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.

Consider a generalized queueing network model that is subject to two types of arrivals. The first type represents the regular customers; the second type represents signals. A signal induces a regular customer already present at a node to leave. Gelenbe [5] showed that such a network possesses a product form solution when each node consists of a single exponential server. In this paper we study a number of issues concerning this class of networks. First, we explain why such networks have a product form solution. Second, we generalize existing results to include different service disciplines, state-dependent service rates, multiple job classes, and batch servicing. Finally, we establish the relationship between these networks and networks of quasi-reversible queues. We show that the product form solution of the generalized networks is a consequence of a property of the individual nodes viewed in isolation. This property is similar to the quasi-reversibility property of the nodes of a Jackson network: if the arrivals of the regular customers and of the signals at a node in isolation are independent Poisson, the departure processes of the regular customers and the signals are also independent Poisson, and the current state of the system is independent of the past departure processes.

This article provides series expansions of the stationary distribution of a finite Markov chain. This leads to an efficient numerical algorithm for computing the stationary distribution of a finite Markov chain. Numerical examples are given to illustrate the performance of the algorithm.

A duality is presented for real-valued stochastic sequences [Vn] defined by a general recursion of the form Vn+1 = f(Vn, Un), with [Un] a stationary driving sequence and f nonnegative, continuous, and monotone in its first variable. The duality is obtained by defining a dual function g of f, which if used recursively on the time reversal of [Un] defines a dual risk process. As a consequence, we prove that steady-state probabilities for Vn can always be expressed as transient probabilities of the dual risk process. The construction is related to duality of stochastically monotone Markov processes as studied by Siegmund (1976, The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes, Annals of Probability 4: 914–924). Our method of proof involves an elementary sample-path analysis. A variety of examples are given, including random walks with stationary increments and two reflecting barriers, reservoir models, autoregressive processes, and branching processes. Finally, general stability issues of the content process are dealt with by expressing them in terms of the dual risk process.

We compare and test statistical estimates of failure propagation in data from versions of a probabilistic model of loading-dependent cascading failure and a power system blackout model of cascading transmission line overloads. The comparisons suggest mechanisms affecting failure propagation and are an initial step toward monitoring failure propagation from practical system data. Approximations to the probabilistic model describe the forms of probability distribution of cascade sizes.

We investigate some new properties of mean inactivity time (MIT) order and increasing MIT (IMIT) class of life distributions. The preservation property of MIT order under increasing and concave transformations, reversed preservation properties of MIT order, and IMIT class of life distributions under the taking of maximum are developed. Based on the residual life at a random time and the excess lifetime in a renewal process, stochastic comparisons of both IMIT and decreasing mean residual life distributions are conducted as well.

In this article, mixture representations of survival functions of residual lifetimes of k-out-of-n systems are obtained when the components are independent but not necessarily identically distributed. Then we stochastically compare the residual lifetimes of k-out-of-n systems in one- and two-sample problems. In particular, the results extend some results in Li and Zhao [14], Khaledi and Shaked [13], Sadegh [17], Gurler and Bairamov [7] and Navarro, Balakrishnan, and Samaniego [16]. Applications in the proportional hazard rates model are presented as well.

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.

The spectral approach to first passage time distributions for Markov processes requires knowledge of the eigenvalues and eigenvectors of the infinitesimal generator matrix. We demonstrate that in many cases knowledge of the eigenvalues alone is sufficient to compute the first passage time distribution.

This paper investigates extensions to feed-forward queueing networks of an algorithm to set staffing levels (the number of servers) to stabilize performance % at Quality of Service (QoS) targets in an Mt/GI/st+GI multi-server queue with a time-varying arrival rate. The model has a non-homogeneous Poisson process (NHPP), customer abandonment, and non-exponential service and patience distributions. For a single queue, simulation experiments showed that the algorithm successfully stabilizes abandonment probabilities and expected delays over a wide range of Quality-of-Service (QoS) targets. A limit theorem showed that stable performance at fixed QoS targets is achieved asymptotically as the scale increases (by letting the arrival rate grow while holding the service and patience distributions fixed). Here we extend that limit theorem to a feed-forward queueing network. However, these fixed QoS targets provide low QoS as the scale increases. Hence, these limits primarily support the algorithm with a low QoS target. For a high QoS target, effectiveness depends on the NHPP property, but the departure process never is exactly an NHPP. Thus, we investigate when a departure process can be regarded as approximately an NHPP. We show that index of dispersion for counts is effective for determining when a departure process is approximately an NHPP in this setting. In the important common case when all queues have high QoS targets, we show that both: (i) the departure process is approximately an NHPP from this perspective and (ii) the algorithm is effective.

