Tmnrpn Rex., Vol. 10, pp. 201407. Pqamm Press 1976. Printed in hat Britain

THE LOCATION OF TWO RING ROADS AND THE CONTROL OF TRAFFIC SPEED WHICH TOGETHER MINIMISES

RADIAL TRAVEL IN A TOWN

M. J. SMITH Department of Mathematics, University of York, England

(Receioed 13 September 1974; in reuisedfonn 29 November 1975)

Ah&act-In a simple theoretical model, we seek the location of two ring roads which minimises the total radial travel in a circular town of unit radius. We assume that drivers choose least time paths and that the speed of vehicles is controlled so as to minimise radial travel. We show that, whatever the distribution of origins and destinations, if an internal ring road miniiises the total radial travel then the radial traffic flow just inside the ring equals the radial traffic flow just outside the ring. We show that if origins and destinations are uniformly and independently distributed over that part of the radials within the town. and a circumferential ring road already exists, then the optimum radius of a sir& inner ring road is a-- 1.

INTRODUCTION

The radial roads in a town generally serve many purposes. They are important traffic arteries for the whole spectrum of town travel, they provide access to private houses, shops, offices and factories, and in historic towns they provide the setting of much of the important architecture.

This diversity of function renders the problems caused by radial traffic flow particularly difficult and important. A possible way of tackling these problems is suggested by the observation that a reduction in the radial traffic flow itself would have a beneficial effect on all the different functions of a radial road. Even the traffic would like less traflic!

In this paper we study a theoretical model in order to determine the location of two ring roads and speed control policy tihich together minimises the total radial traffic flow in a town.

A suggested road network and trajic control Our theory suggests that the following provisions may,

in practice, substantially reduce the radial traffic flow in towns:

(a) A high-capacity high-speed outer ring road close to the town.

(b) A low-capacity low-speed inner ring road around the central area.

(c) Linked traffic lights to guarantee that average speeds are slow within the town, particularly on the inner ring road.

(d) Radial roads which are unconnected at the town- centre.

COMMENTS ON THE THEORETICAL MODEL.

A description of the model, and its relationship to other work

We suppose that a town of radius 1 has only radial roads, and that these radials are unconnected at the town-centre. We suppose that two ring roads, of radii r and R, are to be added to the radial network and seek the values of r and R which minim& the total radial travel in

the town. We assume that the speed control imposed, for any r and R, is that which minimises the total radial travel.

By “optimum” we shall mean “minimises the total radial travel”.

The best location of two ring roads, using different criteria, and different speed control, has been studied by Blumenfeld and Weiss (1970) and Pearce (1974). Smeed (1971) has considered the location of a single ring road and he seeks to minimise, like us, a measure of the radial traffic flow.

Our present approach is characterised by the following main features:

1. We have a criterion function which is appropriate for both the environment and the traffic; this is to be the total radial distance travelled within the town.

2. Speed control is employed specifically to affect route choice.

3. No assumptions are made about the distribution of origins and destinations, except that it is fixed.

4. We only assume the existence of an arbitrary number (finite or infinite) of radial roads.

We shall now comment in more detail on the first two of these features, but we note in passing that our theory gains substantial generality from its independence of both the distribution of origins and destinations and of the number of radials.

The criterion junction We shall seek to minimise the total radial travel within

the town. As remarked earlier, both the traffic and the environment would welcome a low radial flow.

This pleasing community of interest allows our theory to work without capacity restraints and without consider- ing external costs; because both internal and external costs are embodied in our criterion function. This is not the case if total journey time is used as the criterion function, so that any theory seeking to minimise total journei time must necessarily involve further balancing factors. These factors may be capacity constraints or the external costs imposed by the trailic flow.

201 T.R. 10/3-E

202 M. J. SMKH

A second important feature of our criterion function is that the optimum speeds will be independent of the distribution of journeys and their number. We are thus able to concentrate on the relationship between the distribution of journeys (pairs of origins and destinations) and the optimum road network. This feature also would not generally be present if another criterion function were used, and so any such theory would have to deal simultaneously with the speed control, the road network, and the distribution of journeys. Results would necessar- ily be more complicated, as they would involve all three of these variables.

Our criterion function does bear some resemblance to the traffic intensity used by Smeed (1%3 and 1971, for example).

