We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

We introduce a new class of lifetime distributions exhibiting a notion of positive ageing, called the ℳ-class, which is strongly related to the well-known ℒ-class. It is shown that distributions in the ℳ-class cannot have an undesirable property recently observed in an example of an ℒ-class distribution by Klar (2002). Moreover, it is shown how these and related classes of life distributions can be characterized by expected remaining lifetimes after a family of random times, thus extending the notion of NBUE. We give examples of ℳ-class distributions by using simple sufficient conditions, and we derive reliability bounds for distributions in this class.

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

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.

In this paper we provide new results about stochastic comparisons of the excess lifetime at different times of a renewal process when the interarrival times belong to several ageing classes. We also provide a preservation result for the new better than used in the Laplace transform order ageing class for series systems.

Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.

We prove that if the renewal function M(t) corresponding to a life distribution F is convex (concave) then F is NBU (NWU), and hence answer two questions posed by Shaked and Zhu (1992). Moreover, based-on the renewal function, some characterizations of the exponential distribution within certain classes of life distributions are given.

Suppose that a device is subjected to shocks governed by a counting process N = {N(t), t ≧0}. The probability that the device survives beyond time t is then H̄(t)=Σk=0∞Q̄ℙ[N(t)=k], where Q̄k is the probability of surviving k shocks. It is known that H is NBU if the interarrivals Uk, ∊ ℕ+, are independent and NBU, and Q̄k+j ≦ Q̄k· Q̄j holds whenever k, j ∊ ℕ. Similar results hold for the classes of the NBUE and HNBUE distributions. We show that some other ageing classes have similar properties.

Age and block replacement policies are commonly used in order to reduce the number of in-service failures. The focus in this paper is on the block replacement policies, about which relatively less is known than age replacement policies. Several new results which connect the properties of block replacement policies with the properties of the corresponding renewal function and the excess lifetimes are obtained. Some applications and the relationships between these new results and some known results are included.

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

We establish a conservation law for G/G/1 queues with any work-conserving service discipline using the equilibrium equations, also called the basic equations. We use this conservation law to prove an extremal property of the first-come firstserved (FCFS) service discipline: among all service disciplines that are work-conserving and independent of remaining service requirements for individual customers, the FCFS service discipline minimizes [maximizes] the mean sojourn time in a G/G/1 queue with independent (but not necessarily identical) service times with a common mean and new better [worse] than used (NBUE[NWUE]) distributions. This extends recent results of Halfin and Whitt (1990), Righter et al. (1990) and Yamazaki and Sakasegawa (1987a,b). In addition we use the conservation law to obtain an approximation for the mean queue length in a GI/GI/1 queue under the processor-sharing service discipline with finite degree of multiplicity, called LiPS discipline. Several numerical examples are presented which support the practical usefulness of the proposed approximation.

In this paper, we study a similar replacement model in which the successive survival times of the system form a process with non-increasing means, whereas the consecutive repair times after failure constitute a process with non-decreasing means. The system is replaced at the time of the Nth failure since the installation or last replacement. Based on the long-run average cost per unit time, we determine the optimal replacement policy N∗ and the maximum of the long-run average reward explicitly. Under additional conditions, the policy N∗ is even optimal among all replacement policies.

It is shown that among all work-conserving service disciplines that are independent of the future history, the first-come-first-served (FCFS) service discipline minimizes [maximizes] the average sojourn time in a G/GI/1 queueing system with new better [worse] than used in expectation (NBUE[NWUE]) service time distribution. We prove this result using a new basic identity of G/GI/1 queues that may be of independent interest. Using a relationship between the workload and the number of customers in the system with different lengths of attained service it is shown that the average sojourn time is minimized [maximized] by the least-attained-service time (LAST) service discipline when the service time has the decreasing [increasing] mean residual life (DMRL[IMRL]) property.

The concept of positive ageing describes the adverse effects of age on the lifetime of units. Various aspects of this concept are described in terms of conditional probability distributions of residual lifetimes, failure rates, equilibrium distributions, etc. In this paper we further analyse this concept and relate it to stochastic dominance of first and higher orders. In the process we gain many insights and are able to define several new kinds of ageing criteria which supplement those existing in the literature.

Conditions are given on a process so that a shock model governed by the process will have nbue survival time. A related condition is given so that survival will be NBU and it is shown that a pure birth process satisfies this condition. A result of A-Hameed and Proschan is obtained as a corollary.

Exponential bounds for the stationary waiting-time distribution of the type ae–θt are considered. These bounds are obtained by the use of Kingman's method of ‘integral inequalities’. Approximations of Θ and a are given which are useful especially if the service and/or inter-arrival time distribution functions are NBUE or NWUE.