Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

It is known that for decreasing hazard rate (DHR) service times the foreground-background discipline (FB) minimizes the mean delay in the M/G/1 queue among all work-conserving and nonanticipating service disciplines. It is believed that a similar result is valid for increasing mean residual lifetime (IMRL) service times. However, on the one hand, we point out a flaw in an earlier proof of the latter result and construct a counter-example that demonstrates that FB is not necessarily optimal within class IMRL. On the other hand, we prove that the mean delay for FB is smaller than that of the processor-sharing discipline within class IMRL, giving a weaker version of an earlier hypothesis.

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to the nth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of the kth moment of the time of ruin in the classical risk model.

Suppose Y is the strength of a system subjected to a sequence of stresses X1, X2, X3, · ··. Let N(t) be the number of stresses occurring by time t and let T be the lifetime of the system. Then the survival distribution of is given by where and . In this note we aim at determining from the above model various non-parametric classes of life distributions which are of interest in reliability theory.

Suppose that a device is subjected to shocks and that denotes the probability of surviving k shocks. Then is the probability that the device will survive beyond t, where is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the pure birth shock model. In this paper we shall prove that is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).