Consider a genetic population evolving in time. In particular, suppose that the population is very large, that there is a large number of genetic types (alleles) possible for the genes making up the population, and that the numbers of the various alleles change stochastically in time, subject to Mendel's law of inheritance. Assume that there are no selective differences among the genotypes, but that mutation may occur among the alleles.

The stationary probabilities for certain networks of queues as defined by Kelly [4] were recently shown by Barbour [1] to depend on the service-time distributions involved only through their means. This type of insensitivity has been studied by König and Jansen [5] for a general class of stochastic processes. Kelly's networks yield special cases of such processes. We point this out in the present paper, thus shedding new light on the insensitivity phenomenon observed in these networks and its connection with the phenomenon of local balance. As a consequence of our recent study [8] we also obtain a new insensitivity result for these networks.

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

Symmetric diffusions on manifolds with boundary are studied. Symmetric diffusions are nicer to work with than non-symmetric diffusions because (1) it is easier to tell if an equilibrium density exists, and (2) it is easier to find the equilibrium density when it does exist. If an equilibrium density exists, a symmetric diffusion is time reversible. On the line essentially all diffusions are symmetric.

Using symmetric diffusions, it is shown that a large family of densities can be realized as equilibrium densities of time-reversible diffusions. Some examples are given.

Consider a pair of single server queues arranged in series. (This is the simplest example of a queuing network.) In an earlier paper [2], a limit theorem was proved to justify a heavy traffic approximation for the (two-dimensional) equilibrium waiting-time distribution. Specifically the waiting-time distribution was shown to be approximated by the limit distribution F of a certain vector stochastic process Z. The process Z was defined as an explicit, but relatively complicated, transformation of vector Brownian motion, and the general problem of determining F was left unsolved.

It is shown in this paper that Z is a diffusion process (continuous strong Markov process) whose state space S is the non-negative quadrant. On the interior of S, the process behaves as an ordinary vector Brownian motion, and it reflects instantaneously at each boundary surface (axis). At one axis, the reflection is normal, but at the other axis it has a tangential component as well. The generator of Z is calculated.

It is shown that the limit distribution F is the solution of a first-passage problem for a certain dual diffusion process Z∗. The generator of Z∗ is calculated, and the analytical theory of Markov processes is used to derive a partial differential equation (with boundary conditions) for the density f of F. Necessary and sufficient conditions are found for f to be separable (for the limit distribution to have independent components). This extends slightly the class of explicit solutions found previously in [2]. Another special case is solved explicitly, showing that the density is not in general separable.

The stationary characteristics of an n-component periodic review system which is subject to stochastic deterioration (but not to failure) are investigated. When the n-component vector which expresses the state of deterioration of the system pierces a certain surface the entire multicomponent system is replaced by items of identical cost structure at the time of the next review. In the absence of this situation nothing is replaced. It is assumed that there is a fixed cost associated with each replacement and that the operating cost of each item is a strictly increasing function of its state of deterioration. The conditions for minimizing the long-term cost of maintaining a system which operates under the stated policy were found through solution of a problem in variational calculus. Two examples are worked. A useful graph which aids in the solution of such problems is provided.

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

The annual premium for a life assurance contract is obtained by equating the expected discounted value of the sum payable on death to the expected discounted value of the series of premiums, and loading the resultant net premium for expenses and contingencies. The approach is essentially deterministic. An adequate stochastic model of mortality is readily available, but it is of limited practical value. A life office writes a large number of policies on independent lives and, apart from the effects of a few very large policies, the overall mortality behaviour of its portfolio is effectively deterministic.

