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Trip assignment to radial major roads
0191-2615/87 U.OO+ .OO 0 1987 Pcrgamon Journals Ltd.
Vol. ZlB, No. 6, pp. 433-442, 1987 Tmmpn. Rcs.3 Primed in Great Britain.
TRIP ASSIGNMENT
TO RADIAL
MAJOR ROADS
GLEN D’ESTE School of General Studies, Tasmanian State Institute of Technology, P.O. Box 1214, Launceston, Tasmania, Australia 7250 (Received 29 Junuary 1986; in revisedform 5 May 1986) Ah&act-The problem of flow-dependent trip assignment is considered for a city with a small number of radial major roads. CBD-based work trips are assigned to these radial major roads on the basis that each commuter seeks to minimise his individual travel time. A system of differential equations is derived for the spatial pattern of trip assignment in a model city with a continuous distribution of home locations and a ring-radial road network. This system is then solved for the special case of a uniform distribution of home locations. INTRODUCTION
In many cities travel to and from the central business district (CBD) takes place predominantly on a small number of radial major roads. This paper looks at the problem of trip assignment to these radial major roads in a certain model city. Here the term “trip assignment” is being used to denote the process of predicting the route followed between a given origin and destination. The assignment criterion that will be used is Wardrop’s 1st Principle which states that the equilibrium assignment is one for which, “the journey times on all routes actually used are equal, and less than those which would be experienced by a single vehicle on any unused route” (Wardrop, 1952). Basically this is a selfish strategy since each commuter seeks to optimise his own route regardless of other road users. It is however the one most likely to be used in practice. Given this behaviour, the spatial pattern of trip assignment will be derived for the following model city. THE
MODEL
CITY
Consider an axially symmetric city of radius R and in particular the group of commuters who travel each day to and from workplaces in the CBD. This study is only concerned with CBD-based work trips however it is recognised that this is only one component of the overall traffic pattern of the city, albeit an important one. It will be assumed that the CBD is sufficiently compact in comparison with the city as a whole to be taken as being concentrated at the city centre. On the other hand, the homes of commuters are widely dispersed throughout the city. Their distribution will be given by an axially symmetric urban density model, f(r), where r is the distance from the city centre. It should be noted that f(r) is not simply a population density model since it only relates to a certain subset of the population. Rather it is a trip-end distribution function, conditional on workplace location. Serving the CBD is a set of n radial major roads. These major roads will be denoted % . . %= and it will be assumed that they are of dual carriageway design and have pri&y g; intersections so that traffic speed is dependent only on local traffic density in the direction of travel. Many functional forms have been proposed to describe the relationship between traffic speed and density, see Branston (1976). Of these the one that will be used for all radials is the Bureau of Roads model u = V”[l + AqE]-‘, where u = traffic speed, free (maximum) traffic speed, traffic flow, and A, B are constants.
V” = q =
433 TRb21/6-b
(1)
G. D’Esrn
434
Table 1. Summary of notation used in a dynamic trip assignment model (a) Geometric R
d,,
aP2,. . * 3” 4i si VO
A
radius of the city
R radial major roads numbered consecutively clockwise from an arbitrary fixed point angle between 92, and %,+, boundary contour which divides commuters who live between those who use 9Zi and those who use .$+, free (maximum) traffic speed on a radial major road nonnalised speed/flow gradient on 2,
9, and %,+, into
(b) Functional
tb r&(r) VI(r) ‘I;(r) P,(r) W) L,(r)
radial distance from the city centre density of trip origins traffic speed on the minor road network at a distance r from the city centre traffic flow on .?Xiat a distance r from the city centre average traffic speed on R, at a distance r from the city centre time taken to travel a distance r from the city centre along Si probability that a commuter lives at a distance greater than r from the city centre and uses 28, angle between 9Zi and Bi at a distance r from the city centre distance from 9Zi to Bi along a minor ring at a distance r from the city centre
This model has three parameters V”, A, and B which can be used to represent the individual characteristics of each radial. However since maximum traffic speed is presumably governed by regulations rather than traffic conditions it will be assumed that V” is the same for all radials. Further in the first instance it will be assumed that the roadway configuration is uniform throughout its length and that the speed/flow relationship on all radials can be well represented by a first order (B = 1) model. Also remembering that the Bureau of Roads model is a local model, we arrive at the form Vi(x) = V”[l + Aiqi(x)]-‘,
(2)
where Vi(x) = traffic speed on %i at a distance x from the city centre, G(X) = traffic flow at x on %i, and Ai = speed/flow gradient on %i. The radial major roads will be assumed to provide the quickest and most convenient means of travelling to and from the CBD. Consequently during the morning and evening peak-hours commuters travelling to or from the CBD always use these radial major roads while other travellers avoid them. Hence, the contribution to traffic density from other trip types and purposes is negligible. Access from the commuter’s home to these radial major roads is provided by an infinitely fine set of ring roads. These are minor roads and it will be assumed that traffic speed on them is dependent only on distance from the city centre. ROUTE
CHOICE
Within the road network described above a commuter travelling from home to a workplace in the CBD first travels circumferentially along the minor ring road that passes his home and continues until reaching a radial major road. The commuter then proceeds along this major road to the city centre. Routes are thus single-turn ring-radial. With this routeing strategy the route of an individual commuter is well defined and has simple geometry. There is however one aspect of the route that is not specified. That is the radial major road on which the commuter chooses to travel. The problem of trip assignment is then to predict the outcome of this choice given that each commuter seeks to minimise his individual travel time. The first step is to identify the range of options available within the framework of
Trip assignment to radial major roads
435
the current model. Consider a CBD-bound commuter whose home lies in the region Conceivably such a commuter could use any radial, however for between gi and S%?;+l* the purposes of the current model it will be argued that the choice is between &; and %i+1 only. This assumption can be justified by appealing to the definition of a major road. With the current routeing discipline it is a simple matter to show (see D’Este, 1983) that at any given distance from the city centre the set of origins that contribute to traffic on a particular radial major road forms a continuous circular arc that intersects that radial. Therefore, if there exists a commuter that lives between 3; and %;+I but finds it quicker to travel to the CBD by %;_I, it follows that amongst other commuters that live at the same distance from the city centre, nobody will use .%$. It is then questionable whether 5%;qualifies as a major road since it is not even used by commuters that live on it. The purpose of this study is to consider trip assignment to genuine radial major roads; these roads need not be identical but none should have a capacity so much lower than its neighbours that it can be completely dominated by them.
