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5Traffic Flow Theory

Sven Maerivoet∗ and Bart De MoorDepartment of Electrical Engineering ESAT-SCD (SISTA)†,

Katholieke Universiteit LeuvenKasteelpark Arenberg 10, 3001 Leuven, Belgium

(Dated: February 2, 2008)

The scientific field of traffic engineering encompasses a richset of mathematical techniques, as well as re-searchers with entirely different backgrounds. This paperprovides an overview of what is currently the state-of-the-art with respect to traffic flow theory. Starting witha brief history, we introduce the microscopic andmacroscopic characteristics of vehicular traffic flows. Moving on, we review some performance indicators thatallow us to assess the quality of traffic operations. A final part of this paper discusses some of the relationsbetween traffic flow characteristics, i.e., the fundamentaldiagrams, and sheds some light on the different pointsof view adopted by the traffic engineering community.

PACS numbers: 02.50.-r,45.70.Vn,89.40.-a

Keywords: xxx

Contents

I. A brief history of traffic flow theory 2

II. Microscopic traffic flow characteristics 3A. Vehicle related variables 3B. Traffic flow characteristics 3

III. Macroscopic traffic flow characteristics 4A. Density 5

1. Mathematical formulation 52. Passenger car units 6

B. Flow 61. Mathematical formulation 62. Oblique cumulative plots 7

C. Occupancy 8D. Mean speed 9

1. Mathematical formulation 92. Fundamental relation of traffic flow theory 10

E. Moving observer method and floating car data 11

IV. Performance indicators 12A. Peak hour factor 12B. Travel times and their reliability 12

1. Travel time definitions 122. Queueing delays 133. An example of travel time estimation using cumulative plots13

4. Reliability and robustness properties 14C. Level of service 14D. Efficiency 15

V. Fundamental diagrams 16A. Traffic flow regimes 16

1. Free-flow traffic 162. Capacity-flow traffic 163. Congested, stop-and-go, and jammed traffic16

†Phone: +32 (0) 16 32 17 09 Fax: +32 (0) 16 32 19 70URL: http://www.esat.kuleuven.be/scd∗Electronic address: sven.maerivoet@esat.kuleuven.be

4. A note on the transitions between different regimes17

B. Correlations between traffic flow characteristics17

1. The historic origin of the fundamental diagram17

2. The general shape of a fundamental diagram18

3. Empirical measurements 21C. Capacity drop and the hysteresis phenomenon 22D. Kerner’s three-phase theory 23

1. Free flow, synchronised flow, and wide-moving jam23

2. Fundamental hypothesis of three-phase traffic theory24

3. Transitions towards a wide-moving jam 244. From descriptions to simulations 25

E. Theories of traffic breakdown 25

VI. Conclusions 27

A. Glossary of terms 271. Acronyms and abbreviations 272. List of symbols 28

Acknowledgements 30

References 30

The scientific field of traffic engineering encompassesa rich set of mathematical techniques, as well as re-searchers with entirely different backgrounds. This paperprovides an overview of what is currently the state-of-the-art with respect to traffic flow theory. Starting witha brief history, we introduce the microscopic and macro-scopic characteristics of vehicular traffic flows. Movingon, we review some performance indicators that allow usto assess the quality of traffic operations. A final part ofthis paper discusses some of the relations between trafficflow characteristics, i.e., the fundamental diagrams, andsheds some light on the different points of view adoptedby the traffic engineering community.

Because of the large diversity of the scientific field (en-

gineers, physicists, mathematicians, . . . all lack a unifiedstandard or convention), one of the principal aims of thispaper is to define both alogical and consistent terminol-ogy and notation. It is our strong belief that such a consis-tent notation is a necessity when it comes to creating or-der in the ‘zoo of notations’ that in our opinion currentlyexists. For a concise but complete overview of all abbre-viations and notations proposed and adopted throughoutthis paper, we refer the reader to appendix A.

I. A BRIEF HISTORY OF TRAFFIC FLOW THEORY

Historically, traffic engineering got its roots as a ratherpractical discipline, entailing most of the time a commonsense of its practitioners to solve particular traffic prob-lems. However, all this changed at the dawn of the 1950s,when the scientific field began to mature, attracting engi-neers from all sorts of trades. Most notably, John GlenWardrop instigated the evolving discipline now known astraffic flow theory, by describing traffic flows using math-ematical and statistical ideas [115].

During this highly active period, mathematics establisheditself as a solid basis for theoretical analyses, a phe-nomenon that was entirely new to the previous, more‘rule-of-thumb’ oriented, line of reasoning. Two ex-amples of the progress during this decade, include thefluid-dynamic model of Michael James Lighthill, Ger-ald Beresford Whitham, and Paul Richards (or theLWRmodelfor short) for describing traffic flows [72, 102], andthe car-following experiments and theories of the club ofpeople working at General Motors’ research laboratory[17, 40, 41, 50]. Simultaneous progress was also madeon the front of economic theory applied to transportation,most notably by the publication of the ‘BMW trio’, Mar-tin J. Beckmann, Charles Bartlett McGuire, and Christo-pher B. Winsten [6].

From the 1960s on, the field evolved even further with theadvent of the early personal computers (although at thattime, they could only be considered as mere computingunits). More control-oriented methods were pursued byengineers, as a means for alleviating congestion at tun-nels and intersections, by e.g., adaptively steering traf-fic signal timings. Nowadays, the field has been kindlyembraced by the industry, resulting in what is calledin-telligent transportation systems(ITS), covering nearly allaspects of the transportation community.

In spite of the intense booming during the 1950s and1960s, all progress seemingly came to sudden stop, asthere were almost no significant results for the next twodecades (although there are some exceptions, such asthe significant work of Ilya Prigogine and Robert Her-man’s, who developed a traffic flow model based on agas-kinetic analogy [101]). One of the main reasons forthis, stems from the fact that many of the involved keyplayers returned to their original scientific disciplines,af-ter exhausting the application of their techniques to thetransportation problem [99]. Note that despite this calmperiod, the application of control theory to transporta-

tion started finding new ways to alleviate local congestionproblems.

At the beginning of the 1990s, researchers found a re-vived interest in the field of traffic flow modelling. On theone hand, researchers’ interests got kindled again by theappealing simplicity of the LWR model, whereas on theother hand one of the main boosts came from the worldof statistical physics. In this latter framework, physiciststried to model many particle systems using simple and el-egant behavioural rules. As an example, the now famousparticle hopping (cellular automata) model of Kai Nageland Michael Schreckenberg [92] still forms a widely-cited basis for current research papers on the subject.

