Elements of Simulation

Byron J.T. Morgan

• Publisher: Chapman and Hall/CRC
• Year: 2018
• ISBN:
  9781351452779
• Edition: 1
• Topic: Business
• Subtopic: Operations
• Index: Business

The use of simulation in statistics dates from the start of the 20th century, coinciding with the beginnings of radio broadcasting and the invention of television. Just as radio and television are now commonplace in our everyday lives, simulation methods are now widely used throughout the many branches of statistics, as can be readily appreciated from reading Chapters 1 and 9. The book has grown out of a fifteen-hour lecture course given to third-year mathematics undergraduates at the University of Kent, and it could be used either as an undergraduate or a postgraduate text. Simulation may either be taught as an operational research tool in its own right, or as a mathematical method which cements together different parts of statistics and which may be used in a variety of lecture courses. In the last three chapters indications are made of the varied uses of simulation throughout statistics. Alternatively, simulation may be used to motivate subjects such as the teaching of distribution theory and the manipulation of random variables, and Chapters 4 and 5 especially will hopefully be useful in this respect.

1

INTRODUCTION

1.1 Simulation

‘Simulation’ is a word which is in common use today. If we seek its definition in the Concise Oxford Dictionary, we find:

‘simulate, verb transitive. Feign, pretend to have or feel, put on; pretend to be, act like, resemble, wear the guise of, mimic, …. So simulation, noun.’

Three examples are as follows:

(a) Following the crash of a DC-10 jet after it had lost an engine, the Observer newspaper, published in Britain on 10 June 1979, reported that, ‘A DC-10 flight training simulator is being programmed to investigate whether the aircraft would be controllable with one engine detached.’

(b) Model cars or aeroplanes in wind-tunnels can be said to simulate the behaviour of a full-scale car, or aeroplane, respectively.

(c) A British television ‘Horizon’ programme, presented in 1977, discussed, with the aid of a simulated Stegosaurus, whether or not certain dinosaurs were hot-blooded. In this case the simulating model was very simply a tube conducting hot water; the tube was also equipped with cooling vanes, the shape of which could be changed. Different shapes then gave rise to differing amounts of cooling, and the vane shapes of the Stegosaurus itself were shown to provide efficient cooling. Thus these shapes could have developed, by natural selection, to cool a hot-blooded creature.

For statisticians and operational-research workers, the term ‘simulation’ describes a wealth of varied and useful techniques, all connected with the mimicking of the rules of a model of some kind.

1.2 What do we mean by a model?

It frequently occurs that we find processes in the real world far too complicated to understand. In such cases it is a good idea to strip the processes of some of their features, to leave us with models of the original processes. If we can understand the model, then that may provide us with some insight into the process itself. Thus in examples (a) and (b) above, it proves much cheaper and easier to investigate real systems through simulated models. In the case of (b) a physical scale model is used, while, in the case of (a), the model would most likely have been a computer simulation. In example (c) one can only employ simulated models because dinosaurs are extinct!

Subjects such as physics, biology, chemistry and economics use models to greater and lesser extents, and the same is true of mathematics, statistics and probability theory. Differential equations and laws of mechanics, for instance, can be viewed as resulting from models, and whenever, in probability theory, we set up a sample-space (see for example ABC^†, p. 62, who use the term ‘possibility space’) and assign probabilities to its elements, we are building a model of reality. Some particular models are described later. When the models are given a mathematical formulation, but analytic predictions are not possible, then quite often simulation can prove to be a useful tool, not only for describing the model itself, but also for investigating how the behaviour of the model may change following a change in the model; compare the situation leading to example (a) above. Abstract discussion of models is best accompanied by examples, and several now follow.

1.3 Examples of models which may be investigated by simulation

(a) Forest management

A problem facing foresters is how to manage the felling of trees. One possible approach is to ‘clear fell’ the forest in sections, which involves choosing a section of forest, and then felling all of the trees in it before moving on to a new section. An alternative approach is to select only mature, healthy trees for felling before moving on to the next section. A disadvantage of the former approach is that sometimes the trees felled will not all be of the same age and size, so that some will only be useful for turning into pulp, for the manufacture of paper, say, while others will be of much better quality, and may be used for construction purposes. A disadvantage of the latter approach that has been encountered in the Eucalypt forests of Victoria, in Australia, is that the resulting tree stumps can act as a food supply for a fungal disease called Armilleria root-rot. Spores of this fungus can be transmitted by the wind, alight on stumps and then develop in the stumps, finally even proceeding into the root-system of the stump, which may result in the transmission of the infection to healthy neighbouring trees from root-to-root contact.

