of the process in terms of the stationary distribution of {An}.

Keywords

Gated service disciplinejob schedulingqueueingregenerative processstabilitystochastic schedulingsubadditive

MSC classification

Primary: 60K25: Queueing theory 90B36: Scheduling theory, stochastic
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 621 - 634
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

Research supported in part by KBN under grant 2 P03A 049 15 (1998–2001).

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