Note on a Moving Single Server Problem

Abstract

B. McMillan and J. Riordan in [1] derived the generating function for the probability distribution of the number of items completed before absorption in a moving single server problem in two special cases. Through an analogy to the work of L. Takacs [2] on busy period problems for a simple queue, McMillan and Riordan postulated a nonlinear integral equation relation for the generating function. In this note the validity of this relation is proved in general by exploiting the analogy more fully, and the generating function in the two special cases is obtained directly from the integral equation. A similar functional relation is established for the Laplace-Stieltjes transform of the distribution of time until absorption, and the transform is obtained for the two special cases.