A Waiting Line Process of Markov Type

Abstract

Waiting-line or queuing processes of the Markov type are studied, the incoming traffic being of Poisson type and having negative-exponential holding time. The parameters are allowed to depend on time. The problem of finding an exact solution for the probability distribution of the waiting-line length as a function of time is reduced to the solution of an integral equation of the Volterra type. When the ratio of the parameters for the incoming and out-going traffic is constant, this equation can be solved explicitly and the required distribution obtained. Using this solution, the behavior of the process for large values of $t$ is studied, particularly for the unstable case with traffic intensity $\geqq 1$.