Environment and Planning, 1969, volume 1, pages 221-227

A kinetic theory of traffic distribution and similar problems S.G.Tomlin Physics Department, University of Adelaide, South Australia Received 8 September 1969

Abstract. A general approach to traffic distribution problems which provides a means of dealing with both equilibrium states and time-dependent situations is presented. It depends upon a knowledge of certain transition coefficients and points out the fundamental importance of these quantities in the analysis of distribution problems. Although the paper is written as a discussion of traffic distribution it is suggested that the method is of much wider significance and may be valuable for dealing with a variety of social, economic, and biological systems. Introduction The problem to be considered is that of the distribution of the traffic flow between a set of origins denoted by subscripts / = 1,2, ... m and a set of destinations denoted by subscripts / = 1,2, ... n, when the generalised cost of the journey / to / is c,y, and when the system is subjected to certain constraints. Assuming that an equilibrium state is reached, it has been shown by Wilson (1967) and Tomlin and Tomlin (1968) how this distribution may be derived by the methods of statistical mechanics, or, more correctly, by arguments from analogy with statistical mechanics. But even if this kind of argument is sound it is necessarily limited to the discussion of the steady state and it is desirable to develop a means of dealing with non-equilibrium situations. In physics such problems lie within the field of kinetic theory and it is proposed to investigate the application of this theory to traffic distribution problems. These problems may be regarded as examples of a class of problems concerned with the achievements of certain goals in the face of deterrents, and a successful method of dealing with the traffic system may be expected to have a number of other fields of application. The fundamental equations These equations are formulated in terms of rates of transitions between routes, for it is only as a result of such transitions that an equilibrium state can be attained, or the system adjust to a new equilibrium after a change of its parameters. We introduce transition coefficients afjpq defined as the probability that a traveller will change in unit time from the route //' to the route pq. Then for a closed system the rate of increase of 7}j, the number of travellers on route // at time r, is equal to the rate of transfer of travellers from all routes into //, less those moving out of // per unit time. That is T\f VI

~

Z*>\®pqij*pq

^ijpq^ij)

>

(1)

pq

where DT^/Dt is used for the total rate of change of 7]y due to all causes, and is to be distinguished from dTtj-/dt which will be used to denote any contribution to the total rate of change of Ttj from outside sources when the system is not closed. For such an open system Dt ~ dt

Hapqif*pq

a

ijpqUj) J

(2)

S. G. Tomlin

222

and this will be applicable to problems of a growing population in any, or all, of the origins. These systems of simultaneous differential equations are the fundamental equations which may be applied to both stationary and time-varying systems, and it may be remarked that whatever may be thought about the validity of the statistical mechanical approach, these equations are certainly correct. They are forms of Boltzmann's transport equation, and obviously their usefulness depends upon a knowledge of the coefficients aijpq. The transition coefficients First it may be remarked that the direct determination of these coefficients should be a relatively straightforward matter involving the asking of appropriate questions in a survey. This appears to be a rare case where it may be easier to deal with people than with particles. It is suggested that the practical determination of these coefficients is of the first importance, but in the absence of such direct information we must attempt to infer their nature. We proceed by thinking of the traveller in the route // as being in a 'cost well', in analogy with a particle in a potential well, and that in order to escape from that well he has to overcome a cost barrier. Reference to an elementary discussion of diffusion in solids, such as is given by Kittel (1956) will indicate the basis of this argument. The probability of escape per unit time, or escape frequency, will be denoted by e,y for route //, and will involve some function of the depth of the 'cost well' for this route. We suppose that there is some level of cost c0 such that, if ctj is the generalised cost associated with route //, then the route corresponds to a 'cost well' of depth c0— c,y. It is then a plausible assumption that e,y

where £2,7 =

MiV

225

Kinetic theory of traffic distribution

Using the condition in Equation (10) to determine TV we have

T =T

(11)

" ^r if

This is a general form for the free case distribution function, whatever the form of the dependence of c and ju on costs. From Equations (3) and (6), if Ma

h

=

,

" v„ n

« ~ ^f(c,y) •

Thus if 0(g//)