TRIP
ASSIGNMENT
Given that a commuter living in the sector between Bi and %;+I will use either 3; or ZX;+lTit follows that at any distance r from the city centre there is a point at which single-turn trips to the CBD by either radial have equal duration. The set of such points will map out a boundary contour S?;, which divides commuters in the sector into those who use !ZZ;and those who use %;+I. The problem of trip assignment is therefore one of deriving an expression for B;Let O;(r) denote the angle between 3; and g; at a distance r from the city centre. A point on B; must then satisfy the condition rei(r)
Ti(r) + -
u(r)
=
Ti+t(r)
+
$j
(4%
-
ei(r)),
where 4; = the angle between 3; and S??;+l, = the traffic speed on a minor ring road at a distance r from the city centre, and 7’;(r) = the time taken to travel from the city centre to a distance r from the city centre (or vice versa) on %!;,
u(r)
which is simply a statement of the equality of trip duration by either radial. Rearranging (3) to make e;(r), the subject of the equation gives O,(r) =
4 +$
{&+1(r) - K;:(r)).
(4)
This is the fundamental equation governing trip assignment but its usefulness depends on finding an appropriate expression for T;(r). Since traffic speed on radials is locally flow dependent, at any point r on S;, the traffic speed is given by (2) and T;(r)
=
I’ -k 0
=
(5)
vi(x)
h r (1 0
+ Aiqi(X)}dx,
(6)
where q;(X) is the flow on the radial B; at a distance x from the city centre. Under equilibrium conditions the traffic flow at x on 3; is proportional to the probability P;(x), that a commuter lives at a distance greater than x from the city centre and uses SF%!;. The
G. D’ESTE
436
region in which home locations satisfy this condition is bounded by Bi, LBi _ 1, and circles of radius x and R, so Pi(X) =
IR
*‘a+;-1+ e,(Z)
j(z)
x
- ei-l(Z)] dr.
(7)
Let the constant of proportionality which in this case is the global trip generation be absorbed into the gradient parameter Ai,then Ti(r) =
dz}dx,
$6 {I +Aj JI”j(z) *z[+i-l
+ e,(z) - ei-l(Z)1
rate
(8)
and finally
+ Ai+18i+l(z)
-
(Ai+
+
Ai)&
+ Aiej_l(z)}
dr dx.
(9)
A similar equation will hold for each radial thus giving rise to a set of n coupled double integral equations in the n variables 8i, . . . , 8,. It must be noted however that because of the circular nature of the problem
e,(r) = e,(r),
(10)
0,+&) = M% and we find that the system is not linearly independent. To close the system and allow a unique solution to be found, another equation is required. This equation can be derived from (4) by summing both sides over i to give
2,
e,(r) = i
i i=l
i=l
(11)
but I: +i = 2~r so
i
O,(r)
=
7F.
i=l
(12)
This is a conservation equation and highlights the interdependence of flows; if the patronage of one radial increases then that of another must decrease. Conceivably (9) and (12) could be solved numerically, however the process of finding that solution can be simplified by first converting to the equivalent differential equation and making the change of variable Lj(r) = r&(r).
