Some Aging Properties of the Residual Life of<tex>$k$</tex>-out-of-<tex>$n$</tex>Systems

Abstract

The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> -out-of- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(n-k+1)$</tex> -out-of- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> system with i.i.d. components, given that the total number of the failures of components is less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l-1(1leq l≪ kleq n)$</tex> at time <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$tgeq0$</tex> . It is shown that the GRL is decreasing in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$l$</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(n-k+1)$</tex> -out-of- <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> systems are conducted given that the lifetime of their components are assumed to be ordered in the hazard rate order.