Consider the random walk {Sn} whose summands have the distribution P(X=0) = 1-(2/π), and P(X = ± n) = 2/[π(4n2−1)], for n ≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean time xk for the random walk starting from S0 = k to exit the interval. The explicit formula yields the limiting behavior of xk as N → ∞ with k fixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By letting N → ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting from k = 0 results. The distributions for some related random variables are also discovered.

Applications to stress concentration calculations in discrete lattices are briefly reviewed.

This paper studies the single-server queueing system with two independent input streams: a GI/G and an M/G stream. A new proof is given of an old result which shows how this system can be transformed into an equivalent ‘single input stream’ GI/G/1 queue, and methods to study that equivalent system numerically are given. As part of the numerical analysis, algorithms are given to compute the moments and the distribution function of busy periods in the M/G/1 queue, and of other related busy periods. Special attention is given to the single-server queue with independent D/G and M/G input streams.

This work is to be used in the modeling of real-time computer systems, which can often be described as a single-server queueing system with independent D/G and M/G input streams, see for example Ott (1984b).