The q-analysis of road intersections

Int. J. Man-Machine Studies (1976) 8, 531-548

The q-analysis of road intersections J. H . JOHNSON

Regional Research Project, University of Essex, Colchester, U.K. (Received 10 June 1976) There are two finite simplicial complexes associated with a road junction and there is a correspondence between the structure of these complexes and an intuitive evaluation of the junction's ability to carry flows. This intuitive feeling is made well-defined by the introduction of an order relation between junctions according to the flows they can accommodate. In comparing junction designs we find that a relatively low dimension structure q-connected only for relatively small q is associated with a superior ability to accommodate flows, and we illustrate the application of our findings to a complicated interchange and an experimental roundabout. The structural analysis was performed on the Essex PDP-10 computer using less than one minute computing time and 2K store for a complicated intersection showing there are no computing problems to be overcome.

1. Introduction It is widely accepted that the greatest source of traffic congestion arises at the intersection o f two or more roads. In the case of simple intersections one can see intuitively why one layout will " p e r f o r m " better than another but in more complicated intersections it is not so clear. By the application of the methodology of q-analysis to the subject of road intersections we attempt to formulate structural reasons to support our intuitive feelings for simple junctions so that these may give insights into more complicated cases. Our structural approach is absent from the literature which is preoccupied with functional relationships, often of a statistical nature. In this work we reject the statistical approach and pursue the path indicated by Atkin's Methodology of q-analysis, where the fundamental ideas are discussed elsewhere (Atkin, 1974a, b, c, 1975) as is the mathematics. In what follows we shall use a second structure vector, P, associated with a complex K Y ( X ) where Pc is the number of elements Yl e Y which are identified with an r-simplex in KY(X) where r>~q. We use the notation P Y to denote the second structure vector of KY(X) and PX to denote that of KX(Y). We extend this notation to the first structure vector Q so that QY is the first structure vector of KY(X) while QX is that o f KX(Y). .

The structure of a t w o - r o a d junction

o

b

c

L1

L2

d

...................... h

L4

L3

g

f

FIG. I. 531

e

532

J.H. JOHNSON

We consider the intersection of two roads illustrated in Fig. 1 where it is divided into four areas L1, L2, La and L~ called links. The intersection has four origins a, c, e, g, four destinations b, d, f, h and twelve origin-destination pairs (O-D pairs) ab, ad, af, cb, ere: We define a route through the intersection for each O-D pair as a set of links, and we may refer to a route by its O-D pair. As a preliminary to investigating flow patterns, we analyse the structure of four designs for a two-road junction. 2.1. NARROW JUNCTION In the case of a narrow junction we assume a left turning vehicle traverses one link, a non-turning vehicle traverses two links while a right turning vehicle traverses all links as illustrated in Fig. 2.

l l it :,l° ~,.

i

. . . . . . . . . . . . . . . . . .

~'@~

.....

Imfl

L

Vehicle Left turn

NO t u r n

Right turn

F I G . 2.

We represent the routes between the O-D pairs by the incidence matrix A of Table 1 which represents the relation 2 c R X L defined as R~ 2 Lj iff Lj is traversed by Rv Thus, for example, Rs = {La, L4} since Asa = A84 = 1. The relation 2 has associated with it the simplicial complexes KR(L, 2) and KL(R, 2-1) and we investigate their structure by q-analysis.

q-analysis of narrow junction KR(L, 2) q=3 (R3, Re, Re, R12) q=2 (R3, Re, Rg, Rx~) q=l (R3, Rs, Re, Rv,, Rl, Rs, R8, RII) q=0 (R3, Re, R~, R12, R~, R 5, R8, Rll, RI, R4, R~, Rio)

Structure vectors 4

0

QR = (1 1 1 1) P R = (4 4 8 12)

KL(R, 2-1)

q----6 (L1), (L,), (L3), (L4) q=5 (LO, (L~), (L3), (L~) q=4 (L1, L,, Ls, L,) All smaller q-values give one component as above.

Structure vectors 6

QL----- 0 4 1 1 1 1 1 ) PL

=

(4444444)

0

533

Q-ANALYSIS OF ROAD INTERSECTIONS

TABLE 1 Incidence matrix for narrow junction Route

O-D pair

Lx

L=

Ls

L,

1

ab

1

0

0

0

2 3 4 5 6 7

ad af cd cf ch ef

1 1 0 0 1 0

1 1 1 1 1 0

0 1 0 1 1 1

0 1 0 0 1 0

8

eh

0

0

1

1

9 10 11 12

eb gh gb gd

1 0 1 1

1 0 0 1

1 0 0 1

1 1 1 1

2.2. WIDE JUNCTION OR MINI-ROUNDABOUT

t !I

i|il

.....

