On the stability of a batch clearing system

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 deenition of \traac intensity under congestion" in this setting is := lim t!1 t ?1 Es(all tasks arriving during time 0; t]): We show that < 1, and nite mean of individual service times, are necessary and suucient to imply stability of the system. A key observation is that the numbers of arrivals during successive service periods form a Markov chain fA n g, enabling us to apply classical regenerative techniques and to express the stationary distribution of the process in terms of the stationary distribution of fA n g.