Speed control We use the control of traffic speed to alfect the flow

pattern. To determine this pattern (for any fixed road network and specification of speeds), we assume that drivers choose least time paths. This assumption has been made by, e.g. Smeed, Bhtmenfeld, Weiss and Pearce. It fits in very well with our theory because the optimum speed control is then independent of the distribution of journeys and is, ‘indeed, optimum for each individual journey. This simplifies our theory and makes our results both sharper and more general.

As Smeed (1971) points out on his page 7, least cost paths may, under certain reasonable conditions, be transformed into least time paths by a small modification of the usual definition of speed. Thus this theory may be tailored, by such a modification, to fit more closely the actual behaviour of the average motorist, who is influenced in his choice of route by the length, as well as the time, of his journey. So our assumption that drivers choose least time paths is not nearly as restrictive as it may at Iirst sight appear.

In practice, journey times are affected by traffic light settings and our theory is intended as a theoretical idealisation of the practical situation in which linked traffic lights control the average speed of vehicles over fairly short distances.

Allsop (1974) considers the possibility of using linked traffic lights to affect route choice, and gives a simple example to show that total journey time may be reduced by this means. He suggests using transport theory exemplified by Bechmann (1956), Gilbert (1968, Murch- land (1%9) and Evans (1973) to determine the equilibrium traflic pattern arising from any fixed signal settings, and then optimising the signal settings. Flow-dependence difficulties affect both steps, but the latter step (particularly) is likely to be difficult.

Our ideas have something in common with those of Allsop’s paper, but we avoid all flow-dependence dillicul- ties by simply supposing that our prescribed speeds are attained whatever the flow. We are only able to do this because we use an appropriate “self-balancing” c$erion function. It is this which prevents our theory from collapsing into triviality. In our model there are no capacity restraints whatsoever.

THE TEEORETICAL MODEL

We suppose that a circular town, of radius 1, has at least two radials which are unconnected and which are to be connected by ring roads of radii r and R with 0 < r CR < 1. We suppose that drivers minimise their journey time and that vehicle speeds are to be given by Table 1.

Table 1. The optimum specifi- cation of travel speeds

Road Speed Ring of radius r Ring of radius R ii Radial 2,

The numbers c and v are constants, and we recall that the radius of the town is our unit of distance.

The choice of speeds given by Table 1 derives from Smith (1974) where we showed that (for any fixed r and R) this choice minimises the distance travelled on radials by every driver. Thus this choice does minimise the total radial distance travelleddur criterion function-very strongly.

We now seek radii r and R such that the total travel on radials is minimised not only with respect to the speeds of vehicles but also with respect to r and R.

The specification of speeds given in Table 1 enables us to use a very simple and useful representation of our town and the journeys within it.

The representation of journeys A journey between the two points (given in polar

co-ordinates whose origin is the town centre) (x, 0) and (y, 4) may be represented by the point (x, 0, y, 4) in a space of four dimensions. We shall find that only the radial distances x and y are important, so we use the point (x, y) to represent such a journey instead. Thus all possible journeys are represented by points in the upper right-hand quadrant of the plane shown in Fig. 1 (we assume x 2 0 and y a 0).

Of course, there can only be a finite subset of this quadrant whose points correspond to journeys which actually occur in any one year (say), and some of these will be more important than others by virtue of the

1 ..

*W)

01 0 1

Fig. 1. Representation of journeys.

Minimising radial travel 203

number, size or weight of vehicles making the journey to which they correspond.

We shall represent the number of passenger car units making a journey by a proportional mass at the corresponding point, but then we shall approximate to this mass distribution by another smoother one with a density f. In this model we shall suppose that the weight m(A) of journeys represented by points in the subset A of the quadrant is given by:

m(A) = II f(x, Y 1 dx dy.

A

Thus the weight of journeys with origin in the annulus of radius x and thickness dx, and destination in the annulus of radius y and thickness dy, is f(x, y)dx dy.

Again because only the radial distance x of a point (x, 0) is important, we represent all points in the town and outside it by the set of non-negative real numbers. This is depicted in the simple Fig. 2.

Fig. 2. Representation of the town.

The classification of journeys The two numbers r and R cut the half-line of Fig. 2 into

the three pieces (ignoring their common edges at r and R) shown in Fig. 3; so that C represents all points inside both

C I E A / n II b

cl r R 1 --

Fig. 3.

ring roads, E represents all points outside both ring roads and I represents all points between the two ring roads. We divide the quadrant of Fig. 1 by agreeing.that (x, y) E EI ifandonlyifx E Eandy E Iandsoon.ThusEIisjust the Cartesian product E x I. This division of the quadrant is shown in Fig. 4, where we have divided II into triangles II, and IZ* for reasons which will appear.