(13)
Therefore, Li(r) is the distance from Bi to!Bi along a minor ring road at a distance r from the city centre. In terms of L,(r) the differential equation equivalent to (9) is
Trip assignment to radial major roads
437
with boundary conditions Li(0) = 0,
L;(R)
(13 Wb)
-
and
Much of the apparent complexity of (14) results from the dependence of traffic speed on minor ring roads on distance from the city centre. However, this dependence is weak and it is not unreasonable to consider traffic speed on minor roads to be independent of location. Therefore, if traffic speed on minor roads has the constant value u, (14) reduces to
L:! +
T$L%+lLi+l -
[Ai+ + Ai]Li + AiLi-,}
= s
[Ai&-
- A,+,+(]
(16)
with Li(0) = 0, L;(R)
(1%
= 442,
(17b)
which in turn can be reduced to the system of 2n coupled first-order differential equations
= T$ I Q;
{(Ai+, + 4)Li - Ai+lLi+l - AiLi-
+ r(Aih-1 - Ai+,&))
t18)
L; = Qi
with Li(0) = 0,
Pa)
and
2 Li(r)
i=l
= TP*
(I9c)
The process of calculating solutions can be further simplified by decoupling the system using an eigenvalue technique described in the Appendix. In the decoupled system, the equations are independent and can be solved separately by standard analytic or numerical techniques. Alternatively the current system (16) and (17) or indeed the original system (14) and (15) can be solved numerically by the “shooting” method or by finite-difference methods. See Conte and de Boer (1972) for details of these techniques. In any case, having manipulated the model into a form solvable by standard techniques, the trip assignment problem is essentially complete. The “shooting” and finite-difference methods are both robust techniques in that the equation coefficients need not be fixed throughout the interval. This quality can be exploited in extending the trip assignment model while retaining both as viable solution techniques. So far the parameter Ai has been considered to have the same value at all points along a radial, but since Ai reflects the roadway configuration, this is an unrealistic assumption. It is unlikely that a radial major road will have the same configuration all
G. D’Esrz
438
the way from the city centre to its outskirts. Hence, Ai should be allowed to vary with location. Inherently this will be a discrete process so there will be annular regions in which the traffic speed/flow gradient @ constant on all radials. The number of such layers will be equal to one plus the timber of distant radii at which there is a change in the value of any A;. Within each layer the conditions governing route choice are identical to those already modelled. Further, the boundary conditions (17) still apply and it follows that (16) and (17) is still a valid representation of the trip assignment pattern. The only additional condition is that solutions are continuous across layer boundaries. Hence, the model extends naturally to a situation in which the roadway configuration is allowed to vary along a radial. This model of trip assignment can be viewed as being complementary to those of Lam and Newell (1967) and Blumenfeld and Weiss (1970). These authors considered a city having a ring-radial routing system and a circular CBD with all trip origins outside the CBD and all destinations inside. Then for a given origin and destination they derived optimum routing strategies for travel within the CBD. In both cases the incoming radial of an individual was assumed to be known; the above model sheds some light on how that radial might have been chosen.
TRIP
ASSIGNMENT
IN A UNIFORM
CIRCULAR
CITY
Consider trip assignment to radial major roads in a uniform circular city with constant traffic speed on minor ring roads. This is a very special case since under these conditions (16) reduces to a system of differential equations with constant coefficients and hence an exact solution can be found. Without loss of generality the city can be taken to have unit radius so that R=l The conservation
f(r)
and
= l/n.
(20)
equation Ll =
TT
-
i
Li
(21)
i=2
can then be used to eliminate L1 from (16) so that the problem reduces to finding the solution to the nonhomogeneous vector-matrix equation a” - Aa
=
TW,
(22)
a(0) = 0,
(23)
a’(1) = +/2,
(24)
where a = (L2, &, . . . 7 L) and A is a constant (n - 1) x (n - 1) array and o, + are constant vectors of length (n - 1) whose values follow from substituting (21) into (16). Since (22) is a linear system the complete solution can be written as the sum of the solution of the corresponding homogeneous system a U = Ao
(25)
and a particular solution to (22). By inspection this particular solution is ap = -t-A-‘W.
(26)
Trip assignment to radial major roads
439
Since all coefficients are constants, solutions to the corresponding homogeneous system (25) will be of the form a = pe"
(27)
where lo,is a constant vector of length (n - 1). Substituting (27) into (25) yields (m21- A)pe"'
= 0
(28)
where I is the identity matrix. It follows immediately that m2 is an eigenvalue of A and p is the corresponding eigenvector. Therefore the complete solution to (25) is the linear combination n-1
aH = 2 Pi[f’i exp(-w) i=l
+ Qi exptmdl
(29)
is over the (n - 1) eigenvalues of A. where Pi, Qi are constants and the su~ation Now the bracketed term in (29) can be written as
Pisinh mir + Qi cash ntir
(30)
and further simplified by applying the initial condition a(0)= 0.Since ap(0) = 0 it follows that Qt = 0 and n-1
aH = C Pi pi sinh mir, i=l
(31)
which can be expressed more concisely as
ati = y(r).?,
(32)
where y is a constant vector of length (n - 1) and Y(r) is the so-called fundamental matrix of the system (25). The fundamental matrix of a system of differential equations is an array of its solutions, so in this case each column of Y is a vector function of the form, p sinh mr.Therefore, the complete solution to (22) is a(r) = Y(r).7 - t-A-‘co.
(33)
Applying the boundary condition (24) gives Y
= Y’(l)-*.{+/2 + A-b}
(34)
and a(r) = Y(t),Y’(l)-r{+/2
+ A-b) - rA-'o.
(35)
Finally Li follows from (21) and the trip problem for a uniform circular city is solved. All that remains is to convert back to the origin variables & using (13). The variable @i is a more instructive measure than L+because the effect of trip assignment is not clouded by any inherent dependence on distance from the city centre. Any variation in 9: with radial distance is derived purely from the effects of commuter choice. An example of trip assignment in a uniform circular city with no layering and 5 radial major roads is illustrated in Figs. 1 and 2. The corresponding model parameters are summarised in Table 2. For this hypothetical city the pattern of trip assignment is investigated for two scenarios, termed Cases 1 and 2, which differ only in the traffic
G. D’ESTE
440
Angular Coverage Case I
Case 2 zc
130.e..... ----
llO-
Radial Radial Radial
----_
--IIO-
I: ii s70ii
Radial 4 Radial 5
----__
---_ --.
