We consider a series system with p components where the failure rate of each component depends on the residual number of defects present. Successive tasks are given to the system with each task completion time independent of each component failure time. Based on the outcomes over a fixed testing period, the asymptotic properties of the estimators of the component parameters, task completion parameters, and eventual system reliability are obtained.

Formulas are derived for moments of the number of refused customers in a busy period for the M/GI/1/n and the GI/M/1/n queueing systems. As an interesting special case for the M/GI/1/n system, we note that the mean number is 1 when the mean interarrival time equals the mean service time. This provides a more direct argument for a result given in Abramov [1].

We obtain necessary and sufficient moment conditions for exponentiality among all class ℒ aging distributions. This result, which solves an open problem, is derived as a consequence of the answer to a more general question of investigating conditions for stochastic equivalence of Laplace ordered nonnegative r.v.'s and its ramifications.

Components which have bathtub shapes are reasonable candidates for burn-in in the sense that if these components are exposed to standard or elevated operating conditions for a short period of time, they tend to improve. In this paper we study the principle which states that burn-in should occur at or before the point at which a bathtub-shaped failure function starts increasing. To this end we develop three sign change results which characterize bathtub-shaped failure rates. These results are used to show that if a failure rate function is bathtub-shaped, then various other related functions have bathtub or upside-down bathtub shape. Finally, we give a framework for determining when the above principle holds.

The GI/M/1 queueing system was long ago studied by considering the embedded discrete-time Markov chain at arrival epochs and was proved to have remarkably simple product-form stationary distributions both at arrival epochs and in continuous time. Although this method works well also in several variants of this system, it breaks down when customers arrive in batches. The resulting GIX/M/1 system has no tractable stationary distribution. In this paper we use a recent result of Miyazawa and Taylor (1997) to obtain a stochastic upper bound for the GIX/M/1 system. We also introduce a class of continuous-time Markov chains which are related to the original GIX/M/1 embedded Markov chain that are shown to have modified geometric stationary distributions. We use them to obtain easily computed stochastic lower bounds for the GIX/M/1 system. Numerical studies demonstrate the quality of these bounds.

Let [script S] be a system that can be in one of two states, up or down. We can interpret the up state as the working state and the down state as the nonworking state for system [script S]. Assume the system [script S] is up at time t = 0. Denote the elapsed time from zero until [script S] enters down state by U1, and the elapsed time from then until [script S] is up again by D1. The interval [0,U1 + D1) is called cycle 1, and, at that point, cycle 2 starts and the sequence is repeated so on and so forth. In general, the up time and down time in the jth cycle are denoted by Uj and Dj, respectively, for j ≥ 1. We can introduce a binary process {X(t), t ≥ 0} to describe the state of [script S] at time t using X(t) = 1(0) to indicate [script S] is up (down). Many problems require finding the probability that [script S] is up at time t, that is, P(X(t) = 1). However, an even more interesting question to answer is what the probability is that [script S] is in the up state in an interval of length w starting at time t, that is, P(X(s) = 1, t ≤ s ≤ t + w). Due to the complexity of this problem, no explicit expressions are available for most systems even in the case where (Uj, Dj), j ≥ 1 are i.i.d. Fortunately, in practice engineers are more interested in the long-run properties of the above two probabilities.

We are interested in E [N], the mean time until the most recent k values of a sequence of independent and identically distributed random variables exceeds a specified constant. Using recent results, we present a simulation procedure for determining E [N]. These results are also used to obtain upper and lower bounds for E [N]. These bounds, however, are in terms of a quantity ω that is not easily calculated. A recursive procedure for evaluating ω when the data distribution is Bernoulli is given. Efficient simulation procedures for estimating ω in the cases of normal and exponential population distributions are also presented, as is a Markov chain monte carlo procedure when the distribution is general.

The move-to-front rule is applied on a model where several records are accessed each time. The records will then be placed in the front positions randomly or with the former relative order between themselves preserved. Equilibrium distributions are explored. Comparison of expected stationary search costs is carried out.

We give a new bound on the spectral gap of an ergodic time reversible Markov chain in term of the conductance of the chain.

In this paper, we provide counterexamples to a conjecture, made by Miyazawa and Tijms (1993), on the upper and lower bounds for the loss probability in finite-buffer queues.

Fill and Holst conjectured for the move-to-front rule that the probability that the search time is greater than c will be Schur concave in the stationary distribution for any value of c. This paper disproves the conjecture but proves some conclusions that would be implied by the conjecture.

This note presents an example indicating for denumerable Markov decision chains that nonexistence of the Laurent expansion of the α-discounted rewards in α = 1 implies also nonexistence of the Puiseux expansion.

We consider a K-node queueing system sharing a setup server. Each node has a node server, a waiting position, and a service position, and it behaves as an M/G/1/2 type queue except that each job in the waiting position requires a setup by the setup server to move to the service position. The service discipline of the setup server is nonpreemptive work-conserving random selection. The setup times have a common exponential distribution. The main purpose of this paper is to derive the stationary distribution for the K-node system. For each node, we construct a corresponding setup server queue (CSQ). The stationary distribution of the K-node system is given by a product form of the stationary distributions of CSQs. This result enables us to obtain the stationary distribution of a K-node system by analyzing individual CSQs.

Discrete-time queues are frequently used in telecommunication networks. This paper proposes a new method which is based only on probabilistic arguments. It derives the distributions of numbers of customers in the discrete-time systems Geom/G/1 and Geom/G/1/M. Though the derivation does not involve use of the transforms, the transforms may be obtained, if desired. Another advantage of this is that the numerical results obtained are stable as there are no negative signs involved in summations. Further, the method can be easily used to solve more complex problems in discrete- and continuous-time queues.

We consider the problem of estimation of the standard deviation σ and the process capability indices Cp and Cpm for a normally distributed process. The problem is addressed with the objective of obtaining optimal estimators under the classic criterion of minimum variance unbiased estimation, as well as under the Pitman measure of closeness.