The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.

For point processes, such that the interarrival times of points are independently and identically distributed, let T(L, m) denote the time until at least points cluster within an interval of length at most L. Let τ (L, m) + 1 be the total number of points observed until the above happens. Simple approximations of Eτ (L, m) and ET(L, m) are derived, as well as lower and upper bounds for their value. Approximations to the variances are also given. In particular the Poisson, Bernoulli and compound Poisson processes are discussed in detail. Some numerical tables are included.

The Pólya process is employed to illustrate certain features of the structure of infinitely divisible stochastic point processes in connection with the representation for the probability generating functional introduced by Milne and Westcott in 1972. The Pólya process is used to provide a counterexample to the result of Ammann and Thall which states that the class of stochastic point processes with the Milne and Westcott representation is the class of regular infinitely divisble point processes. So the general representation problem is still unsolved. By carrying the analysis of the Pólya process further it is possible to see the extent to which the general representation is valid. In fact it is shown in the case of the Pólya process that there is a critical value of a parameter above which the representation breaks down. This leads to a proper version of the representation in the case of regular infinitely divisible point processes.

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r1, r2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

Let be a continuous homogeneous stochastic process with independent increments. A review of the recent work on the characterization of Wiener and stable processes and connected results through stochastic integrals is presented. No proofs are given but appropriate references are mentioned.

It is shown that many properties of Markov processes can be extended to the case where the time parameter is multidimensional (a partially ordered set). This includes Kolmogorov's backward and forward equations, and semi-group treatment of homogeneous processes.

The non-parametric discrete-time estimation of the covariance function R(t) of stationary continuous-time processes is considered. The characteristics of the sampling instants necessary for the consistent estimation of R(t) are explored. A class of covariance estimates is introduced and its asymptotic statistics are derived.

We consider two types of particles performing simple random walks on the integers. When two particles of different types collide they are immediately annihilated. All particles of one type are started off to the left of the origin, and all particles of the other type are started off to the right of the origin. We determine the asymptotic rate of annihilations and the fluctuations, around this rate, of the numbers annihilated at large times.

We present some general results related to first-passage percolation. These results involve families of random variables arranged in independent subfamilies. Applications are given to the study of random networks and to first-passage percolation.

Queues with delayed feedback have been little studied in queueing theory. Presented here is a rather complete discussion of such problems including queue length processes, busy period processes and several customer flow processes (e.g., departure processes). The case in which the delay mechanism is an M-server queue is studied in detail but it is shown later that many of the results carry over to a more general delay mechanism.

This paper is concerned with the assessment of the statistical efficiency of proposed regenerative simulation methods. We compare the efficiency of the ‘marked job' and ‘labelled jobs' methods for estimation of passage times in multiclass networks of queues with general service times. Using central limit theorem arguments, we show that the confidence intervals constructed for the expected value of a general function of the limiting passage time using the labelled jobs method are shorter than those obtained from the marked job method. This is consistent with intuition since the labelled jobs method extracts more passage-time information from a fixed-length simulation run.

In this paper we generalize the minimal repair replacement model introduced by Barlow and Hunter (1960). We assume that there is available information about the underlying condition of the system, for instance through measurements of wear characteristics and damage inflicted on the system. We assume furthermore that the system failure rate and the expected cost of a repair/replacement at any point of time are adapted to this information. At time t = 0 a new system is installed. At a stopping time T, based on the information about the condition of the system, the system is replaced by a new and identical one, and the process is repeated. Failures that occur before replacement are rectified through minimal repair. We assume that a minimal repair changes neither the age of the system nor the information about the condition of the system. The problem is to find a T which minimizes the total expected discounted cost. Under appropriate conditions an optimal T is found. Some generalizations and special cases are given.

We prove a conjecture of Arnold (1968) which simplifies the determination of an optimal bound on absorption probability originally due to Moran (1960).

It is possible to deduce some information about the time constant μ without using Wierman and Reh's (1978) renewal theorem.

Let X1, X2, · ·· be a stationary sequence of random variables and E1, E2, · ··, EN mutually exclusive events defined on k consecutive X's such that the probabilities of the events have the sum unity. In the sequence Ej1, Ej2, · ·· generated by the X's, the mean waiting time from an event, say Ej1, to a repetition of that event is equal to N (under a mild condition of ergodicity). Applications are given.

In [1], a deterministic counting rate condition is shown to be necessary and sufficient for a counting process induced on a Markov step process Z to be multivariate Poisson. We show here that the result continues to hold without Z being a Markov step process.

In the mathematical learning literature, reward–penalty rules have been studied in various decision-theoretic and game-theoretic contexts, the multi-armed bandit problem included. Here we propose an elaboration of Bather's randomised allocation indices which yields rules for the multi-armed bandit which are both reward-penalty and asymptotically optimal.