On the Mean Residual Life Function of Coherent Systems

We consider a coherent structure consisting of
n
components having the property that if it is known that at most
r
components (
r
<
n
) have failed, the system is still operating with probability 1. Some examples of the systems having this property are (
n
-
k
+ 1)-out-of-
n
, some parallel-series, and some series-parallel structures. Depending on the structure, and the number of active components of the coherent systems at time
t
, the mean residual life function of the system is studied, by several authors. This paper investigates more properties of the mean residual life function of the coherent systems sharing the described property. We will show that, when the components of the system have increasing failure rate, the mean residual life function of the system is decreasing in time. Several examples, and illustrative graphs are also provided.