One of the most fundamental results in inventory theory is the optimality of (s, S) policy for inventory systems with setup cost. This result is established under a key assumption of infinite ordering/production capacity. Several studies have shown that, when the ordering/production capacity is finite, the optimal policy for the inventory system with setup cost is very complicated and indeed, only partial characterization for the optimal policy is possible. In this paper, we consider a continuous review production/inventory system with finite capacity and setup cost. The demand follows a Poisson process and a demand that cannot be satisfied upon arrival is backlogged. We show that the optimal control policy has a very simple structure when the holding/shortage cost rate is quasi-convex. We also develop efficient algorithms to compute the optimal control parameters.

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.

We consider a memoryless loss system with servers = {1, …, J}, and with customer types = {1, …, I}. Servers are multi-type: server j works at rate μj, and can serve a subset of customer types C(j). An arriving customer will go to the longest idling server which can serve him, or be lost. We obtain a simple explicit steady-state distribution for this system, and calculate various performance measures of this system in steady state. We provide some illustrative examples. We compare this system with a similar system discussed recently by Adan, Hurkens, and Weiss [1]. We also show that this system is insensitive, the results hold also for general service time distributions.

Discrete-time renewal sequences play a fundamental role in the theory of stochastic processes. This article considers periodic versions of such processes; specifically, the length of an interrenewal is allowed to probabilistically depend on the season at which it began. Using only elementary renewal and Markov chain techniques, computational and limiting aspects of periodic renewal sequences are investigated. We use these results to construct a time series model for a periodically stationary sequence of integer counts.

A class of Markov chains we call successively lumbaple is specified for which it is shown that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a(typically much) smaller state space and this yields significant computational improvements. We discuss how the results for discrete time Markov chains extend to semi-Markov processes and continuous time Markov processes. Finally, we will study applications of successively lumbaple Markov chains to classical reliability and queueing models.

We consider a general polling model with N stations. The stations are served exhaustively and in cyclic order. Once a station queue falls empty, the server does not immediately switch to the next station. Rather, it waits at the station for the possible arrival of new work (“wait-and-see”) and, in the case of this happening, it restarts service in an exhaustive fashion. The total time the server waits idly is set to be a fixed, deterministic parameter for each station. Switchover times and service times are allowed to follow some general distribution, respectively. In some cases, which can be characterized, this strategy yields a strictly lower average queuing delay than for the exhaustive strategy, which corresponds to setting the “wait-and-see credit” equal to zero for all stations. This extends the results of Peköz [12] and of Boxma et al. [4]. Furthermore, we give a lower bound for the delay for all strategies that allow the server to wait at the stations even though no work is present.

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 a two-person red-and-black game, each player wants to maximize the probability of winning the entire fortune of his opponent by gambling repeatedly with suitably chosen stakes. We find the multiplicativity (including submultiplicative and supermultiplicative) of the win probability function is important for the profiles (bold, timid) or (bold, bold) to be a Nash equilibrium. Surprisingly, a Nash equilibrium condition for the profile (bold, any strategy) is also given in terms of multiplicativity. Finally, we search for some suitable conditions such that the profile (timid, timid) is also a Nash equilibrium.

The total time on test transform (ttt) order was introduced in Kochar, Li, and Shaked (2002) for nonnegative random variables, which has a close connection to the location independent riskier and the excess wealth orders. In this article, the ttt order is redefined for comparing not necessarily nonnegative random variables. Such an extension will lead us to conveniently derive several new intrinsic properties of the ttt order. In particular, we establish an interesting separation result on the connection between the ttt and the excess wealth orders. As two applications of this separation result, we obtain the generating process of the ttt order in terms of mean-decreasing right stretches and the closure property of the ttt order under convolutions. The generating process of the ttt order parallels those of the location independent riskier and the excess wealth orders established by Landsberger and Meilijson (1994).

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.

A cascading failure is a failure in a system of interconnected parts, in which the breakdown of one element can lead to the subsequent collapse of the others. The aim of this paper is to introduce a simple combinatorial model for the study of cascading failures. In particular, having in mind particle systems and Markov random fields, we take into consideration a network of interacting urns displaced over a lattice. Every urn is Pólya-like and its reinforcement matrix is not only a function of time (time contagion) but also of the behavior of the neighboring urns (spatial contagion), and of a random component, which can represent either simple fate or the impact of exogenous factors. In this way a non-trivial dependence structure among the urns is built, and it is used to study default avalanches over the lattice. Thanks to its flexibility and its interesting probabilistic properties, the given construction may be used to model different phenomena characterized by cascading failures such as power grids and financial networks.

In this paper, we study a traffic intersection with vehicle-actuated traffic signal control. Traffic lights stay green until all lanes within a group are emptied. Assuming general renewal arrival processes, we derive exact limiting distributions of the delays under heavy traffic (HT) conditions. Furthermore, we derive the light traffic (LT) limit of the mean delays for intersections with Poisson arrivals, and develop a heuristic adaptation of this limit to capture the LT behavior for other interarrival-time distributions. We combine the LT and HT results to develop closed-form approximations for the mean delays of vehicles in each lane. These closed-form approximations are quite accurate, very insightful, and simple to implement.

The purpose of this study is two-fold. First, we investigate further properties of the second-order regular variation (2RV). These properties include the preservation properties of 2RV under the composition operation and the generalized inverse transform, among others. Second, we derive second-order expansions of the tail probabilities of convolutions of non-independent and identically distributed (i.i.d.) heavy-tail random variables, and establish second-order expansions of risk concentration under mild assumptions. The main results extend some ones in the literature from the i.i.d. case to non-i.i.d. case.

Here, we discuss the stochastic comparison of residual lifetimes of parallel systems and inactivity times of series systems by means of the reversed hazard rate order when the components of the systems are independent but not necessarily identically distributed. We also establish some monotonicity properties of such residual lifetimes of parallel systems and inactivity times of series systems. These results extend some of the recent results in this direction due to Zhao, Li, and Balakrishnan [21], Kochar and Xu [12], and Saledi and Asadi [16].

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 study some stochastic comparisons of the maxima in two multiple-outlier geometric samples based on the likelihood ratio order, hazard rate order, and usual stochastic order. We establish a sufficient condition on parameter vectors for the likelihood ratio ordering to hold. For the special case when n = 2, it is proved that the p-larger order between the two parameter vectors is equivalent to the hazard rate order as well as usual stochastic order between the two maxima. Some numerical examples are presented for illustrating the established results.

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.

When launching a new product, a firm has to set an initial price with incomplete information about demand. However, after observing the demand over a period of time, the firm might decide to mark down the price, especially when the Bayesian updated belief about demand is lower than originally anticipated. We consider the case in which the manufacturer makes three decisions: initial price, when to mark down the price, and the markdown price. Modeling the cumulative demand as a Brownian motion with an unknown drift, we compute the posterior probability distribution of the unknown drift. We then show that it is optimal to mark down the price when the posterior probability is below a computable threshold. This threshold policy enables us to determine the optimal (a) regular price, (b) markdown price, and (c) markdown time. Additionally, we examine the impact of demand volatility and evaluate the value of learning.

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9].

We analyze the asymptotic number of items chosen in a selection procedure. The procedure selects items whose rank among all previous applicants is within the best 100p percent of the number of previously selected items. We use analytic methods to obtain a succinct formula for the first-order asymptotic growth of the expected number of items chosen by the procedure.