In this paper we analyze an asymmetric 2 × 2 buffered switch, fed by two independent Bernoulli input streams. We derive the joint equilibrium distribution of the numbers of messages waiting in the two output buffers. This joint distribution is presented explicitly, without the use of generating functions, in the form of a sum of two alternating series of product-form geometric distributions. The method used is the so-called compensation approach, developed by Adan, Wessels, and Zijm.

We consider two single-server queues in tandem. The customers receive service at the first station and then they proceed to the second. There are no intermediate waiting positions and, so, the blocking phenomenon is observed. The arriving customers who find the first station unavailable (busy or blocked) behave like retrial customers. We consider the case of “constant retrial policy.” For such a system, we obtain the system state probabilities in steady state in terms of generating functions. We present also some interesting measures of the system performance and directly deduce results for the corresponding ordinary (without retrials) model. We consider, finally, the special case of exponential service times as an example.

In most conventional settings, the events caused by an external shock are initiated at the moments of its occurrence. In this paper, we study the new classes of shock models: (i) When each shock from a nonhomogeneous Poisson processes can trigger a failure of a system not immediately, as in classical extreme shock models, but with delay of some random time. (ii) When each shock from a nonhomogeneous Poisson processes results not in an ‘immediate’ increment of wear, as in classical accumulated wear models, but triggers its own increasing wear process. The wear from different shocks is accumulated and the failure of a system occurs when it reaches a given boundary. We derive the corresponding survival and failure rate functions. Furthermore, we study the limiting behavior of the failure rate function where it is applicable.

Control limit type policies are widely discussed in the literature, particularly regarding the maintenance of deteriorating systems. Previous studies deal mainly with stationary deterioration processes, where costs and transition probabilities depend only on the state of the system, regardless of its cumulative age. In this paper, we consider a nonstationary deterioration process, in which operation and maintenance costs, as well as transition probabilities “deteriorate” with both the system's state and its cumulative age. We discuss conditions under which control limit policies are optimal for such processes and compare them with those used in the analysis of stationary models.

Two maintenance models are examined: in the first (as in the majority of classic studies), the only maintenance action allowed is the replacement of the system by a new one. In this case, we show that the nonstationary results are direct generalizations of their counterparts in stationary models. We propose an efficient algorithm for finding the optimal policy, utilizing its control limit form. In the second model we also allow for repairs to better states (without changing the age). In this case, the optimal policy is shown to have the form of a 3-way control limit rule. However, conditions analogous to those used in the stationary problem do not suffice, so additional, more restrictive ones are suggested and discussed.

Let X1:n ≤ X2:n ≤ ··· ≤ Xn:n denote the order statistics of a set of independent and not necessarily identically distributed random variables X1,..., Xn. Under mild assumptions, it is shown that Xk−1:n−1 ≤lrXk:n for k = 2,..., n if X1 ≤lrX2 ≤lr ··· ≤lrXn and that Xk:n ≤lrXk:n−1 for k = 1,..., n − 1 if X1 ≥lrX2 ≥lr ··· ≥lrXn, where ≤lr denotes the likelihood ratio order. Concerning the mean residual life order (≤mrl), it is shown that Xn−1:n−1 ≤mrlXn:n if Xj ≤mrlXn for j = 1,..., n − 1. Two counterexamples are also given to illustrate that Xk−1:n−1 ≤mrlXk:n in this case is, in general, not true for k < n.

A set of stochastic jobs is to be processed on a single machine that is subject to breakdown. All jobs make progress as they are processed in the absence of machine breakdowns. However, breakdowns cause setbacks to (possibly) all jobs in the system, except those that have already been completed. With machines subject only to fairly mild restrictions on this “damage” process, we demonstrate the existence of a nonpreemptive policy that is optimal in the class of all preemptive ones.

We relate two variants of the inspection paradox by showing that both represent the length of a renewal interval that contains a chosen point, the difference being in the way that point is chosen. We show, in both cases, that the length of the interval is likelihood ratio larger than that of an ordinary renewal interval.

We examine the problem of optimally maintaining a stochastically degrading system using preventive and reactive replacements. The system's rate of degradation is modulated by an exogenous stochastic environment process, and the system fails when its cumulative degradation level first reaches a fixed deterministic threshold. The objective is to minimize the total expected discounted cost of preventively and reactively replacing such a system over an infinite planning horizon. To this end, we present and analyze a Markov decision process model. It is shown that, for each environment state, there exists an optimal threshold-type replacement policy. Additionally, empirical evidence suggests that, when the environment process is monotone, and the state-dependent degradation rates are totally ordered, the optimal threshold is monotone. Lastly, we derive closed-form bounds on the optimal thresholds.

In this paper we show how the permutation Monte Carlo method, originally developed for reliability networks, can be successfully adapted for stochastic flow networks, and in particular for estimation of the probability that the maximal flow in such a network is above some fixed level, called the threshold. A stochastic flow network is defined as one, where the edges are subject to random failures. A failed edge is assumed to be erased (broken) and, thus, not able to deliver any flow. We consider two models; one where the edges fail with the same failure probability and another where they fail with different failure probabilities. For each model we construct a different algorithm for estimation of the desired probability; in the former case it is based on the well known notion of the D-spectrum and in the later one—on the permutational Monte Carlo. We discuss the convergence properties of our estimators and present supportive numerical results.

