Optimal control of stochastic bandwidth-sharing networks is typically difficult. In order to facilitate the analysis, deterministic analogues of stochastic bandwidth-sharing networks, the so-called fluid models, are often taken for analysis, as their optimal control can be found more easily. The tracking policy translates the fluid optimal control policy back to a control policy for the stochastic model, so that the fluid optimality can be achieved asymptotically when the stochastic model is scaled properly. In this work we study the efficiency of the tracking policy, that is, how fast the fluid optimality can be achieved in the stochastic model with respect to the scaling parameter. In particular, our result shows that, under certain conditions, the tracking policy can be as efficient as feedback policies.

The problem of the optimal control of the Wiener process in Rn is considered. The optimal value of the control is obtained from the mathematical expectation of a quantity defined in terms of the moment and the place where the uncontrolled process hits the boundary of the continuation region for the first time. Explicit results are presented.

An array of points {Z1, Z2, …, Zn–1} in the interval V = [0, L], is such that 0 ≦ Z1, ≦ Z2 ≦ … ≦ Zn–1, ≦ L. One of the points is chosen at random (Zk, say, with probability pk) and displaced to a new position within the interval [Zk–1, Zk+ 1], the position again chosen at random according to a probability distribution Gk. We derive some results concerning the limiting distribution of the array after a succession of such displacements. If Gk is a uniform distribution, it appears that the number of displacements necessary to open up a gap between at least one pair of adjacent points of size at least γ is O(ρ n), n →∞, where ρ = L/(L – γ).

A random stream of convex particles in , all moving uniformly in the same direction, is given. A convex test body is brought into the stream by means of a random rigid motion, independent of the stream. The number of particles that hit the test body at a preassigned boundary set within unit time is a random variable. Its expected value is determined under different model assumptions.

This paper gives necessary and sufficient conditions for the convergence in distribution of sums of the 0–1 Markov chains to a compound Poisson distribution.

In this article we examine the behavior of Mm(τ), the number of susceptibles remaining uninfected at time τ in a carrier-borne epidemic as the initial number of susceptibles m tends to infinity for fixed initial number of carriers n. We show that as m → ∞ the proportion of susceptibles process {Mm(τ)/m : ≧ 0} converges weakly to a limiting process. The limiting process has sample functions almost surely of the form exp (– h(τ)), where the form of h(τ) is determined by the removal times of the carriers.

Random digits are collected one at a time until a pattern with given digits is obtained. Blom (1982) and others have determined the mean waiting time for such a pattern. It is proved that when a given pattern has larger mean waiting time than another pattern, then the waiting time for the former is stochastically larger than that for the latter. An application is given to a coin-tossing game.

We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned with counts of the number of vertices of a particular containment level.

For the local containment profile, we obtain the exact mean, variance, and probability distribution in terms of standard combinatorial quantities such as generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution.

As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers. The orders in the variance–covariance matrix establish concentration laws.

The purpose of this paper is to explore the properties of a new model which can describe the interaction among individuals. Various other models have been proposed, but all assume that each individual acts independently of all others. Our model permits the assumption that individuals can be influenced and dominated by other individuals. We also demonstrate how this model can describe the interaction of various species in an ecosystem. This model will lead naturally to the definition of a population index of diversity. A means of estimating the values of the model will also be given.

A drug addiction process in which individuals in a closed population can become addicts or pushers is modelled. Expressions for the state probabilities and factorial moments are obtained. A generalized model is also developed. This generalized drug addiction process can be viewed as a particular case of what may be called a competing death process.

We present a new and straightforward algorithm that simulates exact sample paths for a generalized stress-release process. The computation of the exact law of the joint inter-arrival times is detailed and used to derive this algorithm. Furthermore, the martingale generator of the process is derived, and induces theoretical moments which generalize some results of [3] and are used to demonstrate the validity of our simulation algorithm.

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

A method for proving the invariance of hitting-time distributions with respect to the control strategy of a Markov chain is presented. The method is applied to resource-sharing problems. It provides a new proof of a known result and extends it. Sufficient conditions for such an invariance are given and are illustrated by examples.

Let {(Jn, Xn), n ≧ 0} be the standard J–X process of Markov renewal theory. Suppose {Jn, n ≧ 0} is irreducible, aperiodic and positive recurrent. It is shown using the strong mixing condition, that if converges in distribution, where an, bn > 0 (bn → ∞) are real constants, then the limit law F must be stable. Suppose Q(x) = {PijHi(x)} is the semi-Markov matrix of {(JnXn), n ≧ 0}. Then the n-fold convolution, Q∗n(bnx + anbn), converges in distribution to F(x)Π if and only if converges in distribution to F. Π is the matrix of stationary transition probabilities of {Jn, n ≧ 0}. Sufficient conditions on the Hi's are given for the convergence of the sequence of semi-Markov matrices to F(x)Π, where F is stable.

We consider a refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulae for the Laplace transform of its distribution. Our formulae are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. In addition, the results are used to price variable annuities with state-dependent fees.

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence.

Importance sampling has become an important tool for the computation of extreme quantiles and tail-based risk measures. For estimation of such nonlinear functionals of the underlying distribution, the standard efficiency analysis is not necessarily applicable. In this paper we therefore study importance sampling algorithms by considering moderate deviations of the associated weighted empirical processes. Using a delta method for large deviations, combined with classical large deviation techniques, the moderate deviation principle is obtained for importance sampling estimators of two of the most common risk measures: value at risk and expected shortfall.

Let $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

A discrete time Markov chain with (n + 1) states is constructed in which there are n states having different decay parameters for geometric ergodicity.

A population of susceptible individuals partitioned into m groups and exposed to a contagious disease is considered.

The progress of this simple epidemic is modeled by an m -dimensional stochastic process, whose components are the number of infective individuals in the respective groups at time t. Exact and approximate formulas for the joint and marginal state probabilities are obtained.