The marvelous thing about traffic flow is the fact that you canjam it anywhere at any time with so little effort.
Siegfried Wache
Abstract In this chapter we discuss different visualizations of microscopic andmacroscopic cross-sectional data and the possible conclusions that one can drawfrom them. Time series of aggregated quantities such as speed, flow, and densityshow temporal developments, while speed-density and flow-density diagrams allowus to make statements about the average driving behavior on the observed roadsegment. Particularly the flow-density diagram contains so much information aboutthe traffic dynamics that its idealized form is also called fundamental diagram oftraffic flow. If single-vehicle data is available, we can also obtain distributions ofmicroscopic quantities (vehicle speeds, time gaps, etc.).
4.1 Time Series of Macroscopic Quantities
One way of representation are time series of some aggregated quantity, which hasbeen measured at a cross-section. Flow, speed, and density time series of a fewhours’ data tell us about traffic breakdowns, types of traffic congestion (oscillatoryor essentially stationary), and the capacity drop after a breakdown (Fig. 4.1).
From the specific daily patterns of traffic demand (Fig. 4.2), the reader can easilyrecognize whether it was recorded on a Monday, Tuesday/Wednesday/Thursday,Friday or on a weekend.1 However, these daily traffic-demand plots are used primarilyin transportation planning and are beyond the scope of this book.
1 School and national holidays as well as holidays and associated “long weekends” are special caseswith their own characteristic patterns.
Fig. 4.1 Time series during the morning peak-hour from one-minute data. From top to bottom:arithmetic mean speed V , flow Q, and estimated density ρ = Q/V (see Sect. 3.3.1)
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Fig. 4.2 Typical daily time series of the traffic flow (demand) on a weekday (Wednesday)
It is very easy to draw incorrect conclusions when interpreting traffic jam dynamicsusing single time series, as the following exercise illustrates:
Why is it wrong to conclude from the time series in Fig. 4.1 that the traf-fic breakdown occurred at around 7 a.m.? Can we at least conclude (fromthe figure) that vehicles near the cross-section at 7 a.m. decelerate, or that
Fig. 4.3 Sketches of speed time series at a cross-section and possible spatiotemporal traffic patternscausing them
vehicles near the cross-section at 8.30 a.m. accelerate? If not, what are alter-native explanations for the observed patterns?
Solution. According to Fig. 4.3 the speed drop shortly before 7 a.m. is an upstreamjam front that is moving upstream. Alternatively, it could be a downstream jam frontmoving downstream (with the driving direction) that is caused by a moving bottle-neck, e.g., by an oversize load. However, this case is rather unlikely, so we assumethat it is an upstream jam front and vehicles are braking to avoid a rear-end collision.
The rise in speed at 8.30 a.m. can be explained by two different scenarios: (i)It is a downstream-moving upstream front, i.e. the traffic jam shrinks. This wouldimply that, after 8.30 a.m., vehicles are braking shortly after passing the detector,while the time series indicates an acceleration. (ii) Alternatively, it could be anupstream-moving downstream front, caused for example by a disappearing temporarybottleneck (road block, traffic light, etc.) as the waiting vehicles subsequently start tomove again. In this case, the vehicles accelerate as indicated by the time series. Forboth scenarios, we can estimate the jam front velocity directly from the fundamentaldiagram (see Sect. 4.4 and Part II).
4.2 Speed-Density Relation
If we plot the aggregated vehicle speed over traffic density we obtain a speed-densitydiagram (cf. Fig. 4.4). We see that the average speed is lower in denser traffic. Fur-thermore, the diagram reflects the average behavior of a (typical) driver-vehicle
28 4 Representation of Cross-Sectional Data
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Fig. 4.4 Speed-density relation obtained from one-minute data collected on the Autobahn A9 nearMunich, Germany, using the average over both lanes (top left), individual averages of both lanes (topright), and individual lane averages conditioned on night (bottom left) and day hours (bottom right)
unit in different densities and external influences such as speed limits, weatherconditions, etc.
