Partial orderings under cumulative damage shock models

Two devices are subjected to common shocks arriving according to two identical counting processes. Let and denote the probability of surviving k shocks for the first and the second device, respectively. We find conditions on the discrete distributions and in order to obtain the failure rate order (FR), the likelihood ratio order (LR) and the mean residual order (MR) between the random lifetimes of the two devices. We also obtain sufficient conditions under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.