In parallel with this kind of modelling approach, manyof the old ‘beliefs’ (e.g., the fluid-dynamic approach totraffic flow modelling) started to get questioned. As aconsequence, a plethora of models quickly found its wayto the transportation community, whereby most of thesemodels didn’t give a thought as to whether or not theirassociated phenomena corresponded to real-life trafficobservations.

We note here that, whatever the modelling approachmay be, researchers should always compare their re-sults to the reality of the physical world. Ignoringthis basic step, reduces the research in our opinion tonothing more than a mathematical exercise !

As the international research community began to spawnits traffic flow theories, Robert Herman aspired to bringthem all together in december 1959. This led to the tri-annual organisation of theInternational Symposium onTransportation and Traffic Theory(ISTTT), by some her-alded as ‘the Olympics of traffic theory’ because the sym-posium talks about the fundamentals underlying trans-portation and traffic phenomena. Another example of theevolution of recent developments with respect to the par-allels between traffic flows and granular media, is thebi-annual organisation of the workshop onTraffic andGranular Flow (TGF), a platform for exchanging ideasby bringing together researchers from various scientificfields.

Nowadays, the research and application of traffic flowtheory and intelligent transportation systems continues.The scientific field has been largely diversified, encom-passing a broad range of aspects related to sociology, psy-chology, the environment, the economy, . . . The globalavidity of the field can be witnessed by the exponentiallygrowing publication output. Keeping our previous com-ment in mind, researchers from time to time just seem to‘add to the noise’ (mainly due to the sheer diversity ofthe literature body), although there occasionally exist ex-ceptions such as the late Newell, as subtly pointed out byMichael Cassidy in [100].

As a final word, we refer the reader to two personalisedviews on the history of traffic flow theory, namely themusings of the late Gordon Newell and Denos Gazis[42, 99]. We furthermore invite the reader to cast a glanceat the ending pages of Wardrop’s paper [115], in which arather colourful discussion on the introduction of mathe-matics to traffic flow theory has been written down.

II. MICROSCOPIC TRAFFIC FLOWCHARACTERISTICS

Road traffic flows are composed of drivers associatedwith individual vehicles, each of them having their owncharacteristics. These characteristics are calledmicro-scopicwhen a traffic flow is considered as being com-posed of such a stream of vehicles. The dynamical as-pects of these traffic flows are formed by the underly-ing interactions between the drivers of the vehicles. Thisis largely determined by the behaviour of each driver, aswell as the physical characteristics of the vehicles.

Because the process of participating in a traffic flow isheavily based on the behavioural aspects associated withhuman drivers [39], it would seem important to includethese human factors into the modelling equations.However, this leads to a severe increase in complexity,which is not always a desired artifact [76]. However,in the remainder of this section, we always considera vehicle-driver combination as a single entity, takingonly into account some vehicle related traffic flowcharacteristics.

Note that despite our previous remarks, we do notdebate the necessity of a psychological treatment oftraffic flow theory. As the research into driver be-haviour is gaining momentum, a lot of attention isgained by promising studies aimed towards driverand pedestrian safety, average reaction times, the in-fluence of stress levels, aural and visual perceptions,ageing, medical conditions, fatigue, . . .

A. Vehicle related variables

Considering individual vehicles, we can say that each ve-hicle i in a lane of a traffic stream has the following in-formational variables:

• a length, denoted byli,

• a longitudinal position, denoted byxi,

• a speed, denoted byvi =dxi

dt,

• and anacceleration, denoted byai =dvi

dt=

d2xi

dt2.

Note that the positionxi of a vehicle is typically takento be the position of its rear bumper. In this first ap-proach, a vehicle’s other spatial characteristics (i.e., itswidth, height, and lane number) are neglected. And inspite of our narrow focus on the vehicle itself, the abovelist of variables is also complemented with a driver’sre-action time, denoted byτi.

With respect to the acceleration characteristics, it shouldbe noted that these are in fact not only dependent on thevehicle’s engine, but also on e.g., the road’s inclination,being a non-negligible factor that plays an important role

in the forming of congestion at bridges and tunnels. Wedo not use the derivative of the acceleration, calledjerk,jolt, or surge(jerk is also used to represent the smooth-ness of theacceleration noise[82]).

Except in the acceleration capabilities of a vehicle, weignore the physical forces that act on a vehicle, e.g., theearth’s gravitational pull, road and wind friction, cen-trifugal forces, . . . A more elaborate explanation of theseforces can be found in [27].

B. Traffic flow characteristics

Referring to Fig. 1, we can consider two consecutive ve-hicles in the same lane in a traffic stream: a followeri andits leaderi + 1. From the figure, it can be seen that vehi-clei has a certainspace headwayhsi

to its predecessor (itis expressed in metres), composed of the distance (calledthespace gap) gsi

to this leader and its ownlengthli:

hsi= gsi

+ li. (1)

By taking, as stated before, the rear bumper as a vehicle’sposition, the space headwayhsi

= xi+1 − xi. The spacegap is thus measured from a vehicle’s front bumper to itsleader’s rear bumper.

(i) (i + 1)

xi xi+1li gsi

hsi

FIG. 1: Two consecutive vehicles (a followeri at positionxi

and a leaderi + 1 at positionxi+1) in the same lane in a trafficstream. The follower has a certain space headwayhsi

to itsleader, equal to the sum of the vehicle’s space gapgsi

and itslengthli.

Analogously to equation (1), each vehicle also has atimeheadwayhti

(expressed in seconds), consisting of atimegapgti

and anoccupancy timeρi:

hti= gti

+ ρi. (2)

Both space and time headways can be visualised in atime-space diagram, such as the one in Fig. 2. Here,we have shown the two vehiclesi andi + 1 as they aredriving. Their positionsxi andxi+1 can be plotted withrespect to time, tracing out twovehicle trajectories. Asthe time direction is horizontal and the space directionis vertical, the vehicles’ respective speeds can be derivedby taking the tangents of the trajectories (for simplicity,we have assumed that both vehicles travel at the sameconstant speed, resulting in parallel linear trajectories).

time

space

(i)

(i + 1)

xi

xi+1

titi+1

hsi

gsi

li

hti

gtiρi

FIG. 2: A time-space diagram showing two vehicle trajectoriesi and i + 1, as well as the space and time headwayhsi

andhti

of vehicle i. Both headways are composed of the spacegapgsi

and the vehicle lengthli, and the time gapgtiand the

occupancy timeρi, respectively. The time headway can be seenas the difference in time instants between the passing of bothvehicles, respectively atti+1 andti (diagram based on [74]).

Accelerating vehicles have steep inclining trajectories,whereas those of stopped vehicles are horizontal.