In very low-density traffic, the drivers are usually not influenced by other vehiclesand we obtain the average free speed V0 for ρ → 0 (cf. Fig. 4.5). This speed is theminimum of (i) the actual desired speed of the drivers, (ii) the physically possibleattainable speed (especially relevant for trucks on uphill slopes), and possibly (iii) anadministrated speed limit (plus the drivers’ average speeding). However, V0 is oftendirectly referred to as the desired speed.
To approximatively obtain the distribution of desired speeds from empirical data,we can use the speed distributions in single-vehicle data of low-density traffic (cf.Sect. 3.1 and Fig. 4.6). In this case, there are few interactions between the drivers andmost of the drivers can be expected to drive at their desired speed. The distributionsof speeds on the left and middle lane are symmetric and approximately Gaussian,while speeds on the right lane are distributed bimodally, showing the superpositionof the different speed distributions of trucks and passenger cars. Figure. 4.7 showsaverage speed differences between lanes. In denser traffic, the speed difference tendstowards zero, leading to a speed synchronization of the lanes.
Speed-density diagrams might show heterogeneous traffic and different externalconditions, which has to be considered when interpreting them. Examples include avarying percentage of trucks at different times of the day, different weather conditions
Fig. 4.5 Speed-density dia-grams, averaged over all lanes,for segments of the Dutch A9(Haarlem to Amsterdam) andthe German A8 (Munich toSalzburg, Austria)
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Fig. 4.6 Probability distrib-utions of the vehicle speed,P(v), in low-density trafficon the German Autobahn A3(three lanes in each direction)[From: Knospe et al., PhysicalReview E 65, S. 56133 (2002)]
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(lighting, precipitation), and time-dependent speed limits issued by traffic controlsystems. This also applies to the flow-density diagrams which will be discussed inSect. 4.4.
(1) In the upper left (V, ρ)-diagram of Fig. 4.4, the average speed decreasesagain for very small densities. Does this imply that drivers are “afraid of thefree road”? Explain this observation statistically.(2) The upper right panel of Fig. 4.4 shows two point clusters, around 100 km/hand 125 km/h, in the left lane (red open circles). Give a possible explanation forthis bimodality. Consider the diagrams in the bottom panels (a traffic controlsystem issuing traffic-dependent speed limits by variable message signs isinstalled on this road segment).
30 4 Representation of Cross-Sectional Data
Fig. 4.7 Difference in aver-age speed between neigh-boring lanes (A9-South nearMunich, Germany)
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4.3 Distribution of Time Gaps
Using single-vehicle data, we can also obtain the distributions of time gaps (cf.Eq. 3.3), as shown in Fig. 4.8 for two different speed ranges corresponding to freeand congested traffic. The time-gap distributions exhibit the following properties:
1. Time gaps are broadly scattered—it is not unusual to see standard deviationslarger than the arithmetic mean 〈T 〉, i.e., a coefficient of variation greaterthan 1.
2. The distributions are strongly asymmetric. Both in free and congested traffic weobserve time gaps longer than 10 s.
3. In free traffic (with speeds larger than some critical speed Vc) the most probabletime gap T̂ (the statistical mode) is significantly smaller than in congested traffic.In both speed regimes, T̂ is significantly smaller than the recommended safe timegap in the USA (“leave one car length for every ten miles per hour of speed”), orin Europe (“safety distance (in meters) equals speed (in km/h) divided by two”,corresponding to 1.8 s).
4. The arithmetic mean is also significantly smaller in dense free traffic than incongested traffic.
The mean flow is equal to the inverse of the arithmetic mean of the time headways.Thus, we can also determine the flow decrease after a traffic breakdown from thedistributions in Fig. 4.8. Traffic jams usually do not dissolve quickly once they haveemerged, due to this capacity drop.
Most of the observed time-gap distributions are not identical to the distribution ofthe drivers’ desired time gaps, but provide an upper bound only. The real time gap islarger in free traffic because most vehicles are not actually following another vehicle.With a flow of, e.g., 360 veh/h per lane (corresponding to a mean headway of 10 s), themode of the time-gap distribution is still below 1 s. There are also dynamic influences,since the followed vehicle might be “getting away” if the following vehicle cannotaccelerate any further (or its driver does not want to). These effects explain, at leastpartially, the strong asymmetry of the distributions.
