Geometric Distribution in Some Two-Dimensional Queuing Systems

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Abstract
The state of the system is a two dimensional random variable N = (N1, N2) with N1 ≧ 0, 1 ≦ N2 ≦ m. Transitions require negative exponential times. The vector Pn of probabilities of being in states for which N1 = n satisfy the general condition λIPn−1 + BPn + CPn+1 = 0. Two arguments are given showing Pn = RPn−1 and an iterative scheme for finding R is constructed.