Some Aging Properties of the Residual Life of$k$-out-of-$n$Systems

The
$k$
-out-of-
$n$
structure is a very popular type of redundancy in fault-tolerant systems. It has been applied in industrial, and military systems. In this paper, we investigate the general residual life (GRL) of a
$(n-k+1)$
-out-of-
$n$
system with i.i.d. components, given that the total number of the failures of components is less than
$l-1(1leq l≪ kleq n)$
at time
$tgeq0$
. It is shown that the GRL is decreasing in
$l$
in terms of the likelihood ratio order; Behavior of IFR, and NBU of life distributions are discussed in terms of the monotonicity of GRL. Finally, comparison of the GRL of two
$(n-k+1)$
-out-of-
$n$
systems are conducted given that the lifetime of their components are assumed to be ordered in the hazard rate order.