A new class of iterated function systems is introduced, which allows for the computation of non-compactly supported invariant measures, which may represent, for example, greytone images of infinite extent. Conditions for the existence and attractiveness of invariant measures for this new class of randomly iterated maps, which are not necessarily contractions, in metric spaces such as , are established. Estimates for moments of these measures are obtained.

Special conditions are given for existence of the invariant measure in the interesting case of affine maps on . For non-singular affine maps on , the support of the measure is shown to be an infinite interval, but Fourier transform analysis shows that the measure can be purely singular even though its distribution function is strictly increasing.

We present a method of approximating Markov jump processes which was used by Fuhrmann [7] in a special case. We generalize the method and prove weak convergence results under mild assumptions. In addition we obtain bounds on the rates of convergence of the probabilities at arbitrary fixed times. The technique is demonstrated using a state-dependent branching process as an example.

Generalized perturbation analysis (PA) estimates to study sensitivity of performance measures of discrete events dynamic systems for discontinuous sample functions are introduced. Their convergence conditions and rate of convergence are given. It is shown that the PA estimates based on a single sample path always converge faster to the unknown sensitivity parameter (vector of parameters) than their counterpart—crude Monte Carlo ones.

We study the effect of perturbations in the data of a discrete-time Markov reward process on the finite-horizon total expected reward, the infinite-horizon expected discounted and average reward and the total expected reward up to a first-passage time. Bounds for the absolute errors of these reward functions are obtained. The results are illustrated for a finite as well as infinite queueing systems (M/M/1/S and ). Extensions to Markov decision processes and other settings are discussed.

Consider a perturbation in the one-step transition probabilities and rewards of a discrete-time Markov reward process with an unbounded one-step reward function. A perturbation estimate is derived for the finite horizon and average reward function. Results from [3] are hereby extended to the unbounded case. The analysis is illustrated for one- and two-dimensional queueing processes by an M/M/1-queue and an overflow queueing model with an error bound in the arrival rate.

How should calls be routed or capacity allocated in a circuit-switched communication network so as to optimize the performance of the network? This paper considers the question, using a simplified analytical model of a circuit-switched network. We show that there exist implicit shadow prices associated with each route and with each link of the network, and that the equations defining these prices have a local or decentralized character. We illustrate how these results can be used as the basis for a decentralized adaptive routing scheme, responsive to changes in the demands placed on the network.

This paper shows how the Laplace transform analysis of Bailey (1954), (1957) can be continued to yield additional insights about the time-dependent behavior of the queue-length process in the M/M/1 model. A transform factorization is established that leads to a decomposition of the first moment as a function of time into two monotone components. This factorization facilitates developing approximations for the moments and determining their asymptotic behavior as . All descriptions of the transient behavior are expressed in terms of basic building blocks such as the first-passage-time distributions. The analysis is facilitated by appropriate scaling of space and time so that regulated or reflected Brownian motion (RBM) appears as the special case in which the traffic intensity ρ equals the critical value 1. An operational calculus is developed for obtaining M/M/1 results directly from corresponding RBM results as well as vice versa. The analysis thus provides useful insight about RBM approximations for queues.

We consider a single station queueing system with several customer classes. Each customer class has its own arrival process. The total service requirement of each customer is divided into a (possibly) random number of service quanta, where the distribution of each quantum may depend on the customer's class and the other quanta of that customer. The service discipline is round-robin, with random quanta.

We prove a heavy traffic limit theorem for the above system which states that as the traffic intensity approaches unity, properly normalized sequences of queue length and sojourn time processes converge weakly to one-dimensional reflected Brownian motion.

The present paper considers a multicompartment storage model with one-way flow. The inputs and outputs for each compartment are controlled by a denumerable-state Markov chain. Assuming finite first and second moments, it is shown that the amounts of material in certain compartments converge in distribution while for others they diverge, based on appropriate first-moment conditions on the inputs and outputs. It is also shown that the diverging compartments under suitable normalization converge to functionals of Brownian motion, independent of those compartments which converge without normalization.

It is shown that an easily calculated ergodicity coefficient of a stochastic matrix P with a unique stationary distribution πT, may be used to assess sensitivity of πT to perturbation of P.