AN ANALITICAL METHOD FOR INTERSECTION ENTERNING TRAFFIC FLOWESTIMATIONby Rui WANG1, Ryota HORIGUCHI2, Shinji TANAKA3, and Masao KUWAHARA4

Abstract
In this paper, we try to propose a simple but analytical approach to describe distribution of intersection entering
traffics on a grid road network. The approach is a continuous model intersection oriented and depending on thenetwork parameters only. This approach has an explicit structure and free of the need of a detailed OD(Origin-Destination) table thus very useful for the planning stage of road network. Using a proved commercial, theproposed approach is finally validated as of acceptable accuracy.
1. Introduction
For the purpose of proper road network planning, it is very important to predict the traffic distribution on a roadnetwork. Currently the most widely used approaches can be roughly divided into two types: the static traffic
assignment models and the dynamic traffic simulation (macroscopic or microscopic) models. However, the traditionalmodels inevitably require detailed OD information, and complicated black-box type of algorithms. As a result, users
often reach to their computation outputs without knowing the underlying logic behind. Furthermore, although there
are some exceptions, the traditional models are all link oriented while intersections play a trivial role (if any) in the
traditional models, despite the fact that it is the intersections rather than the basic sections of road links, decide theperformance of urban road networks in modern cities.
In this paper we assume the passing probability of an intersection equals to the utilization of the intersection among
all routes between an OD pair within a road network. Under this assumption, an explicit analytical approach isdeveloped to estimate the distribution of traffic within a grid network from a single pair of OD. It is done firstly by adiscrete model and then by a continuous model that can be subjected to the method of integral in the future. Finally theproposed approach is validated by using a proved commercial simulation model to validate. This approach can beregarded as a simplified estimating method of traffic distribution and is also helpful for the researches on the road
network density. We have to make clear that through our research we do not consider about the influence ofcongestion since our objective is limited to the planning stage only.
8D7

2. A Discrete Intersection Utilizing Model of a Single OD Pair
A road network consists of many links (roads) and nodes (intersections).
For most modern urban road networks, the common knowledge is thatthe smoothness of traffic movement is more likely decided byintersections, rather than basic road sections. This research concentrates
on the intersections, by assuming all origins and destinations of all OD
trips are from or to the intersections, instead of the traditional zones.
Similarly, routes between any ODs are also described by a sequence ofintersections, rather than links.
16543210O0 1 2 3 4 5 6 7 8Figure 1: A Road Network ModelDr. Eng., Research Associate, Dept of Civil Eng., Yokohama National Univ., 79-5 Tokiwadai, Hodogaya,Yokohama, 240-8501 JAPAN2Dr. Eng., President, i-Transport Lab. Co., Ltd., 1-4 Jinbo-cho, Chiyoda-ku, Tokyo, JAPAN3Dr. Eng., Lecturer, Institute of Industrial Science, Univ. of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, JAPAN
4Ph.D., Prof., Institute of Industrial Science, Univ. of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, JAPAN

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ii+14567IO0i-1ii+1a) Ro(i,j), route numbers from OIi,jb) Estimating Directional Traffics

Figure 2: An Intersection Choice Model(1) Road Network Model
A road network that consists of intersections and the connecting roads is shown in Fig. 1. In this paper, we assume aroad hierarchy1) with the high-hierarchical roads shown as solid lines on a uniform background of low-hierarchicalroads (grey shade), as suggested by H. Lehmanh2).
(2) Discrete Intersection Choice Model for Estimating Directional Intersection Entering Traffics
As stated earlier, in this paper, we assume the percentage of traffic crossing a certain intersection equals to theutilizing probability of the intersection in the routes between an O and D.
a) Number of routes from O
Route choice is defined as choices of intersections in this paper, for the purpose to estimate directional intersection
entering traffics. For a network of IJ with O and D at two opposite corners, the total number of route s from O to all
the intersections are shown in Figure 2-a). If we define R(O,I) as the number of routes from O to the intersection Ii,j,we can get the Equation 1.

R(O, I ij ) [(i  0)  ( j  0))]! (i  j )!
(i  0)!( j  0)!
i! j!

(1)b) Estimating Directional Traffics
As shown in Figure 2-b), at the intersection Ii,j, the total entering traffic at time t, fij(t), can be estimated by thefollowing equation (2).

f ij (t )   OD (t )R(O, I ij )  R( I ij , D)
R(O, D)  OD (t )[(i  j )! ( I  i  J  j )! ( I  J )!
/]   OD (t )  pij
i! j! ( I  i )!( J  j )! I ! J !

(2)

where R(Iij, D) is number of routes from Iij to D; R(O,D) is the total number of routes from O to D; and  OD (t ) is
the trip generating/attracting rate between O and D.
 Definition of Intersection Utilizing Probability (pij),: here we define pij as the intersection utilizing probabilityfrom O to D. It can be calculated by the equation (3). Obviously, the value of pij depends on the network itself, thus
independent from the traffic demand or the OD pattern.

pij R(O, I ij )  R( I ij , D)
R(O, D)(i  j )! ( I  i  J  j )! ( I  J )!
/i! j! ( I  i)!( J  j )! I! J !

