New Bounds for All Types of Multi-state Consecutive $k$ -Out-of- $r$ -From- $n$ : F System Reliability

A multi-state consecutive k-out-of-r-from-n: F system (named MS C(k,r,n:F)) comprises of n linearly ordered components. The MS C(k,r,n:F) and its components have more than 2 states: 0, 1,
_⋯
,H. The state of MS C(k, r, n:F) is less than j if any r consecutive components contains k
_l
or more, for all j ≤ l ≤ H. This system is a model for a large number of applications. The existing methods for system-reliability evaluation are suitable for some special cases only. In this paper, we suggest bounds for MS C(k, r, n:F) reliability. These bounds are suitable for all system types whether in case of equal or unequal components probabilities. The suggested bounds are examined by previously published examples, when available. Also, illustration examples for the new bounds and system modelling are presented. Furthermore, we studied the cases that make the proposed bounds are sharp.