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.

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.

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.

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.

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.

A serial production line with an arbitrary number of stations is analyzed using the holding-time model. In contrast to Markovian state models, the holding-time model does not require the service times to be exponentially distributed. The distribution of the interdemand time at the first station is obtained as a function of the distribution of station service times which are of a general nature. The throughput rate of the line is the reciprocal of the mean interdemand time. For service times of phase type, two numerical solution methods, one exact and the other approximate, are presented. Numerical results are given for up to ten stations.

This paper considers a dam (or storage) model of the GI/G/I type with a finite capacity K. An arriving input being larger than the unfilled capacity of the dam causes an overflow where the excess amount is lost. Important performance measures for this system are the overflow probability and the long-run fraction of input that is lost. We give asymptotic expansions for these measures for large K both for the case of a load factor less than 1 and for the case of a load factor larger than 1. Also, related results are obtained for the impatient customer model of the M/G/l type.

A sequential procedure to select optimal prices based on maximum likelihood estimation is considered. Asymptotic properties of the pricing scheme and the concommitant estimation problem are examined. For small sample sizes, simulation results show that the proposed procedure has high efficiency relative to the best procedure when the parameter is known.