An application of Riordan arrays to the transient analysis of M/M/1M/M/1 queues

Riordan arrays are used extensively in many contexts as a combinatorial tool for solving enumeration problems such as lattice path counting problems. In this paper, we first show how Riordan array method can be applied to the transient analysis of M/M/1M/M/1 queue with zero customer at the initial point. Second, we extend this method to the generalized Riordan array with multiple support functions in order to deal with the transient analysis of M/M/1M/M/1 queue with non-zero customers at the initial point. Numerical examples are also given to show how easy and quick the transient probability obtained from the Riordan method can be computed.