In this article, we consider deterministic (both fluid and discrete) polling systems with N queues with infinite buffers and we show how to compute the best polling sequence (minimizing the average total workload). With two queues, we show that the best polling sequence is always periodic when the system is stable and forms a regular sequence. The fraction of time spent by the server in the first queue is highly noncontinuous in the parameters of the system (arrival rate and service rate) and shows a fractal behavior. Moreover, convexity properties are shown and are used in a generalization of the computation of the optimal control policy (in open loop) for the stochastic exponential case.

We study the optimal admission of arriving customers to a Markovian finite-capacity queue (e.g., M/M/c/N queue) with several customer types. The system managers are paid for serving customers and penalized for rejecting them. The rewards and penalties depend on customer types. The penalties are modeled by a K-dimensional cost vector, K ≥ 1. The goal is to maximize the average rewards per unit time subject to the K constraints on the average costs per unit time. Let Km denote min{K,m − 1}, where m is the number of customer types. For a feasible problem, we show the existence of a Km-randomized trunk reservation optimal policy, where the acceptance thresholds for different customer types are ordered according to a linear combination of the service rewards and rejection costs. Additionally, we prove that any Km-randomized stationary optimal policy has this structure.

In the present article, we consider a class of reliability structures that can be efficiently described through a finite Markov chain (Markov chain imbeddable systems) and investigate its closeness with respect to the increasing failure rate (IFR) property. More specifically we derive a sufficient condition for the system's lifetime to have increasing failure rate when the identical and independent components comprising it own this property. As an application of the general theory, we establish an alternative proof of the IFR property preservation for the k-out-of-n system and derive some related results for the family of weighted k-out-of-n systems.

We consider the problem of estimating passage times in stochastic simulations of Markov chains. Two types of estimator are considered for this purpose: the “simple” and the “overlapping” estimator; they are compared in terms of their asymptotic variance. The analysis is based on the regenerative structure of the process and it is shown that when estimating the mean passage time, the simple estimator is always asymptotically superior. However, when the object is to estimate the expectation of a nonlinear function of the passage time, such as the probability that the passage time exceeds a given threshold, then it is shown that the overlapping estimator can be superior in some cases. Related results in the Reinforcement Learning literature are discussed.

In this article we first give a characterization of a class of probability transition matrices having closed-form solutions for transient distributions and the steady-state distribution. We propose to apply the stochastic comparison approach to construct bounding chains belonging to this class. Therefore, bounding chains can be analyzed efficiently through closed-form solutions in order to provide bounds on the distributions of the considered Markov chain. We present algorithms to construct upper-bounding matrices in the sense of the ≤st and ≤icx order.

In this article we study a firm that is facing demand from two sources: demand for new items and demand to replace failed items under warranty. We model this setting as a multiperiod single-product inventory problem where the demands for new items in different periods are independent and the demands for replacing failed items depend on the number of the items under warranty. We consider backlogging and emergency supply cases and study both discounted-cost and average-cost criteria. We prove the optimality of the w-dependent base stock ordering policy, where the base stock level is a function of w, the number of items currently under warranty. For the special case where the demand for new products is stationary, we prove the optimality of a stationary w-dependent base stock policy for the finite-horizon discounted-cost and the infinite-horizon discounted- and average-cost cases. We compare the integrated inventory policy with the one that neglects demands from items under warranty.

Chebyshev inequality estimates the probability for exceeding the deviation of a random variable from its mathematical expectation in terms of the variance of the random variable. In modern probability theory, the Chebyshev inequality is the most frequently used tool for proving different convergence processes; for example, it plays a fundamental role in proofs of various forms of laws of large numbers. The mathematical expression of the bound on the probability in the Chebyshev inequality is very simple and can be modified easily for different kinds of sequence of random variables (e.g., for the case of sums of independent random variables). This fact lies behind these frequent applications. In this setting, the Chebyshev inequality has pure theoretical “applications” in probability theory and its role is to provide “a guarantee” of convergence but not to give a bound on concrete probability content.

In the present article we consider the Chebyshev inequality as a probability bound that is essential for the translation from its conventional theoretical applications to the practical setting if easy-to-compute multivariate generalizations are derived.

Such an inequality for the random vectors having multivariate Normal distribution is proved. The new inequality gives a lower bound in terms of variances on the probability that the random vector in question falls into an Euclidean ball with center at mean vector. The need and importance of consideration of this kind of multivariate Chebyshev inequality stemmed from several problems in engineering and informational sciences (Hassibi and Boyd [9], Jeng [10], Jeng and Woods [11], Molina, Katseggelos, Mateos, Hermoso, and Segall [13]). Jeng [10] derived an inequality that gives an upper bound for the probability in question. The simultaneous application of the established multivariate Chebyshev inequality and Jeng's inequality is useful in practical problems by providing lower and upper bounds on the probability content.

The inequality is attractive by its being easy to compute and its similarity to the original Chebyshev inequality, in contrast to well-known complicated multivariate Chebyshev inequalities. The present article also gives some insights into the very origin of the Chebyshev inequality, which makes the article self-contained.

Software reliability is one of important characteristics of software quality, and software release time is an important application of the software reliability model. In this article we consider a software release policy based on a Gamma-Gamma-type Kalman filter as well as the risk cost due to software failures and the cost for debugging in software systems. Under this model, the optimal release time that minimizes the expected cost in every test-debugging stage subject to a reliability constraint is discussed. An example to illustrate the framework of our model is given.

The equilibrium distribution arises as the limiting distribution of the forward recurrence time in a renewal process. The purpose of this article is to study the relationships between the equilibrium distributions (including its higher derivates) and the original distributions. Some stochastic order relations and the relations between their aging properties are investigated and some applications in the field of insurance and financial investments are given. In addition, the relation between the equilibrium distribution of a series system and the series system of equilibrium distribution is investigated. Bivariate equilibrium distribution whose reliability properties are consistent with those of the univariate equilibrium distribution is defined.