Mean Residual Life Function, Associated Orderings and Properties

Abstract

Although the shape of the failure rate function plays an important role in repair and replacement strategies, the mean residual life ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mrl</i> ) function is more relevant as the latter summarizes the entire residual life function, whereas the former considers only the risk of instantaneous failure. In this paper, we give a review of the different partial ordering results related to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mrl</i> order, and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">proportional mrl</i> model, with some characterization results. Some properties of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">shifted mrl</i> order have been developed along with the closure of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">shifted mrl</i> order under mixture-type operation. We show that redundancy at the component level is not superior to that at the system level, even when the lifetimes of the original and the spare components are iid, although it is well known that the result holds for usual stochastic order. The <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">shifted mrl</i> order is characterized in terms of the DMRL class. Some other characterizations are also presented here followed by the closure of the order under Poisson shock model.