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

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 := lim_t→∞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 {A_n}, enabling us to apply classical regenerative techniques and to express the stationary distribution of the process in terms of the stationary distribution of {A_n}.