We consider the problem of establishing the existence of stationary distributions for continuous-time Markov chains directly from the transition rates Q. Given an invariant probability distribution m for Q, we show that a necessary and sufficient condition for m to be a stationary distribution for the minimal process is that Q be regular. We provide sufficient conditions for the regularity of Q that are simple to verify in practice, thus allowing one to easily identify stationary distributions for a variety of models. To illustrate our results, we shall consider three classes of multidimensional Markov chains, namely, networks of queues with batch movements, semireversible queues, and partially balanced Markov processes.

A stochastic process {Xt(s)} is viewed as a collection of random variables parameterized by time (t) and the initial state (s). {Xt(s)} is termed spatiotemporally increasing and convex if, in a sample-path sense, it is increasing in s and t and satisfies a directional convexity property, which implies that it is increasing and convex in s and in t (individually) and is supermodular in (s, t). Simple sufficient conditions are established for a uniforniizable Markov process to be spatiotemporally increasing and convex. The results are applied to study the convex orderings in GI/M(n)/l and M(n)/G/1 queues and to solve the optimal allocation of a joint setup among several production facilities. For a counting process that possesses a stochastic intensity, we show that its spatiotemporal behavior can be characterized by its conditional intensity via a birth process.

In reliability theory, Cox's proportional hazard model is quite popular and widely used. In many situations, it is observed that failure rates under consideration are not proportional, rather they cross each other. In such situations, an alternative to Cox's proportional hazard model may be monotone hazard ratio model (provided the ratio exists). A notion of relative aging based on increasing hazard ratio was introduced by Kalashnikov and Rachev [19]. Sengupta and Deshpande [40] further explored this model and posited two other notions of relative aging based on increasing reversed failure rate ratio and increasing mean residual life ratio. In this study, for two life distributions, we derive sufficient conditions under which a life distribution ages faster than the other with respect to notions of relative aging described above. These sufficient conditions are easy to verify and can be used in practical applications where one is interested in studying relative aging of two life distributions. Applications of these results to relative aging of weighted distributions have also been illustrated. We also introduce a new relative aging ordering in terms of mean inactivity time order and study its fundamental properties.

It is well known that various characteristics in risk and queuing processes can be formulated as Markov renewal functions, which are determined by Markov renewal equations. However, those functions have not been utilized as they are expected. In this article, we show that they are useful for studying asymptotic decay in risk and queuing processes under a Markovian environment. In particular, a matrix version of the Cramér–Lundberg approximation is obtained for the risk process. The corresponding result for the MAP/G/1 queue is presented as well. Emphasis is placed on a straightforward derivation using the Markov renewal structure.

We study deviation matrices of birth–death processes. This is relevant to the control of multidimensional queueing systems. We give an algorithm for computing deviation matrices for birth–death processes. As an application, we compute them explicitly for the M/M/s/N and M/M/s/∞ queues.

This article presents Monte Carlo techniques for estimating network reliability. For highly reliable networks, techniques based on graph evolution models provide very good performance. However, they are known to have significant simulation cost. An existing hybrid scheme (based on partitioning the time space) is available to speed up the simulations; however, there are difficulties with optimizing the important parameter associated with this scheme. To overcome these difficulties, a new hybrid scheme (based on partitioning the edge set) is proposed in this article. The proposed scheme shows orders of magnitude improvement of performance over the existing techniques in certain classes of network. It also provides reliability bounds with little overhead.

Simple and insensitive lower and upper bounds are proposed for the call congestion of M/GI/c/n queues. To prove them we establish the general monotonicity property that increasing the waiting room and/or the number of servers in a /GI/c/n queue increases the throughput. An asymptotic result on the number of busy servers is obtained as a consequence of the bounds. Numerical evidence as well as an application to optimal design illustrates the potential usefulness for engineering purposes. The proof is based on a sample path argument.

For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function . For such models, we study some properties of multivariate aging of that are described by means of the multivariate aging function , which is a useful tool for describing the level curves of . Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals.

We consider the loss probability of a customer in a single-server queue with finite buffer and partial rejection and show that it can be identified with the tail distribution of the cycle maximum of the associated infinite-buffer queue. This equivalence is shown to hold for the GI/G/1 queue and for dams with state-dependent release rates. To prove this equivalence, we use a duality for stochastically monotone recursions, developed by Asmussen and Sigman (1996). As an application, we obtain several exact and asymptotic results for the loss probability and extend Takács' formula for the cycle maximum in the M/G/1 queue to dams with variable release rate.

Reliability theory and survival analysis, the residual entropy is known as a measure suitable to describe the dynamic information content in stochastic systems conditional on survival. Aiming to analyze the variability of such information content, in this paper we introduce the variance of the residual lifetimes, “residual varentropy” in short. After a theoretical investigation of some properties of the residual varentropy, we illustrate certain applications related to the proportional hazards model and the first-passage times of an Ornstein–Uhlenbeck jump-diffusion process.

This article provides us with some simple criteria to compare Birnbaum reliability importance measure of components in a general binary coherent system. Such criteria are particularly useful in the absence of information concerning component reliabilities. We also find several simple (necessary and sufficient) conditions concerning system structure, under which such comparison is possible. Examples are given to illustrate our results.