We consider time-inhomogeneous Markov chains on a finite state-space, whose transition probabilitiespij(t) = cijε(t)Vij are proportional to powers of a vanishing small parameter ε(t). We determine the precise relationship between this chain and the corresponding time-homogeneous chains pij= cijε(t)vij, as ε ↘ 0. Let {} be the steady-state distribution of this time-homogeneous chain. We characterize the orders {ηι} in = θ(εηι). We show that if ε(t) ↘ 0 slowly enough, then the timewise occupation measures βι := sup { q > 0 | Prob(x(t) = i) = + ∞}, called the recurrence orders, satisfy βi — βj = ηj — ηi. Moreover, : = { ηι|ηι = minj} is the set of ground states of the time-homogeneous chain, then x(t) → . in an appropriate sense, whenever η(t) is “cooled” slowly. We also show that there exists a critical ρ* such that x(t) → if and only if = + ∞. We characterize this critical rate as ρ* = max.min min max. Finally, we provide a graph algorithm for determining the orders [ηi] [βi] and the critical rate ρ*.

We consider stopping times associated with sequences of non-negative random variables and Poisson processes. With sufficient conditions on the dependence property between the sequences of non-negative random variables (or the Poisson processes) and the stopping times, we develop easily computable stochastic bounds for the stopping times. We use these bounds to develop approximations within the class of generalized phase-type distribution functions for arbitrary distribution functions. The computational tractability of generalized phase-type distribution functions facilitate the computational analysis of complex stochastic systems using these approximate distribution functions. Applications of these approximations to renewal processes are illustrated.

Consider the problem in which impatient customers have to be routed to or scheduled from a set of parallel homogeneous service stations with finite buffer capacities. When the service times and the deadlines of customers are exponentially distributed, we identify the control policies that stochastically minimize the total number of losses by any time t. We study systems with deadlines to the beginning or to the end of service. The results regarding the individual types of losses, i.e., losses due to deadline misses and losses due to buffer overflow, vary across different systems. In all cases we state explicitly the properties and the limitations of the optimal control policies.

This article considers separation for a birth-death process on a finite state space S = [1,2,…, N]. Separation is defined by si(t) = 1 – minj∈sPij(t)/πj, as in Fill [5,6], where Pij(t) denotes the transition probabilities of the birth-death process and πj the stationary probabilities. Separation is a measure of nonstationarity of Markov chains and provides an upper bound of the variation distance. Easily computable upper bounds for si-(t) are given, which consist of simple exponential functions whose parameters are the eigenvalues of the infinitesimal generator or its submatrices of the birth-death process.

Using the Laplace transform, we characterize, by means of necessary and sufficient conditions, the property that two life distributions are ordered in the sense of stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

In 1989, Boland, Proschan, and Tong [2] introduced the notion of criticality ranking among nodes and developed a procedure for obtaining an optimal assignment of components in coherent systems. In this article we obtain characterizations of the criticality ranking in terms of minimal cut (path) sets for coherent systems. Furthermore, utilizing the characterizations, it is shown that the criticality ranking defined by Boland et al. [2] is consistent with the cut-importance ranking introduced by Butler in 1979 [4]. A relationship between the criticality ranking and the well-known and widely used Birnbaum reliability importance measure is also derived.

Consider a system consisting of a number of modules. The probability that a particular module is among the ones that have already failed by the time of system failure can be used as a measure of the importance of that module. In El-Neweihi and Sethuraman [6], this probability was called the role of a module in the failure of a system and its properties were developed. In this article we propose the number of failed modules from among a particular subgroup of modules as the basis for measuring the role of that subgroup of modules. We study some monotonicity properties of the distribution of that random variable, in some second order r-out-of-k systems. We illustrate the possible extensions of these results to multistate systems. Yet another measure of the role of a particular subgroup of modules can be based on the number of failed components from this subgroup by the time of system failure. We derive some monotonicity properties of the expected value of the number of such failed components.

For n-dimensional survival functions, we study some probabilistic aspects of the Schur-constant property. The latter is of interest in that it extends the “lack-of-memory” property in a Bayesian context. Some general facts are studied in detail, and related results about interdependence, aging, and extendibility are presented.

Motivated by stochastic allocation problems, we study sufficient conditions that imply a general class of stochastic arrangement inequalities. This class includes many known and useful arrangement inequalities in the literature. Some illustrative examples in reliability theory are also given.

To date, research in shock models has been primarily concerned with the classpreserving properties of certain shock and damage kernels. This article focuses on the order-preserving properties of those kernels. We show that they preserve stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

Techniques of exploratory data analysis are used in the construction of simple approximations for the percentage points of the chi-square distribution. The approximations are direct functions of the degrees of freedom and the tail probability. They are more accurate than Hoaglin's approximations.

In asynchronous transfer mode (ATM), networks, buffering is required to reduce cell loss and increase bandwidth utilization by the bursty traffic. Several types of traffic with different performance requirements on delay and loss will share buffer resources. We focus on the fully shared buffer with push-out and a randomized service priority rule. Some conservation relations are obtained which can be used to tune the parameters of the buffering and service disciplines.

The following corrections for “A Globally Gated Polling System with Server Interruptions, and Applications to the Repairman Problem” by O.J. Boxma, J.A. Weststrate, and U. Yechiali, which appeared in Volume 7, Number 2, of Probability in the Engineering and Informational Sciences, missed the press date.

p. 192, (3.3): Xi should be Xi

p. 194, (3.12):should be

p. 195, (4.1): N should be n in the five summations over n (15x)

p. 195, (4.2): should be moved to the next line (after the multiplication sign)

p. 195, below (4.3): Add: so that EY1m = EXm = λmEC

p. 195, (4.4): Add: for m < n

p. 195, I. –3: Add: for m < n

p. 196, (4.5): Add: for m < n

p. 196, I. 5: Add: Again cov(Xm, Xn) = λm λn var(C), independent of the server interruption process.

p. 196, (4.8): m ≤ n should be m < n

p. 197, 1st formula: [λi should be

p. 199, I. 6: 5.2 The should be The

p. 199, I. 14: 5.3 The should be The

p. 200, I. 10: [E should be E(

p. 204: The last line of subsection 6.2 should be changed to: Moreover, when sj → 0, the index rules (6.2) and (6.4) approach the well-known cμ rule in both the static and dynamic optimization.

p. 207, I. –1: (to appear) should be: 85–90

p. 208, I. 1: 92 should be 93 (2×)

p. 208, I. 5: (to appear) should be 35: 187–208