On Random Walks and Stable GI/G/1 Queues

Abstract

Let X, X1, X2, … be independent and identically distributed random variables with E{X} < 0, let S0 ≡ 0, Sn = X1 + ⋯ + Xn for n ≥ 1, and M = sup(Sn, n ≥ 0). If r > 0 then a well-known result in random walk theory slates that M has a finite moment of order r if and only if the positive part of X has a finite moment of order r + 1. Utilizing the intimate relationship between random walks and single server queues, this paper presents a new and simple proof for this basic result. Identities involving the distribution functions and the integer moments of M and X are then derived by simple, probabilistic arguments. Bounds are then obtained for the expected values of some important variables associated with the random walk (Sn, n ≥ 1). All of these random-walk results are applied to the stable queue GI/G/1, and some standard queueing results are subsequently obtained in an effortless manner.