Mixing of Two Renewal Processes and Its Applications to Reliability Theory

The probabilistic behavior of repairable two-state systems is studied. It is assumed that the reliability of the system can be measured by the amount of time the system has been operative. The operating and repair states are a pair of renewal processes, a particular mixture of which describes the statistical behavior of the system. The object of this contribution is to extend the results of Muth who has earlier obtained the average value of the downtime and its variance, when one of the constituent renewal processes has its interval lengths distributed exponentially. This paper, by the repeated use of the method of regeneration points, obtains the mean and mean-square values of the uptime distribution. In addition the correlation of the uptime for different times has been derived and a proof of Takacs´ theorem is provided. Since the criteria for the reliability of the system include the associated cost, it is worthwhile investigating the operating cost over any period of time for arbitrary distribution of the two states. In particular a demonstration of the calculation of the first two moments of the total cost is given.