Redundancy with recoveries and unreliable switches (Q914545)

A redundant system consists of \((n+1)\) identical elements and switches. If an active element fails, a switch is changed so that the next element becomes active. A switch can fail with probability \(\alpha\). If so, the next operational pair switch/element is chosen. Repair time distributions of elements and switches are known. A system fails if there is no operational switch/element. It is to find an asymptotic distribution of systems time to first failure. This distribution is considered in an equivalent problem with an infinite number of spare elements and a threshold n for system failure. A stochastic process modelling a number of failed elements is a service process of an M/G/\(\infty\) queueing system with Poisson group arrivals having parameter \(\lambda\) and a geometric distribution of the number of elements of the group. An explicit formula for the distribution of time to first failure is given in an unproved theorem.