Maximizing mean-time to failure in k-resilient systems with repair

A k-resilient system with N components can tolerate up to k component failures and still function correctly. We consider k-resilient systems where the number of component failures is a constant fraction of the total number of components, that is k=N/c and c is a constant such that 2⩽c<∞. Under a Markovian assumption of constant failure and repair rates, we compute the system size N_max at which the mean-time to failure (MTTF) for such a system is maximized. Our results indicate that N_max can be expressed in terms of constant c and parameter ρ as N_max=K(c,ρ)/ρ, where ρ=λ/μ and K(c, ρ) is a function of c,ρ. In addition, we have found that the variation of N_max over the whole range of c is remarkably small, and as a result, even if the resilience k of a system as a function of N varies widely, the system size at which the MTTF is maximized is within the range 0.36/ρ and 0.5/ρ. We validate our results through event-driven simulation, and, in addition, examine the behavior of systems with Weibull distributed failure times