90-
130-
1 2 3
----_
_
.............................~*~**~~~~
m-
e
n
c-50- ________------- ___--- -
50-
307
I
I
I
307
0
.25
.5
.7s
r
1
Distance
I 5
I .25
0
I .75
l1
from the city center
Fig. 1.
speed/flow gradient of 9X1. Therefore, this example can be interpreted as examining the effect of road upgrading on trip assignment; with Case 1 modelling conditions before the upgrading and Case 2 after the upgrading. Figure 1 shows the angular coverage of each of the five radials as a function of distance from the city centre. Here the term “angular coverage”, denoted n,(r), is being used to indicate the total angle spanned by the catchment area of LRi at a distance r from the city centre, in which case Cl,(r) = 6,(r) + +i-l
- @j-l(r)*
(36)
In Fig. 2 the effect of the upgrading is highlighted by directly comparing the area of the region served by each radial, before and after the upgrading.
Catchmant
Area as o Percentage of the City
Compartson of Cases 1 ond 2
”
Radial 1
Radial 2
Radial 4
Radial 3 Radial
I
Fig. 2.
Major
Roads
Radial 5
Trip assignment to radial major roads
441
Table 2. Summary of model parameters used in trip assignment examples (a) Characteristics
of the City
1. Radius of the city R=l 2. Density of trip origins f(r) = l/lr 3. Relative location of radial major roads I$* = & = I$3 = +, = +s = 2nl5 (b) Characteristics of Traffic Flow Speed of travel on minor ring roads u(r) = 30 kmph Free speed on radial major roads VO = 60 kmph Norma&d speed/flow gradient on radial major roads Case 1 Case 2 3, 15 6 & 10 10 33 15 15 2. 4 4 33 12 12
CONCLUSION
The value of the model developed in this paper lies in its potential use as a strategic planning tool. Given a city with a strong development of radial commuter routes, it is possible to model the pattern of trip assignment to radial major roads in terms of a small number of meaningful parameters. There is considerable scope for tailoring these parameters to reflect the existing characteristics of the city or for allowing them to vary in accordance with proposed changes in the road network. In terms of radial major roads this may involve the construction of a new radial or the significant upgrading of an existing route. Either option involves considerable expense and disruption to the traffic flows. The proposed trip assignment model can be used to estimate, at least at the macroscopic level, the change in trip patterns that would result from a given development option. In modelling the city as an entity the planner can gauge not only local effects of a proposal but also the long-term adjustment in trip patterns that would spread throughout the city. Acknowledgements-The author wishes to acknowledge the contribution of C. E. M. Pearce who suggested the decoupling process described in the Appendix.
REFERENCES Blumenfeld D. E. and Weiss G. H. (1970) Routing in a circular city with two ring roads. Tr~nrpn. Res. 4, 235-256. Branston D. M. (1976) Link capacity functions: A review. Trapn. Res. 10,223-236. Conte S. D. and de Boer C. (1972) Elementary Numerical Analysis-An Algorithmic Approach. McGrawHill, New York. D’Este G. M. (1983) Time-dependence in Continuous Urban Transport Modelling. PhD thesis, University of Newcastle. Lam T. N. and Newell G. F. (1%7) Flow dependent traffic assignment on a circular city. Trans. Sci. 1,318361. Wardrop J. G. (1952) On some theoretical aspects of road traffic research. Proc. ofthe Inst. of Civil Engineers 1,325-362.
APPENDIX The coupled system of nonhomogeneous differential equations (16) with boundary conditions (17) can be simplified by a series of transformations. First let B(r) = uf(r)12V”,
G. D’ESTE
442 then (16) can be rewritten as C’ + PM+,(L+,
- L, + 74,) - A,& c
which suggests the change of variable Z(r)
- L,
(Al)
+ r&-1)1 = 0,
r
= L(r)
- L-,(r)
W)
+ r&1
with Z.,, = Z, and inverse L, = (lln){Z,+,
+ 2Z,+, + 3Z,+, + ... + nZ, + r(n - +, - 2+,+, + ..’
+
Substituting into (Al) gives C’ + PtA,+,Z,+, - AZ,} = 0 so that (L, - I+,)”
+ f3{A,+lZ,,, - 2A,Z, + Aj_,Zi_,}
= 0
(W
Or
C’
+
P{A+,Z+,
2A;Z, + A,_,Z,_,}
-
= 0.
(A6)
From (17) and (A2) it follows that the pertinent boundary conditions are Z,(O) = 0, Z:(R) = t(+; + dL,).
(A7)
The system (A6) can be conveniently expressed in the vector-matrix
form
z” + PzA = 0 where z is the row vector (Z,, Z,,
. . , Z.) and A is the coefficient matrix A,
0
...
A,
AZ
-2A2
AZ
1..
0
0
A,
-2A,
...
0
...
.
-2A,
A=
I... A.
0
0
-2A.
.
I
If Xi is one of the (n - 1) nonzero eigenvalues of A and p, is the associated right eigenvector then z”t.~t + @Ap, = 0
(A9)
(zlk)” + PA,(ZPi) = 0
(AlO)
x:’ + PA,X, = 0
(All)
X, = CLitz, + P,*Zz + ... + )L,.Z”
(Al2)
becomes
or
where X,(r) is the scalar function
and u,, is the jth component of pi. The zero eigenvalue, corresponding returns the conservation equation
to the eigenvector
X. = Z, + Z2 + ... + Z. = 2ar.
(1, 1, . . . , 1)
(A13)
Hence, the system has been decoupled and reduced to a set of tractable second-order differential equations that are amenable to solution by standard analytic or numerical techniques. Each equation can be solved separately with boundary conditions
X,(O) = 0, X:(R)
= i i
(A14) cLoj(+j +
+,-I),
1-l
then the original variables L,(r), can be constructed by inverting the transformations.
