Reliability of Combined <i>m</i>-Consecutive-<i>k</i>-out-of-<i>n</i>:F and Consecutive <i>k_c</i>-out-of-<i>n</i>:F Systems

Abstract

A combined m-consecutive-k-out-of-n:F & consecutive-k(c)-out-of-n:F system consists of linearly ordered components, and fails iff there exist at least k(c) consecutive failed components, or at least m non-overlapping runs of k consecutive failed components. This structure has applications for modeling systems such as infrared detecting and signal processing, and bank automatic payment systems. In this paper, we derive a combinatorial equation for the number of path sets of this structure including a specified number of working components. This number is used to derive a reliability function, and a signature based survival function formulae, for the system consisting of i.i.d. components. We also obtain a combinatorial equation for the reliability of a system with Markov dependent components.