Let Sn(a1, …, an) be the sum of n independent exponential random variables with respective hazard rates a1, …, an or the sum of n independent geometric random variables with respective parameters a1, …, an. In this article, we investigate sufficient conditions on parameter vectors (a1, …, an) and under which Sn(a1, …, an) and are ordered in terms of the increasing convex and the reversed hazard rate orders for both exponential and geometric random variables and in terms of the mean residual life order for geometric variables. For the bivariate case, all of these sufficient conditions are also necessary. These characterizations are used to compare fail-safe systems with heterogeneous exponential components in the sense of the increasing convex and the reversed hazard rate orders. The main results complement several known ones in the literature.

In this article we analyze the MX/GY/1/K+B bulk queue. For this model, we consider three rejection policies: partial acceptance, complete rejection, and complete acceptance. For each of these policies, we are interested in the loss probability for an arriving group of customers and for individual customers within a group. To obtain these loss probabilities, we derive a numerically stable method to compute the limiting probabilities of the queue length process under all three rejection policies. At the end of the article we demonstrate our method by means of a numerical example.

In this article we introduce a new method of mitigating the problem of long wait times for low-priority customers in a two-class queuing system. To this end, we allow class 1 customers to be upgraded to class 2 after they have been in queue for some time. We assume that there are ci servers at station i, i=1, 2. The servers at station 1 are flexible in the sense that they can work at either station, whereas the servers at station 2 are dedicated. Holding costs at rate hi are accrued per customer per unit time at station i, i=1, 2. This study yields several surprising results. First, we show that stability analysis requires a condition on the order of the service rates. This is unexpected since no such condition is required when the system does not have upgrades. This condition continues to play a role when control is considered. We provide structural results that include a c-μ rule when an inequality holds and a threshold policy when the inequality is reversed. A numerical study verifies that the optimal control policy significantly reduces holding costs over the policy that assigns the flexible server to station 1. At the same time, in most cases the optimal control policy reduces waiting times of both customer classes.

This article analyzes a generic class of queuing systems with server vacation. The special feature of the models considered is that the duration of the vacations explicitly depends on the buffer content evolution during the previous active period (i.e., the time elapsed since the previous vacation). During both active periods and vacations, the buffer content evolves as a Lévy process. For two specific classes of models, the Laplace–Stieltjes transform of the buffer content distribution at switching epochs between successive vacations and active periods, and in steady state, is derived.

This article considers an important class of discrete time restless bandits, given by the discounted multiarmed bandit problems with response delays. The delays in each period are independent random variables, in which the delayed responses do not cross over. For a bandit arm in this class, we use a coupling argument to show that in each state there is a unique subsidy that equates the pulling and nonpulling actions (i.e., the bandit satisfies the indexibility criterion introduced by Whittle (1988). The result allows for infinite or finite horizon and holds for arbitrary delay lengths and infinite state spaces. We compute the resulting marginal productivity indexes (MPI) for the Beta-Bernoulli Bayesian learning model, formulate and compute a tractable upper bound, and compare the suboptimality gap of the MPI policy to those of other heuristics derived from different closed-form indexes. The MPI policy performs near optimally and provides a theoretical justification for the use of the other heuristics.

We consider the steady-state simulation output analysis problem for a process that satisfies a functional central limit theorem. We construct an estimator for the time-average variance constant that is based on excursions of a process above the minimum. The resulting estimator does not require a fixed run length, and the memory requirement can be dynamically bounded. Standardized time series methods based on excursions are also described.

In most decision models dealing with unobservable stochastic congested environments, one looks for a (Nash) equilibrium behavior among customers. This is a strategy that, if adopted by all, then under the resulting steady-state conditions; the best response for an individual is to adopt this strategy too. The purpose of this article is to look for a simple decision problem but where the assumption of steady-state conditions is removed. Specifically, we consider an M/M/N/N loss model in which one pays for trying to get service but is rewarded only if one finds an available server. The initial conditions at time 0 are common knowledge and each customer possesses his arrival time as his private information. The equilibrium profile tells each arrival whether to try (randomization allowed) given his time of arrival. We show that all join up to some point of time. At this point, there is a quantum drop in the joining probability from one to some fraction. From then on, their joining probability continuously converges to the equilibrium joining probability under the model that assumes steady state.

In this article, we consider the two-node fork-join model with a Poisson arrival process and exponential service times of heterogeneous service rates. Using a mapping from the queue lengths in the parallel nodes to the join queue length, we first derive the probability distribution function of the join queue length in terms of joint probabilities in the parallel nodes and then study the exact tail asymptotics of the join queue length distribution. Although the asymptotics of the joint distribution of the queue lengths in the parallel nodes have three types of characterizations, our results show that the asymptotics of the join queue length distribution are characterized by two scenarios: (1) an exact geometric decay and (2) a geometric decay with the prefactor n−1/2.

We consider a two-player game in which the first player (the Guesser) tries to guess, edge-by-edge, the path that second player (the Chooser) takes through a directed graph. At each step, the Guesser makes a wager as to the correctness of her guess and receives a payoff proportional to her wager if she is correct. We derive optimal strategies for both players for various classes of graphs, and we describe the Markov-chain dynamics of the game under optimal play. These results are applied to the infinite-duration Lying Oracle Game, in which the Guesser must use information provided by an unreliable Oracle to predict the outcome of a coin toss.

In this article we develop techniques for applying Approximate Dynamic Programming (ADP) to the control of time-varying queuing systems. First, we show that the classical state space representation in queuing systems leads to approximations that can be significantly improved by increasing the dimensionality of the state space by state disaggregation. Second, we deal with time-varying parameters by adding them to the state space with an ADP parameterization. We demonstrate these techniques for the optimal admission control in a retrial queue with abandonments and time-varying parameters. The numerical experiments show that our techniques have near to optimal performance.

