A formula is proved for the expected value of sum of a function of gene frequency along a sample path in the past, given the present frequency. The proof and the explanation of the formula is based on the general theory of time reversal of Markov processes. Moreover, a relation between time reversal and conditional processes is discussed, and it is shown that the fictitious drift term appears when one looks back at the history of mutants given the present frequency.

Various classifications and representations of multilocus recombination structures are delimited. A class of recombination distributions called the count–location chiasma process is parametrized by a distribution of the number of crossover events and for such crossover events by a conditional distribution of crossover locations. A number of properties and examples of this recombination structure are developed connecting orderings among the recombination mapping functions and the nature of interference.

An axiomatic derivation of the Wright–Fisher–Kimura (wfk) diffusion model for genetic drift is given using a variational principle. This is analogous to the characterization of the standard normal distribution in terms of an isoperimetric problem in the calculus of variations where the integrand is Shannon's information measure. We, on the other hand, give a Fisher-type information-theoretic interpretation of the variational principle largely motivated by the geometric approach to statistical likelihood theory due to S. V. Huzurbazzar, B. R. Rao and A. W. F. Edwards. In our process theory, the Ricci curvature tensor plays the role of information matrix. Ultimately, it is proved to be a normalized matrix of second partial derivates of the pseudodensity associated with the Christoffel velocity field. The proofs use classical projective differential geometry and depend on previous work in this series on the geometry of random drift.

We consider an individual which performs a diffusive motion in a certain state space and dies according to a state-dependent death rate. An integral equation for the survival probability is derived, and finally a sufficient criterion for the existence of an initial state is given, for which the corresponding individual dies with probability one.

The effect on cable reliability of random cyclic loading such as that generated by the wave-induced rocking of ocean vessels deploying these cables is examined. A simple model yielding exact formulas is first explored. In this model, the failure time of a single fiber under a constant load is assumed to be exponentially distributed, and the random loadings are a two-state stationary Markov process. The effect of load on failure time is assumed to follow a power law breakdown rule. In this setting, exact results concerning the distribution of bundle or cable failure time, and especially the mean failure time, are obtained. Where the fluctuations in load are frequent relative to bundle life, such as may occur in long-lived cables, it is shown that randomness in load tends to decrease mean bundle life, but it is suggested that the reduction in mean life often can be restored by modestly reducing the base load on the structure or by modestly increasing the number of elements in the bundle.

In later pages this simple model is extended to cover a broader range of materials and random loadings. Asymptotic distributions and mean failure times are given where fibers follow a Weibull distribution of failure time under constant load, and loads that are general non-negative stationary processes subject only to some mild condition of asymptotic independence. When the power law breakdown exponent is large, the mean time to bundle failure depends heavily on the exact form of the marginal probability distribution for the random load process and cannot be summarized by the first two moments of this distribution alone.

After preliminaries on Markov chains, supermartingales and potential theory (Section 1), the energy of a potential supermartingale generated by an increasing process is defined. The paper examines some properties of the energy of potentials of the form Ut = p(Xt) where p is a purely excessive function (which is also a potential of a charge) for a Markov chain (Xt). Also, the mutual energy of two potentials associated with the same Markov chain is discussed. Finally, several applications and examples are worked out in detail (Section 3).

The concept of ‘coefficient of ergodicity’, τ(P), for a finite stochastic matrix P, is developed from a standpoint more general and less standard than hitherto, albeit synthesized from ideas in existing literature. Several versions of such a coefficient are studied theoretically and by numerical examples, and usefulness in applications compared from viewpoints which include the degree to which extension to more general matrices is possible. Attention is given to the less familiar spectrum localization property: where λ is any non-unit eigenvalue of P. The essential purpose is exposition and unification, with the aid of simple informal proofs.

We showed in [2] that if an object of initial size x (x large) is subjected to a succession of random partitions, then the object is decomposed into a large number of terminal cells, each of relatively small size, where if Z(x, B) denotes the number of such cells whose sizes are points in the set B, then there exists c, (0 < ≦ 1), such that Z(x, B)x−c converges in probability, as x → ∞, to a random variable W. We show here that if a parent object of size x produces k offspring of sizes y1, y2, ···, yk and if for each k x - y1 - y2 - ··· - yk (the ‘waste’ or the ‘cover’, depending on the point of view) is relatively small, then for each n the nth cumulant, Ψn (x, B), of Z(x, B) satisfies Ψn (x, B)x-c → κn (B), as x → ∞, for some κn(B). Thus, writing N = xc, Z(x, B) has approximately the same distribution as the sum of N independent and identically distributed random variables (The determination of the distribution of the individual appears to be a difficult problem.) The theory also applies when an object of moderate size is broken down into very fine particles or granules.

In an earlier paper (Bartholomew (1977)) the problem of maintaining a grade (or age) structure by controlling the recruitment flows was discussed. Here we carry out a parallel analysis for the case where control is by promotion. We show how to calculate the probability that a structure can be maintained and investigate the performance of several strategies for keeping a structure as near as possible to that desired. As before, the results demonstrate the limitations of a deterministic analysis of the problem.

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process.

Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue.

In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

This paper is devoted to the practical implications of the theoretical results obtained in Part I [1] for queueing systems consisting of two single-server queues in series in which the service times of an arbitrary customer at both queues are identical. For this purpose some tables and graphs are included. A comparison is made—mainly by numerical and asymptotic techniques—between the following two phenomena: (i) the queueing behaviour at the second counter of the two-stage tandem queue and (ii) the queueing behaviour at a single-server queue with the same offered (Poisson) traffic as the first counter and the same service-time distribution as the second counter. This comparison makes it possible to assess the influence of the first counter on the queueing behaviour at the second counter. In particular we note that placing the first counter in front of the second counter in heavy traffic significantly reduces both the mean and variance of the total time spent in the second system.

We show that, if the input process of a general GI/G/1 queue is a positive recurrent Markov renewal process then the output process, too, is a positive recurrent Markov renewal process (the conservation property). As an application we consider a general tandem queue and prove a total variation limit theorem for the associated waiting and service times.