Reliability of a consecutive ( r, s)-out-of-( m, n):F lattice system with conditions on the number of failed components in the system

Abstract

A consecutive( r, s)-out-of-( m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of- n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an ( m, n) matrix and fails iff the system has an ( r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of- mn:F and a consecutive ( r, s)-out-of-( m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive ( r, s)-out-of-( m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.