Figure 4 is reminiscent of Table 3 (say) in Smith (1974); but it also has, under our present conditions, a geometrical significance which we shall now explore. It is important to

1 I CE IE EE

t k I

0 1

Fig. 4. Classification of journeys.

remember that all the regions of Fig. 4 depend on r or R (or both). For these regions we shall sometimes write EE = H?(R), 1Z2 = II& R) and so on.

The geometrical significance of Fig. 4 Suppose that r and R are given and 0 < r < R < 1.

Since the speed of vehicles is given by Table 1, and since drivers minimise their journey time, it follows that the distance s(r, R, x, y) travelled on radials within the town by the journey (x, y) with origin and destination in different radials is given by:

Cmin()x-rl+(y-rl,IR-xl+lR-yl)

s(r,R,x,y)= I ifxslandysl 1-RttR-yJ ifx>landysl (1) 2(1-R) ifx>landy>l.

An example is given in Fig. 5, in which the radial distance s(r, R, x, y) travelled by the journey (x, y) is given by the length of the L-shaped path.

KY)

L cm

01 0 1

Fig. 5. The length s(x, y, r, R) in a particular case.

We define D, to be alar and 4 to be alaR. By slightly varying r in Fig. 5 it is easy to see that, for

this particular (x, y), Ix - rl t Iy - rJ is locally constant (as r varies), and so the partial derivative as/ar(r, R, x, y) = D,s(r, R, x, y) = 0. In a similar way it is easy to see what Dls and D2s are for all (x, y) B II

It follows from (1) that a journey (x, y) E II is attracted to the inner ring if

or x-r+y-r<R-xtR-y

xty<rtR.

Similarly (x, y) E II is attracted to the outer ring if

x+y>Rtr.

Hence II, = ZZ,(r, R) of Fig. 4 is the set of all journeys in II attracted to the inner ring and Z&(r, R) is the set of all journeys in II attracted to the outer ring. The (diagonal) dividing line between 11, and II, is given by:

xty=R+r.

204 M. J. SMITH

A little thought shows that

Ddr, R, x, Y) = -2 if (x,y) E III

Oif (x,y) E 112

and similarly

DzS(r, R X, Y) = Oif(x,y) E zz, 2 if tx, y) E zz2.

The complete specification of derivatives, for all (x, y), is given in Figs. 6 and 7.

We are interested in minimising. the total radial distance, S(r, R), travelled by all journeys (x, y) taking account of their weight. This is given by:

S(r, R)= II

s(r,R,x,y)f(x,y)dxdy (2)

where the integral is taken over all x, y 30. The derivatives specified in Figs. 6 and 7 enable us to

0 0

0 I\ 0

-2

2 0 0

Fig. 6. The partial derivatives D,s. 40--l= &+I.

But this argument fails unless we can justify differentiat- ing under the integral sign. We perform this justification in the Appendix. We note here that inequality (7) depends crucially on our choice of travel speed, and that (3) would be false, in general, for other choices.

A similar argument yields:

D&r, R)=2m(ZZ&, R))-2m(EE(R)). (4)

Hence if r is fixed and R < 1 minimises S then

m(&(r, R)) = m(EE(R)) (9

and also if R is fixed and r mhimises S then

m(CC(r)) = m(ZZ,(r, R)). (6)

The relationships (3), (4), (5) and (6) show the geometrical significance of Fig. 4, and constitute our main results.

We may now easily derive a test to show when an internal ring road is not optimum.

A simple test Consider the inner ring road. If we double each side of

(6) and add

m(CZ)+m(CE)+m(ZC)+m(EC),

then on the left-hand-side we obtain the total radial flow q (r -) just inside the ring, while on the right-hand-side we have the total radial flow q(r+) just outside the ring. Hence (6) holds if and only if

It follows that if q (r -) # q( r +) then the internal ring at r is not optimum.

A similar test applies to any larger ring of radius R within the town.

Optimum value of R An important application of this theory is to suggest

when any R < 1 would not be optimum (even ignoring ring travel). This will certainly be so if through traffic is reasonably important. More precisely, for an inner ring of radius r the larger ring should be outside the town if 0

m (EE(R )) > m (ZZdr, R )) Fig. 7. The partial derivatives Dzs.

for all r < R < 1. This will generally be the case if

differentiate S(r, R). It is natural to write:

D,S(r, R) = I

Ddr, R, x, ylfk y) dx dy

= I I 2f(x, y)dx dy + II

(- 2)f (x, Y 1 dx dy

CCV) I1lV.R)

(3)

=2m(CC(r))-2m(ZZdr, R)).

m(EE(1)) > m(ZZdr, 1)).