Fig. 4.8 Distribution of thetime gaps in two speed regimes(free and congested traffic),measured on the Dutch A9
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4.4 Flow-Density Diagram
The flow-density diagram, i.e., plotting traffic flow against density, allows us to makea number of statements on the macroscopic (i.e., average) behavior of a driver-vehicleunit. In its idealized form, i.e., steady state equilibrium of identical driver-vehicleunits, it is also called fundamental diagram. The following quantities can be derivedfrom the fundamental diagram:
1. The desired speed equals the asymptotic gradient Q′(0) of the fit Q(ρ) for ρ = 0.This quantity can be more accurately determined using speed-density diagrams(cf. Sect. 4.2).
2. The actual mean speed for a defined density is given by the slope Q(ρ)/ρ of thesecant through (0, 0) and (ρ, Q(ρ)).
3. The maximum value of Q(ρ) is the road capacity per lane.4. The inverse of the smallest nonzero density ρmax for which Q(ρmax ) = 0 equals
the average vehicle length plus the average gap between stopped vehicles.5. The mean time gap T can be determined from the (negative) slope of Q(ρ) at
large densities (see Chap. 8).6. The slopes of flow-density diagrams also allow to read off the propagation
velocities of jam fronts and variations of macroscopic quantities (this is alsodiscussed in Chap. 8).
Bias with respect to the fundamental diagram. It is important to carefully distin-guish between measured flow-density data and the fundamental diagram.
The fundamental diagram describes the theoretical relation between densityand flow in stationary homogeneous traffic, i.e., the steady state equilibrium ofidentical driver-vehicle units. The flow-density diagram represents aggregatedempirical data that generally describes non-stationary heterogeneous traffic,i.e., different driver-vehicle units far from equilibrium.
Fig. 4.9 Flow-density dia-gram (averaged over all lanes)for sections of the Dutch A9(Haarlem to Amsterdam) andthe German A8-East (Munichto the Austrian border) nearIrschenberg
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There are multiple reasons for flow-density data not to coincide with the funda-mental diagram:
• The measurements process induces systematic errors (Sect. 3.3).• The traffic flow is not at equilibrium.• The traffic flow has spatial inhomogeneities or contains non-identical driver-
vehicle units.
The statements on traffic jam dynamics and driving behavior derived in the aboveenumeration are exact for the fundamental diagram, only. Since each of the afore-mentioned factors can cause significant differences between the density obtainedfrom Eq. (3.14) and the theoretical expectation in the fundamental diagram (it isnot unusual to see discrepancies by a factor of two), deriving statements from flow-density data is quite error-prone. In the following examples of empirical flow-densityrelations shown in the Figs. 4.9, 4.11 and 4.12 (upper left panel), the maximum trafficdensity obtained by extrapolation is unrealistically small, while the front propaga-tion velocities derived from the trend of flow-density point clouds of congestedregions are too large in magnitude (and the point clouds do not always show a cleartrend).
To estimate the effects of the errors mentioned above, we can use traffic sim-ulations that also simulate the measurement process using virtual cross-sectionaldetectors. Fig. 4.10 shows that the flow-density diagram depends strongly on themethod of averaging for obtaining the macroscopic speed and the flow (cf. Sect. 3.2),at least at large densities. Particularly, all methods yield estimated densities thatstrongly deviate from the actual density, which is, of course, available in the simula-tion. Remarkably, plotting the flow Q against the density estimate
Fig. 4.10 a Microscopic simulation of a traffic breakdown and stop-and-go waves caused by anon-ramp. Shown is the local speed. b–e flow-density data where the measurement process wassimulated using data of “virtual” detectors and different aggregation methods. b Flow Q = 1/〈�tα〉versus density Q/V (the standard procedure), c flow Q versus density Q/VH , d flow Q∗ = 〈1/�tα〉versus density Q∗/VH , e flow Q versus density Q∗/VH . For comparison, plot f displays the pointcloud obtaining by using the actual local values of flow and density, and the fundamental diagramis plotted as solid line in b–f
(Fig. 4.10e) consistently yields the least biased result in the simulations althoughthe unbiased flow is given by the harmonic mean Q (Eq. 3.6) of the microscopicflow, and not by the arithmetic average Q∗ (Eq. 3.12). In any case, the differencebetween the true flow-density points (f) and the data shown in (b)–(e) is caused bythe measurement process. The difference between the flow-density data (f) and thefundamental diagram, however, is due solely to non-equilibrium effects. This canbe concluded since identical driver-vehicle units were simulated (for details, seeFig. 11.4 in Part II where this simulation is discussed in detail).
