We consider a modulated fluid system with a finite state-space Markov chain Jt as modulating process and general state-dependent net input rates. We derive differential equations for the transient and the stationary distribution of (Wt, Jt), where Wt is the content process, and the corresponding Laplace transforms with respect to time. Moreover, we study the level hitting times of Wt. Our results lead to explicit formulas in the case of two modulating states.

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.

This paper addresses characteristics of finite-buffer Markov-modulated fluid processes, particularly those related to their deviant behavior. Our aim in this paper is to find rough asymptotics for the probability of a loss cycle. Apart from that, we derive some properties of the fluid process in case of the buffer contents reaching a high level (a process we call the conjugate of the original process). Our main goal is to obtain practicable methods to find the rate matrix of this conjugate process. For this purpose we use large deviations techniques, but we consider the governing eigensystem, as well, and we discuss the relation between these two approaches. We extend the analysis to the multiple source case. Finally, we use the obtained results in simulation. We examine variance reduction by importance sampling in a multiple source example. The new statistical law of the fluid process is based on the conjugate rate matrices.

In this paper, we generalize the Gaussian Variance Approximation (GVA), developed by Massey and Pender [16], to Jackson networks with abandonment. We approximate the queue length process with a multivariate Gaussian distribution and thus, we are able to estimate the mean and covariance matrix of the entire network with more accuracy than the associated fluid and diffusion limits of Mandelbaum, Massey, and Reiman [14]. We also show how the GVA method can be used to construct staffing schedules that approximately stabilize salient performance measures such as the probability of delay and the abandonment probabilities for the entire network. Unlike the work of Feldman et al. [5] which uses Monte Carlo simulation to stabilize the delay probabilities, our method does not require simulation and only requires the numerical integration of ${1 \over 2}(N^2 + 3N)$ differential equations for an N-dimensional network, which is more computationally efficient. Lastly, to confirm our approximations are accurate, we perform several numerical experiments for a wide range of parameter settings.

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.

We consider a Markov decision chain with countable state space, finite action sets, and nonnegative costs. Conditions for the average cost optimality inequality to be an equality are derived. This extends work of Cavazos-Cadena [8]. It is shown that an optimal stationary policy must satisfy the optimality equation at all positive recurrent states. Structural results on the chain induced by an optimal stationary policy are derived. The results are employed in two examples to prove that any optimal stationary policy must be of critical number form.

In this paper we develop a stochastic differential equation to describe the dynamic evolution of the congestion window size of a single TCP session over a network. The model takes into account recovery from packet losses with both fast recovery and time-outs, boundary behavior at zero and maximum window size, and slow-start after time-outs. We solve the differential equation to derive the distribution of the window size in steady state. We compare the model predictions with the output from the NS simulator.

This article deals with Rényi entropies for the residual life and the inactivity time. Monotonic properties of the entropy in order statistics, record values, and weighted distributions are investigated, and the comparison on weighted random variables is studied in terms of residual Rényi entropy as well.

In this paper, we consider optimal components grouping in series–parallel and parallel–series systems composed of k subsystems. All components in each subsystem are drawn from a heterogeneous population consisting of m different subpopulations. Firstly, we show that when one allocation vector is majorized by another one, then the series–parallel (parallel–series) system corresponding to the first (second) vector is more reliable than that of the other. Secondly, we study the impact of changes in the number of subsystems on the system reliability. Finally, we study the influence of the selection probabilities of subpopulations on the system reliability.

Models of life time observations that have conditionally independent generalized gamma distributions are investigated. Positive dependency among the unconditional joint distribution is discussed. General formula for moment inequalities are given. Conditional hazard rate, mean residual life functions, and bounds for system reliabilities are also discussed. An application was given.

In this paper, a new concept called generalized stochastic convexity is introduced as an extension of the classic notion of stochastic convexity. It relies on the well-known concept of generalized convex functions and corresponds to a stochastic convexity with respect to some Tchebycheff system of functions. A special case discussed in detail is the notion of stochastic s-convexity (s ∈ [real number symbol]), which is obtained when this system is the family of power functions {x0, x1,..., xs−1}. The analysis is made, first for totally positive families of distributions and then for families that do not enjoy that property. Further, integral stochastic orderings, said of Tchebycheff-type, are introduced that are induced by cones of generalized convex functions. For s-convex functions, they reduce to the s-convex stochastic orderings studied recently. These orderings are then used for comparing mixtures and compound sums, with some illustrations in epidemic theory and actuarial sciences.

Given a collection of n points in the plane, the Euclidean matching problem is the task of decomposing the collection into matched pairs connected by line segments in such a way as to minimize the sum of all the segment lengths. The greedy heuristic provides an approximate solution to the Euclidean matching problem by successively matching the two closest unmatched points. We study the behavior of Gn, the sum of the lengths of the segments produced by the greedy heuristic.