~bl ..... /L._ . ~!

Ill

.

/

.

. r~i

iBi

I

'Vehlc le Left t u r n

No turn

Right turn

FIG. 3.

In the case of a wide junction or mini-roundabout we assume the paths of left, right and non-turning vehicles illustrated in Fig. 3 giving the route-link incidence matrix o f Table 2.

q-analysis of wide junction

KR (L, 2) q=2

(Rs), (Re), (R9), (R1,) q=l

(R3, Rf, Rg, R12, R2, Rs, R8, Rll) q:0

Structure vectors 2

0

QR = (4 1 1) P n = (4 8 12)

(R3, Rs, Rg, R12, R2, Rs, R8, Rn, Rx, R4, RT, Rio) KL(R, ~-1) q=5 (L0, (L,), (L3), (L,) q=4 (L:), (L2), (L3), (L,) q=3 (La), (L,), (La), (L,)

Structure vectors 5

0

Q L = (4 4 4 1 1 1) P L = (4 4 4 4 4 4)

534

J.H. JOHNSON

q=2 (Lx, L2, L3, L,) All smaller q-values give one component as above. TABLE 2

Incidence matrix for wide junction Route

O-D pair

Lx

L2

L3

L,

1

ab

1

0

0

0

2 3 4 5 6 7

ad af cd cf ch ef

1 1 0 0 0 0

1 1 1 1 1 0

0 1 0 1 1 1

0 0 0 0 1 0

8

eh

0

0

1

1

9 10 11 12

eb gh gb gd

1 0 1 1

0 0 0 1

1 0 0 0

1 1 1 1

2.3. JUNCTION WITH A CENTRAL RESERVATION b

c

f

L1

L2

d

0 L6

h

Ls

"-..

L4

9

L3

f

FIo. 4. I f there is sufficient r o o m new links may be introduced into the intersection to form a central reservation illustrated in Fig. 4. This arrangement has the route-link incidence matrix of Table 3.

KR q=3

(R~), (R12) q=2

(Re, R~), (R12, Ru) q----1 (Re, Rs, Rs), (RI,, R n , R~), (R3), (Rg) q=0

(Rs, Rs, Rs, R12, Rn, R,, R3, Rg, R1, R4, RT, Rio)

Structure rectors 3

0

Q R = (2 2 4 1) PR = (24812)

Q-ANALYSIS OF ROAD INTERSECTIONS

535

KL q=4 (L1), (L2), (L3), (L4) q=3 (L1), (L2), (Ls), (L4) q=2 (L0, (L2), (La), (L4), (Ls), (L6) q=l (Lx, L2,/-,8, I_,4, Ls, L6) q=O (L~, L,, La, L,, Ls, L,)

Structure vectors 4

0

QL = (44611) PL = (44666)

TABLE 3

Incidence matrix for central reservation junction Route

O-D pair

Lz

Lt

Ls

L,

L5

L.

1

ab

1

0

0

0

0

0

2 3

ad af

1 0

1 0

0 1

0 0

0 0

0 1

4

cd

0

1

0

0

0

0

5 6 7

cf ch ef

0 0 0

1 1 0

1 1 I

0 1 0

1 1 0

0 0 0

8

eh

0

0

1

1

0

0

9 10 11 12

eb gh gb gd

1 0 1 1

0 0 0 1

0 0 0 0

0 1 1 1

1 0 0 0

0 0 1 1

2.4. J U N C T I O N WITH UNDERPASS

r"-'

°/t

rb c]/ L1

.

.

.

.

i,.~ .

.

.

.

"LI ...... [GI

.

.

