Some structural properties of a Markovian storage/production system

Abstract

We consider a storage/production system with state-dependent production rate and state-dependent demand arrival rate. Every arriving demand gives rise to a `peak' in the trajectory of the content process. We characterize the processes N-0(x) and N-1(x), defined as the number of peaks and the number of record peaks, respectively, before the content reaches the level x. The results are applied to the virtual waiting time process W(t) of a M/G/1 queue. Assuming that W(0) = x(0), M(x) is defined to be the number of arrivals before the virtual waiting time drops from x(0) to x(0-x) (0 less than or equal to x less than or equal to x(0)); in particular, M(x(0)) is the number of customers arriving during the first busy period. It is shown that (M(x))(0 less than or equal to x less than or equal to x0) is a compound Poisson process, and its jump size distribution is derived in closed form.