Discrete Models of Traffic Flow (1-1)

Quantitative Description of Traffic Flow 1

Why Study Traffic Flow

The automobile is the primary means of human transportation in modern era. Because of the high cost of automobiles, the first priority for social planners in serach of the efficient transportation is to spread them to larger portion of a society. The automobile alone, on the other hand, is useless without a system of well connected road whose development requires more sophisticated social organization. When a good road system is develpped, the transportation nirvana is still further away, and it will arrive only after successful tackling the difficult problem of traffic jam.

The traffic jam is epidemic in all major cities all over the world. Cities have avenues and boulvards whih fill considerable portion of city area, yet have traffic conjestion in every blocks at the rush hours. Is there any solution to this problem barring a fantastical one insisting on leveling the cities and paving all area to have a single gigantic road. To solve the traffc jam, we naturally have to start to understand the phenomenon traffic jam and that comes only through the understaning of the dynamics of traffic flow itself in quantitative manner. Our goal is to find the elements that control the traffic flow in order to alleviate the traffic jam.

We cannot, however, hope to do the real life experiment with city blocks and highway system which are emptied for that specific purpose even for a day. That would be too costly and with sufficient budget, that kind of experiments would be permited probably in a country with the Dear Leader, but urely not in modern democratic countries. The study of traffic flow, therefore, has to be performed virtually, on computers as numerical simulation. It becomes crucial to build models of traffic flow that capture the characteristics of real traffc yet sufficiently simple to allow efficient numerical treatment.

Traffic on One Dimensional Ring

Let us consider a single-lane road which is specified as one-way traffic. The traffic on this road is a uni-directional flow. The road starts at certain location and ends at another location. Acordingly, the flow has two ends. There is an incoming flow at one of them, and outgoing flow at the other. These incoming and outgoing flow has to be balanced on average. Otherwise, there would be either overflow of cars or the empty road. Assuming that the essential characteristics of traffic flow is independent on the precise specification of flows at the edges, we consider a simple model in which outgoing cars at one end immediately enters from the other end. In other words, what we have is a circular traffic on a ring road instead of a linear one. In physics, this type of treatment of edges is called periodic boundary condition.

One dimensional traffic on a closed ring is the basic model that is a starting point for the models of all other forms of traffic. In this note, we limit ourselves to the one-way single-lane ring traffic, from which realistic two-way multiple-lane traffic with crosses can be developped.