A special case; R = 1 and f(x, y) = (i i~~eyW’~e

The density in our special case corresponds to the variable distribution of trip ends in Smeed (1971), and the assumption that R = 1 means that, with our speed control and determination of route, both through traffic and cross-cordon traffic travel a minimum possible radial

Minimising radial travel 205

distance. Hence, putting f(x, y) = 0 if either x or y 3 1 merely has the effect of removing from our consideration journeys whdse radial travel has been already minim&d, irrespective of the location of any inner ring road.

Now any journey from (x, 0) to (y, 4) must travel at least a distance Ix - y 1 on radials, whatever ring roads are provided. Hence the total radial travel, for our particular f, must always be at least So where

so =

It is clear that, under our assumptions, this value may be approached arbitrarily closely by having many ring roads.

We shall use So as a reference, and define P(r, l), which may be thought of as the radial penalty incurred by having only a single interior ring road at r (as well as the circumferential ring), instead of a very large number of ring roads, by

P(r, 1) = S(r, 1) -So.

The quantities S(r, l), So and P(r, 1) may be evaluated by integrating the integrands shown in Figs. 8-10 over their corresponding regions and adding. In Figs. 8 and 9 these integrands give the radial distance travelled by the journey (x, y) belonging to the corresponding set, while Fig. 10 is obtained by subtraction.

0 0 r 1

Fii. 8. Integraads for S(r, 1).

1

Y-X

I

X-Y

OW Fig. 9. Intcgrands for S,.

It follows from Fig. 10 that P(r, 1) is given by the volumes of two pyramids; one has base area r* and height 2r and the other has base area (1 - rp and height 1 - r. Hence

P(r ,)=2r)+U--r)’ 1 3 3

-=~(l-3r+3r2+r3).

1 2(1-y)

0 2(x-r) 2WX)

r

L!iIEi

2(y-r)

2(r- 0

0 2(r-

0 r 1

Fig. 10. Integrands for P(r, I).

As may be checked using (6), the optimum value of r is d-1 and we then have P(fi-l,l)= [(6-4ti)/3] +O.ll.

The difference in the shape of the two pyramids indicated by Fig. 10 suggests that a very small ring (functioning at a very slow average speed) should be considered. If we do this we obtain Fig. 11 instead.

1 20-Y)

0 2(x-r) 2(1-x)

2(y-r)

r litJ?sJ r- 2x XI 0

0 2 0 r 1

Fig. 11. Integrands for P(0, r, 1).

The radial penalty in this case is

P(O,r,l)=f(r’+(l-r)‘)

=$3rZ-3r+l).

This is a minimum if r = i, and

P(0, 1, 1) = t + 0.083.

Noting that P( 1) = P( 1,l) is the raeal penalty incurred by having no ring road within the town, we list the P’s we have:

P(l)=;+O.67,

P(d2-1,1)=2-+-0.11,

P(O,+, 1) = $ + 0.083.

Since So = f, the corresponding values of S are:

S(1) = 1,

S(V2-1,1)+%0.44,

S(0, :, 1) = ; + 0.42.

206 M. J.

A single internal ring road is therefore surprisingly good at reducing the radial flow, and very little is gained by having even the optimum “leakage” through the town-centre.

Remarks 1. A special case of our central results (3), (4), (5) and

(6) does appear as a limiting case in Pearce (1974). 2. Similar arguments may be applied to determine the

optimum location of a single ring road; a problem which has been considered by Smeed (1971).

CONCLUSION

The most important effect of our ring road systems, which derives from our speed control, is the reduction of the radial travel by through and cross-cordon journeys. Each system wifh R = 1 reduces this to an absolute minimum since, under our assumptions, the circumferen- tial road is fast atid any inner ring road is slow enough to be unattractive to these categories of traffic. This effect is independent of the size (or existence!) of the inner ring provided that our speed control is employed.

We have also shown here that the radial travel of the totally internal traffic may, under our assumptions, be substantially reduced by just one interior ring road (together with the circumferential ring). This reduction is, perhaps, surprisingly close to the maximum possible reduction.