The theoretical analyses of the extensive air showers developing from the cosmic radiation has its origins in the work of Carlson and Oppenheimer (1937) and Bhabha and Heitler (1937), at a time when it was thought that such showers were initiated by electrons. The realization that protons and other nuclei were the primary particles led to a reformulation of the theory by Heitler and Janossy (1949), Messel and Green (1952) and others, in which the production of energetic pions and the three-dimensional development of air showers were accounted for. But as the soft (electromagnetic) component of the cosmic radiation is the most prominent feature of air showers at sea level, there has been a sustained interest in the theory of this component. Most of the more recent work, such as that by Butcher and Messel (1960) and Thielheim and Zöllner (1972) has relied on computer simulation; but this method has disadvantages in terms of accuracy and presentation of results, especially where a simultaneous analysis of the development of air showers in terms of several physical variables is required. This is so for instance when the time of arrival is one of the variables. Moyal (1956) played an important part in the analytical formulation of a stochastic theory of cosmic ray showers, with time as an explicit variable, and it is essentially this approach which will be adopted in the following. The actual distribution of arrival times is cosmic ray showers, for which results are obtained, is of current experimental interest (McDonald, Clay and Prescott (1977)).

Many species of birds have a characteristic clutch size which is either fixed at k or is of the form k or k + 1 for some appropriate integer k. For example, the ground-living pigeons lay two eggs, the white stork four and the sulphur-crested cockatoo two and rarely three. The spotted nightjar lays one egg and the white-tailed nightjar lays two. The (European) swift lays two or three eggs, the (Australian) wood swallows lay three or four eggs and a number of (Australian) finches lay five or six eggs. These clutch sizes should be the result of evolutionary forces and it could be expected that the clutch size, for a particular species and environment, is the one which results in maximization of the probability of survival of the species to each time point t. It is consequently of interest to see whether it is possible to prescribe a population process which provides a reasonable description of a bird population and which exhibits this maximization behaviour when the clutch size is either k or is k or k + 1. Such a model has been suggested by Heyde and Schuh (1978) and it is the object of the present note to indicate how this model may be extended while still retaining the desired characteristic features.

Consider a set S of countable non-negative matrices satisfying the property that for any two indices i, j, for some n ≧ 1 there are matrices M1, M2, · · ·, Mn in S with (M1M2 · · · Mn)ij >0. For non-negative vectors x set Tx = supM∈SMx, where the supremum is taken separately in each coordinate. Assume that for each x with Tx finite in each coordinate there is a matrix in S which achieves the supremum simultaneously for all coordinates. With these two assumptions on S, the R-theory for a countable irreducible matrix is extended to the operator T. The results are used to consider the existence of stationary optimal policies for Markov decision processes with multiplicative rewards.

Let qn(t) be the conditioned probability of finding a birth-and-death process in state n at time t, given that absorption into state 0 has not occurred by then. A family {q1(t), q2(t), · · ·} that is constant in time is a quasi-stationary distribution. If any exist, the quasi-stationary distributions comprise a one-parameter family related to quasi-stationary distributions of finite state-space approximations to the process.

The paper studies the problem of optimal adaptation of an M/M/1 queueing station, when the arrival rate λ0 of customers shifts at unknown epoch, τ, to a known value, λ1. The service intensity of the system starts at μ0 and can be increased at most N times to μ1 < μ2 < · · · < μN. The cost structure consists of the cost changing μi to μj (i + 1 ≦ j ≦ N); of maintaining service at rate μ (per unit of time) and of holding customers at the station (per unit of time). Adaptation policies are constrained by the fact that μ can be only increased. A Bayes solution is derived, under the prior assumption that τ has an exponential distribution. This solution minimizes the total expected discounted cost for the entire future.

Starting with a stationary point process on the line with points one unit apart, simultaneously replace each point by a point located uniformly between the original point and its right-hand neighbor. Iterating this transformation, we obtain convergence to a limiting point process, which we are able to identify. The example of the uniform distribution is for purposes of illustration only; in fact, convergence is obtained for almost any distribution on [0, 1]. In the more general setting, we prove the limiting distribution is invariant under the above transformation, and that for each such transformation, a large class of initial processes leads to the same invariant distribution. We also examine the covariance of the limiting sequence of interval lengths. Finally, we identify those invariant distributions with independent interval lengths, and the transformations from which they arise.