Discrete Models of Traffic Flow (4-4)

Standard Model of Nagel and Schreckenberg 4

Next, we look at the case of non-zero probabilistic non-accerelation R.

In the right, The cases of U = 4 and varying R = 0.0, 0.2, 0.4 are shown. The main characteristics of finite R, as compared to zero R, is the smaller critical value of \rho for congestion, and aslo the harder congestion expressed as low F for \rho lager than the critical value. At the smaller \rho before the critical value, the traffic is that of free flow, and the asymptotic behavior at \rho \to 0 is same as R = 0 case;

There is a difference in the behavior toward the limit \rho \to 1, which is given by

This type of limiting behavior is called "asymtotics" in mathematical term.

Close inspection of the graph reveals that the flux in the region above the critical density is not straight but a concave line, and the value of F it self tends to be much smaller than extrapolated from (4.8).

What we learn here is that non-zero value of R, even when the value is very small, greatly affects the position of the critical density, and also reduces the value of F in the jamming pahse cinsiderably.

The deceleration rate R can depend on the road condition and also on the skill and personality of drivers. Driving habit differs in every country, and indeed every region of a country. What is suggested from the above fundamental diagram is that a land with small R where drivers are prudent tends to have higher critical density for the traffic jam, and also the higher traffic flux even with the traffic jam compared to a land with higher R where drivers are hot tempered. Below the critical density, however, difference in R has practically no cosequence. This is just very natural, and also goes along well with the fact that drivers in cities with higher car density tend to behave better on the road.