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

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.