L7

f--t ° " ~L 8

FIG. 5. We represent in Fig. 5 a junction having a central reservation with a two-way underpass beneath it. The incidence matrix of this arrangement is given as Table 4.

q-analysis of junction with underpass KR q=3 (Re), (RI~)

J.H. JOHNSON

536 q=2

(Re, Rs), (RI~, RIO q=l Structure vectors

(Re, Rs, R12, R~0, (Rs), (Ra) q=0 (Re, Rs, R12, Rn, R3, Re, R1, Rv RT, Rio), (R2), (R~)

3

0

QR = ( 2 2 3 3 ) PR = ( 2 4 6 1 2 )

KL q=3 (L0, (Lz), (La), (L~) q=2

Structure vectors

(L~), (L2), (La), (L4), (Ls), (Le)

8

q=l (LI, L,, 1.3, 1,4, Ls, Le) q=0

0

Q L ----- ( 4 6 1 3 ) PL = (4668)

(El, L,, 1,3, L,, L~, Le), (L0, (Le) TABLE 4 Incidence matrix for underpass junction Route

O-D pair

LI

L2

L8

L~

L5

L6

L7

L8

1

ab

1

0

0

0

0

0

0

0

2 3 4 5 6 7

ad af cd cf ch ef

0 0 0 0 0 0

0 0 1 1 1 0

0 1 0 1 1 1

0 0 0 0 1 0

0 0 0 1 1 0

0 1 0 0 0 0

1 0 0 0 0 0

0 0 0 0 0 0

8

eh

0

0

0

0

0

0

0

1

9 10 11 12

eb gh gb gd

1 0 1 1

0 0 0 1

0 0 0 0

0 1 1 1

1 0 0 0

0 0 1 1

0 0 0 0

0 0 0 0

2.5. COMPARING THE STRUCTURES With the notation J1 = narrow junction, Jz = wide junction, Js = junction with central reservation, J~ = junction with underpass, we can see that as we proceed from Jx to J4 the structures tend to decrease in dimension and become less connected at high q-values. Let Q and Q ' be the structure vectors of the abstract complexes KY(X, p) and KY(X, p') and suppose there is a value o f q, q = p, such that Qq/> Q~ for q > / p and Qq ~< Q~ for q < p. For high values of q, q > p, Q~ > Q~ will usually be associated with some of the Y, having greater dimension, and hence being q-connected at higher values of q, in K than in K'. For low values of q, q < p, Qq < Q~ will usually mean that more pairs, Yt and Yj, are not q-connected (and hence not q + l connected, not q + 2 connected, etc.) in K ' than in K. This motivates the definition of a crude comparison of the connectivity structures of two complexes based on their structure vectors.

537

Q..ANALYSIS OF ROAD INTERSECTIONS

If dim K ' ~< dim K we make the structure vectors of equal length by defining Qq = 0 and P~ = 0 for dim K >~ q > dim K'. We define theflipover value of Q with respect to Q', when it exists, as the lowest value o f p such that Q~ >~ Q~ for q ~> p and Q~ ~< Q~ for q < p. If Q # Q' and there exists a flipover value .of Q with respect to Q' we write Q' < Q and K ' < o K. We interpret K ' Pq for all q. A weaker dimension comparison to allow for the introduction of low dimension Yl in K ' (such as the introduction of new links) may be made by defining the flipover value, r, of P with respect to P' as the lowest value of q such that P~/> P~, q ~> r and Pq ~< P~, q < r. If P ~ P' and there exists a flipover value of P with respect to P', (this possibly being different from the tiipover value of Q with respect to Q' when it exists) we write P ' < P and K ' < r K. In the case K and K ' are associated with junctions J and J' we shall write J' < 0 J, J' < e J and J' < p J as suitable. Collecting the results of our analysis we have: KR

8

0

Jt J2 Ja J4

QR QR QR QR

= (1 1 1 1) = (0411) ----- (2 2 4 1) = (2233)

J1

PR PR PR PR

----- ( 4 4 8 1 2 ) = (04812) = (24812) = (24612)

QL QL QL QL PL PL PL PL

= = = = = = = =

3

J2 Ja J4 KL

Jz J~ Ja J4 J1 J~ J3 J4

J~.
J1
J2
Jz < rR Jx, J~ < rR J4,

J~ < PR J1 J~ < PR Ja

J2
Ja
J4
J2
-1"8< eL J1, J4 ~< eL J=,

J4 ~ PL Jt J4 ~< eL Ja

0

e o (4411111) (0444111) (004461 I) (0004613) (4444444) (0444444) (0044666) (0004668)

JS
These results suggest that link dimension and the connectivity structure of K L are particularly relevant in the design of junctions.

3. Flow through an intersection 3.1. THE FLOW PATTERNf We define a flow pattern, f , on the complex KL(R, 2 -1) by defining the zero p a t t e r n f ° on the set of routes as vertices (using the inner product notation) as 0 ~< ( < R t > , f °) = f ~ -----number of vehicles traversing route Rl in z units of time.

