A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

As an alternative to the embedding technique of T. E. Harris, S. Karlin and J. McGregor, we show that given a critical Galton–Watson process satisfying some mild assumptions, we can always construct a continuous-time Markov branching process having the same asymptotic behaviour as the given process. Thus, via the associated continuous process, additional information about the original process is obtained. We apply this technique to the study of extinction probabilities of a critical Galton–Watson process, and provide estimates for the extinction probabilities by regularly varying functions.

One way of analysing complicated Markov population processes is to approximate them by a diffusion about the deterministic path. This approximation alone may not, however, answer all the questions which might reasonably be asked. Many processes have phases, for example near boundaries, where a different approximation is required; such processes are better described by a succession of diffusion and special approximations alternately.

This paper looks at the special treatment required near a point where the deterministic equations are in equilibrium. When the equilibrium is unstable, the process will eventually wander off, and possibly follow a diffusion around another deterministic path. If the equilibrium is stable, the process will behave as if in stable equilibrium about it for a very much longer time, but may at last be trapped away from it, for instance in an absorbing state. The results presented describe the distributions of the time and place of leaving neighbourhoods of the equilibrium point. The neighbourhoods considered are large enough, in the unstable case, to make possible the link with the next phase of motion. In the stable case, exit times are shown to be so long that the possibility of exit can often be ignored in practice, and the quasi-equilibrium distribution treated as a true equilibrium. A more detailed result, showing how closely the normal approximation holds in this situation, is also provided.

We consider a birth and death process with Q-matrix of rates qm,m + 1 = mβ, qm,m − 1 = mδ, qm,m = – m(β + δ) and qm,n = 0 otherwise. We assume that 0 < δ < β and β – δ = 1. The asymptotic behavior of first-arrival time at state n given that the process is at state m at time zero is expressed in terms of polynomials and it is shown that if m < n and m is large that the first-arrival time is close to log n/m with high probability.

This paper presents proofs of the strong law of large numbers and the central limit theorem for estimators of the parameters in quite general finite-parameter linear models for vector time series. The estimators are derived from a Gaussian likelihood (although Gaussianity is not assumed) and certain spectral approximations to this. An important example of finite-parameter models for multiple time series is the class of autoregressive moving-average (ARMA) models and a general treatment is given for this case. This includes a discussion of the problems associated with identification in such models.

A dynamic co-operative two-person game with imperfect information is considered, in which the conflicting aspects of an optimal policy (choice of actions to minimise ‘own cost’, and choice of actions to optimise communication of information) can to a large extent be decoupled and solved separately. In deriving a strategy which is asymptotically optimal (in a sense explained) we derive formulae for the capacity of a vector Gaussian channel and the rate-distortion of a vector Gaussian source.

It is shown that the optimal strategy is in general non-linear in observables, but a numerical example shows that, at least in one rather special but non-trivial case, the best linear strategy may be very nearly optimal. This conclusion is made explicit by the analysis of Section 7.

In this paper, we consider the problem of controlling the intensity of a point process in order to maximize the probability that the number of points in a fixed interval equals a given integer, under the constraint that the intensity belong to some closed interval of R+.

The problem is stated as a problem of optimization on the set of probabilities over the basic measurable space of point processes, and shown to be equivalent to a problem of deterministic control. Structural results concerning the set of optimal solutions are given. The existence of the latter is proven; the control is shown to be bang-bang and a complete solution can be obtained by application of Pontryagin's Maximum Principle.

The paper reviews various aspects, mostly mathematical, concerning the output or departure process of a general queueing system G/G/s/N with general arrival process, mutually independent service times, s servers (1 ≦ s ≦ ∞), and waiting room of size N (0 ≦ N ≦ ∞), subject to the assumption of being in a stable stationary condition. Known explicit results for the distribution of the stationary inter-departure intervals {Dn} for both infinite and finite-server systems are given, with some discussion on the use of reversibility in Markovian systems. Some detailed results for certain modified single-server M/G/1 systems are also available. Most of the known second-order properties of {Dn} depend on knowing that the system has either Poisson arrivals or exponential service times. The related stationary point process for which {Dn} is the stationary sequence of the corresponding Palm–Khinchin distribution is introduced and some of its second-order properties described. The final topic discussed concerns identifiability, and questions of characterizations of queueing systems in terms of the output process being a renewal process, or uncorrelated, or infinitely divisible.

The behaviour in equilibrium of networks of queues is studied. Equilibrium distributions are obtained and in certain cases it is shown that the state of an individual queue is independent of the state of the rest of the network. The processes considered in this paper are irreversible; however, the method used to establish equilibrium distributions is one which has previously only been used when dealing with reversible processes. Results are obtained for models of communication networks, machine interference and birth-illness-death processes.