Scheduling starting times for an active redundant system with non-identical lifetimes

Here we examine an active redundant system with scheduled starting times of the units. We assume availability of n non-identical, non-repairable units for replacement or support. The original unit starts its operation at time s1 = 0 and each one of the (n − 1) standbys starts its operation at scheduled time si (i = 2, …, n) and works in parallel with those already introduced and not failed before si. The system is up at times si (i = 2, …, n), if and only if, there is at least one unit in operation. Thus, the system has the possibility to work with up to n units, in parallel structure. Unit-lifetimes Ti (i = 1, …, n) are independent with cdf Fi, respectively. The system has to operate without inspection for a fixed period of time c and it stops functioning when all available units fail before c. The probability that the system is functioning for the required period of time c depends on the distribution of the unit-lifetimes and on the scheduling of the starting times si. The reliability of the system is evaluated via a recursive relation as a function of the starting times si (i = 2, …, n). Maximizing with respect to the starting times we get the optimal ones. Analytical results are presented for some special distributions and moderate values of n.