Asymptotic analysis of a heterogeneous renewable complex system with random environments

Abstract

This paper deals with an asymptotic analysis of a complex renewable system with N heterogeneous elements looked after by n repairmen. Each element and the repair facility are assumed to operate in independent random environments governed by ergodic Markov chains. The operating and repair times of an element are supposed to be exponentially distributed random variables with parameter depending on the index of the element, the state of the corresponding random environment, and the indices of the failed elements. The repair is carried out according to a First Come, First Served (FCFS) discipline. Assuming that the repair rates are many times greater than the corresponding failure rates (“fast” repair), it is shown that the time to the first system failure converges in distribution, under appropriate norming, to an exponentially distributed random variable. Some numerical examples illustrate the effectiveness of the method proposed by comparing the approximate characteristics to the exact ones.