Preservation of some partial orderings under Poisson shock models

Suppose each of the two devices is subjected to shocks occurring randomly as events in a Poisson process with constant intensity ℝi>l. Let nPk denote the probability that the first device will survive the first k shocks and let nQk denote such a probability for the second device. Let nF(t) and nG(t) denote the survival functions of the first and the second device respectively. In this note the authors show that some partial orderings, namely likelihood ratio ordering, failure rate ordering, stochastic ordering, variable ordering and mean residual-life ordering between the shock survival probabilities nPk and nQk are preserved by the corresponding survival functions nF and nG.