(3)

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Oa) Number of Routesb) Utilizing Probability

Figure 3: Intersection Utilizing Probability of a Symmetrical Network

3. Development of a Continuous Intersection Utilizing Model
Here, we try to develop a continuous model on the basis of the earlier discrete model under the condition of asymmetrical road network pattern. Such a symmetrical grid network is shown in Figure 4, the number of routes andthe utilizing probability can be easily estimated by using the earlier model.
As for the new r-n coordinate system, when the traffic from O to D crossing the blue lines, prn the utilizing
probability of an intersection Irn on these blue lines can be calculated by the following equation (4) since it consist of aperfect binomial distribution.

n rnR(O, I rn )n!
1prn  r n r n     / 2 n ( )nrr!(n  r )! 2nR(O, I rn )     r 0r 0  r (4)where R (O, I rn ) is the number of routes from O to Irn.
As n→∞, by assuming x r n/2
, and applying the Stirling Equation of n! 2 e  n n n1 / 2 , an appropriaten /2

continuous equation can be obtained3).
x2p(r ) 1e 22 

where  (5)n /2

4. Validation of the Proposed ModelIn order to validate the proposed models, a 9 by 9 grid road network is constructed under the simulation package ofSOUND/4U©, which is a macroscopic model developed under the cooperation between the Industrial ScienceInstitute, University of Tokyo and i-Transport Lab. Co., Ltd (ITL).
For the purpose of comparison, assuming  OD (t )  10000 , intersection entering traffics of the grid network areestimated by both equation (4) and the simulation package. The results can be found in the table 1 and table 2.
(1) Estimation Results
Table 1: Estimation Results of the Proposed Models

r00 10000
1 50002 25003 12504625
53136156
778
8
39n

1

2

3456

7

8

5000
5000
3750
2500
1563
938
547
313

2500
3750
3750
3125
2344
1641
1094

1250
2500
3125
3125
2734
2188

625
1563
2344
2734
2734

313
938
1641
2188

156
547
1094

78
313

39

1

2

3

4

5

6

78

5491
3961
3119
2164
1399
793
385
121

3209
3344
3295
2778
2128
1381
811

1782
2442
1893
2937
2693
2099

904
1627
2301
2727
2893

408
919
1538
2146

156
462
939

47
151

7

Table 2: Estimation Results of SOUNDr
n0
1
2
34
5
6
78Sum
10000
10000
10000
10000
10000
10000
100001000010000

09906
4324
2553
1386
735
343
122
326

Sum
9906
9815
9723
9631
9540
8448
9356
9265
9166

(2) Traffic Distribution
Distributions by Simulation

Theoretical DistributionnTraffic (veh/h)

5000
4000
3000
2000
1000

6000
n1n2n3n4n5n6n7n8

5000
Traffic (veh/h)

n1n2n3n4n5n6n7n8

6000

4000
3000
2000
1000

00r0r1r2r3r4rr5r6r7a) Distribution by Proposed Modelr8r0r1r2r3r4rr5r6r7r8b) Distribution by Simulation

Figure 4: Comparison of Distributions of Intersection Entering Traffics

General speaking, the proposed model can give out an acceptable estimation result. Although for the simulation
results, since many vehicles stay on the roads when the calculation finished, the further the intersections, the smaller
the intersection entering traffics are estimated.
5. Concluding Remarks
In this research, an analytical approach is proposed to estimate intersection entering traffic distributions on a grid
network. Both a discrete and a continuous model are developed. The estimation results are validated as acceptable bya mature commercial simulation platform. The proposed approach is a totally analytical model that applying network
parameters only. Such kinds of approaches are especially important to the planning stage when detailed OD data is notavailable and the planners have the need to fully understand relationship between the network parameters and thetraffic distributions. Its influence on both the academic and practical fields is expected.

In future, the author will concentrate on two parts: 1) the case of a full All To All OD distribution, and 2) the case of
the non grid networks. The major purpose of the future research will also be consistent, to describe a network using
the network parameters only.
Acknowledgements
The SOUND/4U Simulation platform kindly provided by the i-Transport Lab. Co., Ltd (ITL) is highly appreciated.
Thank you very much, Mr. Hanabusa, for all the kindly supports! The kindly suggestions and encouragement fromProf. Nakamura, Prof. Okamura and other members of the transportation lab at Yokohama National University is also
highly appreciated.
References
1) Wang, R, Wako, M, Kuwahara, M: Simplification of macroscopic simulation by combination of an analyticalmethod for estimating intra-zone traffic, Proceedings of the 15th World Congress on ITS, CD-rom, New York, 2008.
2) Lehmann, H: Directed and diffusive contributions to urban traffic flow patterns, EuroPhysics Letters, Vol64 (2),pp.288-294, 2003.
3) Yamamoto, K.: Permutation, Combination and Probability (in Japanese), pp.234-239, Iwanami Shoten Publishers,Tokyo, 1986.