Server farms, consisting of a collection of hosts and a front-end router that dispatches incoming jobs to hosts, are now commonplace. It is well known that when job service requirements (job sizes) are highly variable, then the Size-Interval task assignment policy is an excellent rule for assigning jobs to hosts, since it provides isolation for short jobs by directing short jobs to one host's queue and long jobs to another host's queue. What is not understood is how to classify a “short” job versus a “long” job. For a long time it was believed that the size cutoff separating “short” jobs from “long” ones should be chosen to balance the load at the hosts in the server farm. However, recent literature has provided empirical evidence that load balancing is not always optimal for minimizing mean response time. This article provides the first analytical criteria for when it is preferable to unbalance load between two hosts using Size-Interval task assignment and in which direction the load should be unbalanced. Some very simple sufficient criteria are provided under which we prove that the short job host should be underloaded, and likewise for the long job host. These criteria are then used to prove that the direction of load imbalance depends on moment index properties related to the job size distribution. For example, under the Bounded Pareto (BP) job size distribution with parameter α and a sufficiently high upper bound (the BP is well known to be a good model of empirical computer system workloads), we show that α determines the direction of load imbalance. For α<1, the short job host should be underloaded; for α=1, load should be balanced; and for α>1, the long job host should be underloaded. Many other job size distributions are considered as well. We end by showing that load unbalancing can have a dramatic impact on performance, reducing mean response time by an order of magnitude compared to load balancing in many common cases.

Under very general assumptions, we prove that the ratio between the average sojourn time in a queue under the processor sharing (PS) and the same queue under the fair sojourn protocol (FSP) can be bounded above, and we derive such a bound in terms of the long-run average system size of the PS queue seen by arrivals.

A random walk that is skip-free to the left can only move down one level at a time but can skip up several levels. Such random walk features prominently in many places in applied probability including queuing theory and the theory of branching processes. This article exploits the special structure in this class of random walk to obtain a number of simplified derivations for results that are much more difficult in general cases. Although some of the results in this article have appeared elsewhere, our proof approach is different.

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process, modeled again by a marked point process. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky asset price process and the arrivals process is allowed. By setting and solving a suitable control problem, the characterization of the minimal entropy martingale measure is obtained. In a particular case, a pricing problem is also discussed.

In this article we consider a noncooperative n-person optimal stopping game of Showcase Showdown, in which each player observes the sum of independent and identically distributed random variables uniformly distributed in [0, 1]. Players can decide to stop the draw in each moment. The objective of a player is to get the maximal number of scores that does is not exceeded level 1. If the scores of all players exceed 1, then the winner is the player whose score is closest to 1. We derive the equilibrium in this game on the basis of the dynamic programming approach.

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

Multipath streaming protocols have recently attracted much attention because they provide an effective means to provide high-quality streaming over the Internet. Most existing multipath streaming schemes also apply forward error correction (FEC) encoding in the stream so as to provide high-quality streaming of prestored or live media content. However, the problem of how to intelligently split the FEC-encoded stream among multiple available paths has not been fully addressed. Most previous work focused on protocol design or heuristic-based engineering approaches. Exact analysis turns out to be hard, as it involves heavy combinatorics computation. In this article, we develop an analytical model and use asymptotic analysis to address the problem of optimal load distribution. Using asymptotic approximations, we propose a closed-form formulation for the optimal load distribution problem. We then develop interesting properties of the optimal solution based on majorization, interchanging argument, and optimization techniques. These results are surprisingly simple yet insightful. We further demonstrate through simulation that our asymptotic solution works quite well in practice.

For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function . For such models, we study some properties of multivariate aging of that are described by means of the multivariate aging function , which is a useful tool for describing the level curves of . Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals.

The goal of this work is twofold. We show the following:

1. 1. In spite of the common consensus on the classic Markov chain Monte Carlo (MCMC) as a universal tool for generating samples on complex sets, it fails to generate points uniformly distributed on discrete ones, such as that defined by the constraints of integer programming. In fact, we will demonstrate empirically that not only does it fail to generate uniform points on the desired set, but typically it misses some of the points of the set.

2. 2. The splitting, also called the cloning method – originally designed for combinatorial optimization and for counting on discrete sets and presenting a combination of MCMC, like the Gibbs sampler, with a specially designed splitting mechanism—can also be efficiently used for generating uniform samples on these sets. Without introducing the appropriate splitting mechanism, MCMC fails. Although we do not have a formal proof, we guess (conjecture) that the main reason that the classic MCMC is not working is that its resulting chain is not irreducible. We provide valid statistical tests supporting the uniformity of generated samples by the splitting method and present supportive numerical results.

We consider in this article an urn and ball problem with replacement, where balls are with different colors and are drawn uniformly from a unique urn. The numbers of balls with a given color are independent and identically distributed random variables with a heavy tailed probability distribution—for instance a Pareto or a Weibull distribution. We draw a small fraction p≪1 of the total number of balls. The basic problem addressed in this article is to know to which extent we can infer the total number of colors and the distribution of the number of balls with a given color. By means of Le Cam's inequality and the Chen–Stein method, bounds for the total variation norm between the distribution of the number of balls drawn with a given color and the Poisson distribution with the same mean are obtained. We then show that the distribution of the number of balls drawn with a given color has the same tail as that of the original number of balls. Finally, we establish explicit bounds between the two distributions when each ball is drawn with fixed probability p.