When the vehicle’s speed is constant, the time gap is theamount of time necessary to reach the current position ofthe leader when travelling at the current speed (i.e., it isthe elapsed time an observer at a fixed location wouldmeasure between the passing of two consecutive vehi-cles). Similarly, the occupancy time can be interpretedas the time needed to traverse a distance equal to the ve-hicle’s own length at the current speed, i.e.,ρi = li/vi;this corresponds to the time the vehicle needs to pass theobserver’s location. Both equations (1) and (2) are fur-thermore linked to the vehicle’s speedvi as follows [27]:

hsi

hti

=gsi

gti

=liρi

= vi. (3)

As the above definitions deal with what is called single-lane traffic, we can easily extend them to multi-lane traf-fic. In this case, four extra space gaps — related tothe vehicles in the neighbouring lanes — are introduced,namelygl,f

siat the left-front,gl,b

siat the left-back,gr,f

siat

the right-front, andgr,bsi

at the right-back. The four cor-responding space headways,hl,f

si, hl,b

si, hr,f

si, andhr,b

si, are

introduced in a similar fashion. The extra time gaps andheadways are derived in complete analogy, leading to thefour time gapsgl,f

ti, gl,b

ti, gr,f

ti, andgr,b

ti, and the four cor-

responding time headwayshl,fti

, hl,bti

, hr,fti

, andhr,bti

.

In single-lane traffic, vehicles always keep their relativeorder, a principle sometimes calledfirst-in, first-out(FIFO) [24]. For multi-lane traffic however, this princi-ple is no longer obeyed due to overtaking manoeuvres,resulting in vehicle trajectories that cross each other.If the same time-space diagram were to be drawn foronly one lane (in multi-lane traffic), then some vehicles’

trajectories would suddenly appear or vanish at the pointwhere a lane change occurred.

In some traffic flow literature, other nomenclature isused:spacefor the space headway,distanceor clear-ance for the space gap, andheadwayfor the timeheadway. Because this terminology is confusing, wepropose to use the unambiguously defined terms asdescribed in this section.

III. MACROSCOPIC TRAFFIC FLOWCHARACTERISTICS

When considering many vehicles simultaneously, thetime-space diagram mentioned in section II B can be usedto faithfully represent all traffic. In Fig. 3 we show theevolution of the system, as we have traced the trajecto-ries of all the individual vehicles’ movements. This time-space diagram therefore provides a complete picture ofall traffic operations that are taking place (accelerations,decelerations, . . . ).

t

x

dt

dx

Rt

Rs

Rt,s

Tmp

K

FIG. 3: A time-space diagram showing several vehicle trajec-tories and three measurement regionsRt, Rs, andRt,s. Theserectangular regions are bounded in time and space by a mea-surement periodTmp and a road section of lengthK. The blackdots represent the individual measurements.

Instead of considering each vehicle in a traffic streamindividually, we now ‘zoom out’ to a more aggregatemacroscopiclevel (traffic streams are regarded e.g., asa fluid). In the remainder of this section, we will mea-sure some macroscopic traffic flow characteristics basedon the shown time-space diagram. To this end, we definethree measurement regions:

• Rt corresponding to measurements at a single fixedlocation in space (dx), during a certain time periodTmp. An example of this is a single inductive loopdetector (SLD) embedded in the road’s concrete.

• Rs corresponding to measurements at a single in-stant in time (dt), over a certain road section oflengthK. An example of this is an aerial photo-graph.

• Rt,s corresponding to a general measurement re-gion. Although it can have any shape, in this casewe restrict ourselves to a rectangular region in timeand space. An example of this is a sequence of im-ages made by a video camera detector.

With respect to the size of these measurement regions,some caution is advised: a too large measurement regioncan mask certain effects of traffic flows, possibly ignoringsome of the dynamic properties, whereas a too small mea-surement region may obstruct a continuous treatment, asthe discrete, microscopic nature of traffic flows becomesapparent.

Using these different methods of observation, we nowdiscuss the measurement of four important macroscopictraffic flow characteristics: density, flow, occupancy, andmean speed. We furthermore give a short discussion onthe moving observer method and the use of floating cardata.

With respect to some naming conventions on road-ways, two different ‘standards’ exist for some of theencountered terminology, namely the American andthe British standard. Examples are: the classic multi-lane high-speed road with on- and off-ramps, whichis called afreewayor asuper highway(American), oranarterial or motorway(British). A main road withintersections is called anurban highway(American)or a carriageway(British). In this dissertation, wehave chosen to adopt the British standard. Finally, incontrast to Great Britain and Australia, we assumethat for low-density traffic, everybody drives on theright instead of the left lane.

A. Density

The macroscopic characteristic calleddensityallows usto get an idea of how crowded a certain section of a roadis. It is typically expressed in number of vehicles perkilometre (or mile). Note that the concept of density to-tally ignores the effects of traffic composition and vehiclelengths, as it only considers the abstract quantity ‘numberof vehicles’.

Because density can only bemeasuredin a certain spatialregion (e.g.,Rs in Fig. 3), it iscomputedfor temporal re-gions such as regionRt in Fig. 3. When density can notbe exactly measured or computed, or when density mea-surements are faulty, it has to beestimated. To this end,several available techniques exist e.g., based on explicitsimulation using a traffic flow propagation model [90],based on a vehicle reidentification system [21], basedon a complete traffic state estimator using an extendedKalman filter [114], or based on a non-linear adaptive ob-server [3], . . .

1. Mathematical formulation

Using the spatial regionRs, the densityk for single-lanetraffic is defined as:

k =N

K, (4)

with N the number of vehicles present on the road seg-ment. If we consider multi-lane traffic, we have to sumthe partial densitieskl of each of theL lanes as follows:

k =

L∑

l=1

kl =1K

L∑

l=1

Nl, (5)

in which Nl now denotes the number of vehicles presentin lanel (equation (5) isnot the same as averaging overthe partial densities of each lane)[122].

In general, density can be defined asthe total time spentby all the vehicles in the measurement region, divided bythe area of this region[27, 36]. This generalisation al-lows us to compute the density at a point using the mea-surement regionRt:

k =

N∑

i=1

Ti

Tmp dx=

1

Tmp��dx

N∑

i=1

��dx

vi

=1

Tmp

N∑

i=1

1vi

, (6)

with Ti the travel time andvi the speed of theith vehicle.Extending the previous derivation to multi-lane traffic isdone straightforward using equation (5):

k =1

Tmp

L∑

l=1

Nl∑

i=1

1vi,l

, (7)

with nowvi,l denoting the speed of theith vehicle in lanel.

