A generalized signature of repairable coherent systems

We present a generalized signature in repairable coherent systems resembling Samaniego's notion for statistically independent and identically distributed lifetimes. The repairable systems are made of different components which can individually fail, and be minimally repaired up to a fixed number of times. Failures occur according to Poisson processes, which might have either the same intensity function for each component, or different ones. The former case is similar to the notion of signature presented by Samaniego for i.i.d. random variables, whereas here statistically independent Poisson processes with identical intensity functions are considered. An explicit expression for computing the generalized signature of repairable series systems is obtained. It is shown that the reliability function of any repairable coherent system can be expressed as a generalized mixture of the probabilities of the number of repairs until system failure. We also establish that the stochastic ordering between the generalized signatures of two repairable systems is preserved by their lifetimes.