Some Theoretical Results on Two Idealized Typical Road Traffic Networks

　　　　　　　　　　　　 Mingzhe Li

Faculty of EconomicsFukuoka University

Contents

1. Introduction a. The history of urban traffic networks b. SPCP and DSPCP

2. Discuss the properties and examine the effectiveness of two typical road networks

based on SPCP

3. The influence of one road segment on two typical road networks based on DSPCP

4. Conclusions

1. Introductiona. The history of urban traffic networks

Vitruvian Man [Da Vinci, 15C]

Miletus Palmanova [Hippodamus, 5BC] [Scamozzi, 16C]

　　　　　　 Paris 　　　　　　　　　　　Philadelphia 　　 [Haussman, 19C] 　　　　　　　 [Penn, 17C]

Changan Kyoto [Tang Dynasty, China] [Now, Japan]

Grid type network G(m, n)

Radial-circular type network T(m, n)

☆ Analyze traffic congestion

☆ SPCP (Shortest Path Counting Problem) [Oyama and Taguchi, 1991]

1. In a network N = (B, E) with the vertex set B and the edge set E, the shortest path problem is to find a path with the shortest length from a specified origin vertex to another specified destination vertex.

2. SPCP is a problem to count the number of the shortest paths in an edge among the shortest paths for all vertex pairs in the network, so as to evaluate congestion of each road segment using this index.

1. Introductionb. SPCP and DSPCP

☆ To determine a unique shortest path between any two vertices in case there exist two or more short paths having the same lengths, the following rules are applied.

1. The number of turns is minimized.

2. The number of left turns is maximized when there exist the shortest paths having an equal number of turns (for traffic conditions where vehicles travel on the left side of a road such as in Japan).

☆ DSPCP 　　　 (Directed Shortest Path Counting Problem) 　　

　　　　　　　　　　　　　 [Li and Fushimi, 1994]

1. Based on SPCP, DSPCP is defined by differentiating the concrete direction in each edge, i.e., DSPCP requires us to count the number of the shortest paths passing each directed edge.

2. To facilitate this, in a grid type network, we define ε+ as a directed edge from left to right or for downwards, and ε- as a directed edge from right to left or for upwards. Moreover, in a radial-circular type network, we define ε+ as a directed one from the center to the circumference or for anti-clockwise, and ε- as a directed one from the circumference to the center or for clockwise.

☆ The theoretical results of SPCP in two networks　　　

☆ The theoretical results of DSPCP in grid type network

　　　

☆ The theoretical results of DSPCP in radial-circular type network

　　　

2. The properties of a grid type network based on SPCP [a]

☆ The maximum weight happens at the center for vertical road segments or horizontal road segments.

☆ The minimum expectation and variance are

obtained when m=n, meaning that the same

number of vertical roads n and horizontal

roads m are desirable when designing a

grid type network by considering SPCP.

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2. The properties of a radial-circular type network based on SPCP [b]

☆ The maximum weight of radial roads happens at the inner side and they move to inside with the increasing of the number of radial roads n, while that of circular roads happens at the inmost side.

☆ The minimum expectation and variance are

obtained when n/m holds between 4.5 and 6

for some realistic values of c such as

and it decreases with the increasing of the

number of the vertices c.

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2. Compare the properties of two typical road networks based on SPCP [c]

☆ In most cases, a radial-circular type network is relatively more effective than a grid type network when considering their expectations, variances of SPCP, and the tendency becomes much more clear with the increasing of the number of the vertices c.

3. The influence of one road segment on two typical road networks based on DSPCP

☆ 　 The influence of the destruction of one directed road segment on the transportation system

☆ Calculate DSPCP results of the road segments with most increased weights accompanied with this destruction.

4. Conclusions

1. Based on SPCP and DSPCP , we theoretically studied the properties of two typical road networks, and examined their effectiveness. We also studied the influence of the destruction of one road segment on the traffic system by giving the road segments with the most increased weights accompanied with this destruction.

2. The results show that the maximum road traffic reaches peak at the center except the radial roads in a radial-circular network, where the peak takes place near the center (inner side) but not at the center.

3. The results show that to minimize the expectation and the variance, the relation 　　 usually stands for a grid type network.

On the other hand, the relation generally stands for some realistic values of c in a radial-circular type network.

4. When fixing the total number of the shortest paths and comparing two road networks with minimum values of the expectations and the variances, the structure of a radial-circular type network seems relatively effective in holding down road traffic.

1nm

4.5 6nm

5. The results show that the most congested places occur surrounding the destructed road segment on both road networks.

6. In details, in a grid type network, one surrounding horizontal edge and one surrounding vertical edge are most affected by the destruction of a directed edge. In a radial-circular type network, two surrounding circular edges, or four surrounding circular edges and two surrounding radial edges are most affected by the destruction of a directed radial edge. While, one surrounding radial edge and one surrounding circular edge placed inside are most affected by the destruction of a directed circular edge.

7. We see that the increased weight of each most affected edge is about or exactly one half of the weights of the destructed edge in both road networks.

8. Compared with grid type network, in a radial-circular type network, a. some more edges are most affected by this destruction, and, b. some edges got most congested in both sides when one edge is destructed in both sides, meaning that a grid type network is considered to be relatively stable when mathematically analyzing the two road networks from the viewpoint of risk management.

9. In real urban planning, the estimation of road traffic flows and the design of a road traffic network should be considered and evaluated from many perspectives. The results can not be used or applied as their present form. However, in the initial stage of decision making, this kind of study is considered to be valuable and meaningful by offering important and useful information before designing a road network.

Appendix：The influence of a radial road segment on the average traveling distance

Numerical analysis

Numerical analysis

n=5

n=10

Appendix：The influence of a circular road segment on the average traveling distance

Numerical analysis

Numerical analysis

n=5

n=10