We consider a queuing system with a workload-dependent service rate. We specifically assume that the service rate is first increasing and then decreasing as a function of the amount of work. The latter qualitative behavior is quite common in practical situations, such as production systems. The admission of work into the system is controlled by a policy for accepting or rejecting jobs, depending on the state of the system. We seek an admission control policy that maximizes the long-run throughput. Under certain conditions, we show that a threshold policy is optimal, and we derive a criterion for determining the optimal threshold value.

This article describes the scientific contributions of Milton Sobel. It motivates his research by considering his family background, his war experiences, and his mentors and fellow students at Columbia University. His research in sequential analysis, selection, ranking, group testing, and probabilistic combinatorics are highlighted.

Numerical inversion of Laplace transforms is a powerful tool in computational probability. It greatly enhances the applicability of stochastic models in many fields. In this article we present a simple Laplace transform inversion algorithm that can compute the desired function values for a much larger class of Laplace transforms than the ones that can be inverted with the known methods in the literature. The algorithm can invert Laplace transforms of functions with discontinuities and singularities, even if we do not know the location of these discontinuities and singularities a priori. The algorithm only needs numerical values of the Laplace transform, is extremely fast, and the results are of almost machine precision. We also present a two-dimensional variant of the Laplace transform inversion algorithm. We illustrate the accuracy and robustness of the algorithms with various numerical examples.

We consider the transformation T that takes a distribution F into the distribution of the length of the interval covering a fixed point in the stationary renewal process corresponding to F. This transformation has been referred to as size-biasing, length-biasing, the renewal length transformation, and the stationary lifetime operator. We review and develop properties of this transformation and apply it to diverse areas.

We start with a simple derivation of an identity connecting the conditional expected residual service time as seen by an arrival and the steady-state tail distribution function of the number of customers in the system, which was previously proven by Mandelbaum and Yechiali. We then show how to use it to obtain bounds on the the stationary distribution of the number of customers in the M/G/1 queue.

This article deals with estimations of probabilities of rare events using fast simulation based on the splitting method. In this technique, the sample paths are split into multiple copies at various stages in the simulation. Our aim is to optimize the algorithm and to obtain a precise confidence interval of the estimator using branching processes. The numerical results presented suggest that the method is reasonably efficient.

In the literature, most of the bounds for the renewal function U(x) corresponding to a lifetime distribution F are given in terms of the first two moments of F only. The best general upper bound of this type is the one given in Lorden (1970). In the present article, we show that improved bounds can be obtained if one exploits the specific form of the distribution F. We derive a bound that improves upon Lorden's, at least on an interval [0,a) with a ≤ ∞, and we give both sufficient and necessary conditions for this improvement to hold uniformly for x ≥ 0. Refined upper as well as lower bounds are given for the case where F belongs to a class of distributions with monotone aging or when the renewal density is monotone.

This article is concerned with a loading-dependent model of cascading failure proposed recently by Dobson, Carreras, and Newman [6]. The central problem is to determine the distribution of the total number of initial components that will have finally failed. A new approach based on a closed connection with epidemic modeling is developed. This allows us to consider a more general failure model in which the additional loads caused by successive failures are arbitrarily fixed (instead of being constant as in [6]). The key mathematical tool is provided by the partial joint distributions of order statistics for a sample of independent uniform (0,1) random variables.

For a system in which arrivals occur according to a Poisson process, we give a new approach for using simulation to estimate the expected value of a random variable that is independent of the arrival process after some specified time t. We also give a new approach for using simulation to estimate the expected value of an increasing function of independent uniform random variables. Stratified sampling is a key technique in both cases.

In this article we introduce isotone differences stochastic ordering of Markov processes on lattice ordered state spaces as a device to compare the internal dependencies of two such processes. We derive a characterization in terms of intensity matrices. This enables us to compare the internal dependency structure of different degradable Jackson networks in which the nodes are subject to random breakdowns and repairs. We show that the performance behavior and the availability of such networks can be compared.

Motivated by various applications in queuing theory, this article is devoted to the monotonicity and convexity of some functions associated with discrete-time or continuous-time denumerable Markov chains. For the discrete-time case, conditions for the monotonicity and convexity of the functions are obtained by using the properties of stochastic dominance and monotone matrix. For the continuous-time case, by using the uniformization technique, similar results are obtained. As an application, the results are applied to analyze the monotonicity and convexity of functions associated with the queue length of some queuing systems.

We consider the sojourn time V in the M/D/1 processor sharing (PS) queue and show that P(V > x) is of the form Ce−γx as x becomes large. The proof involves a geometric random sum representation of V and a connection with Yule processes, which also enables us to simplify Ott's [21] derivation of the Laplace transform of V. Numerical experiments show that the approximation P(V > x) ≈ Ce−γx is excellent even for moderate values of x.

For the GI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

We study a single-product periodic-review inventory model in which the ordering quantity is either zero or at least a minimum order size. The ordering cost is a linear function of the ordering quantity, and the demand in different time periods are independent random variables. The objective is to characterize the inventory policies that minimize the total discounted ordering, holding, and backorder penalty costs over a finite time horizon. We introduce the concept of an M-increasing function. These functions allow us to characterize the optimal inventory policies everywhere in the state space outside of an interval for each time period. Furthermore, we identify easily computable upper bounds and asymptotic lower bounds for these intervals. Finally, examples are given to demonstrate the complex structure of the optimal inventory policies.

Suppose that there are n families with children attending a certain school and that the number of children in these families are independent and identically distributed random variables, each with probability mass function P{X = j} = pj, j ≥ 1, with finite mean μ = [sum ]j≥1jpj. If a child is selected at random from the school and XI is the number of children in the family to which the child belongs, it is known that limn→∞P{XI = j} = jpj /μ,j ≥ 1. Here, asymptotic expansions for P{XI = j} are developed under the condition E|X|3 < ∞.

Recently, Pellerey, Shaked, and Zinn [6] introduced a discrete-time analogue of the nonhomogeneous Poisson process. The purpose of this article is to provide some results for stochastic comparisons of the epoch times and the interepoch times of those processes. Also, we show the relationships between these processes and discrete record values and we provide several results for discrete weak record values.

This article discusses a real-world application of a terminating two-person stochastic game. The problem comes from a Dutch television game show in which two finalists play a dice game. Each player chooses a number of dice to be rolled. The score of the roll is added to the player's total provided that none of the dice showed the outcome one. The first player reaching a prespecified number of points is the winner. This article discusses the computation and the structure of an optimal strategy.

The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to investigate conditions on the distributions and the parameters on which the generalized order statistics are based to establish the likelihood ratio ordering of general p-spacings and the hazard rate and the dispersive orderings of (normalizing) simple spacings from two samples. We thus strengthen and complement some results in Franco, Ruiz, and Ruiz [7] and Belzunce, Mercader, and Ruiz [5]. This article is a continuation of Hu and Zhuang [10].

Compared to the known univariate distributions for continuous data, there are relatively few available for discrete data. In this article, we derive a collection of 16 flexible discrete distributions by means of conditional Poisson processes. The calculations involve the use of several special functions and their properties.

In a communication network, one might attempt to route calls from an origin to a destination through n paths that will be tried one by one, each having a success probability pi ∈ (0,1), i = 1,2,…,n. The order of trying is controlled by a routing table. The number of attempts made is defined as the cost of the routing table. Move-forward self-organizing rules are applied to the routing table and comparisons of expected equilibrium costs are performed when p2 = p3 = … = pn. Stationary distributions for a subset of f-swap rules are obtained for general pi's.