The stochastic cash balance problem is a periodic review inventory problem faced by a firm in which the customer demands might be positive or negative. At the beginning of each time period, the firm may decide to replenish the inventory or return excess stock. Both the ordering cost and the return cost include a fixed component and a variable component. A holding or penalty cost is charged depending on whether the inventory level is positive or negative. The objective of the firm is to find an ordering and return policy so as to minimize the total expected cost over the entire planning horizon. We show how the concept of symmetric K-convexity introduced by Chen and Simchi-Levi [2,3] and the concept of (K, Q)-convexity introduced by Ye and Duenyas [13] can be used to characterize the optimal policy for this problem.

In this article we investigate less restrictive conditions on the model parameters and the underlying distribution function upon which the generalized order statistics are based, which enable one to establish the likelihood ratio and the hazard ratio orderings for p-spacings of generalized order statistic. Some previous works in the literature are extended.

Congestion and its uncertainty are big factors affecting customers’ decision to join a queue or balk. In a queueing system, congestion itself is resulted from the aggregate joining behavior of other customers. Therefore, the property of the whole group of arriving customers affects the equilibrium behavior of the queue. In this paper, we assume each individual customer has a utility function which includes a basic cost function, common to all customers, and a customer-specific weight measuring sensitivity to delay. We investigate the impacts on the average customer utility and the throughput of the queueing system of different cost functions and weight distributions. Specifically, we compare systems where these parameters are related by various stochastic orders, under different information scenarios. We also explore the relationship between customer characteristics and the value of information.

We consider a network of parallel service stations each modeled as a single-server queue. Each station serves its own dedicated customers as well as generic customers who are routed from a central controller. We suppose that the cost incurred by a customer is an increasing function of her time spent in the system. In a significant advance on most previous work, we do not require waiting costs to be convex, still less linear. With the objective of minimizing the long-run average waiting cost, we develop two heuristic routing policies, one of which is based on dynamic programming policy improvement and the other on Lagrangian relaxation. In developing the latter policy, we show that each station is “indexable” under mild conditions for customers’ waiting costs and also prove some structural results on the admission control problem that naturally arises as a result of the Lagrangian relaxation. We then test the performance of our heuristics in an extensive numerical study and show that the Lagrangian heuristic demonstrates a strong level of performance in a range of traffic conditions. In particular, it clearly outperforms both a greedy heuristic, which is a standard proposal in complex routing problems, and a recent proposal from the heavy traffic literature.

Let X1, …, Xn be independent exponential random variables with their respective hazard rates λ1, …, λn, and let Y1, …, Yn be independent exponential random variables with common hazard rate λ. Denote by Xn:n, Yn:n and X1:n, Y1:n the corresponding maximum and minimum order statistics. Xn:n−X1:n is proved to be larger than Yn:n−Y1:n according to the usual stochastic order if and only if with . Further, this usual stochastic order is strengthened to the hazard rate order for n=2. However, a counterexample reveals that this can be strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo and Khaledi and Kochar.

We develop a general framework for understanding the nonparametric (aging) properties of nonnegative random variables through the notion of intrinsic aging. We also introduce some new notions of aging. Many classical and more recent results are special cases of our general results. Our general framework also leads to new results for existing notions of aging, as well as many results for our new notions of aging.

In this article we study Markov-modulated queues with and without jumps. These queuing models arise naturally in production-inventory systems with and without an external supplier. We show an interesting decomposition property that relates the equilibrium state distributions in these two systems and present an integrated warranty-inventory management model as an application.

Consider a continuous nonnegative random variable X with mean μ and hazard function h. Assume further that a≤h(t)≤b for all t≥0. Under these constraints, we obtain sharp two-sided bounds for . An application to birth and death processes is discussed.

We consider order statistics corresponding to X1, …, Xn, where , i = 1, …, n, ℰ1, …, ℰn are independent and identically distributed exponentials with mean 1, and λ1, …, λn are possibly dependent, possibly nonidentically distributed, positive random variables, with . Thus, λ1, …, λn, can be interpreted as random failure rates, and their dependency might be due to common environmental factors.

In this article, we investigate the sufficient and/or necessary conditions in order to stochastically compare the order statistics and their spacing vectors of two random vectors X and Y with special symmetric distributions. The conditions are imposed on the sample ranges Xn:n–X1:n and Yn:n–Y1:n or on (X1:n, Xn:n–X1:n) and (Y1:n, Yn:n–Y1:n). In particular, we consider the multivariate usual stochastic order, the convex order, the increasing convex order, and the directionally convex order. Several examples are also given to illustrate the power of the main results.

We study the effects of different pricing strategies available to a production–inventory system with capacitated supply, which operates in a fluctuating demand environment. The demand depends on the environment and on the offered price. For such systems, three plausible pricing strategies are investigated: static pricing, for which only one price is used at all times, environment-dependent pricing, for which price changes with the environment, and dynamic pricing, for which price depends on both the current environment and the stock level. The objective is to find an optimal replenishment and pricing policy under each of these strategies. This article presents some structural properties of optimal replenishment policies and a numerical study that compares the performances of these three pricing strategies.