Fig. 4.11 Flow-density dia-gram describing hysteretictraffic dynamics. Time seriesof these data are shown inFig. 4.1
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We finally notice that quantities that are derived purely from measurements ofthe flow, such as the capacity and the hysteresis effects to be discussed in the nextparagraph, are less subjected to errors.
Capacity drop and hysteresis. Sometimes, a sudden drop of the maximum possi-ble traffic flow (capacity drop) is observed with a traffic breakdown (cf. Fig. 4.11and 4.12). In this case the traffic shows hysteresis effects, i.e., the dynamics does notonly depend on the traffic demand but also on the history of the system. When thetraffic breaks down, the system state switches from the “free branch” onto the “con-gested branch”, lowering the maximum possible flow. This implies that once a trafficjam has emerged, the traffic demand has to fall to a much lower value to dissolve thejam. The flow-density diagram describing this phenomenon is also said to have aninverse-λ form (due to its resemblance of a mirrored Greek letter lambda, λ).
Wide scattering. The strong variation of time gaps (cf. Sect. 4.3) partially explainsthe strong scattering of the flow-density data in congested traffic: While in free trafficthe variations of density and time gaps both cause variations of the flow-density dataalong the one-dimensional curve Q ≈ ρV0, variations of density in congested trafficlead to changes in the flow-density data which are orthogonal to those caused byvariation in the time gaps. Both effects combined lead to a chaotic behavior of theflow-density data in congested traffic (cf. Figs. 4.11 and 4.12).
Finally, variations in the time gaps are not only caused by heterogeneous traffic(i.e., different desired time gaps of the individual drivers), but also by non-equilibriumtraffic dynamics (i.e., the actual time gap is not equal to the desired time gap) andthe systematic aggregation errors discussed above (Fig. 4.10).
4.5 Speed-Flow Diagram 35
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Fig. 4.12 Flow-density, speed-density, and speed-flow diagrams of the 1-minute data captured onthe Autobahn A5 near Frankfurt, Germany using harmonic mean speed. The lines show the fit of atraffic-stream model (see Sect. 6.2.2)
4.5 Speed-Flow Diagram
Plotting vehicle speed against traffic flow is also possible, of course. However, thisdiagram is not as fundamental for modeling as the flow-density diagram and not asdemonstrative as the speed-density diagram. It does have the advantage of showingonly directly observed quantities, Nevertheless, it is also affected by the systematicerrors in the speed aggregation. By the hydrodynamic relation Q = ρV , all threediagram types are equivalent (cf. Fig. 4.12).
Problems
4.1 Analytical fundamental diagramDerive and sketch both the speed-density diagram and the fundamental diagram,subject to the following idealized assumptions: (i) All vehicles are of length l = 5 m.(ii) In free traffic (speed does not depend on other vehicles), all vehicles drive attheir desired speed V0 = 120 km/h. (iii) In congested traffic (speed is the same asthe speed of the leading vehicle), drivers keep a gap of s(v) = s0 + vT to the leadingvehicle, with the minimum gap s0 = 2 m and the time gap T = 1.6 s.
4.2 Flow-density diagram of empirical dataConsidering the speed-density diagram (Fig. 4.5) and flow-density diagram (Fig. 4.9)of the German A8-East and the Dutch A9, determine the desired speed V0, time gapT , maximum density ρmax, and the capacity drop on both highways from the fitted
curves. Which statements can you make about the driving behavior of German andDutch drivers (at least on these specific highways at the time of measurement)?
Further Reading
• Hall, F.: Traffic stream characteristics. Traffic Flow Theory. US Federal HighwayAdministration (1996)
• Daganzo, C.: Fundamentals of Transportation and Traffic Operations. Pergamon-Elsevier, Oxford, U.K. (1997)