Our theoretical systems encourage the radial tratlic to organise itself for the benefit of both the trtic and the environment. It is reasonable to suggest that a similar organisation may be achieved in practice by using linked traflic lights to control the average speed of vehicles.

Linked traffic lights would also segregate (in time) the pedestrians and the road vehicles in a clear and unambiguous manner (spatial segregation is not, in general, feasible on radial roads). It has been shown by Hillier and Arnold (1958) that road casualties may be significantly reduced by using linked traffic lights even when no reduction of trallic volume is involved, and it seems likely that substantial practical benefit would derive from both the reduction in traffic volume and the increase in organisation of the remaining traffic, including pedestrians, if our suggested road network and traflic control were employed in practice.

The study of theoretical models to gain insight into the traffic problems of actual towns has been pursued by Smeed in a series of papers (see Smeed (l%l, 1963,1963, 1964, 1971)). We have here introduced vehicle speed as a new control variable; it is this which brings the theory closer to the practical situation and makes the theory considerably easier. We are also confronted with a number of new theoretical and practical questions, which merit further study.

Acknowledgements-I am grateful for the referees’ comments on an earlier version of this paper.

REFERENCES

Allsop R. E. (1974) Some possibilities for using traffic control to influence trip distribution and route choice. Transportation and Trafic Theory (Proceedings of the Sixth International Sym-

SMITH

posium on Transportation and TmUiti Theory, Sydney, Au- stralia, August 1974.) Elsetier, Amsterdam, 345-374.

Beckmann M., McG@re C. B. and Winsten C. B. (1956) S[udies in the Economics of Transportation Yale University Press, New Haven.

BlumenfeId D. E. and Weiss G. H. (1970) Routing in a circular city with two ring road.& Tmnspn Res. 4, 235-242.

Evans S. P. (1973) Some applications of mathematical optimisa- tion theory in transport planning. Ph.D. Thesis, University of London.

Gilbert A. (1%8) A method for the trafic assignment problem when demand is elastic. London Graduate School of Business Studies, Transport Network Theory Unit, Report LBS-TNT-85, London.

Hillier J. A. and Arnold M. J. (1958) Linked fixed time traflic signals. Surveyor 117 (3472), London, 108%.

Mu&land J. D. *(1%9) Road network tra5c distribution in equilibrium. Methods of Operations Research, Vol. 8 Obenvolfach-T@mg iiber Gperat$ns Research October 1%9, Verlag Anton Han, Mersenheim 1971.

Pearce C. E. M. (1974) Lociting concentric ring roads in a city. Transpn Sci. 8, 142-168.

Smeed R. J. (1961) The Trafic Problem in Towns. Manchester Statistical Society.

Smeed R. J. (1963) The effect of some kinds of routing systems on the amount of traffic in the central areas of towns. J. Inst. Highw. Engnrs. 10(l), 5-30.

Smeed R. J. (1%3) The road space required for trallic in towns. Town Plannina Review 33. 279-292.

Smeed R. J. (19&l) The tra& problem in towns. Town Planning Review 35, 133-158.

Smeed R. J. (1971) The effect of the design of road network on the intensity of traflic movement in diRereat parts of a town with special reference to the effects Of ring roads. Technical Note 17, Construction Industry Research and Information Associa- fion (CIRIA), London. (Presented to the- Tewkesbury Sym- posium, 1970, University of Melbourne, Australia.)

Smith M. J. (1974) Traflic control in a town with two ring roads. Trafic Engineering and Control April/May, 563-565.

APPENDIX

Justification of differentiation under the integral sign in (3) We let, for any non-zero h with - r < h <R - r,

6(x,y,h)=s(r+h,R,x,y)-s(r,R,x,y)

and

so that

A(h) = S(r t h, R)- S(r, R),

A(h) -= II 6(x, Y, h) h h fb y)dx dy.

Now, as h + 0,

Sk Y> h) h

+ D,s(f, R, x, y)

for almost all (x, y). Moreover our aqsumptions about route choice and travel speed imply tha\

IWy,h)l~2lhl I\ (7)

for all (x, y), and so

I I hY,h) s2

h

Minimking radial travel

where

If 2f(x, y) dx dy < tm

207

A(h) -= h I I Wx,y,h)

h fk y)dx dy

+ I I Ddr, R, xv ylf(x, Y) dx dy

since m is finite. Lebesque’s dominated convergence as h j-0, as required. The nub of the argument is theorem now implies that inequality (7).