Availability and reliability analysis of a two-unit parallel system subject to failure and repair operating in a multi-level environment

Abstract

In this paper, a two-unit parallel system is considered subject to failure and repair in a multi-level environment. It is assumed that there are N levels of the environment. The failure time of any operating unit in the k−th level of the environment is exponentially distributed with parameter µk, k = 1, 2, · · · , N. There is a single repair facility. The system is in down-state when both units are in the repair facility. The down-state is designated as level 0 of the environment. The repairs are done under the ‘first-come-first serve’ policy. The repair time of a failed unit in the k−th level of the environment is exponentially distributed with parameter γk, k = 0, 1, 2, · · · , N. When the system is in level 0 of the environment, one failed unit is undergoing repair and the other failed unit is waiting for repair. Upon completion of the repair of the unit, the system is immediately switched to operate in r−th level of the environment with positive probability pr, r = 1, 2, · · · , N and the repair for the other failed unit immediately starts with rate γr. Using the techniques of renewal theory and Laplace transforms, transient state probability distribution, steady-state probability distribution, availability and reliability functions and mean down-time of the system are explicitly found. A numerical illustration is provided to highlight the system performance.