As we now can obtain the density in both spatial and tem-poral regions,Rs andRt respectively, it would seem alogical extension to find the density in the regionRt,s.In order to do this, however, we need to know the traveltimesTi of the individual vehicles, as can be seen in equa-tion (6). Because this information is not always available,and in most cases rather difficult to measure, we use a dif-ferent approach, corresponding to the temporal averageof the density. Assuming that at each time stept, duringa certain time periodTmp, the densityk(t) is known inconsecutive regionsRs, the generalised definition leadsto the following formulation:

k =

1Tmp

∫ Tmp

t=0k(t) dt (continuous),

1Tmp

Tmp∑

t=1

k(t) (discrete).

(8)

For multi-lane traffic, combining equations (5) and (8)results in the following formula for computing the densityin regionRt,s using measurements in discrete time:

k =1

Tmp K

Tmp∑

t=1

L∑

l=1

Nl(t), (9)

whereNl(t) denotes the number of vehicles present inlanel at timet.

There exists a relation between the macroscopic trafficflow characteristics and those microscopic characteristicsdefined in section II B. For the densityk, this relation isbased on the average space headwayhs [27, 115]:

k =N

K=

NN∑

i=1

hsi

=1

1N

N∑

i=1

hsi

=1

hs

, (10)

with hs

−1the reciprocal of the average space headway.

2. Passenger car units

When considering heterogeneous traffic flows (i.e.,traffic streams composed of different types of vehicles),operating agencies usually don’t express the macroscopictraffic flow characteristics using the raw number of vehi-cles, but rather employ the notion ofpassenger car units(PCU). These PCUs, sometimes also calledpassengercar equivalents(PCE), try to take into account the spatialdifferences between vehicle types. For example, bydenoting one average passenger car as 1 PCU, a truck inthe same traffic stream can be considered as 2 PCUs (oreven higher and fractional values for trailer trucks).

Let us finally note that, because density is essentiallydefined as a spatial measurement, it is one of themost difficult quantities to obtain. It is interesting tonotice that at this moment, it is theoretically possiblefor video cameras to measure density over a shortspatial region. However, to our knowledge there cur-rently exists no commercial implementation.

B. Flow

Whereas density typically is a spatial measurement,flowcan be considered as a temporal measurement (i.e., re-gion Rt). Flow, which we use as a shorthand for rateof flow, is typically expressed as an hourly rate, i.e., innumber of vehicles per hour. Note that sometimes othersynonyms such asintensity, flux, throughput, current, orvolume[123] are used, typically depending on a person’sscientific background (e.g., engineering, physics, . . . ).

1. Mathematical formulation

Measuring the flowq in regionRt for single-lane traffic,is done using the following equation, which is based onraw vehicle counts:

q =N

Tmp, (11)

with N the number of vehicles that has passed the detec-tor’s site. For multi-lane traffic, we sum the partial flowsof each of theL lanes:

q =L∑

l=1

ql =1

Tmp

L∑

l=1

Nl, (12)

with nowNl denoting the number of vehicles that passedthe detector’s site in lanel. Note that we assume that eachlane has its own detector, otherwise we would be dealingwith an average flow across all the lanes.

Generally speaking, flow can defined asthe total distancetravelled by all the vehicles in the measurement region,divided by the area of this region[27, 36]. In analogywith equation (6), this generalisation allows us to com-pute the flow using the spatial measurement regionRs:

q =

N∑

i=1

Xi

K dt=

1K ��dt

N∑

i=1

vi ��dt =1K

N∑

i=1

vi, (13)

with nowXi the distance travelled by theith vehicle dur-ing the infinitesimal time intervaldt. The extension tomulti-lane traffic is straightforward:

q =1K

L∑

l=1

Nl∑

i=1

vi,l. (14)

Considering consecutive flow measurements in regionRt,s, we can derive a formulation corresponding to thetemporal average of the flow, similar to that of equation(8). Assuming that at each time stept, during a certaintime periodTmp, the flowq(t) is known in consecutive re-gionsRs, the generalised definition leads to the followingequations:

q =

1Tmp

∫ Tmp

t=0q(t) dt (continuous),

1Tmp

Tmp∑

t=1

q(t) (discrete),

(15)

For multi-lane traffic, combining equations (14) and (15)results in the following formula for computing the flowin regionRt,s using measurements in discrete time:

q =1

Tmp K

Tmp∑

t=1

L∑

l=1

Nl(t)∑

i=1

vi,l(t), (16)

wherevi,l(t) denotes the speed of theith vehicle in lanelat timet.

In analogy with equation (10), there exists a relationbetween the flowq, and the average time headwayht

[27, 115]:

q =N

Tmp=

NN∑

i=1

hti

=1

1N

N∑

i=1

hti

=1

ht

, (17)

with ht

−1the reciprocal of the average time headway.

2. Oblique cumulative plots

As stated before, flows are always expressed as a rate.In contrast to this, we can also consider the raw vehi-cle counts at a certain location (i.e., measurement regionRt). If we plot the cumulative number of passing ve-hicles (denoted byN ) with respect to time for differentregions (e.g., inductive loop detectors), we get a set ofcurves such as the one in the left part of Fig. 4. Thesecurves are calledcumulative plots(or (t, N) diagrams),and although their origins date back as far as 1954 withthe work of Karl Moskowitz [83], it was Gordon Newellwho applied them later on to their full potential (initiallyin the context of queueing theory) [95, 96, 97, 98] (a sim-ilar method was applied by John Luke, in the field of con-tinuum mechanics [25, 75]).

The key benefit of these cumulative plots, comes whencomparing observations stemming from multiple detectorstations at a closed section of the road that conserves thenumber of vehicles (i.e., no on- or off-ramps), in whichcase we also speak ofinput-output diagrams. If there aretwo detector stations, then the upstream and downstreamstations measure theinput, respectivelyoutput, of thesection. Similarly like in queueing theory, the upstreamcurve is sometimes called thearrival function, whereasthe downstream one is called thedeparture function[95].As the method is based on counting the number of in-dividual vehicles at each observation location (wherebyeach vehicle is numbered with respect to a single refer-ence vehicle), this results in a monotonically increasingfunctionN(t) (sometimes called theMoskowitz function,after its ‘inventor’), which increases each time a vehiclespasses by. At each time instantt, the cumulative count is

06:30 08:00 09:30 11:00 12:30 14:00 15:30 17:00 18:30 20:000

1

2

3

4

5

6x 10

4

Time

Cum

ulat

ive

N(t

)

