Some validation of the STCA

Despite somewhat unrealistic features on the level of individual vehicles, these models describe aspects of the macroscopic behavior correctly. If we assume the values given above, i.e. a cell size of $\ell = 7.5~m$ and a time step of $\Delta t = 1~sec$, then speeds are given in multiples of $7.5~m/sec = 27~km/h = 16.875~mph$. More correctly, average free speed is given by $(1 - p_{noise}) \, v_{max}$. With $p_{noise} = 0.2$, one obtains the following possible average link speeds:

$v_{max}$	$v_{max} - p_{noise}$	$m/sec$	$km/h$	$mph$
1	0.8	6.0	21.6	13.500
2	1.8	13.5	48.6	30.375
3	2.8	21.0	75.6	47.250
4	3.8	28.5	102.6	64.125
5	4.8	36.0	129.6	81.000
6	5.8	43.5	156.6	97.875
7	6.8	51.0	183.6	114.750

Since drivers typically do not observe speed limits exactly, it is uncritical that these speeds do not correspond to any ``round'' numbers. Also, there is enough flexibility to model differences between, e.g., residential streets, urban arterials, freeways with speed limits, and freeways without speed limits. There is however not enough resolution to model, say, the difference between a speed limit of 60 vs. 65 mph. If such differences are of interest, a different model needs to be selected.

A typical measurement for real-world traffic is the flow-density fundamental diagram. For this, one measures flow and density at a fixed location over fixed periods of time, for example over 5 minutes. The resulting data is plotted with density on the x-axis and flow on the y-axis (see Fig. 17.1). There are some subtleties involved with measuring fundamental diagrams, which are discussed in Sec. 27.2. For the purposes of this section, let us assume that the two quantities are measured in the CA as follows:

• Flow: Count the number of vehicles, $N_q$, that cross a given location during time $T$. Flow $q_T$ is given as
  
  $$q_T = \frac{N_q}{T} \ .$$
  
  

• Density: Assume a ``measurement area'' which spreads across $v_{max}$ contiguous cells. Sum up the number of vehicles on the measurement area over $T$ time steps. This includes that a vehicle that spends more than one time step on the measurement area is counted several times. If this number is $N_\rho$, then density is given as
  
  $$\rho_T = \frac{N_\rho}{T \, v_{max}} \ .$$
  
  
  Note that using $v_{max}$ cells makes sure that every vehicle is counted at least once.

  The result is the density in ``number of vehicles per cell'', corresponding to ``number of vehicles per 7.5 meters''. Multiplying by $1000/7.5$ converts this into ``number of vehicles per kilometer''.

Flow-density fundamental diagrams, as in Fig. 17.1, start at zero flow when the density is zero (no cars on the road), and eventually come back to zero flow when the jam density is reached. In between, they show a roughly tri-angular shape as can be seen in Fig. 17.1. Theoretical discussions will be postponed until Chap. [[cha:traffic-flow-theory]], but it is important to note that there is some value of maximum flow, about $2000~veh/h$ in Fig. 17.1. For the STCA, this value depends mostly on $p_{noise}$: Larger $p_{noise}$ leads to smaller maximum flows. These maximum flow values, also called capacities, need to come out approximately correctly if one wants a model that is useful for reality. 2000 vehicles per hour and lane is a plausible value. Regional differences could be accomodated by different values of $p_{noise}$; this could even be made a function of the link. One however has to note that changes in $p_{noise}$ also change the average acceleration of vehicles, which will, for example, change signal timing requirements or emissions. This is the reason why the CA approach can only be seen as a first, relatively rough starting point for a regional model. Once all other problems (such as demand generation) are sufficiently solved, the CA driving logic should be replaced by a model with continuous coordinates such as the ones discussed in Chap. [[maps]].

Figure 17.1: One-lane fundamental diagram as obtained with the standard cellular automata model for traffic using $p_{noise} = 0.2$. From (88).