Geometric Distribution in Some Two-Dimensional Queuing Systems

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 P_n of probabilities of being in states for which N₁ = n satisfy the general condition λIP_n−1 + BP_n + CP_n+1 = 0. Two arguments are given showing P_n = RP_n−1 and an iterative scheme for finding R is constructed.