Some monotonicity properties and bounds on expectations of certain functions of minimal repair times and interminimal repair times are obtained under assumptions on the aging characteristics of the equipment under maintenance.

Consider a single machine that can process multiple jobs in batch mode. We have n jobs and the processing time of job j is a random variable Xj with distribution Fj. Up to b jobs can be processed simultaneously by the machine. The jobs in a batch all have to start at the same time and the batch is completed when all jobs have finished their processing (i.e., at the maximum of the processing times of the jobs in that batch). We are interested in two objective functions, namely the minimization of the expected makespan and the minimization of the total expected completion time. We first show that under certain fairly general conditions, the minimization of the expected makespan is equivalent to specific deterministic combinatorial problems, namely the Weighted Matching problem and the Set Partitioning problem. We then consider the case when all jobs have the same mean processing time but different variances. We show that for certain special classes of processing time distributions the Smallest Variance First rule minimizes the expected makespan as well as the total expected completion time. In our conclusions we present various general rules that are suitable for the minimization of the expected makespan and the total expected completion time in batch scheduling.

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

We derive a simple recursion to compute moments of arbitrary order of the duration of busy periods of MX/G/1/n systems starting with an arbitrary number of customers in the system.

To find zeros or locate maximum values of a regression function with noisy measurements, a commonly used algorithm is the RM or KW procedure. In various applications, the dimensionality of the problems involved might be quite large. As a result, enormous memory space and extensive computation time may be needed. Motivated by the recent progress in stochastic approximation methods for decentralized and distributed computing, a parallel stochastic approximation algorithm is developed in this paper. The essence is to take advantage of state-space decompositions, and to exploit the opportunities provided by parallel processing and asynchronous communication. In lieu of utilizing a single processor as in the classical cases, a number of parallel processors are employed to solve the underlying problem in a cooperative way. First, the large dimensional vector is partitioned into a number of subvectors with relatively small dimension, then each of the subvectors is assigned to one of the processors. The processors compute and communicate in an asynchronous manner and at random times. Under rather weak conditions, the global convergence of the parallel algorithm is obtained via the methods of randomly varying truncations.

In this paper, we consider exponentiated location-scale model and obtain several ordering results between extreme order statistics in various senses. Under majorization type partial order-based conditions, the comparisons are established according to the usual stochastic order, hazard rate order and reversed hazard rate order. Multiple-outlier models are considered. When the number of components are equal, the results are obtained based on the ageing faster order in terms of the hazard rate and likelihood ratio orders. For unequal number of components, we develop comparisons according to the usual stochastic order, hazard rate order, and likelihood ratio order. Numerical examples are considered to illustrate the results.

We consider a class of stochastic models for systems subject to random regulation. We derive expressions for the distribution of the intervals between regulating instants and for the transient and equilibrium properties of the process. Some of these are evaluated explicitly for some models of interest.

We consider a stochastic process with an embedded point process in a stationary and ergodic context. Under a “lack of anticipation” assumption for the evolution of the process vis-à-vis the point process, a new better (worse) than used expectation property for the point process, and a monotonicity assumption for the behavior of the process between points, inequalities between event and time averages are obtained. Sample path monotonicity between points is not required (as is the case with existing approaches) and can be replaced with a simple monotonicity requirement for the expected value of the process between points. Inequalities between conditional event and time averages are also examined via a novel argument involving a conditional version of the Palm inversion formula.

In this paper, we consider a history-dependent mixed shock model which is a combination of the history-dependent extreme shock model and the history-dependent $\delta$-shock model. We assume that shocks occur according to the generalized Pólya process that contains the homogeneous Poisson process, the non-homogeneous Poisson process and the Pólya process as the particular cases. For the defined survival model, we derive the corresponding survival function, the mean lifetime and the failure rate. Further, we study the asymptotic and monotonicity properties of the failure rate. Finally, some applications of the proposed model have also been included with relevant numerical examples.

Construction of a certain class of component tests for verification of series system reliability requires the use of some properties of the Poisson distribution that are not commonly known or used. This paper states and proves these results.

A consecutive k−out−of-n: F system is a coherent system of n-ordered components that functions if and only if there is no consecutive run of k failures among the components. Examples of such systems exist in telecommunications, oil pipelines, and integrated circuitry. Most of the research in reliability of consecutive k−out-of-n: F systems assumes that the components function independently of one another. In this paper, we develop a model incorporating positive dependence between adjacent components and show that for k ≥ (n + l)/2, the reliability of the system is a decreasing function of this dependence.

