The following excerpt is taken from Chapter 2 (pp. 5-11) of the Transportation Research Board Special Report on Traffic Flow Theory, published on the website http://www.tfhrc.gov/its/tft/tft.htm.
In general, traffic streams are not uniform, but vary over both space and time. Because of that, measurement of the variables of interest for traffic flow theory is in fact the sampling of a random variable. . . . In reality, the traffic characteristics that are labeled as flow, speed, and concentration are parameters of statistical distributions, not absolute numbers.
Flow rates are collected directly through point measurements, and by definition require measurement over time. They cannot be estimated from a single snapshot of a length of road. Flow rates and time headways are related to each other as follows. Flow rate, q, is the number of vehicles counted, divided by the elapsed time, T:
q = N/T
. . . Flow rates are usually expressed in terms of vehicles per hour, although the actual measurement interval can be much less. Concern has been expressed, however, about the sustainability of high volumes measured over very short intervals (such as 30 seconds or one minute) when investigating high rates of flow. The 1985 Highway Capacity Manual (HCM 1985) suggests using at least 15-minute intervals, although there are also situations in which the detail provided by five minute or one minute data is valuable. . . .
Measurement of the speed of an individual vehicle requires observation over both time and space. . . . In the literature, the distinction has frequently been made between different ways of calculating the average speed of a set of vehicles. . . . The first way of calculating speeds, namely taking the arithmetic mean of the observation,
is termed the time mean speed because it is an average of observations taken over time.
The second term that is used in the literature is space mean speed, but unfortunately there are a variety of definitions for it, not all of which are equivalent. . . . Regardless of the particular definition put forward for space mean speed, all authors agree that for computations involving mean speeds to be theoretically correct, it is necessary to ensure that one has measured space mean speed, rather than time mean speed. . . . Under conditions of stop-and-go traffic, as along a signalized street or a badly congested freeway, it is important to distinguish between these two mean speeds. For freely flowing freeway traffic, however, there will not be any significant difference between the two. . . . When there is great variability of speeds, as for example at the time of breakdown from uncongested to stop and go conditions, there will be considerable difference between the two. Wardrop (1952) provided an example of this kind (albeit along what must certainly have been a signalized roadway Western Avenue, Greenford, Middlesex, England), in which speeds ranged from a low of 8 km/h to a high of 100 km/h. The space mean speed was 48.6 km/h; the time mean speed 54.0 km/h. . . . For relatively uniform flow and speeds, the two mean speeds are likely to be equivalent for practical purposes. Nevertheless, it is still appropriate to specify which type of averaging has been done, and perhaps to specify the amount of variability in the speeds (which can provide an indication of how similar the two are likely to be).
. . . At least for freeways, the practical significance of the difference between space mean speed and time mean speed is minimal. However, it is important to note that for traffic flow theory purists, the only correct way to measure average travel velocity is to calculate space-mean speed directly.
Only a few freeway traffic management systems acquire speed information directly, since to do so requires pairs of presence detectors at each of the detector stations on the roadway, and that is more expensive than using single loops. Those systems that do not measure speeds, because they have only single-loop detector stations, sometimes calculate speeds from flow and occupancy data, using a method first identified by Athol (1965). . . .
Concentration has in the past been used as a synonym for density. For example, Gerlough and Huber (1975, 10) wrote, "Although concentration (the number of vehicles per unit length) implies measurement along a distance." In this chapter, it seems more useful to use concentration as a broader term encompassing both density and occupancy. The first is a measure of concentration over space; the second measure concentration over time of the same vehicle stream.
Density can be measured only along a length. If only point measurements are available, density needs to be calculated, either from occupancy or from speed and flow. Gerlough and Huber wrote (in the continuation of the quote in the previous paragraph), that " . . .traffic engineers have traditionally estimated concentration from point measurements, using the relationship
. . . The difficulty with using this equation to estimate density is that the equation is strictly correct only under some very restricted conditions, or in the limit as both the space and time measurement intervals approach zero. If neither of those situations holds, then use of the equation to calculate density can give misleading results, which would not agree with empirical measurements. These issues are important, because this equation has often been uncritically applied to situations that exceed its validity . . . .
Real traffic flows, however, are not only made up of finite vehicles surrounded by real spaces, but are inherently stochastic (Newell 1982). Measured values are averages taken from samples, and are therefore themselves random variables. Measured flows are taken over an interval of time, at a particular place. Measured densities are taken over space at a particular time. Only for stationary processes (in the statistical sense) will the time and space intervals be able to represent conditions at the same point in the time-space plane. Hence it is likely that any measurements that are taken of flow and density (and space mean speed) will not be very good estimates of the expected values that would be defined at the point of interest in the time space plane. . . .
Speeds within a lane are relatively constant during uncongested flow. Hence the estimation of density from occupancy measurements is probably reasonable during those traffic conditions, but not during congested conditions. . . . In short, once congestion sets in, there is probably no good way to estimate density; it would have to be measured.
Temporal concentration (occupancy) can be measured only over a short section (shorter
than the minimum vehicle length), with presence detectors, and does not make sense over a
long section. Perhaps because the concept of density has been a part of traffic
measurement since at least the 1930s, there has been a consensus that density was to
be preferred over occupancy as the measure of vehicular
It would be fair to say that the majority opinion at present remains in favor of density, but that a minority view is that occupancy should begin to enter theoretical work instead of density. There are two principal reasons put forward by the minority for making more use of occupancy. The first is that there should be improved correspondence between theoretical and practical work on freeways. If freeway traffic management makes extensive use of a variable that freeway theory ignores, the profession is the poorer. The second reason is that density, as vehicles per length of road ignores the effects of vehicle length and traffic composition. Occupancy, on the other hand, is directly affected by both of these variables, and therefore gives a more reliable indicator of the amount of a road being used by vehicles. There are also good reasons put forward by the majority for the continued use of density in theoretical work. Not least is that it is theoretically useful in their work in a way that occupancy is not. . . .