Vol. ZlB, No. 6, pp. 433-442, 1987 Tmmpn. Rcs.3 Primed in Great Britain.
TRIP ASSIGNMENT
TO RADIAL
MAJOR ROADS
GLEN D’ESTE School of General Studies, Tasmanian State Institute of Technology, P.O. Box 1214, Launceston, Tasmania, Australia 7250 (Received 29 Junuary 1986; in revisedform 5 May 1986) Ah&act-The problem of flow-dependent trip assignment is considered for a city with a small number of radial major roads. CBD-based work trips are assigned to these radial major roads on the basis that each commuter seeks to minimise his individual travel time. A system of differential equations is derived for the spatial pattern of trip assignment in a model city with a continuous distribution of home locations and a ring-radial road network. This system is then solved for the special case of a uniform distribution of home locations. INTRODUCTION
In many cities travel to and from the central business district (CBD) takes place predominantly on a small number of radial major roads. This paper looks at the problem of trip assignment to these radial major roads in a certain model city. Here the term “trip assignment” is being used to denote the process of predicting the route followed between a given origin and destination. The assignment criterion that will be used is Wardrop’s 1st Principle which states that the equilibrium assignment is one for which, “the journey times on all routes actually used are equal, and less than those which would be experienced by a single vehicle on any unused route” (Wardrop, 1952). Basically this is a selfish strategy since each commuter seeks to optimise his own route regardless of other road users. It is however the one most likely to be used in practice. Given this behaviour, the spatial pattern of trip assignment will be derived for the following model city. THE
MODEL
CITY
Consider an axially symmetric city of radius R and in particular the group of commuters who travel each day to and from workplaces in the CBD. This study is only concerned with CBD-based work trips however it is recognised that this is only one component of the overall traffic pattern of the city, albeit an important one. It will be assumed that the CBD is sufficiently compact in comparison with the city as a whole to be taken as being concentrated at the city centre. On the other hand, the homes of commuters are widely dispersed throughout the city. Their distribution will be given by an axially symmetric urban density model, f(r), where r is the distance from the city centre. It should be noted that f(r) is not simply a population density model since it only relates to a certain subset of the population. Rather it is a trip-end distribution function, conditional on workplace location. Serving the CBD is a set of n radial major roads. These major roads will be denoted % . . %= and it will be assumed that they are of dual carriageway design and have pri&y g; intersections so that traffic speed is dependent only on local traffic density in the direction of travel. Many functional forms have been proposed to describe the relationship between traffic speed and density, see Branston (1976). Of these the one that will be used for all radials is the Bureau of Roads model u = V”[l + AqE]-‘, where u = traffic speed, free (maximum) traffic speed, traffic flow, and A, B are constants.
V” = q =
433 TRb21/6-b
(1)
G. D’Esrn
434
Table 1. Summary of notation used in a dynamic trip assignment model (a) Geometric R
d,,
aP2,. . * 3” 4i si VO
A
radius of the city
R radial major roads numbered consecutively clockwise from an arbitrary fixed point angle between 92, and %,+, boundary contour which divides commuters who live between those who use 9Zi and those who use .$+, free (maximum) traffic speed on a radial major road nonnalised speed/flow gradient on 2,
9, and %,+, into
(b) Functional
tb r&(r) VI(r) ‘I;(r) P,(r) W) L,(r)
radial distance from the city centre density of trip origins traffic speed on the minor road network at a distance r from the city centre traffic flow on .?Xiat a distance r from the city centre average traffic speed on R, at a distance r from the city centre time taken to travel a distance r from the city centre along Si probability that a commuter lives at a distance greater than r from the city centre and uses 28, angle between 9Zi and Bi at a distance r from the city centre distance from 9Zi to Bi along a minor ring at a distance r from the city centre
This model has three parameters V”, A, and B which can be used to represent the individual characteristics of each radial. However since maximum traffic speed is presumably governed by regulations rather than traffic conditions it will be assumed that V” is the same for all radials. Further in the first instance it will be assumed that the roadway configuration is uniform throughout its length and that the speed/flow relationship on all radials can be well represented by a first order (B = 1) model. Also remembering that the Bureau of Roads model is a local model, we arrive at the form Vi(x) = V”[l + Aiqi(x)]-‘,
(2)
where Vi(x) = traffic speed on %i at a distance x from the city centre, G(X) = traffic flow at x on %i, and Ai = speed/flow gradient on %i. The radial major roads will be assumed to provide the quickest and most convenient means of travelling to and from the CBD. Consequently during the morning and evening peak-hours commuters travelling to or from the CBD always use these radial major roads while other travellers avoid them. Hence, the contribution to traffic density from other trip types and purposes is negligible. Access from the commuter’s home to these radial major roads is provided by an infinitely fine set of ring roads. These are minor roads and it will be assumed that traffic speed on them is dependent only on distance from the city centre. ROUTE
CHOICE
Within the road network described above a commuter travelling from home to a workplace in the CBD first travels circumferentially along the minor ring road that passes his home and continues until reaching a radial major road. The commuter then proceeds along this major road to the city centre. Routes are thus single-turn ring-radial. With this routeing strategy the route of an individual commuter is well defined and has simple geometry. There is however one aspect of the route that is not specified. That is the radial major road on which the commuter chooses to travel. The problem of trip assignment is then to predict the outcome of this choice given that each commuter seeks to minimise his individual travel time. The first step is to identify the range of options available within the framework of
Trip assignment to radial major roads
435
the current model. Consider a CBD-bound commuter whose home lies in the region Conceivably such a commuter could use any radial, however for between gi and S%?;+l* the purposes of the current model it will be argued that the choice is between &; and %i+1 only. This assumption can be justified by appealing to the definition of a major road. With the current routeing discipline it is a simple matter to show (see D’Este, 1983) that at any given distance from the city centre the set of origins that contribute to traffic on a particular radial major road forms a continuous circular arc that intersects that radial. Therefore, if there exists a commuter that lives between 3; and %;+I but finds it quicker to travel to the CBD by %;_I, it follows that amongst other commuters that live at the same distance from the city centre, nobody will use .%$. It is then questionable whether 5%;qualifies as a major road since it is not even used by commuters that live on it. The purpose of this study is to consider trip assignment to genuine radial major roads; these roads need not be identical but none should have a capacity so much lower than its neighbours that it can be completely dominated by them.
