Transient Behavior of Single-Server Queuing Processes with Recurrent Input and Exponentially Distributed Service Times

Customers arrive at a counter at the instants τ1, τ2, …, τn …, where the inter-arrival times τn − τn−1(n = 1, 2, …, τ0 = 0) are indentically distributed, independent, random variables. The customers will be served by a single server. The service times are identically distributed, independent, random variables with exponential distribution. Let ξ(t) denote the queue size at the instant t. If ξ(τn − 0) = k then a transition Ek → Ek+1 is said to occur at the instant t = τn. The following probabilities are determined \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\rho^{(n)}_{ik}=\mathbf{P}\{\xi(\tau_{n}-0)=k\mid\xi(0)=i+1\},\quad P^{\ast}_{ik}(t)=\mathbf{P}\{\xi(t)=k\mid\xi(0)=i\},$$ \end{document} Gn(X) = the probability that a busy period consis...