First-passage time and busy period distributions of discrete-time Markovian queues: Geom(n)/Geom(n)/1/N

This paper gives simple explicit solutions of various first-passage-time distributions for a general class of discrete-time queueing models under arbitrary initial conditions, state-dependent transition probabilities and the finite waiting room. Explicit closed form expressions are obtained in terms of roots. Explicit closed-form expressions are also deduced for the continuous-time models including the busy-period distributions. The analysis is then extended to cover the case of two absorbing states.
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