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

Customers arrive at a counter at the instants (tau) 1 , (tau) 2 , ..., (tau) n ..., where the inter-arrival times (tau) n - (tau) n -1 ( n = 1, 2, ..., (tau) 0 = 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 (xi)( t ) denote the queue size at the instant t . If (xi)((tau) n - 0) = k then a transition E k (rightarrow) E k +1 is said to occur at the instant t = (tau) n . 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} G n ( X ) = the probability that a busy period consists of n services and its length is at most x , and further the distribution of the number of transitions E k (rightarrow) E k +1 occurring in the time interval (0, t ).