06:30 08:00 09:30 11:00 12:30 14:00 15:30 17:00 18:30 20:00−10000

−9000

−8000

−7000

−6000

−5000

−4000

−3000

Time

Obl

ique

N(t

) w

ith b

ackg

roun

d flo

w =

409

9 ve

hicl

es/h

our

10000 20000 30000 40000 50000 60000

incident

FIG. 4: Left: a standard cumulative plot showing the numberof passing vehicles at two detector locations; due to the graph’sscale, both curves appear to lie on top of each other.Right:the same data but displayed using an oblique coordinate sys-tem, thereby enhancing the visibility (the dashed slanted lineshave a slope corresponding to the subtracted background flowqb ≈ 4100 vehicles per hour). We can see a queue (probablycaused due to an incident) growing at approximately 11:00, dis-sipating some time later at approximately 12:30. The shown de-tector data was taken from single inductive loop detectors [113],covering all three lanes of the E40 motorway between Erpe-Mere and Wetteren, Belgium. The shown data was recorded atMonday April 4th, 2003 (the detectors’ sampling interval wasone minute, the distance between the upstream and downstreamdetector stations was 8.1 kilometres).

defined as:

N(t) =

t∑

t′=t0

q(t′) = N(t − 1) + q(t). (18)

The time needed to travel from one location to anothercan easily be measured as the horizontal distance be-tween the respective cumulative curves. Similarly, thevertical distance between these curves allows us to derivethe accumulation of vehicles on the road section, which

gives an excellent indication of growing and dissipatingqueues (i.e., congestion). Furthermore, if we computethe slope of this function at each time instantt, we ob-tain the flowq(t) = [N(t + ∆t) − N(t)]/∆t. Finally,becauseN(t) essentially is a step function, we can definea smooth approximationN(t). This results in an every-where differentiable function, allowing us to compute in-stantaneous flows and local densities asq = ∂N(t, x)/∂t

andk = −∂N(t, x)/∂x, respectively [27].

The main disadvantage of the method is the fact thatthese cumulative functions increase very rapidly, therebymasking the subtle differences between different curves.Cassidy and Windover therefore proposed to subtracta background flowqb from these curves, resulting infunctionsN(t) − t qb [13]. Based on this; Munoz andDaganzo furthermore introduced enhanced clarity byoverlaying this cumulative plot with a set of obliquelines with slope−qb [88]. Choosing an appropriate valuefor qb, allows us to nicely enhance the characteristicundulations that are expressed by the different obliquecurves.

Note that before using these oblique plots, the cumu-lative plots from different detectors stations need tobesynchronised. To understand this, suppose a ref-erence vehicle passes an upstream detector station ata certain time instanttup; after a certain time period,the vehicle reaches the downstream detector stationat a later time instanttdown. The amounttdown− tup

is the time it takes to cross the distance betweenboth detector stations, allowing the synchronisationmechanism to shift the respective cumulative curvesover this time period (i.e., initialising them with thepassing of the reference vehicle).

One way to achieve this, is by looking at the re-spective shapes of both cumulative curves duringlight traffic conditions (e.g., the early morning pe-riod when free-flow conditions are prevailing). Theidea now is to shift one curve such that the dif-ference between the two curves’ shapes is minimal[86, 89, 118]. Note that other corrections may benecessary, as both detector stations can count a dif-ferent number of vehicles (i.e., a systematic bias).

An example of an oblique plot can be seen in the right partof Fig. 4: the cumulative count at each time instant canbe read from an axis that is perpendicular to the oblique(slanted) overlayed dashed lines (e.g., we can see a countof some 30000 vehicles at 14:00). Note that the accu-mulation can still be measured by the vertical distancebetween two curves (i.e., at a specific time instant), butthe travel time should now be measured along one of theoverlayed oblique lines. Such a pair of cumulative curvescan be thought of as a flexible plastic garden hose: when-ever there is an obstruction on the road, the outflow ofthe section will be blocked, resulting in a local thicken-ing of this ‘hose’ (i.e., the accumulation of vehicles onthe section).

Using these oblique cumulative plots, we can now inspectthe traffic dynamics in much more detail than was previ-

ously possible. For example, looking again at the rightpart of Fig. 4, we can see how the specific traffic streamcharacteristics propagate from one detector station to an-other. Even more visible, is a queue that starts to growat approximately 11:00 (i.e., the time of the appearanceof a ‘bulge’), dissipating at approximately 12:30. Asdata curves from upstream detectors lie above data curvesfrom downstream detectors, we see a decrease in the roadsection’s output. Careful investigation of the traffic datarevealed that the detector stations recorded a rather lowflow (approximately 2500 vehicles per hour as opposedto a nominal flow of 4500 vehicles per hour), whereby allvehicles drove at a low speed (between 20 and 60 km/has opposed to 110 km/h). This gives sufficient evidenceto conclude that an incident probably occurred shortly af-ter 11:00, consequently obstructing a part of the road andleading to a build up of vehicles in the section.

Let us finally note that although oblique cumulative plotscurrently are not a mainstream technique used by the traf-fic community, we predict their rising popularity: theyare one of the most simple, yet powerful, techniques forstudying local traffic phenomena, giving traffic engineerspractical insight into the formation of bottlenecks. Somerecent examples include the work of Munoz and Daganzo[85, 86, 87, 89], Cassidy and Bertini [7, 15], Cassidy andMauch [16], and Bertini et al. [8].

C. Occupancy

Notwithstanding the importance of measuring traffic den-sity, most of the existing detector stations on the road areonly capable of temporal measurements (i.e., regionRt).If individual vehicle speeds can be measured, by doubleinductive loop detectors (DLD) for example, then densityshould be computed using equation (6).

However, in many cases these vehicle speeds are notreadily available, e.g., when using single inductive loopdetectors. The detector’s logic therefore resorts to a tem-poral measurement called theoccupancyρ, which corre-sponds to the fraction of time the measurement locationwas occupied by a vehicle:

ρ =1

Tmp

N∑

i=1

oti. (19)

In the previous equation,otidenotes theith vehicle’son-

time, i.e., the time period during which it is present abovethe detector (it corresponds to the shaded area swept bya vehicle at a certain locationxi in Fig. 2). Note thatthis on-time actually corresponds to the effective vehiclelength as seen by the detector, divided by the vehicle’sspeed [20]:

oti=

li + Kld

vi

, (20)

with li the vehicle’s true length andKld > dx the finite,non-infinitesimal length of the detection zone. If we de-fineot as the average on-time (based on the vehicles thathave passed the detector during the observation period),then we can establish a relation between the occupancyand the flow [27] using equations (11) and (19):

ρ =

(N

Tmp

) (1N

N∑

i=1

oti

)= q ot. (21)

