On interval of faultless work for a system of two independent alternating renewal processes

A system of two independent alternating renewal processes with states $0$ and $1$, and an initial shift $t_0$ of one process relative to another one is considered. An integral equation with respect to an expectation of time $T$ (the first time when both processes have state $0$) is derived. For deriving a method of so called minimal chains of overlapping $1$-intervals is used. 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 when lengths of $1$-intervals have exponential distributions and lengths of $0$-intervals have distributions of common view. For more general distributions of $1$-intervals the Monte Carlo method is applied when both processes are simulated numerically by computer. Histograms for estimates of the expectation of $T$ as a function of $t_0$ are demonstrated.