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.

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 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.

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.

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.

This paper studies consecutive-2 systems on trees. We show that given a set of probability values, the optimal assignment for binary trees depends on the particular values, whereas for k−regular trees, which are closely related to binary trees, it depends only on their relative ordering. In our proof, we introduce the notion of first-term-invariance, which might have further applications. We also show that the problem of computing the failure probability of consecutive-2 system is #P-complete in general, though the problem on trees is shown to be strongly polynomial.

We address the problem of assigning multiple copies of n independently developed versions of a program to a set of m(m > n) possibly heterogeneous processors to maximize system reliability. This problem is viewed as a partition and assignment problem. We first partition the set of processors into n clusters or subgroups. A program version is then assigned to be executed on all the processors in the cluster. This means that each processor in the cluster will execute a copy of the assigned version. The cluster's unreliability is the probability of failure of all its processors. Component i of this system is composed of the copies of version i and the assigned cluster of processors.

This paper explores a new, more optimal policy for planning terminal bays for the provisioning of circuits in a telephone operating company. This new policy regularly monitors the terminal bay inventory and orders K units when the spare reaches a certain parameter, R. For particular office characteristics the optimal (R, K) pair which minimizes cost is determined, subject to a performance constraint. A probabilistic circuit demand model is formulated and a combination of simulation and analytic approximation is used to calculate the cost and performance measure. In order to prove that the new policy is more economical, the new policy is compared, via another simulation, to the present policy of a typical telephone operating company.

Two procedures for the group-testing problem based on the Shannon-entropy criteria are proposed. The model considered is that the N units are realizations of N Bernoulli independent and identically distributed (i.i.d.) chance variables with common, known probability q of an arbitrary unit being good and p =1 – q of it being defective. Both the algorithms introduced have low design complexity and yet provide near-optimal result. For N ≤ 5, one of the procedures introduced is optimal for selected values of q.