Suppose that a system consists of n independent and identically distributed components and that the life lengths of the n components are Xi, i = 1, …, n. For k ∈ {1, …, n - 1}, let X(k)1, …, X(k)n-k be the residual life lengths of the live components following the kth failure in the system. In this paper we extend various stochastic ordering results presented in Bairamov and Arnold (2008) on the residual life lengths of the live components in an (n - k + 1)-out-of-n system, and also present a new result concerning the multivariate stochastic ordering of live components in the two-sample situation. Finally, we also characterize exponential distributions under a weaker condition than those introduced in Bairamov and Arnold (2008) and show that some special ageing properties of the original residual life lengths get preserved by residual life lengths.

In this paper we identify a monotonicity in all countable-state-space reversible Markov chains and examine several consequences of this structure. In particular, we show that the return times to every state in a reversible chain have a decreasing hazard rate on the subsequence of even times. This monotonicity is used to develop geometric convergence rate bounds for time-reversible Markov chains. Results relating the radius of convergence of the probability generating function of first return times to the chain's rate of convergence are presented. An effort is made to keep the exposition rudimentary.