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.

Suppose that there are n types of coupons and that each new coupon collected is type i with probability pi. Suppose, further, that there are m subsets of coupon types and that coupons are collected until all of the types of at least one of these subsets have been collected. When these subsets have no overlap, we derive expressions for the mean and variance of the number of coupons that are needed. In the general case where the subsets can overlap, we derive the mean of the number that are needed. We also note that this number is an increasing failure rate on average random variable and we present a conjecture as to a sufficient condition for it to be an increasing failure rate random variable.

Let X1, … , Xn be independent random variables with Xi having survival function λi, i = 1, … , n, and let Y1, … ,Yn be a random sample with common population survival distribution , where = ∑i=1nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Xn:n − X1:n is larger than Yn:n − Y1:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively.

In this article we present a new representation for the steady-state distribution of the workload of the second queue in a two-node tandem network. It involves the difference of two suprema over two adjacent intervals. In the case of spectrally positive Lévy input, this enables us to derive the Laplace transform and Pollaczek–Khintchine representation of the workload of the second queue. Additionally, we obtain the exact distribution of the workload in the case of Brownian and Poisson input, as well as some insightful formulas representing the exact asymptotics for α-stable Lévy inputs.

Under a suitable condition on the conditional moment generating function of the martingale differences, an exponential supermartingale is used to generalize certain martingale inequalities due to Blackwell and Ross.

In this article, we derive a tight closed-form upper bound on the expected value of a three-piece linear convex function E[max(0, X, mX − z)] given the mean μ and the variance σ2 of the random variable X. The bound is an extension of the well-known mean–variance bound for E[max(0, X)]. An application of the bound to price the strangle option in finance is provided.

This article investigates reliability properties of a flexible extended linear failure-rate family of distributions generated using the relation F¯(x) = αF¯0(x)/[1 − α¯F¯0(x)], where α > 0, α¯ = 1 − α, and F¯0(x) is the reliability function of the linear failure-rate distribution.

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.

Let S denote the collection of all finite subsets of . We define an operation on S that makes S into a positive semigroup with set inclusion as the associated partial order. Positive semigroups are the natural home for probability distributions with exponential properties, such as the memoryless and constant rate properties. We show that there are no exponential distributions on S, but that S can be partitioned into subsemigroups, each of which supports a one-parameter family of exponential distributions. We then find the distribution on S that is closest to exponential, in a certain sense. This work might have applications to the problem of selecting a finite sample from a countably infinite population in the most random way.

In this article we determine a formula for fractal and resistance dimensions of two models of uniformly bounded random trees. The type (transient or recurrent) of the random walk on such trees is ascribed, to some extent, to these dimensions. The results presented in this article generalize some of the results of [6] and [7].

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.

Viterbi training (VT) provides a fast but inconsistent estimator of hidden Markov models (HMM). The inconsistency is alleviated with a little extra computation when we enable VT to asymptotically fix the true values of the parameters. This relies on infinite Viterbi alignments and associated with them limiting probability distributions. First in a sequel, this article is a proof of concept; it focuses on mixture models, an important but special case of HMM where the limiting distributions can be calculated exactly. A simulated Gaussian mixture shows that our central algorithm (VA1) can significantly improve the accuracy of VT with little extra cost. Next in the sequel, we present elsewhere a theory of the adjusted VT for the general HMMs, where the limiting distributions are more challenging to find. Here, we also present another, more advanced correction to VT and verify its fast convergence and high accuracy; its computational feasibility requires additional investigation.

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.

We study the stochastic optimal control for an assemble-to-order system with multiple products and components that arrive at the system in random batches and according to renewal reward processes. Our purpose is to maximize expected infinite-horizon discounted profit by selecting product prices, component production rates, and a dynamic sequencing rule for assembly. We refine the solution of some static planning problem and a discrete review policy to batch arrival environment and develop an asymptotically optimal policy for the system operating under heavy traffic, which indicates that the system can be approximated by a diffusion process and exhibits a state space collapse property.

We consider weighted path lengths to the extremal leaves in a random binary search tree. When linearly scaled, the weighted path length to the minimal label has Dickman's infinitely divisible distribution as a limit. By contrast, the weighted path length to the maximal label needs to be centered and scaled to converge to a standard normal variate in distribution. The exercise shows that path lengths associated with different ranks exhibit different behaviors depending on the rank. However, the majority of the ranks have a weighted path length with average behavior similar to that of the weighted path to the maximal node.

Considered are semi-Markov decision processes (SMDPs) with finite state and action spaces. We study two criteria: the expected average reward per unit time subject to a sample path constraint on the average cost per unit time and the expected time-average variability. Under a certain condition, for communicating SMDPs, we construct (randomized) stationary policies that are ε-optimal for each criterion; the policy is optimal for the first criterion under the unichain assumption and the policy is optimal and pure for a specific variability function in the second criterion. For general multichain SMDPs, by using a state space decomposition approach, similar results are obtained.

Motivated by hydrological applications, the exact distributions of R = X + Y, P = XY, and W = X/(X + Y) and the corresponding moment properties are derived when X and Y follow Block and Basu's bivariate exponential distribution. An application of the results is provided to drought data from Nebraska.