# Reliability and Failure Probability Functions of the m-Consecutive-k-out-of-n: F Linear and Circular Systems

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## Abstract

The *m*-consecutive-*k*-out-of-*n*: F linear and circular system consists of *n* sequentially connected components; the components are ordered on a line or a circle; it fails if there are at least *m* non-overlapping runs of consecutive-*k* failed components. This paper proposes the reliability and failure probability functions for both linearly and circularly *m*-consecutive-*k*-out-of-*n*: F systems. More precisely, the failure states of the system components are separated into two collections (the working and the failure collections); where each one is defined as a collection of finite mutual disjoint classes of the system states. Illustrative example is provided.

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