TRIP
ASSIGNMENT
Given that a commuter living in the sector between Bi and %;+I will use either 3; or ZX;+lTit follows that at any distance r from the city centre there is a point at which single-turn trips to the CBD by either radial have equal duration. The set of such points will map out a boundary contour S?;, which divides commuters in the sector into those who use !ZZ;and those who use %;+I. The problem of trip assignment is therefore one of deriving an expression for B;Let O;(r) denote the angle between 3; and g; at a distance r from the city centre. A point on B; must then satisfy the condition rei(r)
Ti(r) + -
u(r)
=
Ti+t(r)
+
$j
(4%
-
ei(r)),
where 4; = the angle between 3; and S??;+l, = the traffic speed on a minor ring road at a distance r from the city centre, and 7’;(r) = the time taken to travel from the city centre to a distance r from the city centre (or vice versa) on %!;,
u(r)
which is simply a statement of the equality of trip duration by either radial. Rearranging (3) to make e;(r), the subject of the equation gives O,(r) =
4 +$
{&+1(r) - K;:(r)).
(4)
This is the fundamental equation governing trip assignment but its usefulness depends on finding an appropriate expression for T;(r). Since traffic speed on radials is locally flow dependent, at any point r on S;, the traffic speed is given by (2) and T;(r)
=
I’ -k 0
=
(5)
vi(x)
h r (1 0
+ Aiqi(X)}dx,
(6)
where q;(X) is the flow on the radial B; at a distance x from the city centre. Under equilibrium conditions the traffic flow at x on 3; is proportional to the probability P;(x), that a commuter lives at a distance greater than x from the city centre and uses SF%!;. The
G. D’ESTE
436
region in which home locations satisfy this condition is bounded by Bi, LBi _ 1, and circles of radius x and R, so Pi(X) =
IR
*‘a+;-1+ e,(Z)
j(z)
x
- ei-l(Z)] dr.
(7)
Let the constant of proportionality which in this case is the global trip generation be absorbed into the gradient parameter Ai,then Ti(r) =
dz}dx,
$6 {I +Aj JI”j(z) *z[+i-l
+ e,(z) - ei-l(Z)1
rate
(8)
and finally
+ Ai+18i+l(z)
-
(Ai+
+
Ai)&
+ Aiej_l(z)}
dr dx.
(9)
A similar equation will hold for each radial thus giving rise to a set of n coupled double integral equations in the n variables 8i, . . . , 8,. It must be noted however that because of the circular nature of the problem
e,(r) = e,(r),
(10)
0,+&) = M% and we find that the system is not linearly independent. To close the system and allow a unique solution to be found, another equation is required. This equation can be derived from (4) by summing both sides over i to give
2,
e,(r) = i
i i=l
i=l
(11)
but I: +i = 2~r so
i
O,(r)
=
7F.
i=l
(12)
This is a conservation equation and highlights the interdependence of flows; if the patronage of one radial increases then that of another must decrease. Conceivably (9) and (12) could be solved numerically, however the process of finding that solution can be simplified by first converting to the equivalent differential equation and making the change of variable Lj(r) = r&(r).