While one can experiment with different management procedures, trees grow slowly, and it could take a lifetime before proper comparisons could be made. One can, however, build a model of the forest, which would include a description of the possible transmission of fungal disease by air and root contact, and then simulate the model under different management policies. One would hope that simulation would proceed very much faster than tree growth. For a related example, see Mitchell (1975).

(b) Epidemics

Diseases can spread through animal, as well as plant populations, and in recent years there has been much interest in mathematical models of the spread of infection (see Bailey, 1975). Although these models are often extremely simple, their mathematical solution is not so simple, and simulation of the models has frequently been employed (see Bailey, 1967).

(c) Congestion

Queues are a common feature of everyday life. They may be readily apparent, as when we wait to pay for goods in a shop, cash a cheque at a bank, or wait for a bus, or less immediately obvious, as, for example, when planes circle airports in holding stacks, waiting for runway clearance; time-sharing computers process computer jobs; or, in industry, when manufacturers of composite items (such as cars) await the supply of component parts (such as batteries and lights) from other manufacturers. The behaviour of individuals in the more apparent type of queue can vary from country to country: Mikes (1946) wrote that the British, for instance, are capable of forming orderly queues consisting of just single individuals!

Congestion in queues can result in personal aggravation for individuals, and costly delays, as when planes or industries are involved. Modifying systems with the aim of reducing congestion can result in unforeseen secondary effects, and be difficult and costly. Here again, models of the systems may be built, and the effect of modifications can then be readily appreciated by modifying the model, rather than the system itself. An example we shall encounter later models the arrival of cars at a toll; the question to be answered here was how to decide on the ratio of manned toll-booths to automatic toll-booths.

Another example to be considered later deals with the queues that form in a doctor’s waiting-room. Some doctors use appointment systems, while others do not, and a simulation model may be used to investigate which system may be better, in terms of reducing average waiting times, for example. (See also Exercise 1.6.)

(d) Animal populations

Simulation models have been used to mimic the behaviour of animal populations. Saunders and Tweedie (1976) used a simulation model to mimic the development of groups of individuals assumed to be colonizing Polynesia from small canoe-loads of migrants, while Pennycuick (1969) used a computer model to investigate the future development of a population of Great Tits. Gibbs (1980) used simulation to compare two different strategies for young male gorillas. The background to this last example is the interesting sociology of the African mountain gorilla, which lives in groups headed by a single male, who is the only male allowed to mate with the group females. In such a society young males who do not head groups have to choose between either biding their time in a hierarchy until the head male and other more dominant males all die, or attempting to set-up a group themselves, by breaking away and then trying to attract females from established groups. Both of these strategies involve risks, and the question to be answered was ‘which strategy is more likely to succeed?’

(e) Ageing

The age-structure of the human populations of many societies is changing, currently with the fraction of individuals that are elderly increasing. The prospect for the future is thus one of progressively more elderly individuals, many of whom will be in need of social services of some kind. Efficient distribution of such services depends upon an understanding of the way in which the population of elderly is composed, and how this structure might change with time.

Harris et al. (1974) developed categories which they called ‘social independence states’ for the elderly, the aim being to simul...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Acknowledgements
• List of notation
• 1 Introduction
• 2 Some probability and statistics revision
• 3 Generating uniform random variables
• 4 Particular methods for non-uniform random variables
• 5 General methods for non-uniform random variables
• 6 Testing random numbers
• 7 Variance reduction and integral estimation
• 8 Model construction and analysis
• 9 Further examples and applications
• Appendix 1 Computer algorithms for generation, testing and design
• Appendix 2 Tables of uniform random digits, and exponential and normal variates
• Solutions and comments for selected exercises
• Bibliography
• Author index
• Subject index