We analyze a mean field game model of SIR dynamics (Susceptible, Infected, and Recovered) where players choose when to vaccinate. We show that this game admits a unique mean field equilibrium (MFE) that consists in vaccinating at a maximal rate until a given time and then not vaccinating. The vaccination strategy that minimizes the total cost has the same structure as the MFE. We prove that the vaccination period of the MFE is always smaller than the one minimizing the total cost. This implies that, to encourage optimal vaccination behavior, vaccination should always be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ($N$) to the MFE. These experiments show that the convergence rate of the cost is $1/N$ and the convergence of the switching curve is monotone.

To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Multivariate Analysis, 2002). It is shown that if two random vectors have a common copula and if their marginal distributions are ordered according to dispersive ordering in the same direction, then the two random vectors are ordered according to this new upper orthant dispersive ordering. Also, it is shown that the marginal distributions of two upper orthant dispersive ordered random vectors are also dispersive ordered. Examples and applications are given.

A repair crew is responsible for the maintenance and operation of N installations. The crew has to perform a collection of preventive maintenance tasks at the various installations. The installations may break down from time to time, generating corrective maintenance requests which have priority over the preventive maintenance tasks. We formulate and analyze this real-world problem as a single-server multi-queue polling model with Globally Gated service discipline and with server interruptions. We derive closed-form expressions for the Laplace-Stieltjes Transform and the first moment of the waiting time distributions of the preventive and corrective maintenance requests at the various installations, and obtain simple and easily implementable static and dynamic rules for optimal operation of the system. We further show that, for the socalled elevator-type polling scheme, mean waiting times of preventive maintenance jobs at all installations are equal.

The control and operation of an electric power system is based on the ability to determine the state of the system in real time. State estimation (SE) has been introduced in the 1960s to achieve this objective. The initial implementation was based on single-phase measurements and a power system model that was assumed to operate under single-frequency, balanced conditions, and a symmetric system model. These assumptions are still prevalent today. The single-frequency, balanced, and symmetric system assumptions have simplified the implementation but have generated practical problems. The experience is that the SE problem does not have 100% performance; that is, there are cases and time periods for which the SE algorithm will not converge. There are practical and theoretical reasons for this and they are explained in the paper. Recent mergers and mandated regional transmission organizations (RTOs) as well as recent announcements for the formation of mega-RTOs will result in the application of the SE in systems of unprecedented size. We believe that these practical and theoretical issues will become of greater importance. There are scientists who believe that the SE problem is scalable, meaning that it will work for the mega-RTOs the same way that it performs now for medium–large systems. There are scientists who believe that this is not true. The fact is that no one has investigated the problem, let alone performed numerical experiments to prove or disprove any claims. This paper identifies a number of issues relative to the SE of mega-RTOs and provides some preliminary results from numerical experiments for the relation between the SE algorithm performance and the power system size.

This paper surveys research on mathematical models that predict the performance of digital devices that operate with intermittent energy sources. The approach taken in this work is based on the “Energy Packet Network” paradigm where the arrival of data to be processed or transmitted, and the energy to operate the system, are modeled as discrete random processes. Our assumption is that these devices will capture energy from intermittent ambient sources such as vibrations, heat or light, and capture it onto electrical energy that may be stored in batteries or capacitors. The devices consume this energy intermittently for processing and for wired or wireless transmission. Thus, both the arrival of energy to the device, and the devices workload, are modeled as random processes. Based on these assumptions, we discuss probability models based on Markov chains that can be used to predict the effective rates at which such devices operate. We also survey related work that models networks of such systems.

We consider a continuous review (s, S) model of perishable items with lost sales. Once items are perished the entire inventory drops instantaneously to zero. The total cost includes the cost of: ordering, unsatisfied demand, units destroyed, holding, and fixed cost of perishability. Both the time to perishability and the lead times are assumed to be exponentially distributed while two cases of demand distribution are considered: Poisson and compound Poisson with general demand sizes. We study the average cost criterion and provide computational results on the problem of finding the optimal re-order level, s, and order up-to level, S. None of the known work on the subject is as general as the model presented here. Our analysis leads to several insights on the optimal (s, S) policies for perishable items in the presence of lead times. For example, we demonstrate that the effectiveness of a heuristic that ignores perishability (and is also analyzed here) decreases with the demand variability and that the cost may either increase or decrease with this variability.

In this article, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of the likelihood ratio order and the hazard rate order. Let X1 and X2 be two independent gamma random variables with Xi having shape parameter r>0 and scale parameter λi, i=1, 2, and let X*1 and X*2 be another set of independent gamma random variables with X*i having shape parameter r and scale parameter λ*i, i=1, 2. Denote by X2:2 and X*2:2 the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ1, λ2) weakly majorize (λ*1, λ*2), then X2:2 is stochastically greater than X*2:2 in the sense of likelihood ratio order. We also establish, among others, that if 0<r≤1 and (λ1, λ2) is p-larger than (λ*1, λ*2), then X2:2 is stochastically greater than X*2:2 in the sense of hazard rate order. The results derived here strengthen and generalize some of the results known in the 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.