## Alan Mercer's statistical papers

Alan Mercer wrote a number of statistical papers. Below we give authors summaries and/or introduction to a number of these. We give the papers in chronological order and for each paper also give Mercer's address at the time and, when appropriate, any acknowledgement.

**Paper.**(with Cyril Samuel Smith) A random walk in which the steps occur randomly in time,*Biometrika***46**(1/2) (1959), 30-35.

**Address.**At the Atomic Weapons Research Establishment at Aldermaston. Formerly at the National Coal Board.

**Acknowledgement.**This paper is published by permission of the National Coal Board.

**Summary.**The steps of a one-dimensional random walk are positive and occur randomly in time at a fixed mean rate. The sizes of the steps are independent and the size of each step has the same given probability distribution. The distribution of the time to reach a fixed barrier is obtained and approximations to its moments are derived. The results are extended to the case in which the barrier and the random walk process converge at a constant rate.

**Introduction.**The problem considered in this paper arose from the study of the wear of conveyor belting. Incidents which damage belting are of various degrees of severity and different types of incident occur with different frequencies. From a given distribution of the life of belting, information is required about the relative importance of different types of incident. In particular, it is desirable to know whether the wear caused by nearly continuous abrasion is more or less important than the damage due to the more destructive but less frequent blows. The model, which has been used, assumes that damage accumulates until the total reaches a certain level, when the belt is considered to be worn out completely. The damage done by each incident may be considered as a step of a random walk process in which the steps occur at a constant mean rate, and the problem is to relate the distribution of the size of steps to the distribution of the time taken to reach a barrier.

**Paper.**Some simple duration-dependent stochastic processes,*J. Roy. Statist. Soc. Ser. B***21**(1959), 144-152.

**Address.**Birkbeck College, University of London and Atomic Weapons Research Establishment at Aldermaston.

**Acknowledgement.**I am indebted to Dr D R Cox for suggesting this problem and for his criticisms during the course of the work.

**Introduction.**Consider a stochastic process with discrete states in continuous time. In certain applications it may happen that the transition probabilities at a particular instant, given say that the ith state is occupied, depend on the total time that the system has spent in the ith state since the start of the process. Thus with three states, corresponding to death, to being alive with a certain disease and to being alive without the disease, it may be approximately true that the transition probabilities at a given time depend on the total length of time for which the individual has had the disease. Processes of this kind do not seem to have been considered before. In the present paper, some properties are obtained of the simplest processes of this type having only two states, and having transition probabilities depending on the length of time spent in just one of the states

**Paper.**A queueing problem in which the arrival times of the customers are scheduled,*J. Roy. Statist. Soc. Ser. B***22**(1960), 108-113.

**Address.**Birkbeck College, University of London and Atomic Weapons Research Establishment at Aldermaston.

**Acknowledgement.**I am indebted to Dr D R Cox for suggesting this problem and for his criticisms during the course of the work.

**Summary.**The queueing problem considered is that in which (i) the customers are scheduled to arrive at equal time intervals but a customer may arrive at any time after the start of the interval during which he was scheduled to arrive, or may not even arrive at all, the lateness distribution being perfectly general; (ii) the total service-time of a customer is supposed to consist of a finite number of stages, such that the times spent in each of the stages are independent and similarly distributed, and the probability of leaving a stage at any time depends on the time since the current scheduling interval began. A method is discussed which enables the non-equilibrium distribution of the queue-length at any time to be derived, and results for the equilibrium distribution are given. Most of the results relate to the case in which there is a single server but extensions of the argument for more than one server are outlined.

**Paper.**Some simple wear-dependent renewal processes,*J. Roy. Statist. Soc. Ser. B***23**(1961), 368-376.

**Address.**Atomic Weapons Research Establishment at Aldermaston, Berks.

**Summary.**The probability that a component fails is assumed to be linearly dependent on a physical wear variable, which can be measured as well as on the age of the component. A wearing mechanism is represented by the extended Poisson process. The alternative strategies of replacement after a given time or when the wear reaches a scheduled level are compared

**Paper.**A queue with random arrivals and scheduled bulk departures,*J. Roy. Statist. Soc. Ser. B***30**(1968), 185-189.

**Address.**University of Lancaster.

**Summary.**The distributions of queue length and waiting time of a customer are considered for the queue where customers arrive at random, wait in the order of arrival and are scheduled to be served in batches by a single server during equal intervals. A general lateness distribution is assumed but a service either takes place in the scheduled interval or not at all; no overlapping of services is allowed.

**Paper.**Queues with scheduled arrivals: a correction, simplification and extension,*J. Roy. Statist. Soc. Ser. B***35**(1973), 104-116.

**Address.**University of Lancaster.

**Summary.**Customers are scheduled to arrive at a queue during equal time intervals but they may be late; the lateness distribution is perfectly general. Bulk arrivals with an exponential service time distribution, single arrivals with staged services and single arrivals with a general service time distribution are considered when customers must arrive in the interval for which they were scheduled or not at all. Results given by the author in a previous paper, for single arrivals who may come in a later interval than the one for which they were scheduled and have an exponential service time distribution, are corrected. It is concluded that when customers are scheduled then the equilibrium distribution of queue length is a truncated sum of weighted geometric distributions, irrespective of the number of arrivals per interval, the lateness distribution, the service time distribution and the number of servers.