538

J.H. JOHNSON

We use the notation ap(L~) to denote the p-simplex identified in K L by Lj. The pattern f " on the r-simplices of KL is defined by the pattern generator z7 as (c.f. Atkin, 1974a). (a,, i f ) = (tr,, ~,f0) in general, and in particular (trp(Lj),f p) = (ap(Lj), ,~pfo) = (f,trp(Lj),fo) =

y. ( < R , > , f °) = ~i a Lj

E A Ri a Lj

is the number of vehicles traversing link Lj in time ~. We definefto be the graded pattern f = f o + f ~ + . . . + f ~ , D = dim KL. As an example let ab(Ll) = ( R a , R2, Rs, Rg, RI~, R12), ~rs(L~) = (R2, Ra, R~, Rs, R6, R12), a(Ls) = (Ra, Rs, Re, RT, Rs, R~) and ~(L~) = (R~, Rs, R~, R~o, R ~ , R ~ ) asinJ~, wide junction. Letf~ = 20,f~ = 30,fa = 20,f~ = 50,f5 = 25,fs = 15,f7 = 10,fs = 40, fa = 35,fro = 5,fxx = 3 0 , f ~ = 10. Then (e~(L0,f ~) = (a(L~),~ ,fo) = ( f , a(L0, fo) = 2 0 + 3 0 + 2 0 + 3 5 + 3 0 + 1 0 = 145 and (trs(L,),f ~) = 150, (tr~(La),f~) = 145, (a~(L~),f~) ---= 135. 3.2. FLOWSACCOMMODATED BY A JUNCTION It is well known in the theory of road traffic that a piece of road has a maximum flow that can pass over it in a given time where this is usually called its capacity. Thus for each link Lj we assume there is a maximum flow, Fj, that can pass over it in time T. The flow pattern f, primarily defined as f0 on the routes between each origin-destination pair, is such that (ap(L~),f) = (ap(Lj) ~pf0) which is the total flow traversing Lj in time v. This flow must be less than or equal to the maximum F~ so we have for each Lj the link

constraint (ap(L~),f) < ~ -> (3.1) We shall say that a junction J can accommodate a flow p a t t e r n f i f f (3.1) is satisfied for every link of J. For simplicity we shall here assume all links have the same maximum flow which we shall write as F ---- Fj for all Lj. Let J and J' be junctions designed for a common set of origin-destination pairs with one route R, in J and R[ in J' between each origin-destination pair. The junction J is flow inferior to J' (J' isflow superior to J) written J ~
Q-ANALYSISOF ROAD INTERSECTIONS

539

Consider J2 and Ja. L e t f b e a flow pattern accommodated by Jz so that on KL(J2), the link-route complex associated with J~, we have (as(L~),f 5) (trb(Lz),f 6) (as(La), ./5) (trs(L4), fa)

= = = =

fx+f~+fs+fs+f~l+f~2 A+fa+A+A+A+flz fa + A + f , +f~ + f s + f , fe + f s +f9 +flo +fix +f~z

<~F ~< F ~< F ~< F

But each tr4(Ll) in KL(Ja) is a face of as(L~) in KL(Jz) so by definition of ,~ on KL(J3) (a,(L~),f) ~< F f o r i = 1, 2, 3, 4. Also on KL(Ja) (a2(Ls), f ~) = A + A +f0 ~ F so that g cannot be accommodated by J,. Thus J2 is strictly inferior to Js, J,
4. Relationships between structure and accommodated flow

patterns Since the flow pattern f of link constraint (3.1) is generated by z~ which depends on the structure of the complex supporting f, we would expect some relationship between the structure of a junction and the flow patterns it can accommodate. As noted previously there happened to be a correspondence between the structure relations
Proposition 1. If dim KL(J) > dim KL (J') there is a flow pattern accommodated by J' but not accommodated by J, i.e. J' ~ / J. Proof. Let D = dim KL(J') and f be a flow pattern with f~ = F/(D+I) for all routes < R~ > . Then for any trr(Ll ) in KL(J') we have p ~< D and (ap(Lt), f ) = ( p + l ) (F/ ( D + I ) ) ~< F, a n d f i s accommodated by J'. However, there is a simplex aq(Lj) in KL(J) with q = dim K > D. For this simplex (trq(Lj),f) = ( q ÷ l ) (F/(D+I)) > F a n d f i s not accommodated by J. Our four crossroad designs had the property that each KL(JI+I) was a subcomplex of KL(JI), i = 1, 2, 3, and in general we have the following. Proposition 2. If KL(J') is a subcomplex of KL(J) every flow pattern accommodated by J is accommodated by J', i.e. J ~$ J'.

