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# On the Stability of a Batch Clearing System with Poisson Arrivals and Subadditive Service Times

David Aldous, Masakiyo Miyazawa and Tomasz Rolski
Journal of Applied Probability
Vol. 38, No. 3 (Sep., 2001), pp. 621-634
Stable URL: http://www.jstor.org/stable/3216116
Page Count: 14
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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 $\rho := \lim{_{t\to\infty}}\, t^{-1}$Es(all tasks arriving during time [0, t]). We show that $\rho < 1$ < 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

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