Stochastic comparisons of linear (circular) consecutive k-out-of-n:F systems with independent components are studied. A sufficient condition is given under which the lifetime of a circular consecutive k-out-of-n:F system with independent and nonidentically distributed components is stochastically decreasing in n for fixed k. Furthermore, the likelihood ratio orderings of the lifetimes of linear (circular) consecutive k-out-of-n:F systems with independent and identically distributed components are also established, and some counterexamples are given to show that these orderings are not true in general.

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

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.

Motivated by large-scale service systems, we consider an infinite-server queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be time varying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavy-traffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavy-traffic approximations for the stationary model and the model with a time-varying arrival rate. In the simulation experiments, we use the Marshall–Olkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of Marshall–Olkin multivariate hyperexponential distributions to model dependent hyper-exponential service times within a batch.

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

This paper describes techniques to characterize the number of singletons in the setting of the generalized birthday problem, that is, the birthday problem in which the birthdays are non-uniformly distributed over the year. Approximations for the mean and variance presented which explicitly indicate the impact of the heterogeneity (expressed in terms of the Kullback–Leibler distance with respect to the homogeneous distribution). Then an iterative scheme is presented for determining the distribution of the number of singletons. The approximations are validated by experiments with demographic data.

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.

We introduce a robot–safety device system characterized by cold stand-by and by an admissible risky state. The system is attended by a single repairman and the robot has overall (break-in) priority in repair with regard to the safety device. We obtain an explicit formula for the point availability of the robot via an integral equation of the renewal-type. The explicit solution requires the notion of effective repair-versus-virtual repair. In order to decide whether the risky state is admissible, we also introduce a risk criterion. The criterion is always satisfied in the case of fast repair. As an example, we consider the case of Weibull–Gnedenko repair and we display a computer-plotted graph of the point availability obtained by a direct numerical solution of a convolution-type integral equation.