An analytical solution for the discrete time single server system with semi-Markovian arrivals

In this paper the authors derive an analytical solution for the stationary distribution of the number of customers and the idle time in a single server system with semi-Markovian arrival processes in discrete time domain (SM/G/1). This kind of arrival process enables them to take autocorrelations into account, with various applications for the modeling of communication and manufacturing systems. It will be shown that the distribution of the customer number can be represented as a linear combination of geometric distributions. Thus a simple calculation of higher moments of the customer number is possible.