Single server queue with uniformly bounded virtual waiting time

Abstract

Summary: In a previous paper [4] the author studied the stochastic process {w_n , n = 1,2, …}, recursively defined by with K a positive constant, τ₁, τ₂, … σ₁, σ₂, …, independent, nonnegative stochastic variables. τ₁,τ₂…, are identically distributed, and σ₁,σ₂,…, are also identically distributed variables. For this process the generating function of the Laplace-Stieltjes transforms of the joint distribution of W_n , σ₂ + … + σ n and τ₁ + … + τ n−1 has been obtained. Closely related to the process {w_n , n = 1, 2,…} is the process {u_n , n = 1, 2,…} with {u_n = K + [w_n + τ n − K]⁻, n = 1,2,…; these are dual processes. In the present paper we study the stationary distributions of the processes {w_n , n= 1,2, …} and {u_n , n = 1,2, …}, and the distributions ot the entrance times and return times of the events “w_n , n = 0” and “u_n = K” for some n, for discrete as well as for continuous time. For these events various taboo probabilities are also investigated. The mathematical descri ption of the processes {w_n , n = 1,2, …} and {u_n , n= 1,2, …} gives all the necessary information about the time-dependent behaviour for the general dam model with finite capacity K, since the process {w_n , n= 1,2, …} is the basic process for such dam models. In Sections 5, 6 and 7 the general theory is applied to the models M/G/1 and G/M/1. Complete explicit solutions are obtained for these models. The present theory also leads to new and important results for the queueing system or dam model G/G/1 with infinite capacity. For instance the joint distribution of the busy period (or wet period) and of the supremum of the dam content dunng this period is obtained.