On stochastic comparisons of residual life time at random time

Let X1,X2,Θ and Θ′ be independent non-negative random variables. The residual life of Xi at random time Θ, that is, XiΘ=Xi−Θ∣Xi>Θ is considered. Some sufficient conditions which lead to the likelihood ratio ordering, the failure rate ordering, the reverse failure rate ordering and the mean residual life ordering between X1Θ and X2Θ are obtained and an application in queuing theory is explained. A set of conditions which lead to the same stochastic orderings between X1Θ and X1Θ′ are also derived.