In this paper we propose a new approach for estimating the transition probabilities and mean occupation times of continuous-time Markov chains. Our approach is to approximate the probability of being in a state (or the mean time already spent in a state) at time t by the probability of being in that state (or the mean time already spent in that state) at a random time that is gamma distributed with mean t.

A semicrystalline polymer fiber is a composite material consisting of difficult-to-deform crystals joined by more easily deformed and more easily broken amorphous materials. The failure process begins at the atomic level in the amorphous regions where random thermal fluctuations cause, at some time, a molecule to slip relative to other molecules or to rupture at one of its atomic bonds. The frequency of such random events is greatly enhanced by small increases in stress. As molecules slip or rupture, neighboring molecules become overloaded, thus increasing their failure rates. Such molecule failures accumulate locally and give rise to growing microcracks, although the exact kinetic mechanisms are not well understood. These growing minute cracks are the irreversible changes in the microstructure of the material that ultimately lead to macroscopic failure of the fiber.

The relaxation time for an ergodic Markov process is a measure of the time until ergodicity is reached from its initial state. In this paper the relaxation time for an ergodic truncated birth-death process is studied. It is shown that the relaxation time for such a process on states {0,1, …, N} is the quasi-stationary exit time from the set {,2, …, N{0,1,…, N, N + 1} with two-sided absorption at states 0 and N + 1. The existence of such a dual process has been observed by Siegmund [15] for stochastically monotone Markov processes on the real line. Exit times for birth- death processes with two absorbing states are studied and an efficient algorithm for the numerical evaluation of mean exit times is presented. Simple analytical lower bounds for the relaxation times are obtained. These bounds are numerically accessible. Finally, the sensitivity of the relaxation time to variations in birth and death rates is studied.

Investors can increase their payoff by acquiring information on rates of return. The value of the information is greater, the greater the amount to be invested. Therefore, information purchased and consequently the expected rate of return increases with initial wealth, and the distribution of final wealth is more unequal than that of initial wealth.

The transient crystal size distribution (CSD) in a continuous mixed suspension, mixed product removal crystallizer has been modeled through a stochastic approach. Effects of the seed size distribution and size-dependent growth rate on both the transient and steady-state CSD have been investigated. It has been found that the seed size distribution causes a maximum in the steady-state CSD and the size-dependent growth rate results in an upward curvature at the lower end in a semilogarithmic plot of steady-state CSD against crystal size. Under appropriate conditions, the mean CSD of the present stochastic model reduces to the results predicted by the corresponding deterministic population balance model.

A renewal process [N(t), t ≥ 0] with interarrival times Xi, for i ≥ 1 and renewal function m (t) is considered. Let Gn, λ denote the gamma distribution with parameters n and λ–that is, dGn, λ(x) = λε–λx(λx)n-1/(n – 1)

In Section 1 we show how m(t) can be approximated by f m(s) dGn, n/t(s). In addition, we show that these approximations constitute an increasing sequence of lower bounds when the interarrival distribution has the decreasing failure rate property. In Section 2 we show how the integrated renewal function can be approximated in a similar fashion by a decreasing sequence of upper bounds. In Section 3 we consider the problem of approximating the residual life (also called excess life) and the renewal age distribution and their means, and in Section 4 we consider the distribution of N(t). Finally, in Section 5 we remark on the relationship between our approximations and the Feller technique for inverting a Laplace transform.

We show that the time to extinction in critical or subcritical Galton–Watson and Markov branching processes has the antiaging property of log-convex density, and therefore has the decreasing failure rate (DFR) property of reliability theory. Apart from providing new insights into the structure of such extinction time distributions, which cannot generally be expressed in a closed form, a consequence of our result is that one can invoke sharp reliability bounds to provide very simple bounds on the tail and other characteristics of the extinction time distribution. The limit distribution of the residual time to extinction in the subcritical case also follows as a direct consequence. A sequel to this paper will further consider the critical case and other ramifications of the log-convexity of the extinction time distribution.

A model for a system whose state changes continuously with time is introduced. It is assumed that the system is modeled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process and increases the state of the system by a random amount if the state is below a threshold α. Explicit expressions are deduced for the distribution function of X(t), the state of the system at time 1, if X(t) ≤ α and for the Laplace transform of the density of X( t). The stationary case is examined in detail.

An interpretation of log-concavity and log-convexity as aging notions is given in this paper. It imitates a stochastic ordering characterization of the NBU (new better than used) and the NWU (new worse than used) notions but stochastic ordering is now replaced by the likelihood ratio ordering. The new characterization of log-concavity and log-convexity sheds new light on these properties and enables one to obtain intuitively simple proofs of the log-convexity and log-concavity of some first passage times of interest in branching processes and in reliability theory.

