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

Abstract

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/spl les/c</spl infin/. Under a Markovian assumption of constant failure and repair rates, we compute the system size N/sub max/ at which the mean-time to failure (MTTF) for such a system is maximized. Our results indicate that N/sub max/ can be expressed in terms of constant c and parameter /spl rho/ as N/sub max/=K(c,/spl rho/)//spl rho/, where /spl rho/=/spl lambda///spl mu/ and K(c, /spl rho/) is a function of c,/spl rho/. In addition, we have found that the variation of N/sub 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//spl rho/ and 0.5//spl rho/. We validate our results through event-driven simulation, and, in addition, examine the behavior of systems with Weibull distributed failure times.