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\leq x\leq x_0)s in particular, M(x_0) is the number of customers arriving during the first busy period. It is shown that (M(x))_{0\leq x\leq x_0} is a compound Poisson process, and its jump size distribution is derived in closed form.