A Note on Warm Standby System

TL;DR

This paper analyzes conditions under which warm standby systems maintain more surviving standby components at the time of the kth failure, using stochastic orders to compare different system configurations.

Contribution

It introduces a stochastic comparison framework for warm standby systems based on various stochastic orders at the time of the kth failure.

Findings

Provides conditions for stochastic dominance of standby components

Uses multiple stochastic orders for comparison

Enhances understanding of system reliability at failure times

Abstract

In this note, we investigate, under what circumstances, a warm standby system (formed by n active components and m warm standby components) has more number of surviving warm standby components than another similar system at the time of $k$th failure of the active component of the respective system. The number of such components being random, the comparison has been done with respect to different stochastic orders, viz. usual stochastic order, hazard rate and reversed hazard rate orders, and likelihood ratio order.