We introduce and investigate a new type of decision problem related to multiclass fluid networks. Optimization problems arising from fluid networks with known parameters have been studied extensively in the queueing, scheduling, and optimization literature. In this article, we explore the makespan problem in fluid networks, with the assumption that the parameters are known only through a probability distribution. Thus, the decision maker does not have complete knowledge of the parameters in advance. This problem can be formulated as a stochastic nonlinear program. We provide necessary and sufficient feasibility conditions for this class of problems. We also derive a number of other structural results that can be used in developing effective computational procedures for solving stochastic fluid makespan problems.

We consider a multiserver delay queue with finite additional waiting spaces and time-varying arrival rates, where the customers waiting in the buffer may abandon. These are features that arise naturally from the study of service systems such as call centers. Moreover, we assume rewards for successful service completions and cost rates for service resources. Finally, we consider service-level agreements that constrain both the fractions of callers who abandon and the ones who are blocked.

Applying the theory of Lagrangian mechanics to the fluid limit of a related Markovian service network model, we obtain near-profit-optimal staffing and provisioning schedules. The nature of this solution consists of three modes of operation. A key step in deriving this solution is combining the modified offered load approximation for loss systems with our fluid model. We use them to estimate effectively both our service-level agreement metrics and the profit for the original queuing model. Second-order profit improvements are achieved through a modified offered load version of the conventional square root safety rule.

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n→∞: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalization of the classical Perron–Frobenius theorem to imprecise Markov chains.

Consider a three-person game that occurs in stages. The state of the game is given by the integral amounts of chips that the players have, say x=(x1, x2, x3) with M=x1+x2+x3 fixed. At a stage of the game, player i places ai chips in the pot, an integer between 1 and xi. (Player i is already eliminated from the game if xi=0.) The winner of the pot is then immediately chosen in such a way that player i wins the pot with probability proportional to the index wiai for i with xi>0. The idea is that if player i bets more, then he is more likely to win, but this is modified by weights that parameterize the players’ abilities.

Each player is trying to maximize his probability of taking all the chips (i.e., reaching xi=M). In the two-person game, it is known that a Nash equilibrium is for each player to adopt strategy σ of playing timidly (ai=1) or boldly (ai=xi) according to whether the game is in his favor or not (assuming the other also plays σ). In this article, we investigate whether this also is the form of a Nash equilibrium in a three-person game when the weights are of the form (w1, w2, w3)=(w, w, 1−2w) with 0<w<1/2. It turns out that this is true if w<1/3, but not true if w>1/3 and M≥8.

In this article we consider a stochastic game in which each player draws one or two random numbers between 0 and 1. Players can decide to stop after the first draw or to continue for a second draw. The decision is made without knowing the other players’ numbers or whether the other players continue for a second draw. The object of the game is to have the highest total score without going over 1. In the article, we will characterize the optimal stopping rule for each player.

Expressions for the distribution, density, and percentiles of weighted sums of Rayleigh random variables are given, including the tilted Edgeworth expansion.

This article offers an approach for studying the time-dependent occupancy distribution for a modest generalization of the GI/G/1 queuing system in which interarrival times and service times, although mutually independent, are not necessarily identically distributed. We develop and explore an analytical model leading to a computational approach that gives tight bounds on the occupancy distribution. Although there is no general closed-form characterization of probability law dynamics for occupancy in the GI/G/1 queue, our results offer what might be termed “near-closed-form” in that accurate plots of the transient occupancy distribution can be constructed with an insignificant computational burden. We believe that our results are unique; we are unaware of any alternative analytical approach leading to a numerical characterization of the time-dependent occupancy distribution for the G/G/1 queuing systems considered here.

Our analyses employ a marked point process that converges to the occupancy process at any fixed time t; it is shown that this process forms a Markov chain from which the transient occupancy law is available. We verify our analytical approach via comparison with the well-known closed-form expressions for time-dependent occupancy distribution of the M/M/1 queue. Additionally, we suggest the viability of our approach, as a computational means of obtaining the time-dependent occupancy distribution, through straightforward application to a Gamma[x]/Weibull/1 queuing system having batch arrivals and batch job services.

We consider a model in which each round consists of a sequence of games, with each game resulting in either a positive or a zero score. If a zero score occurs, then the current round is ended with no points being accumulated during that round. If a game ends with a positive score, then the player can either end that round or play another game in the round. If she elects to end the round, then the sum of all scores earned in games played during that round are added to her cumulative score and a new round begins.

Under the assumption that successive game scores are independent and identically distributed random variables whose conditional distribution, given that it is positive, is exponential, we consider this problem under such objectives as minimizing the expected number of rounds until a cumulative score exceeds a given goal g and maximizing the probability that a cumulative score of at least g is obtained by the end of round n. We present the model in the hypothetical context of a clinical trial of a treatment for reducing glycated hemoglobin in diabetic patients.