Title: On the stability of a batch clearing system with
Poisson arrivals and subadditive service times
Author: David Aldous and Masakiyo Miyazawa and Tomasz Rolski
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 $s(\cdot)$ is subadditive.
A natural definition of ``traffic intensity under congestion"
in this setting is
\[ \rho :=
\lim_{t \to \infty} t^{-1} E s(\mbox{all tasks arriving during time } [0,t]) .\]
We show that $\rho < 1$, and 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 $\{A_n\}$.