Reliability of Combined $m$-Consecutive- $k$-out-of- $n$:F and Consecutive $k_{c}$-out-of- $n$:F Systems

A combined m-consecutive-k-out-of-n:F & consecutive k<;sub>;c<;/sub>;-out-of-n:F systems consists of linearly ordered components, and fails iff there exist at least k<;sub>;c<;/sub>; consecutive failed components, or at least m nonoverlapping 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.