We suggest a simple algorithm for Monte Carlo generation of uniformly distributed variables on a compact group. Example include random permutations, Rubik's cube positions, orthogonal, unitary, and symplectic matrices, and elements of GLn over a finite field. the algorithm reduces to the “standard” fast algorithm when there is one, but many new example are included.

Diversity of bugs or faults in a software system is a factor contributing to software unreliability which has not yet been appropriately emphasized. This paper is written with the intention of demonstrating the impact of fault diversity on the time to detection of software bugs. A new discrete software reliability model based on the multinomial distribution is introduced. It is shown that for models of this type, the more diverse the fault probabilities are, the longer (stochastically) it takes to detect or eliminate any n faults, while the smaller (stochastically) will be the number of faults detected or eliminated during a given amount of time (or during a given number of inputs to the system). The impact of fault diversity is also demonstrated for the Jelinski–Moranda model.

A consecutive-2 graph is a graph where each vertex is associated with a failure probability and the graph is considered failed if any two adjacent vertices both fail. Recently, the problem of computing reliability for general consecutive-2 graph was shown to be #P-complete while polynomial algorithms exist for trees. In this paper, we give a linear time algorithm for a class of graphs including forests and cycles.

For a given set of failure probabilities qi, the assignment of qi to the vertices of a given graph is optimal if it maximizes the reliability of that graph. It is known that optimal assignments for trees require messy computations while linear algorithms exist for lines and stars. In this paper, we prove that the optimal reliability of any n−tree is bounded between those of an n−line and an n−star.

Resources are to be allocated sequentially to activities to maximize the total expected return, where the return from an allocation is the product of the value of the resource and the value of the activity. The set of activities and their values are given ahead of time, but the resources arrive according to a Poisson process and their values are independent random variables that are observed upon arrival. It is assumed that either there is a single random deadline for all activities, which is the same as discounting the returns, or the activities have independent random deadlines. The model has applications machine scheduling, packet switching, and kidney allocation for transplant. It is known that the optimal policy in the discounted case has a very simple form that does not depend on the activity values. We show that this is also true when the deadlines are independent and in this case the solution can expressed in terms of solutions to single activity models. These results also hold when there are batch arrivals of resources. The effects of pooling separate identical systems with a single activity into a combined system is investigated for both models. When activities have independent deadlines it is optimal to reject a resource in the combined system if and only if it is optimal to reject it in the single activity system. However, when returns are discounted, it is sometimes optimal to accept a resource in the combined system that would be rejected in the single activity system.

Uniform distributions on complicated combinatorial sets can be simulated by the Markov chain method. A condition is given for the simulations to be accurate in polynomial time. Similar analysis of the simulated annealing algorithm remains an open problem. The argument relies on a recent eigenvalue estimate of Alon [4]; the only new mathematical ingredient is a careful analysis of how the accuracy of sample averages of a Markov chain is related to the second-largest eigenvalue.

For nonnegative random variables, the weighted distributions have been compared with the original distributions with the help of partial orderings of probability distributions. Bounds on the moments of the weighted distributions have been obtained in terms of the moments of the original distributions for some nonparametric classes of aging distributions.

A model for random trees is given which provides an embedding of the uniform model into an exponential family whose natural parameter is the expected number of leaves. The model is proved to be analytically and computationally tractable. In particular, a central limit theorem (CLT) for the number of leaves of a random tree is given which extends and sharpens Rényi's CLT for the uniform model. The method used is general and is shown to provide tractable exponential families for a variety of combinatorial objects.

We consider the impact that the introduction of dependent component failures has on the computational difficulty of system reliability analysis. We develop general strategies for computing system reliability when components are subject to dependent failures. In the process, we identify a significant number of cases where computing the system reliability in the presence of dependent failures is not significantly harder than writing down the joint probability distribution and computing the system reliability when the components fail independently.

An inventory whose stock decreases linearly with time is considered. The inventory may be replenished at the instants at which a deliveryman arrives provided that the level of the inventory does not exceed a certain threshold; deliveries are made according to a Poisson process. A partial differential equation for the distribution function of the level of the inventory is solved to yield a formula for the corresponding Laplace–Stieltjes transform. The evaluation of the transform is discussed and explicit results are obtained for the stationary case.

Launer [6] introduced the class of life distributions having decreasing (increasing) variance residual life, DVRL (IVRL). It is shown that the DVRL (IVRL) distributions are intimately connected to the behavior of the mean residual life function of the equilibrium distribution. Some counter examples are presented to demonstrate the lack of relationship between DVRL (IVRL) and NBUE (new better than used in expectation) (NWUE; new worse than used in expectation) distributions. Finally, we obtain bounds on moments and survival functions of DVRL (IVRL) distributions. These bounds turn out to be improvements on the previously known bounds for decreasing (increasing) mean residual life (DMRL (IMRL)) distributions.