Furthermore, it is as before possible to define the oc-cupancy for generalised measurement regions, using thetotal space consumed by the shaded areas of vehiclesin a time-space diagram (e.g., Fig. 2), divided by thearea of the measurement region [14, 27, 36]. Continu-ing our discussion, assume that individual vehicle lengthsand speeds are uncorrelated; it can then be shown that[20, 27]:

ρ = l k =⇒ k =ρ

l, (22)

with l the average vehicle length (note that this can corre-spond to the concept of passenger car units defined in sec-tion III A). Multiplying equation (22) by 100, allows usto express the occupancy as a percentage. For multi-lanetraffic, the occupancy is derived in analogy to equation(7):

ρ =L∑

i=1

ρl =1

Tmp

L∑

l=1

Nl∑

i=1

oti,l, (23)

with nowoti,lthe on-time of theith vehicle in lanel. Note

that the total occupancy derived in this way, can exceed1 (but is bounded byL); if desired, it can be normalisedthrough a division byL to obtain theaverage occupancy.

Note that if we apply equation (22) to measurement re-gion Rs based on the density in equation (4), then theoccupancyρ can be written as:

ρ =

(1

��N

N∑

i=1

li

)��N

K=

1K

N∑

i=1

li. (24)

So the occupancy now represents the ‘real density’ of theroad, i.e., the physical space that all vehicles occupy.

In the past, density was sometimes referred to asconcentration. Nowadays however, concentration isused in a more broad context, encompassing bothdensity and occupancy whereby the former is meantto be a spatial measurement, as opposed to the latterwhich is considered to be a temporal measurement[39].

D. Mean speed

The final macroscopic characteristic to be considered, isthe mean speedof a traffic stream; it is expressed inkilometres (or miles) per hour (the inverse of a vehicle’sspeed is called itspace). Note that speed is not to beconfused withvelocity; the latter is actually a vector, im-plying a direction, whereas the former could be regardedas the norm of this vector.

1. Mathematical formulation

If we base our approach on direct measurements of theindividual vehicles’ speeds, we can generally obtain themean speed asthe total distance travelled by all the vehi-cles in the measurement region, divided by the total timespent in this region[27, 36]. This gives the followingderivations for the spatial and temporal regions,Rs andRt respectively:

vs =

N∑

i=1

Xi

N∑

i=1

Ti

=

N∑

i=1

vi ��dt

N ��dt=

1N

N∑

i=1

vi (regionRs),

N ��dxN∑

i=1

��dx

vi

=1

1N

N∑

i=1

1vi

(regionRt),

(25)

with nowXi andTi the distance, respectively time, trav-elled by theith vehicle andN the number of vehiclespresent during the measurement. The mean speed com-puted by the previous equations, is called theaveragetravel speed(the computation also includes stopped vehi-cles), which is more commonly known as thespace-meanspeed(SMS); we denote it withvs (note that in some en-gineering disciplines, the sole letteru is used to denote amean speed, however, this is ambiguous in our opinion).

It is interesting to see that the spatial measurement isbased on anarithmetic averageof the vehicles’instan-taneous speeds, whereas the temporal measurement isbased on theharmonic averageof the vehicles’spotspeeds. If we instead were to take the arithmetic averageof the vehicles’ spot speeds in the temporal measurementregionRt, this would lead to what is called thetime-meanspeed(TMS); we denote it byvt:

vt =1N

N∑

i=1

vi (regionRt). (26)

Similarly, we can compute the time-mean speed for mea-surement regionRs, by taking the harmonic average ofthe vehicles’ instantaneous speeds. With respect to both

space- and time-mean speeds, Wardrop has shown thatthe following relation holds [115]:

vt = vs +σ2

s

vs

, (27)

with σ2s the statistical sample variance defined as follows:

σ2s =

1N − 1

N∑

i=1

(vi − vs)2, (28)

in whichvi denotes theith vehicle’s instantaneous speed.One of the main consequences of equation (27), is that thetime-mean speed always exceeds the space-mean speed(except when all the vehicles’ speeds are the same, inwhich case the sample variance is zero and, as a conse-quence, the time- and space-mean speeds are equal). So astationary observer will most likely see more faster thanslower vehicles passing by, as opposed to e.g., an aerialphotograph in which more slower than faster vehicles willbe seen [27]. Despite this mathematical quirk, the practi-cal difference between SMS and TMS is often negligiblefor free-flow traffic (i.e., light traffic conditions); how-ever, under congested traffic conditions both mean speedswill behave substantially differently (i.e., around 10%).

Using equation (27), we can also estimate the space-meanspeed, based on the time-mean speed and approximatingthe variance of the SMS with that of the TMS [9]:

vs = vt −σ2

s

vs

,

≈ vt −σ2

t

vs

,

⇓

vs − vt ≈ −σ2

t

vs

,

v2s − vsvt ≈ −σ2

t ,

v2s − 2 vs

vt

2+

v2t

4≈

v2t

4− σ2

t ,

(vs −

vt

2

)2

≈v2

t

4− σ2

t ,

⇓

vs ≈vt

2+

√v2

t

4− σ2

t ∀ vt ≥ 2 σt.(29)

In general, using the space-mean speed is preferred to thetime-mean speed. However, in most cases only this lattertraffic flow characteristic is available, so care should betaken when interpreting the results of a study (unless ofcourse when SMS and TMS are negligibly different).

The extension of equation (25) to multi-lane is straight-forward; for example, the space-mean speed is computed

as follows:

vs =

L∑

l=1

Nl∑

i=1

vi,l

/L∑

l=1

Nl (regionRs),

1

1L∑

l=1

Nl

L∑

l=1

Nl∑

i=1

1vi,l

(regionRt), (30)

with now vi,l the instantaneous (or spot) speed of theith vehicle in lanel.

2. Fundamental relation of traffic flow theory

There exists a unique relation between three of the previ-ously discussed macroscopic traffic flow characteristicsdensityk, flow q, and space-mean speedvs [115]:

q = k vs. (31)

This relation is also called thefundamental relation oftraffic flow theory, as it provides a close bond betweenthe three quantities: knowing two of them allows us tocalculate the third one (note that the time-mean speed inequation (26) does not obey this relation). In generalhowever, there are two restrictions, i.e., the relation isonly valid for (1) continuous variables[124], or smoothapproximations of them, and (2) traffic composed of sub-streams (e.g., slow and fast vehicles) which comply to thefollowing two assumptions:

Homogeneous trafficThere is a homogeneous composition of thetraffic substream (i.e., the same type of vehi-cles).

