We consider a system of two independent alternating renewal processes with states 0 and 1 and an initial shift t0 of one process relative to the other one. An integral equation with respect to the expectation of time T (the first time when both processes have state 0) is derived. To derive this equation, we use the method of so-called minimal chains of overlapping 1-intervals. Such a chain generates some breaking semi-Markov process of intervals composing the interval (0, T ). A solution of the integral equation is obtained for the case where the lengths of 1-intervals have exponential distributions and lengths of 0-intervals have arbitrary distributions. For more general distributions of 1-intervals, the Monte Carlo method is applied when both processes are simulated numerically by a computer. A histogram for estimates of the expectation of T as a function of t0 is demonstrated. Bibliography: 4 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 442, 2015, pp. 143–165.
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Harlamov, B.P., Prourzin, O.V. On an Interval of Faultless Work for a System of Two Independent Alternating Renewal Processes. J Math Sci 225, 818–832 (2017). https://doi.org/10.1007/s10958-017-3498-x
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DOI: https://doi.org/10.1007/s10958-017-3498-x