Optimal age and block replacement policies for a multi-component system with a shock type failure interaction

Abstract

Generalized age and block replacement policies for a multi-component system with shock failure, interaction are considered. The ith component (1 ≤ i ≤ N) is subject to shocks that arrive according to a non-homogeneous Poisson process {N_i(t), t ≥ 0}. As shocks occur the ith component has two types of failure. Type I failure (minor failure) is removed by a minimal repair, whereas type II failure (catastrophic failure) induces a total failure of the system (i.e. failure of all other components in the system) and is removed by an unplanned (or unscheduled) replacement of the system. The choice of these two possible actions is based on some random mechanism which depends on the number of shocks each component suffered since the last replacement. For an age replacement maintenance policy, planned (or scheduled) replacements occur whenever an operating system reaches age T, whereas in the block replacement case planned replacements occur every T units of time. The aim of this paper is to derive the expressions for the expected long-run cost per unit time and the total a discounted cost for each policy. The optimal T* which would minimize the cost rate or the total ∝-discounted cost is discussed. Various special cases are detailed. A numerical example is given to illustrate the method.