A run shock-erosion model

Abstract

Suppose that a system is subject to independent random shocks over discrete time periods, with the shocks occurring each period with a probability of p, and suppose that $k_1$ and $k_2\mathrm{ }(1\le k_1<k_2)$ are two critical levels. Assume that the occurrence of a shock run of length $l\mathrm{ }(k_1\le l<k_2)$ causes a partial failure, leading to a decrease in the efficiency of the system, and that a complete failure will happen when a shock run of length $k_2$ occurs. The occurrence of shocks under the mentioned conditions makes a multi-state system, which leads to the full active working and the partially working, with their distributions being under study here. In addition, the present study discusses the mean of full active working and that of the partially working and obtains the joint distribution for the number of partial failures and the failure time of the system. There will also be a discussion of the mean failure time of the system and some of its properties under different conditions of occurrence or nonoccurrence of partial failures. Finally, the present study examines the mean failure time of the system in a different scheme in which, erosion can also cause failure of the system.

Keywords:

• Multi-state system
• run shock model
• type I waiting time distribution of order $\mathrm{k}$

MATHEMATICS SUBJECT CLASSIFICATION:

• 60C05
• 60E05
• 62N05