Stationary trafficWhen observing the traffic substream at dif-ferent times and locations, it ‘looks the same’.Putting it a bit more quantitatively, all thevehicles’ trajectories should be parallel andequidistant [27]. A stationary time periodcan be seen in a cumulative plot (e.g., Fig. 4)where the curve corresponds to a linear func-tion.

The latter of the above two conditions, is also referredto as traffic operating in asteady stateor atequilibrium.Based on equations (5) and (12) using partial densitiesand flows for different substreams (e.g., vehicle classeswith distinct travel speeds, macroscopic characteristicsof different lanes, . . . ), we can now calculate the space-mean speed, using relation (31), in the following equiva-lent ways:

vs = q / k,

=

C∑

c=1

qc

/C∑

c=1

kc, (32)

=

C∑

c=1

qc

/C∑

c=1

qc

vsc

, (33)

=

C∑

c=1

kc vsc

/C∑

c=1

kc, (34)

in whichC denotes the number of substreams,qc, kc, vsc,

andvtcthe flow, density, space, and time-mean speed re-

spectively of thec-th substream. In the above derivations,equation (32) should be used when both the flows anddensities are known, equation (33) should be used whenboth the flows and space-mean speeds are known, andequation (34) should be used when both the densities andspace-mean speeds are known.

As can be seen in equation (34), the space-mean speedis calculated by averaging the substreams’ space-meanspeeds using their densities as weighting factors. Simi-larly, the time-mean speed can be derived by using theflows as weighting factors for the substreams’ time-meanspeeds:

vt =

C∑

c=1

qc vtc

/C∑

c=1

qc, (35)

Because density can not always be easily measured, wecan compute it using the fundamental relation (31). Den-sity can then be directly derived from flow and space-mean speed measurements, or if the latter are not avail-able, they can be estimated from occupancy measure-ments; in [20, 21, 22], Coifman provides a nice set oftechniques for dealing with these difficulties.

E. Moving observer method and floating car data

When measuring and/or computing the macroscopic traf-fic flow characteristics in the previous sections, we al-ways assumed a fixed measurement region. There existshowever yet another method, based on what is called amoving observer[116]. The idea behind the techniqueis to have a vehicle drive in both directions of a trafficflow, each time recording the number of oncoming vehi-cles and the net number of vehicles it gets overtaken by,as well as the times necessary to complete the two trips.Note that the assumption of stationary traffic still has tohold, i.e., the round trip should be completed before traf-fic conditions change significantly.

Using this method, it is then possible to derive the flowand density of the traffic stream in the direction of interest[27, 39]. However, the main disadvantage of this methodis that, in order to obtain an acceptable level of accuracy

on a road with a low flow, a very large number of trips arerequired [39, 84, 116].

One of the techniques that has entered the picture dur-ing the last decade, is the use of so-calledfloating carsor probe vehicles. They can be compared to the mov-ing observer method, but in this case, the vehicles areequipped with GPS and GSM(C)/GPRS devices that de-termine their locations based on the USA’s NAVSTAR-GPS (or Europe’s planned GNSS Galileo), and transmitthis information to some operator. Initially, this allows anagency, e.g., a parcel delivery service, to track its vehiclesthroughout a network, based on their locations. Nowa-days, the technique has evolved, resulting in several com-pleted field tests of which the main goal was to estimatethe traffic conditions based on a small number of probevehicles. During field measurements, floating cars canmimic several types of behaviour, most notably by travel-ling at the traffic flows’ mean speed, or by trying to travelat the road’s speed limit, or even by chasing another ran-domly selected vehicle from the traffic stream.

Some examples of studies and experiments with floatingcar data (FCD) are given in the following. Firstly, Fasten-rath gives an overview of a telematic field trial (VEhicleRelayed Dynamic Information, VERDI) that addressesissues such as economical, political, and technical con-straints [38]. Secondly, Westerman provides an overviewof available techniques for obtaining real-time road traf-fic information, with the goal of controlling the trafficflows through telematics [118], and Wermuth et al. de-scribe a ‘TeleTravel System’ used for surveying individ-ual travel behaviour [117]. Then, Taale et al. comparetravel times from floating car data with measured traveltimes (using a fleet of sixty equipped vehicles drivingaround in Rotterdam, The Netherlands), concluding thatthey correspond reasonably well [107]. Next, Michlerderives the minimum percentage of vehicles necessary,in order to estimate traffic stream characteristics for cer-tain traffic patterns (e.g., free-flow and congested traffic)based on rigid statistical grounds [81], and Linauer andLeihs measure the travel time between points in a roadnetwork, based on a high number of users that submit alow number of GSM hand-over messages [73]. In addi-tion, Demir et al. accurately reconstruct link travel timesduring periods of traffic congestion, using only a verylimited number of FCD-messages with a small numberof users [33]. A final, more regional, example is thefounding of the government-supported Belgian ‘Telem-atics Cluster’[125], a platform for encouraging the useof telematics solutions for ITS. The initiative already in-cludes some 57 members, stimulating the cooperation be-tween users, telecommunication companies, and the au-tomotive industry.

In conclusion, we can state that the use of probe vehiclesprovides an effective way to gather accurate currenttravel times in a road network, thereby allowing good up-to-date estimations of traffic conditions. The techniquewill continue to grow and evolve, already by introducingpersonalised traffic information to drivers, based on theirlocation and the surrounding traffic conditions. Thisdevelopment is furthermore stimulated by the fact thatGSM market penetration still rises above 70% [73], and

it is our belief this will also be the case for personal GPSdevices and in-vehicle route planners.

Despite the obvious major advantage of obtainingaccurate information on the traffic conditions, thetechnique suffers from a jurisprudential battle, in thatthere are many delicately privacy concerns involvedwith respect to the mobile operator that wants totrack individual people’s units (not to mention themonetary cost associated with the numerously in-duced communications).

IV. PERFORMANCE INDICATORS

After considering the previously mentioned macroscopictraffic flow characteristics, we now take a look at somepopular performance indicators used by traffic engineerswhen assessing the quality of traffic operations. We con-cisely discuss the peak hour factor, the reliability of traveltimes, the levels of service, and a measure of efficiencyof a road. For a more complete overview, we refer thereader to [105].

A. Peak hour factor

During high flow periods in the peak hour, a possible in-dicator for traffic flow fluctuations is the so-calledpeakhour factor(PHF). It is calculated for one day as the aver-age flow during the hour with the maximum flow, dividedby the peak flow rate during one quarter hour within thishour [79]:

PHF=q|60

q|15. (36)

For example, suppose we measure flows on a main unidi-rectional road with three lanes, during a morning peak:from 07:00 to 08:00 we measure consecutively 3500,6600, 6200, and 4500 vehicles/hour during each quarter.The total average flowq|60 is 5200 vehicles/hour, with apeak 15 minute flow rateq|15 = 6600 vehicles/hour. The

PHF is therefore equal to 5200/6600= 0.78.