(13)
Therefore, Li(r) is the distance from Bi to!Bi along a minor ring road at a distance r from the city centre. In terms of L,(r) the differential equation equivalent to (9) is
Trip assignment to radial major roads
437
with boundary conditions Li(0) = 0,
L;(R)
(13 Wb)
-
and
Much of the apparent complexity of (14) results from the dependence of traffic speed on minor ring roads on distance from the city centre. However, this dependence is weak and it is not unreasonable to consider traffic speed on minor roads to be independent of location. Therefore, if traffic speed on minor roads has the constant value u, (14) reduces to
L:! +
T$L%+lLi+l -
[Ai+ + Ai]Li + AiLi-,}
= s
[Ai&-
- A,+,+(]
(16)
with Li(0) = 0, L;(R)
(1%
= 442,
(17b)
which in turn can be reduced to the system of 2n coupled first-order differential equations
= T$ I Q;
{(Ai+, + 4)Li - Ai+lLi+l - AiLi-
+ r(Aih-1 - Ai+,&))
t18)
L; = Qi
with Li(0) = 0,
Pa)
and
2 Li(r)
i=l
= TP*
(I9c)
The process of calculating solutions can be further simplified by decoupling the system using an eigenvalue technique described in the Appendix. In the decoupled system, the equations are independent and can be solved separately by standard analytic or numerical techniques. Alternatively the current system (16) and (17) or indeed the original system (14) and (15) can be solved numerically by the “shooting” method or by finite-difference methods. See Conte and de Boer (1972) for details of these techniques. In any case, having manipulated the model into a form solvable by standard techniques, the trip assignment problem is essentially complete. The “shooting” and finite-difference methods are both robust techniques in that the equation coefficients need not be fixed throughout the interval. This quality can be exploited in extending the trip assignment model while retaining both as viable solution techniques. So far the parameter Ai has been considered to have the same value at all points along a radial, but since Ai reflects the roadway configuration, this is an unrealistic assumption. It is unlikely that a radial major road will have the same configuration all
G. D’Esrz
438
the way from the city centre to its outskirts. Hence, Ai should be allowed to vary with location. Inherently this will be a discrete process so there will be annular regions in which the traffic speed/flow gradient @ constant on all radials. The number of such layers will be equal to one plus the timber of distant radii at which there is a change in the value of any A;. Within each layer the conditions governing route choice are identical to those already modelled. Further, the boundary conditions (17) still apply and it follows that (16) and (17) is still a valid representation of the trip assignment pattern. The only additional condition is that solutions are continuous across layer boundaries. Hence, the model extends naturally to a situation in which the roadway configuration is allowed to vary along a radial. This model of trip assignment can be viewed as being complementary to those of Lam and Newell (1967) and Blumenfeld and Weiss (1970). These authors considered a city having a ring-radial routing system and a circular CBD with all trip origins outside the CBD and all destinations inside. Then for a given origin and destination they derived optimum routing strategies for travel within the CBD. In both cases the incoming radial of an individual was assumed to be known; the above model sheds some light on how that radial might have been chosen.
TRIP
ASSIGNMENT
IN A UNIFORM
CIRCULAR
CITY
Consider trip assignment to radial major roads in a uniform circular city with constant traffic speed on minor ring roads. This is a very special case since under these conditions (16) reduces to a system of differential equations with constant coefficients and hence an exact solution can be found. Without loss of generality the city can be taken to have unit radius so that R=l The conservation
f(r)
and
= l/n.
(20)
equation Ll =
TT
-
i
Li
(21)
i=2
can then be used to eliminate L1 from (16) so that the problem reduces to finding the solution to the nonhomogeneous vector-matrix equation a” - Aa
=
TW,
(22)
a(0) = 0,
(23)
a’(1) = +/2,
(24)
where a = (L2, &, . . . 7 L) and A is a constant (n - 1) x (n - 1) array and o, + are constant vectors of length (n - 1) whose values follow from substituting (21) into (16). Since (22) is a linear system the complete solution can be written as the sum of the solution of the corresponding homogeneous system a U = Ao
(25)
and a particular solution to (22). By inspection this particular solution is ap = -t-A-‘W.
(26)
Trip assignment to radial major roads
439
Since all coefficients are constants, solutions to the corresponding homogeneous system (25) will be of the form a = pe"
(27)
where lo,is a constant vector of length (n - 1). Substituting (27) into (25) yields (m21- A)pe"'
= 0
(28)
where I is the identity matrix. It follows immediately that m2 is an eigenvalue of A and p is the corresponding eigenvector. Therefore the complete solution to (25) is the linear combination n-1
aH = 2 Pi[f’i exp(-w) i=l
+ Qi exptmdl
(29)
is over the (n - 1) eigenvalues of A. where Pi, Qi are constants and the su~ation Now the bracketed term in (29) can be written as
Pisinh mir + Qi cash ntir
(30)
and further simplified by applying the initial condition a(0)= 0.Since ap(0) = 0 it follows that Qt = 0 and n-1
aH = C Pi pi sinh mir, i=l
(31)
which can be expressed more concisely as
ati = y(r).?,
(32)
where y is a constant vector of length (n - 1) and Y(r) is the so-called fundamental matrix of the system (25). The fundamental matrix of a system of differential equations is an array of its solutions, so in this case each column of Y is a vector function of the form, p sinh mr.Therefore, the complete solution to (22) is a(r) = Y(r).7 - t-A-‘co.
(33)
Applying the boundary condition (24) gives Y
= Y’(l)-*.{+/2 + A-b}
(34)
and a(r) = Y(t),Y’(l)-r{+/2
+ A-b) - rA-'o.
(35)
Finally Li follows from (21) and the trip problem for a uniform circular city is solved. All that remains is to convert back to the origin variables & using (13). The variable @i is a more instructive measure than L+because the effect of trip assignment is not clouded by any inherent dependence on distance from the city centre. Any variation in 9: with radial distance is derived purely from the effects of commuter choice. An example of trip assignment in a uniform circular city with no layering and 5 radial major roads is illustrated in Figs. 1 and 2. The corresponding model parameters are summarised in Table 2. For this hypothetical city the pattern of trip assignment is investigated for two scenarios, termed Cases 1 and 2, which differ only in the traffic
G. D’ESTE
440
Angular Coverage Case I
Case 2 zc
130.e..... ----
llO-
Radial Radial Radial
----_
--IIO-
I: ii s70ii
Radial 4 Radial 5
----__
---_ --.
