In this paper we study m-discount optimality (m ≥ −1) and Blackwell optimality for a general class of controlled (Markov) diffusion processes. To this end, a key step is to express the expected discounted reward function as a Laurent series, and then search certain control policies that lexicographically maximize the mth coefficient of this series for m = −1,0,1,…. This approach naturally leads to m-discount optimality and it gives Blackwell optimality in the limit as m → ∞.

We introduce the idea of controlling branching processes by means of another branching process, using the fractional thinning operator of Steutel and van Harn. This idea is then adapted to the model of alternating branching, where two Markov branching processes act alternately at random observation and treatment times. We study the extinction probability and limit theorems for reproduction by n cycles, as n → ∞.

Félix Pollaczek was born on 1 December 1892 in Vienna, Austria; he was the descendant of a well-known mid-European Jewish family which for several centuries had counted many learned men amongst its members. He became a Frenchman by naturalization in 1947. He obtained his scientific education and training at the Latin grammar school of Vienna, and at the Technical Universities of Vienna and Brno (Czechoslovakia); he later studied mathematics at the University of Berlin. A master's degree in electrical engineering in 1920 from Brno was followed by a doctoral degree in mathematics in 1922 from Berlin with a thesis on number theory. This was concerned with the fields generated by the Ith and 12th roots of one, I being an irregular prime number. His first paper, on number theory, dates from 1917.

A Malthusian parameter for the generation-dependent general branching process is introduced and some asymptotics of the expected population size, counted by a general non-negative characteristic, are discussed. Processes both with and without immigration are considered.

A number of multi-priority jobs are to be processed on two heterogeneous processors. Of the jobs waiting in the buffer, jobs with the highest priority have the first option of being dispatched for processing when a processor becomes available. On each processor, the processing times of the jobs within each priority class are stochastic, but have known distributions with decreasing mean residual (remaining) processing times. Processors are heterogeneous in the sense that, for each priority class, one has a lesser average processing time than the other. It is shown that the non-preemptive scheduling strategy for each priority class to minimize its expected flowtime is of threshold type. For each class, the threshold values, which specify when the slower processor is utilized, may be readily computed. It is also shown that the social and the individual optimality coincide.

This paper considers a certain class of continuous-time Markov processes, whose time-dependent and stationary distributions are studied. In the stationary case, the analogy with Whittle's relaxed Markov process is pointed out. The derivation of the probability generating functions of the general process provides useful results for the analysis of some population and queueing processes.

We analyze the dynamics of a stochastic process driven by binomial random variables, where the probability of success depends on the past realization. We study the limit behavior when the group size is fixed but the number of iterations increases. It will become apparent that the so-called policy function and its fixed point play an outstanding role. Some applications to a statistical analysis of gender bias are also briefly discussed.

For each positive integer n, let Sn be the nth partial sum of a sequence of i.i.d. random variables which assume the values +1 and −1 with respective probabilities p and 1 – p, having mean μ= 2p − 1. The exact distribution of the random variable , where sup Ø= 0, is given for the case that λ > 0 and μ+ λ= k/(k + 2) for any non-negative integer k. Tables to the 99.99 percentile of some of these distributions, as well as a limiting distribution, are given for the special case of a symmetric simple random walk (p = 1/2).

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij. Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

When the interarrival and service time distributions of a queue are interchanged a new queue is obtained which can be considered as the dual of the original. Another dual system, a dam, can also be associated with the original queue. Events defined for the original system can be transformed into events defined for the duals and conversely, and hence, probabilities obtained for one system can be extended to the others. In this paper several duality relations are derived, with particular emphasis on results pertaining to a single busy period. Examples are given, most of which refer to the M/G/1 – G/M/l queues.

In the general epidemic model we study the first two moments of the final size. Beginning with the backwards equation, algebraic methods are used to find their asymptotic series expansions as the population size increases.

Realizations of the autoregressive series considered by Kendall are generated using both Gaussian and uniformly distributed random variables. Estimates of the autocorrelation and power spectrum of the process obtained from these realizations are compared with the theoretical expressions derived by Bartlett. Agreement is well within the theoretical variance.

In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed.

The standard ANOVA models with random effects for multi-indexed arrays of random variables with an arbitrary nesting structure on the indices are considered from the viewpoint of symmetry. It is found that the covariance matrix of such an array has sufficient symmetry to permit viewing the usual components of variance as a generalised spectrum and the linear models of random effects as a generalised spectral decomposition.

In a previous paper the author derived some results describing the tail-behavior of the busy time distribution function B of a stable M/G/1 queue in heavy traffic, i.e., some results were given for B (t), t large. In this paper some complementary results are given, in the same heavy-traffic situation, for B (t), t small.

Faddy (1990) has conjectured that the variability of a pure birth process is increased, relative to the linear case, if the birth rates are convex and decreased if they are concave. We prove the conjecture by relating variability to the correlation structure of certain more informative versions of the process. A correlation inequality due to Harris (1977) is used to derive the necessary positive and negative correlation results.

Bartlett has proposed two alternative forms of spatial-temporal Markov model, each leading to the standard spatial equilibrium distribution for nearest-neighbour systems on a multi-dimensional lattice. The validity of one of these forms is discussed with the aid of a lemma.

This paper describes two one-for-one ordering (S − 1, S) inventory models in which the time required for order replenishment is state-dependent. The demand is assumed to follow a compound Poisson distribution, and that portion of the leadtime corresponding to the actual filling of orders is assumed to depend on the number of outstanding orders. Since the orders placed are assumed to go into a single-server queue, queuing results are used to obtain the expected inventory cost as a function of S in order to obtain an optimal value of S.

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) (T) covered by the process in the time interval [0,T]. The Laplace transform of (T) follows as a consequence. The main proof involves a double Laplace transform of (T) and is based on excursion theory and local time for Brownian motion.