540

J.H. JOHNSON

PrOof. L e t f b e a flow pattern accommodated by J. For every a(L~) in KL(J') there is a simplex a in KL(J) with tr(L~) a face of a. By definition of 3, (tr(Lt), f ) <<,(tr,f ) <<.F so that f i s accommodated by J'. Corollary. The structure of J1, J2, Ja and J4 require Jx <$J2 < $ J a <.¢ J4. We see from Proposition 2 that the introduction of new links in the design of a junction will not limit the accommodation of flow patterns provided the new links are faces of an existing link, and by Proposition 1 new patterns will be accommodated if the introduction of new links reduces the dimension of KL. If two routes are p-near (Atkin, 1976) their flows are mutually interfering on p + 1 links and their flow values are involved together in p + l link constraints (3.1). A reduction in q-nearness means they are mutually involved in less of these constraint equations and their flows are less constrained.

Proposition 3. Let R, and Rj bep-near in KR(J) and q-near in KR(J'), q < p. Let < Lk > be a vertex of Rt and Rj in KR(J) but not a vertex of R~in KR(J'). If there exists a flow patternfaccommodated by J and J' with ( a ( L , ) , f ) = F and for I ~ k (tr(Ll),f) < F on KL(J) and KL(J'), then J' ~ : J.

Proof. Let c~ be the smallest of F--(a(L~),f) over all l =/: k, and let g be a pattern generated on gO = f 0 except for g~ ----f~+~. Then g is accommodated by J' since for all Lz :/= Lk, (a(L,), g) ---- (a(L,), f q - a f ) ~< (a(L,), f ) q-a, <~(a(L,),f)q-F--(a(L,),f) = F, while (a(Lk), g) = (a(Lk),f) because < Rl > is not a vertex of tr(L,) in KL(J'). But g is not accommodated by J since on KL(J) we have (a(Lk), g) = ( a ( L , ) , f ) + a i -----F + a i > F. This proposition suggests that it is advantageous to disconnect routes, and if we disconnect two routes altogether we obtain the next, stronger, result.

Proposition 4. If a(R~) and a(R:) are q-near in KL(J), q >~ 0, but share no vertices in KR(J'), i.e. they are disjoint, then J' <~: J.

Proof. Letfi = fj but not by J.

----- F ,

and all other route flows be zero. T h e n f i s accommodated by J'

5. Traveltime patterns The time a vehicle spends on a link will depend on the characteristics of that link and the route the vehicle is travelling. We define t~j to be the mean traveltime for a vehicle on route R~ to traverse link Lj and obtain the quantities Tlj ----fihj -- time taken for thef~ vehicles on R~ to traverse Lj :~ 0 iff R~ 2 Lj and f~ > 0. 5.1. ROUTE TRAVELTIME PATTERNS For each R~ we can define on K R a pattern T~ = T~i+ T ~ + . . . ( < Lj >,fo.) _-- Tij and T{ = ,~PT~iso that (~(RD, T 9 ----

~,

+T~, D -----dim KR, with

T~j = a ~

Rk ~ Lj

is the time spent by thef~ vehicles on Rl traversing the links it shares with Rk, and a , is the total time spent on the junction by thef~ vehicles on route R~.

Q-ANALYSIS OF ROAD INTERSECTIONS

541

I f for every flow patternfaccommodated by J and J' we have a t" ~< a , for all Rl with inequality holding for some i we shall say that J is time inferior to J', J < t J'5.2. I N T E R F E R I N G F L O W S

As the flow on a piece of road increases, the vehicles interfere with each other by mutual obstruction and in the more difficult conditions the drivers slow down because they feel a greater likelihood of accidents. It is often assumed that vehicle traveltime on a link increases with the flow on that link, so that tij are not constants but increasing functions o f link flow, i.e. we suppose that

t,j = t,j( (crp(L,), f ) ) is an increasing function for each R~ and each Lj. A complete study of this pattern relationship is beyond our scope here, but we obtain some useful special results by pursuing our study of the relationship between junction structure and patterns defined on it.

Proposition 5. If a route R~ is related to the same set o f Lj in J' as in J, but the dimension o f these links in KL(J') is in some cases less than and other cases equal to their dimension in J, then J' ~ t J.