,

,

a

^y=exp(-fc y ) the statistical mechanical formula of Equation (7) results, with an interpretation of the weighting factors cj,y in terms of the vif and A,y. Particularly simple forms for the functions and f are exponentials such that etj=

vifexp(k(3cif)

M//= Aiyexp[(fc-l)j3czy] and these, as we now see, do result in the statistical mechanical formulae. These expressions, simplified for convenience by putting k = \ will be used elsewhere in the applications of this theory to other problems. The semi-free case First consider the situation where there are constraints on the origins only, so that YtTq = Of (7 = 1, 2, ... m), where Ot is the prescribed number of travellers leaving source i. This case is not only of theoretical interest but corresponds to the real situation where a number of alternative routes end in a given area and there are no restrictions on the capacities of the routes. The free probabilities must now be modified and it seems reasonable to suggest that one should write a

ijpq

oc e

iffXpqn

9JL >

where Op is the prescribed number leaving source p, and Opf the number that would be leaving that source if there were no constraints at all. This latter is obtained from the free distribution in Equation (11) and is

oPf = TT 1

Inpq rpJL pq

it is

0Pf~

S. G. Tomlin

226

The transition coefficient, properly normalised, may then be written

a

'-=e*iXCi'

where tp =

"

*p^pVpq

PQ

(?°«) •

With this expression for the coefficients the equilibrium Equations (9) give, for all //, €

Lepq*pq

r q

-TV" = r

„

= N (a constant) .

PQ

In this case the condition in Equation (10) makes TV = 1, and the distribution is Tif = AfitSlq

with ,4, = ( i f y )

.

Again, if

and

the statistical mechanical result in Equation (8) is obtained. Evidently for the semi-free case in which the constraints are i

the transition coefficients are B D =

9 Q^PQ

e

Z.BqDqfxpq PQ

and the resulting distribution function is Tv = BjDjSlij ,

withBj = \I.Sly)

.

The fully constrained case In this case, where the constraints are both X70 = O, a n d - l T ^ D , , we may infer from the above discussion, and our knowledge of the statistical mechanical result, that the transition coefficients are given by A'pOpB'qDqlxpq i:ApOpB'qDqnpq

Zi,

'

PQ

where A'p = \j[lB'qDqapt\ B'q = (l,A'pOpSl„^

, \ .

Kinetic theory of traffic distribution

227

It then follows from Equations (9), using the previous arguments, that the equilibrium distribution is Tif = A\OtB)Dfln

(13)

which again coincides with the statistical mechanical result if Slif = coiyexp(-j3c,y). A possibly useful modification of this result may be obtained by relaxing some of the constraints. For example if there are some destinations with more jobs available than there are people to fill them the destination constraints are

I

The application of the Lagrange multiplier method to the optimisation of the entropy then leads to the results in Equations (12) and (13) except that when Z Tgf < Dj then B}Df = 1 . i

This case may be of interest in connection with growth problems. Comments This method of approach to distribution problems is obviously applicable to other systems, as well as the traffic problem, if the behaviour of the system is determined by exchanges within it for which transition coefficients can be defined and measured. It is again emphasised that these coefficients are of fundamental importance for distribution theory, and studies of these quantities by appropriate surveys will provide the information which will make it possible to apply the general Equations (1) and (2) to the prediction of systems behaviour. In particular the cost dependence of the coefficients is of great interest, for if it should be of exponential form this would justify the statistical mechanical approach to equilibrium distributions. The kinetic method not only allows the determination of equilibrium conditions but provides a general method for the discussions of time-varying situations such as the effects of changing costs, or, by means of Equation (2), the effects of population growth. Some problems of this kind will be examined elsewhere, and the significance of the relaxation time of a system will be demonstrated. Acknowledgements. I am indebted to the University of Adelaide for the grant of study leave during which this work was done at the Centre for Environmental Studies, London. It is a pleasure to thank the Director of the Centre for the opportunity to work there and to benefit from exchanges with its members. References Kittel, C, 1956, Introduction to solid state physics (John Wiley, New York). Tomlin, J. A., and Tomlin, S. G., 1968, "Traffic distribution and entropy", Nature, 220, 974-976. Wilson, A. G., 1967, "A statistical theory of spatial distribution models", Transportation Research, 1, 3, 253-269.