Improved Ordering Results for Fail-Safe Systems with Exponential Components

Abstract

Let X_{2: n} and Y_{2: m} be the second order statistics from n independent exponential variables with hazards λ₁, …, λ_n, and an independent exponential sample of size m with hazard change to λ, respectively. When m ⩾ n, we obtain necessary and sufficient conditions for comparing X_{2: n} and Y_{2: m} in mean residual life, dispersive, hazard rate, and likelihood ratio orderings based on some inequalities between λ_i’s and λ. The established results show how one can compare an (n − 1)-out-of-n system consisting of heterogeneous components with exponential lifetimes with any (m − 1)-out-of-m system consisting of homogeneous components with exponential lifetimes.

Keywords:

• Bounds
• Exponential distribution
• Fail-safe systems
• Mean residual life ordering
• Hazard rate ordering
• Likelihood ratio ordering
• Order statistics
• Usual stochastic ordering

Mathematics Subject Classification:

• Primary 90B25; Secondary 60K10.