Let [NA(t), t≥0] and [NB(t), t≥0] be two counting processes with independent interarrivals Ai and Bi, respectively, i ∈ + ≡ [1,2,… ], and let [Xi, i ∈ +] and [Yi, i ∈ +] be two sequences of independent non-negative random variables. Consider the two compound processes [U(t), t ≥ 0] and [V(t), t ≥ 0] defined as U(t) = Xi, t ≥ 0, and V(t) = Yi, t ≥ 0. Then it is well known (and not hard to verify) that if the Ai's are independent of the Xi's, and if the Bi's are independent of the yi's, and if Ai ≥st Bi and Xi ≤st Yi, for all i ∈ +, then [U(t), t ≥ 0] ≤st [V(t), t ≥ 0]. In this paper we provide conditions under which this stochastic order relation holds even if Ai ≤st Bi and/or Xi ≥st Yi for all i ∈ +. In fact, we derive the stochastic order relationships among processes that are much more general than the cumulative ones described above. For example, it is not necessarily assumed that the Ai's are independent of the Xi's or that the Bi's are independent of the Yi's. Examples that illustrate the applications of the theory are included.

In this article, we study the order statistics from heterogenous negative binomial random variables. Sufficient conditions are provided for comparing the extreme order statistics according to the usual stochastic order. For the special case of geometric distribution, a sufficient condition is established for comparing order statistics in the sense of multivariate stochastic order. Applications in the Poisson–Gamma shock model and redundant systems have been described as well.

We study a multi-item two-stage production system subject to Markov-modulated demands and production quantity requirements. The demand distribution for each item in each period is governed by a discrete Markov chain. The products are manufactured in two stages. In the first stage, a common intermediate product is manufactured, followed by product differentiation in the second stage. Lower and upper production limits, also known as production smoothing constraints, are imposed on both stages for all items. We propose a close-to-optimal heuristic to manage this system. To do so, we develop a lower bound problem and show that a state-dependent, modified base-stock policy is optimal. We also show when and why the heuristic works well. In our numerical study, the average optimality gap was 4.34%. We also establish some monotonicity results for policy parameters with respect to the production environment. Using these results and our numerical observations, we investigate the joint effect of (i) the two-stage production process, (ii) the production flexibility, and (iii) the fluctuating demand environment on the system's performance. For example, we quantify the value of flexible production as well as the effect of smoothing constraints on the benefits of postponement. We show that a redesign of the production process to allow for delayed product differentiation is more effective and valuable when it is accompanied by an investment in production flexibility.

In this paper, some results on the complete moment convergence for arrays of rowwise negatively associated (NA, for short) random variables are established. The results obtained in this paper correct the corresponding one obtained in Ko [13] and also improve and generalize the corresponding ones of Kuczmaszewska [14] and Ko [13]. As an application of the main results, we present a result on complete consistency for the estimator in a non-parametric regression model based on NA errors. Finally, we provide a numerical simulation to verify the validity of our result.

Numerical inversion of a one-sided z transform, corresponding to causal positive sequence, is considered. The numerical inversion requires the availability of a finite number of the transform's derivatives. The approximate analytical form is obtained by resorting to the maximum entropy principle. Entropy and then L1-norm convergence are proved.

In this paper, we study the almost sure convergence for sequences of asymptotically negative associated (ANA) random variables. As a result, we extend the classical Khintchine–Kolmogorov convergence theorem, Marcinkiewicz strong law of large numbers, and the three series theorem for sequences of independent random variables to sequences of ANA random variables without necessarily adding any extra conditions.

Skellam's name is traditionally attached to the distribution of the difference of two independent Poisson random variables. Many bivariate extensions of this distribution are possible, e.g., through copulas. In this paper, the authors focus on a probabilistic construction in which two Skellam random variables are affected by a common shock. Two different bivariate extensions of the Skellam distribution stem from this construction, depending on whether the shock follows a Poisson or a Skellam distribution. The models are nested, easy to interpret, and yield positive quadrant-dependent distributions, which share the convolution closure property of the univariate Skellam distribution. The models can also be adapted readily to account for negative dependence. Closed form expressions for Pearson's correlation between the components make it simple to estimate the para-meters via the method of moments. More complex formulas for Kendall's tau and Spearman's rho are also provided.

Aleliunas et al. [3] proved that for a random walk on a connected raph G = (V, E) on N vertices, the expected minimum number of steps to visit all vertices is bounded by 2|E|(N - 1), regardless of the initial state. We give here a simple proof of that result through an equality involving hitting times of vertices that can be extended to an inequality for hitting times of edges, thus obtaining a bound for the expected minimum number of steps to visit all edges exactly once in each direction.

Unimodality and strong unimodality of the distribution of ascendingly ordered random variables have been extensively studied in the literature, whereas these properties have not received much attention in the case of descendingly ordered random variates. In this paper, we show that log concavity of the reversed hazard rate implies that of the density function. Using this fundamental result, we establish some convexity properties of such random variables. To do this, we first provide a counterexample showing that a claim of Basak & Basak [7] about the lower record values is not valid. Then, we provide conditions under which unimodality properties of the distribution of lower k-record values would hold. Finally, some extensions to dual generalized order statistics in both univariate and multivariate cases are discussed.