NDLI: On a Stochastic Process Concerning Some Waiting Time Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Takcs, Lajos
Copyright Year	1957
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.
Starting Page	90
Ending Page	103
Page Count	14
File Format	PDF
ISSN	0040585X
DOI	10.1137/1102004
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	1
Volume Number	2
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-28
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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