be made. Essentially the proposed new system is being simulated instead of being put into operation and then tested, when it may be too late to reverse the decision.

Example 3.2

British Gas transport natural gas under high pressure around Britain prior to its local distribution. When North Sea natural gas was first available, they had the problem of guaranteeing to customers, both industrial and domestic, that sufficient gas would always be available. There was no available pipe network which could be used to transmit the gas. Another complication is that the demand for gas fluctuates with seasonal and daily weather conditions.

As in Example 3.1 the answers were needed before putting large resources into building the system. A mathematical model was formulated, based upon the known physical properties of gas and how it would flow along large, very long pipes. The model was then operated under various conditions to simulate the real system. A number of different pipe layouts were tried to obtain an optimal design. Again, only after many simulations had led to a confident prediction that gas demand could be met in all circumstances was the construction of the pipe network authorised.

In both the above examples it can be appreciated that, having formulated quite complicated models, the model would be translated into a computer program ready for subsequent simulation. The problems described in these two examples may seem far removed from the work of chapter 2 but, to describe the terminology adequately, it is necessary to raise our horizons a little.

The description deterministic model is usually used in cases where the outcome is a direct consequence of the initial conditions of the problem. This directness is not affected by any arbitrary external factors or, in particular, random factors. Very often, but not always, this kind of mathematical model involves differential equations in which time is the independent variable. In fact the gas flow model from British Gas mentioned above is one such deterministic model where differential equations are used. The gas flow received at some

remote location will depend entirely on the initial state at a North Sea terminal and on the pipe network. Many models described in this book will be found to involve deterministic differential equations and chapter 6 is devoted to this particular topic.

The term stochastic model, on the other hand, is reserved for those situations where a random effect plays a central role in the problem investigation. We have already seen examples in chapter 2 (Examples 2.9 and 2.12). Many models of this kind are essentially ‘next-event’ models often involving queues and services. Random arrivals at bus queues or random service times at the supermarket are common events for everyone. In these situations the outcome is not fixed in the sense that it is unique because we have to allow for the random variability of arrivals and departures. This of course was the case in Example 3.1, where repeated simulations have to be statistically appraised before any sort of answer can be given. Stochastic models form the subject of chapter 7.

To sum up, it is often useful to divide our mathematical models into two categories, deterministic and stochastic, but care is necessary because there are many situations where, within the same model, some features are random and others deterministic.

3.3 Methodology and modelling flow chart

As has been said earlier, one of the main purposes of this guide is to teach how mathematical modelling is done in practice. One of the important conclusions from the previous section is that the activity of modelling is a process which involves a number of clearly identifiable stages. The most helpful way of representing these stages is by means of a modelling flow chart. Our particular version of this flow

chart is illustrated in Figure 3.1. We do not claim complete originality for this chart and most books on mathematical modelling will have something similar. Experience shows that the flow chart does help in developing the right attitudes, leading to successful model building. There is perhaps some danger in trying to fit all situations onto the flow chart in a rigid manner but the comforting framework which it provides usually outweighs these dangers. The main point to remember is that when faced with a modelling problem, you should not be disconcerted if you feel ‘lost’ and wonder where to start. This is a perfectly normal reaction, even for experienced modellers. The point of the flow chart is that it gives us a framework to refer to and acts as a channel for our thoughts and ideas.

In this flow chart, each clear stage in the modelling process is represented by a box. We shall now amplify each ‘box’ with a series of questions and hints which should indicate what is intended.

Figure 3.1

Box 1: Identify the real problem

What do we want to know? What is the purpose and objective? How will the outcome be judged? What are the sources of facts and data, and are they reliable? Is there one particular unique answer to be found? Classify the problem: is it essentially deterministic, or stochastic? Do we need to use simulation?

Box 2: Formulate a mathematical model

Look first for the simplest model. Draw diagrams where appropriate. Identify and list the relevant factors. Collect data and examine them for information explaining the behaviour of the variables.