Proof. L e t f b e accommodated by J and J' with allf~ equal. Then for each link Lj of R~ we have (trp(Lj),f0 = ( p + l ) f i in KL(J) and (trq(L1),fq) = ( q + 1)fi on KL(J'), where p >~ q and inequality holds in son~ cases. Thus the links of R~ have greater than or equal flow on them in J than in J', and the traveltimes on the links of R~ in J' are less than or equal to the traveltime on those links in J and have a smaller sum in J' than J. Thus for this f we have a ,i ~< a , . For the following results we assume the functions t~j are the same for all links and routes and rewrite them as t.

Proposition 6. Let tr,(Rt) and as (R j) be p-near in KR(J) and q-near in KR(J'), q < p. Then J' ~Zt J.

Proof. Letf~ = f j = 17/2 and let the flow on all other routes be zero. On K R we have the traveltime pattern (tr,(R,), T[) =

~ R i ALk

T,k =

~

f~ t(tr(Lk),f))

R i A Lk

= (r--x)(fi t(f~))-}-(x+l)(fl t(fi-}-fj)), x = dim tr,(Rt) n trs (Rj). Since t is an increasing functionfi t(f~) < fl t(f~ +fj), so that when x = q in J' the traveltime is less than when x = p > q in J. Proposition 7. Let Rl be associated with a,(Rl) in KR(J') andtrs (R~) in KR(J) with r ~< s. Let p(Rt) be the number of links of Rt in KL(J) with dimension greater than or equal to p, and p'(R~) be the number of links of R~ in KR(J') with dimension greater than or equal

542

J . H . JOHNONS

to p. If p'(Rl)~
aii ~ ~li"

Proof. We can define a 1-1 mapping, 0, of the set of links of R, in J' into the set of links of Rl in J with 0 : Lj --* 0(Lj) and q = dim tr~(L~) ~< dim ~rp(0(Ll)) = p, where inequality holds for some Lj. Under conditions of equal flow (trq(Lj),fq) = (q+l)fi ~< (pW1)fi = (trp(O(Lj)),fP). Comparing < Lj > in KL(J') and < 0(Lj) > in KL(J) we have ( < Lj > , = f ~ t(q+l)fi) ~ , T~) since t is an increasing function of flow. Thus for each link L: of R, in J' there is a unique link 0(L~) of R~ in J, i.e. 0(L~) = 0(L~) iffj -----k, with traveltime greater than or equal on 0(Lj) than Lj. Over R~ 2 Lj in J' the sum of the ( < Lj > , T~) is less than the sum of the ( < 0(L~) > , T~) which is less than or equal to the traveltime on Rl in J. Hence a[~ < art.

6. q-transmission Let K R and K L be complexes associated with a junction and consider in K R a chain of q-connection trp(R~), trp(R~), trp(R3) with trp(Rx) n ap(Ra) = o. Let {a(L~)}, (a(Lb)} and {tr(Lc)} be the sets of simplices in K L having < Rx > , < R2 > , and < R 3 > respectively as vertices, so that {a(Lo)} n {a(Lc)} = e. We illustrate this in Fig. 6. KR

(

¢(R1)

or(R2 )

~(R 3 )

st°, o--,-sreo 6r°,o-,-6r3
l

+

i

i

, o. . . . . 6t>o st2o . . . . sf, o

lo'(u,)}

{*'cu~)}

{,'(a)}

Ro. 6. Consider an increase in flow on R1 as a force (v. Atkin, 1974a) Jfl > 0 on < Rx > in KL. By definition of J this requires a force Jft > 0 on all the links of {or(L=)}. For each link L= traversed by R2 this requires Jt2= > 0 and JT2° > 0 on < L° > in KR, which requires JT2 > 0 on tr(R~) with Jaa2 > 0, this being the sum of the increased traveltimes f2 Jt2° on the links it shares with a(R1). The greater the value of q, tr(Rx)and tr(R2)being q-near the greater the value of Ja2z will be, in other words the more links Ra shares with Rx the more will the increased flow on Rx increase the traveltime on Rz. The increased traveltime on R= means that fewer vehicles will travel it in time ~ so there is a force

