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Published online by Cambridge University Press: 14 July 2016
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}.
Research supported in part by KBN under grant 2 P03A 049 15 (1998–2001).