On a Stochastic Process Concerning Some Waiting Time Problems

Abstract

Let us consider the following probability model: Suppose that m machines are serviced by a single repairman. Normally the machines work continuously during the time interval $0 < t < \infty $, but at any moment any of the machines may break down and require service. We assume that the probability that a machine working at time t will require service in the time interval $(t,t + \Delta t)$ is equal to $\mu \Delta t + O(\Delta t)$. The machines work independently of each other. If a machine breaks down, then it is serviced immediately, unless the repairman is busy servicing another machine. In the latter case the repair of the machines takes place in the order in which they break down. The periods of time necessary for servicing the machines are random, independent, equally distributed variables. Let $\eta (t)$ be the number of machines working at time t, and let $\chi (t)$ be the time from the moment t to the end of any servicing being done at that moment. It is proved in this article that for $t \to \infty $ the quantities $\eta (t)$ and $\chi (t)$ have limiting distributions which are independent of the initial conditions. The obvious formulae for these limiting distributions are found. A limiting distribution is also found for the number of machines which are working at the moment when the repair of a machine is begun. Finally, for a stationary process, we consider the distribution function for the time from the breakdown of a machine to the beginning of its repair.