Q-ANALYSIS OF ROAD INTERSECTIONS

543

8f~ < 0 on KL. This in turn requires a force 8 f < 0 on the links {a(Lb)} which is thus experienced by the links which Rs shares with Ra. For these links Lb, 8t3b < 0 so that 8 T ° < 0 on those < Lb > which are links of R3. This requires 8Ts < 0 on a(Rs) with 8a~ < 0 where this decrease in traveltime on Rs gets larger with q, a(R~) and a(Rs) being q-near. If the links of Rs not belonging to {a(Lo)} have a flow less than F, and there is sufficient demand to travel Rs it is possible for more vehicles to traverse it in time as an increase in flow 6fs > 0 on < Rs > in K L which requires a force 8 f > 0 on the simplices {a(L~)}. By this argument we see that, although Rx and Rsshare no links, a force on R~ can be q-transmitted along a chain of q-connection resulting in forces on Rs, and the greater the dimension of q the greater will be the value of the q-transmitted forces. The obstruction vector, and hence the structure vector can be seen to offer obstruction to the q-transmission of such forces since q-transmission can only occur within a q-connected component. We now see another advantage in a(R,) and tr(R~) being disjoint, viz. only if this is the case can an increase in flow 6f~ > 0 be transmitted as ~f~ > 0. These transmitted forces are associated with flows mutually interfering and the more highly connected K R and K L the more flows interfere. For these reasons we see obstruction to q-transmission manifest by large values o f Q, in the structure vectors Q as desirable, particularly for low values of q. Example Let J be a crossroads (right turns prohibited) having four links as in Fig. 1. Let ero(R1) = < L1 > , erl(Ra) = < Lz, L2 > , ao(Rs) = < L~ > , al(R4) = < L~, I-3 > , ao(Rs) = < L3 > , at(Re) = < 1-3, L4 > , a0(R~) = < 1-4 > and a~(Rs) = < L4, L1 > . Consider a force ~f~ > 0 as an increase in flow on Re shown in Fig. 7.

KR

KL

sf2>O l

6r2< 0

S~> 0 Sf
~ST1
sTs>o . . . . . . .

-I,~t~f,(R,d~- . . . .

I ";,7~ I

~sf.
ST7>O <0

=~Sf7

ST3 <° ~ ../sf 3 >o

i

,t.~

" ~"

'L,3 ~,

s~>o=,,

Sfa>O

",,R1

Re

R~~

sf,
ST5> 0 ,-I~

61'5<0

R7

6f>O ~ 6f>O

Sf6> 0 ~O. 7.

This requires a force g f > 0 on a,.(Ls) and a2(L,) and hence forces ~T,, ~Ts, ~TT, 6T s > 0 o n O'l(R~) , ao(Rs), ao(RT), at(Rs) respectively. These require forces 6f~, cSfs, ~f~, ~fs < 0 on < R4 > , < R5 > , < R7 > , < R8 > respectively and hence a force ~. < 0 on era(L1) and aa(L~). This requires forces ~T1, ¢~T2, 6T3 < 0 on ero(R1), al(R~), a0(Rs) respectively, and assuming the flows on these routes were previously inhibited by the other flows we have ~f~ > 0 for i = 1, 2, 3. In this example the forces are 0-transmitted

544

J. rI. JOHNSON

around the 0-object (Atkin, 1976) R2-R4-Re-Rs-R z in KR and the 0-object Lx-Lz-LaL4-L1 on KL.

7. Illustrative examples We have argued that the flow patterns a junction can accommodate depends on the multidimensional structure inherent in its design. We now show how our structural observations may be applied in evaluating different designs for two types of interchange. 7.1. MULTIDIMENSIONALSPAGHETTI Consider an interchange combining a "T" motorway junction with a large roundabout, the motorway having origins a, c, e and destinations b, d, f while the roundabout has origins g, i, k, m and destinations h, j, 1,n. With these origins and destinations there are 42 OD-pairs so the interchange will necessarily be complicated. In Figures 8 and 9 we illustrate two possible designs, A and B respectively. hg b

~

39

!i

19 20

FIG. 8.

By defining a route for each OD-pair in each design we obtain the complexes KR(A), KL(A) associated with design A and KR(B), KL(B) associated with design B. These complexes were q-analysed and the structure vectors are compared below. 11

5

0

KR(A)

Q =

(1 3 7 1 1 11 8 811 7 8 3 1)

KR(B)

Q =

(1261010

KR(A)

P

=

(138141923293337424242)

KR(B) P

=

( 1 2 6 1 1 15 19 25 31 37 42 42 42)

II II 11

17

KL(A)

Q =(555555

ii

111

5

8 8131111

0

4 1)

5

5

5

0

0

0

2 2 2 8 6 1 1 1 I)

Q - A N A L Y S I S OF R O A D INTERSECTIONS 15

11

KL(B)

Q = (0055555311

K'L(A)

P

= (5555557778

KL(B)

P

= (0055555577

17

545 5

1 212

II

15

5

o

7 3 2 1 1) 0

1010253238383843)

11

5

0

8 9242934364049) ji

hg 33

0

4

35

"

.Y 2 43

FIG. 9.