Very few stochastic systems are known to have closed-form transient solutions. In this article we consider an immigration birth and death population process with total catastrophes and study its transient as well as equilibrium behavior. We obtain closed-form solutions for the equilibrium distribution as well as the closed-form transient probability distribution at any time t ≥ 0. Our approach involves solving ordinary and partial differential equations, and the method of characteristics is used in solving partial differential equations.

It is well known that mixtures of decreasing failure rate (DFR) distributions have the DFR property. A similar result is, of course, not true for increasing failure rate (IFR) distributions. In a recent note, Gurland and Sethuraman (1994, Technometrks36(4): 416–418) presented two examples where mixtures of IFR distributions show DFR property. In this paper, we present a general approach to study the mixtures of distributions and show that the failure rates of the unconditional and conditional distributions cross at most at one point. Mixtures of Weibull distribution with a shape parameter greater than 1 are examined in detail. This also enables us to study the monotonic properties of the mean residual life function of the mixture. Some examples are provided to illustrate the results.

We consider a controlled queueing system that is a generalization of the M/M/c/W queue. There are m types of customers that arrive according to independent Poisson processes. Service times are exponential and independent and do not depend on the customer type. There is room in the system for a total of N customers; if there are N customers in the system, new arrivals are lost. Type j customers are more profitable than type (j + 1 ) customers, j = 2,…, m —, and type 1 customers are at least as profitable as type 2 customers. The allowed control is to accept or reject customers at arrival. No preemption of customers in service is allowed. The goal is to maximize the average reward per unit of time subject to a constraint that the blocking probability of type 1 customers is no greater than a given level.

For an M/M/c/c system without a constraint, Miller [12] proved that an optimal policy has a simple threshold structure. We show that, for the constrained problem described above, an optimal policy has a similar structure, but one of the thresholds might have to be randomized. We also derive an algorithm that constructs an optimal policy and describe other forms of optimal policies.

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.

In the present paper, we introduce a generalization of the cumulative entropy proposed by Di Crescenzo and Longobardi [8]. This new notion is related to the lower records and the reversed relevation transform. Dynamic version of the newly proposed measure is considered. Several properties including the effect of linear transformations, a two-dimensional version, a normalized version, bounds, stochastic ordering, etc. are studied for the generalized cumulative entropy (GCE). Similar results are obtained for the dynamic GCE. Various relationships with other functions are derived. A class of distributions is introduced and several properties are studied. Finally, empirical GCE is proposed to estimate the newly proposed information measure.

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 paper we propose a new approach for estimating the transition probabilities and mean occupation times of continuous-time Markov chains. Our approach is to approximate the probability of being in a state (or the mean time already spent in a state) at time t by the probability of being in that state (or the mean time already spent in that state) at a random time that is gamma distributed with mean t.

To protect the Dutch polders against flooding, more than 2500 km of dikes have been constructed. Due to settlement, subsoil consolidation, and relative sea-level rise, these dikes slowly sink “away into the sea” and should therefore be heightened regularly (at present, every 50 years). In this respect, one is interested in safe and cost-optimal dike heightenings for which the sum of the initial cost of investment and the future (discounted) cost of maintenance is minimal.

For optimization purposes, a maintenance model has been developed for dikes subject to uncertain crest-level decline. On the basis of engineering knowledge, crest-level decline has been modeled as a monotone stochastic process with expected decline being either linear or nonlinear (i.e., linear after transformation) in time. For both models and for a particular unit time, the increments are distributed according to mixtures of exponentials.

In a case study, the maintenance decision model has been applied to the problem of heightening the Dutch “Oostmolendijk.”

In this paper, we carry out stochastic comparisons of largest order statistics from multiple-outlier exponential models according to the likelihood ratio order (reversed hazard rate order) and the hazard rate order (usual stochastic order). It is proved, among others, that the weak majorization order between the two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between largest order statistics, and that the p-larger order between the two hazard rate vectors is equivalent to the hazard rate order (usual stochastic order) between largest order statistics. We also extend these results to the proportional hazard rate models. The results established here strengthen and generalize some of the results known in the literature.

We examine several approaches to derive lower and upper bounds on tail probabilities of discrete distributions. Numerical comparisons exemplify the quality of the bounds.