Discrete Models of Traffic Flow (1-3)

Quantitative Description of Traffic Flow 3

Flux of Traffic Flow

Up to now, we identified ourselves with drivers in search of the quantity measuring the traffic jam. The situation might look rather different from the view point of those who administer the road traffic. Let's assume, for argument's sake, that you are suddenly appointed to the president of the Tokyo Metropolitan Highway Authority. The income of the president is decided on his mandarinic seniority, so you will have very little interest in the problem of traffic jam. This is no good for this story, So we go thatcherite, and privatize the Highway Authority. You are now the president of the Metropolitan Highway Company (Please do something about Hakozaki, Hamazakibashi and Kosuge bottlenecks, Sir!) Your income is now coupled to the income of the company, and you start looking at the way to increase it. The important thing for you here is not the velocity of the traffic, but the efficiency of the traffic measured in terms of the number of cars passing throughthe toll gate in a given time period.

Supose you stand at a certain point of a road, and count the average number of cars passing through. We call this quantity flux of the traffic. The road administrator would like to raise the flux which result in the raising of the income from the toll gate. So the traffic jam, defined for them as the hindrance to their income, should be measured by the flux.

Relations among Density, Average velocity and Flux

So the argument seems to be headed for the diversion of the interest of drivers and the administrators. But fortunately, the two quantities characterizing both interests are closely related, or, in a sense, almost identical.

We have to make the concept of the fux slightly more precise. The literal interpretation of the above definition gives the varying number depending on the time and location of the measurement. So let us assume that flux at a given location is measured as an average over a sufficiently long period of time. Then the flux should be identical at every point of the road, since all the cars eventually pass through all locations of the road.

If a collection of cars of density \rho advances with average velocity V_av, it will move the distance V_av in unit time, and the number of cars include within this distance is \rho times V_av. This is precisely the number of cars passing through a point in unit time, which is the definition of the flux F. Therefore, we have

Once the density is fixed, there is no essential difference between judging something based on average velocity V_av or on flux F.