Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T03:25:04.543Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Operations research and management science Special processes

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

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

References

Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, Berlin.Google Scholar
Baccelli, F., and Hong, D. (2000). Analytic expansions of max-plus Lyapunov exponents. Ann. Appl. Prob. 10, 779–827.Google Scholar
Bambos, N., and Walrand, J. (1991). On the stability and performance of parallel processing systems. J. Assoc. Comput. Mach. 38, 429–452.Google Scholar
Brémaud, P. (1999). Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, New York.Google Scholar
Durrett, R. (1991). Probability: Theory and Examples. Wadsworth and Brooks, Pacific Grove, CA.Google Scholar
Fiat, A., and Woeginger, G. J. (eds) (1998). Online Algorithms (Lecture Notes Comput. Sci. 1442). Springer, Berlin.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Meyn, S. P., and Tweedie, R. D. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Miyazawa, M. (1994). Rate conservation laws: a survey. Queueing Systems 15, 1–58.Google Scholar
Pakes, A. G. (1971). A branching process with a state dependent immigration component. Adv. Appl. Prob. 3, 301–314.CrossRefGoogle Scholar
Perry, D., and Stadje, W. (2001). A stochastic clearing model with a Brownian and a compound Poisson component. Unpublished manuscript.Google Scholar
Resing, J. A. C. (1993). Polling systems and branching processes. Queueing Systems 13, 409–426.Google Scholar
Sharma, V. (1998). Some limit theorems for regenerative queues. Queueing Systems 30, 341–363.CrossRefGoogle Scholar
Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice Hall, New Jersey.Google Scholar