Traffic flow measurements

It was already pointed out in Sec. 17.3 that important real world quantities for traffic are flow and density. A third quantity is speed. In fact, there are two different ways to measure traffic: space-averaged measurements, and point ($=$ spot) measurements. The space-averaged measurements are done at specific points in time, and they correspond to what one is used to from, say, fluid-dynamics. The point measurements are closer to what is measured in reality: A sensor, e.g. an induction loop, usually covers only a small amount of space. It is common use to average point measurements over sometime $T$, for example $T=60~sec$ or $T=5~min$.^27.1 These differences are not particularly intereresting, but they are necessary to avoid some caveats.

Speed

The two measurements are:

• Space-mean speed, also called travel velocity:
  

  $$v_L = {1 \over N_L} \, \sum_{i=1}^{N_L} v_i \ .$$ (27.1)

  
  
  Thus, one averages over a stretch of road of length $L$.

• Point velocity, also called spot speed or instantaneous velocity. We observe at a fixed position, and we average over the velocities of all vehicles that pass by. When $N_T$ is the number of vehicles that passed by, then spot speed is
  

  $$\tilde v_T = {1 \over N_T} \sum v_i \ .$$ (27.2)

  
  

One can immediately see that there is a difference between space-mean speed and spot speed by noting that space-mean speed includes vehicles of speed zero into the average while spot speed does not. If, however, all vehicles always have the same velocity, then both measurements lead to the same result. The formal relationship is a bit more complicated.^27.2

Travel velocity $v$ is the more relevant quantity since $L/v$ is the time an average traveller needs for a distance $L$. It is also the quantity which is relevant for fluid-dynamical relations, for example $q = \rho \, v$.

Flow

(also throughput). This is traditionally the most important quantity, since it is easy to measure (one just has to count the number of passing vehicles at a fixed location), and it is important for the performance of the transportation system as a whole. In order to allow comparison, it is often useful to divide flow by the number of lanes. Say that during time $T$ we have measured $N_T$ vehicles. Flow then is

$$q_T = {N_T \over T \, N_{lanes}} \ .$$ (27.4)

A typical unit of flow is ``(number of) vehicles per hour and lane''.

Transportation science also uses the term volume. According to Gerlough and Huber (48), this should be reserved to hourly flows (i.e. measured over one hour and expressed in ``vehicles per hour''). Maximum flow is also called capacity.

There is no direct way to measure space-mean flow. However, sometimes it is useful to use the relation $q = \rho v$. We then have

$$q_L = \rho_L \, v_L = {1 \over L \, N_{lanes}} \, \sum_{i=1}^{N_{veh}} v_i$$ (27.5)

where $\rho_L$ is taken from the next section.

Density

Space-averaged density $\rho_L$ is the number of vehicles on a certain stretch of road, divided by the length $L$ of that stretch. In order to allow comparison, it is useful to also divide by the number of lanes:

$$\rho_L = {N_{veh} \over L \, N_{lanes}} \ .$$ (27.6)

The resulting density is for example given in ``(number of) vehicles per km and lane''.

Point density has no natural measurement. One can use $\rho_T = q_T/v_T$.

An alternative method for point density is the ``fraction of time that a sensor is covered by a vehicle'', also called occupancy. Unfortunately, this quantity is difficult to obtain from a time-discrete simulation. Since the duration a sensor is covered by a vehicle is $\ell_i/v_i$, the correct measurement in a simulation would be [[check]]

$$\rho_T = {1 \over T} \sum \ell_i/v_i \ .$$ (27.7)

In the CA context, $\ell_i = const =1$. In field measurements, it is usually impossible to obtain $\ell_i$ for each vehicle, which means that an exact translation of occupancy into density is impossible.

Figure 27.1: Time series of speed, flow, and density.