90-
130-
1 2 3
----_
_
.............................~*~**~~~~
m-
e
n
c-50- ________------- ___--- -
50-
307
I
I
I
307
0
.25
.5
.7s
r
1
Distance
I 5
I .25
0
I .75
l1
from the city center
Fig. 1.
speed/flow gradient of 9X1. Therefore, this example can be interpreted as examining the effect of road upgrading on trip assignment; with Case 1 modelling conditions before the upgrading and Case 2 after the upgrading. Figure 1 shows the angular coverage of each of the five radials as a function of distance from the city centre. Here the term “angular coverage”, denoted n,(r), is being used to indicate the total angle spanned by the catchment area of LRi at a distance r from the city centre, in which case Cl,(r) = 6,(r) + +i-l
- @j-l(r)*
(36)
In Fig. 2 the effect of the upgrading is highlighted by directly comparing the area of the region served by each radial, before and after the upgrading.
Catchmant
Area as o Percentage of the City
Compartson of Cases 1 ond 2
”
Radial 1
Radial 2
Radial 4
Radial 3 Radial
I
Fig. 2.
Major
Roads
Radial 5
Trip assignment to radial major roads
441
Table 2. Summary of model parameters used in trip assignment examples (a) Characteristics
of the City
1. Radius of the city R=l 2. Density of trip origins f(r) = l/lr 3. Relative location of radial major roads I$* = & = I$3 = +, = +s = 2nl5 (b) Characteristics of Traffic Flow Speed of travel on minor ring roads u(r) = 30 kmph Free speed on radial major roads VO = 60 kmph Norma&d speed/flow gradient on radial major roads Case 1 Case 2 3, 15 6 & 10 10 33 15 15 2. 4 4 33 12 12
CONCLUSION
The value of the model developed in this paper lies in its potential use as a strategic planning tool. Given a city with a strong development of radial commuter routes, it is possible to model the pattern of trip assignment to radial major roads in terms of a small number of meaningful parameters. There is considerable scope for tailoring these parameters to reflect the existing characteristics of the city or for allowing them to vary in accordance with proposed changes in the road network. In terms of radial major roads this may involve the construction of a new radial or the significant upgrading of an existing route. Either option involves considerable expense and disruption to the traffic flows. The proposed trip assignment model can be used to estimate, at least at the macroscopic level, the change in trip patterns that would result from a given development option. In modelling the city as an entity the planner can gauge not only local effects of a proposal but also the long-term adjustment in trip patterns that would spread throughout the city. Acknowledgements-The author wishes to acknowledge the contribution of C. E. M. Pearce who suggested the decoupling process described in the Appendix.
REFERENCES Blumenfeld D. E. and Weiss G. H. (1970) Routing in a circular city with two ring roads. Tr~nrpn. Res. 4, 235-256. Branston D. M. (1976) Link capacity functions: A review. Trapn. Res. 10,223-236. Conte S. D. and de Boer C. (1972) Elementary Numerical Analysis-An Algorithmic Approach. McGrawHill, New York. D’Este G. M. (1983) Time-dependence in Continuous Urban Transport Modelling. PhD thesis, University of Newcastle. Lam T. N. and Newell G. F. (1%7) Flow dependent traffic assignment on a circular city. Trans. Sci. 1,318361. Wardrop J. G. (1952) On some theoretical aspects of road traffic research. Proc. ofthe Inst. of Civil Engineers 1,325-362.
APPENDIX The coupled system of nonhomogeneous differential equations (16) with boundary conditions (17) can be simplified by a series of transformations. First let B(r) = uf(r)12V”,
G. D’ESTE
442 then (16) can be rewritten as C’ + PM+,(L+,
- L, + 74,) - A,& c
which suggests the change of variable Z(r)
- L,
(Al)
+ r&-1)1 = 0,
r
= L(r)
- L-,(r)
W)
+ r&1
with Z.,, = Z, and inverse L, = (lln){Z,+,
+ 2Z,+, + 3Z,+, + ... + nZ, + r(n - +, - 2+,+, + ..’
+
Substituting into (Al) gives C’ + PtA,+,Z,+, - AZ,} = 0 so that (L, - I+,)”
+ f3{A,+lZ,,, - 2A,Z, + Aj_,Zi_,}
= 0
(W
Or
C’
+
P{A+,Z+,
2A;Z, + A,_,Z,_,}
-
= 0.
(A6)
From (17) and (A2) it follows that the pertinent boundary conditions are Z,(O) = 0, Z:(R) = t(+; + dL,).
(A7)
The system (A6) can be conveniently expressed in the vector-matrix
form
z” + PzA = 0 where z is the row vector (Z,, Z,,
. . , Z.) and A is the coefficient matrix A,
0
...
A,
AZ
-2A2
AZ
1..
0
0
A,
-2A,
...
0
...
.
-2A,
A=
I... A.
0
0
-2A.
.
I
If Xi is one of the (n - 1) nonzero eigenvalues of A and p, is the associated right eigenvector then z”t.~t + @Ap, = 0
(A9)
(zlk)” + PA,(ZPi) = 0
(AlO)
x:’ + PA,X, = 0
(All)
X, = CLitz, + P,*Zz + ... + )L,.Z”
(Al2)
becomes
or
where X,(r) is the scalar function
and u,, is the jth component of pi. The zero eigenvalue, corresponding returns the conservation equation
to the eigenvector
X. = Z, + Z2 + ... + Z. = 2ar.
(1, 1, . . . , 1)
(A13)
Hence, the system has been decoupled and reduced to a set of tractable second-order differential equations that are amenable to solution by standard analytic or numerical techniques. Each equation can be solved separately with boundary conditions
X,(O) = 0, X:(R)
= i i
(A14) cLoj(+j +
+,-I),
1-l
then the original variables L,(r), can be constructed by inverting the transformations.