We see that q = 5 is the flipover value for B
546

J.H. JOHNSON

of attractive forces, allows a shorter traveltime on some routes, can accommodate all the flow patterns accommodated by design A and can accommodate flow patterns which design A cannot. For these reasons we judge design B to be superior to design A. 7.2. EXPERIMENTALROUNDABOUT The problem of increasing the capacity of roundabouts has been studied at the Road Research Laboratory over a number of years and their imaginative experiments may be experienced all over the country. We choose to study the A414/A416 Plough Roundabout (G.B. Road Research Laboratory, 1973) at Hemel Hempstead where this experiment was begun in June 1973 and uses the novel idea of having two-way traffic on the weaving sections as shown in Fig. 10. The complexes associated with the pre-experimental design Q

k--AT;,1

J---7"4~o/ \~16s 4 ~2' ~

2

7

1

6/

/ I"'

•

,

x(-~--~-~

L:'.°-Y g

,, 5 \ ~

"

g

f

(a)

-

-

t

(b)

FIo. 10. (a) Pre-expenment;(b) post-experiment. are denoted KR(C) and KL(C) while those of the experimental design are denoted KR(D) and KL(D). The structure vectors of these complexes are compared below. 12

8

4

0

KR(C)

Q = (6 6 1 2 1 2 1 3 1 3

KR(D)

Q --(0

KR(C)

P

= (6 6 1 2 1 2 2 4 2 4 2 4 3 0 3 0 3 0 3 0 3 0 3 0 )

KD(D)

P

= (0 0 0 0 1 2 1 2 1 8 2 4 3 0 3 0 3 0 3 0 3 0 )

KL(C)

Q = ( 12 12 12 7 7 7 1 1 1 1 1 1)

KL(D)

Q = (

KL(C)

P

= ( 12 12 12 30 30 30 30 30 30 30 30 30)

KL(D)

P

= (

12

7 2 2 1 1 1 1)

8

0 0 0121218

12

8

12

8

II

4

4

8

0 0 0181818

ii

4

8

11

11

4

8

8

0

2 8 2 2 1 1)

4

0

0

0

0

7 7 7 1 1 1) 4

4

0

0

0 0 0181818303030303030)

Q-ANALYSIS OF ROAD INTERSECTIONS

547

We see that q ---- 7 is the flipover value for D
8. Conclusions Of the many ways of organizing road intersections some are inherently superior to others because of their connectivity structures. The Methodology of q-analysis permits a unified view of this structure with the flow and traveltime patterns it supports and provides a rich vocabulary for the study of intersection design. In general we find that a low dimensional structure which is only q-connected for low values ofq is superior to a higher dimensional structure q-connected at higher values of q, where this comparison can be made using the results of q-analysis and the associated structure vectors. The application requires only a modest computer capability and provides a simple way of comparing complicated intersection designs.

References AaroN, R. H. (1974a). Mathematical Structure in Human Affairs. London: Heinemann. ATKIN, R. H. (1974b). An approach to structure in architectural and urban design, 1: introduction to mathematical theory. Environment and Planning, B, 1, 51-67.

548

J. H. JOHNSON

ATKIN, K. H. (1974c). An approach to structure in architectural and urban design, 2: algebraic representation and local structure. Environment and Planning, B, 1, 173-191. ATKIN, R. H. (1975). An approagh to structure in architectural and urban design, 3 : illustrative examples. Environment and Planning, B, 2, 21-57. ATION, R. H. (1976). An algebra for patterns on a complex, II. International Journal of ManMachine Studies, 8, 483-498. G.B. MINISTRYOF TRANSPORT(1966). Roads in Urban Areas. London: H.M.S.O. G.B. ROAD RESEARCHLABORATORY(1965). Research on Road Traffic. London: H.M.S.O. G.B. ROAD RESEARCHLABORATORY(1973). Ring Junction Experiment at the A414/A416 Roundabout, Hemel Hempstead, Herts. T. & R R L Leaflet LF 367. JOHNSON, J. H. (1975). A multidimensional analysis of urban road traffic. Ph.D. thesis, University of Essex.