The classical Reed-Frost process is generalized by allowing infection probabilities to depend on current epidemic size. Such a process can be imbedded in a simple Markov process derived from i.i.d. waiting times. The final size of the epidemic has the same distribution as the time for the first crossing of a certain linear barrier of the imbedding process. The asymptotic distribution of the final size can be derived from some weak convergence results for the imbedding process. The existence of a distribution determining set of harmonic functions for these chain-binomial processes is also established.

A mathematical model for the process of recombinational repair of DNA damage is presented. Based on the model, a method is proposed for analyzing fragment mass distributions from experiments designed to detect recombinational repair in cells. The procedures developed can be used to analyze experiments involving sucrose-gradient measurements of mass distributions. The model also provides a framework for discussion of various molecular models of this repair process.

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite state n onwards, the birth and death rates are rational functions of n. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

We address the question of controlling the Brownian path in several dimensions (d≧2) by continually choosing its drift from among vectors of the unit ball in ℝd. The past and present of the path are supposed to be completely observable, while no anticipation of the future is allowed. Imposing a suitable cost on distance from the origin, as well as a cost of effort proportional to the length of the drift vector, ‘reasonable’ procedures turn out to be of the following type: to apply drift of maximal length along the ray towards the origin if the current position is outside a sphere centred at the origin, and to choose zero drift otherwise. It is shown just how to compute the radius of such a sphere in terms of the data of the problem, so that the resulting procedure is optimal.

The necessary and sufficient conditions for various modes of interactions of the upper and lower schemes with headstarts are derived. The expression for the Laplace transform of the run-length distribution (for both interacting and non-interacting schemes) is obtained and used to develop a method of analysis for general two-sided cumulative sum schemes with headstarts. The results are shown to be relevant in the case when the schemes are supplemented by Shewhart’s control limits.

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).

The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

A static model is proposed for a string thrown at random on a horizontal plane. The curvature of such a string is described by a Markov field that is derived from the Brownian bridge.

Suppose Xi≧0 are i.i.d., i = 1, 2, ···. We derive a saddlepoint approximation for P{∑N(t)k=1Xk> y} as y→∞ and t is fixed, where N(t), t≧0, is either a Poisson or a Pólya process. These results are then compared and contrasted with the well-known Esscher approximation.

In this paper upper and lower bounds for the availability and unavailability, to any level, in a fixed time interval are arrived at for multistate monotone systems based on corresponding information on the multistate components. These are assumed to be maintained and interdependent. Such bounds are of great interest when trying to predict the performance process of the system, noting that exact expressions are obtainable just for trivial systems. The bounds given generalize the existing bounds known in traditional binary theory, and represent improvements of the ones now being developed in multistate theory.

We have shown that within the setting of a difference equation it is possible to link ergodicity with stability via the physical notion of energy in the form of a Lyapunov function.

Goldstein (1974) derived an upper bound on the variance of certain non-negative functions when the first two moments of the underlying random variable are known. This bound is compared to a simple and fundamental variance bound which requires only that the range of the function be known. It is shown that Goldstein’s bound frequently exceeds the simpler bound. Finally, an interpretation of such bounds in the context of economic risk analysis is given.