Note that some manuals express the peak 15 minute flowrate as the number of vehicles during that quarter hour,necessitating an extra multiplication by 4 in the denomi-nator of equation (36) to convert the flow rate to an hourlyrate.

We can immediately see that the PHF is constrained tothe interval [0.25,1.00]; the higher the PHF, the flatter thepeak period (i.e., a longer sustained state of high flow).Typically, the PHF has values around 0.7 – 0.98. Notethat two of the obvious problems with the PHF are, onthe one hand, the question of when to pick the correct 15minute interval, and on the other hand the fact that somepeak periods may last longer than one hour.

B. Travel times and their reliability

When travelling around, people like to know how long aspecific journey will take (e.g., by public transport, car,bicycle, . . . ). This notion of an expected travel time, isone of the most tangible aspects of journeying as per-ceived by the travellers. When people are travelling totheir work, they are required to arrive on time at their des-tinations. Based on this premise, we can naturally statethat people reason with a built-in safety margin: they con-sider theaverage timeit takes to reach a destination, anduse this to decide about their departure time.

Aside from the above obvious human rationale, there isalso an increased interest in obtaining precise informa-tion with respect to travel times in the context ofad-vanced traveller information systems(ATIS). Here, anessential ingredient is the accurate prediction of futuretravel times. Coupled with incident detection for exam-ple, drivers can obtain correct travel time information,thereby staying informed of the actual traffic conditionsand possibly changing their journey. The requested in-formation can reach the driver by means of a cell-phone(e.g., as a feature offered by the mobile service provider),it can be broadcasted over radio (e.g., theTraffic MessageChannel– TMC), or it can be displayed usingvariablemessage signs(VMS) above certain road sections (e.g.,dynamic route information panels– DRIPs), . . .

1. Travel time definitions

The travel time of a driver completing a journey, can bedefined as ‘the time necessary to traverse a route betweenany two points of interest’ [111]. In this context, theex-perienced dynamic travel time, starting at a certain timet0, over a road section of lengthK is defined as follows[9]:

T (t0) =

∫ K

0

1v(t, x)

dx ∀ t ≥ t0, (37)

for which it is assumed that all localinstantaneousve-hicle speedsv(t, x) are known at all points along theroute, and at all time instants (hence the termdynamictravel time). In most cases however, we do not know allthe v(t, x), but only a finite subset of them, defined bythe locations of the detector stations (demarcating sectionboundaries). The travel time can then be approximatedusing the recorded speeds at the beginning and end of asection (there is an underlying assumption here, namelythat vehicles travel at a more or less constant speed be-tween detector locations). As stated earlier, the experi-enced travel time requires the knowledge of local vehiclespeeds atall time instants afterT0. Because this is notalways possible, a simplification can be used, resulting inthe so-calledexperienced instantaneous travel time:

T (t0) =

∫ K

0

1v(t0, x)

dx, (38)

In general, we can derive the travel time using equation(25), i.e., the total distance travelled by all the vehicles,divided by their space-mean speed:

T (t0) =K

vs(t0), (39)

in which an accurate estimation of the space-mean speedvs(t0) at timet0 is necessary (e.g., by taking the harmonicaverage of the recorded spot speeds).

2. Queueing delays

Traffic congestion nearly always leads to the build up ofqueues, introducing an increase (i.e., thedelay) in theexperienced travel time. The congestion itself can haveoriginated due to traffic demand exceeding the capacity,or because an incident occurred (e.g., road works, a traf-fic accident, . . . )[126]. This can createincidental(non-recurrent) orstructural (recurrent) congestion. Conges-tion can thus be seen as a loss in travel time with respectto some base line reference. Two such commonly usedreferences are the travel time under free-flow conditions,and the travel time under maximum (i.e., capacity) flow.The delay is typically expressed in vehicle hours. Asstated earlier, there are several ways to inform a driverof the current and predicted travel time. Using DRIPs itis possible to advertise the extra travel time (the delay isnow typically expressed in vehicle minutes), as well asqueue lengths. We note that in our opinion it is more in-tuitive to advertise a temporal estimation (i.e., the traveltime or the delay), than a spatial estimation (e.g., thequeue length on a motorway).

3. An example of travel time estimation using cumulativeplots

There exist several techniques for estimating the cur-rent travel time; one method for directly ‘measuring’ thetravel time, is by using a probe vehicle (we refer thereader to section III E for more details). This way, itis possible to extract actual travel times from a trafficstream. Note that as traffic conditions get more con-gested, more probe vehicles are required in order to ob-tain an accurate estimation of the travel time.

Another method for measuring the travel time, is basedon historical data, namely cumulative plots (introducedin section III B 2). As mentioned earlier, the travel timecan then be measured as the distance along the horizontal(or oblique) time axis; any excess due to delays can theneasily be spotted on a set of oblique cumulative plots.

Based on cumulative plots of consecutive detector sta-tions, we can calculate the travel time between the up-stream and downstream end of a road section. To illus-trate this, let us reconsider the cumulative curves shownin Fig. 4 of section III B 2. The evolution of the travel

time during the day for these curves, is depicted in thetop part of Fig. 5. The derived histogram (indicative ofthe underlying travel time probability density function),in the bottom part of the figure, shows that the mean traveltime during the day is approximately 4 minutes.

08:00 09:30 11:00 12:30 14:00 15:30 17:00 18:30 20:000

2

4

6

8

10

Time

Tra

vel t

ime

[min

utes

]

3 3.5 4 4.5 5 5.5 6 6.5 7 7.50

0.1

0.2

0.3

0.4

Travel time [minutes]

Nor

mal

ised

trav

el ti

me

freq

uenc

y

incident slower traffic

FIG. 5: Top: The evolution of the travel time during one day,based on the cumulative plots from section III B 2. As can beseen, an incident likely occurred at 11:00, increasing the traveltime from 4 to 7 minutes. Furthermore, at approximately 18:45in the evening, all traffic seemed to simultaneously slow downfor a period of some 10 minutes.Bottom: Based on the calcu-lated travel times during the day, we can derive a histogram thatis an approximation of the underlying travel time probabilitydensity function. The mean is located around 4 minutes.

We already mentioned the likely occurrence of an inci-dent at 11:00, resulting in the formation of a queue. Dur-ing this period, the travel time shot up, reaching first 5,then 7 minutes. Looking at the top part of Fig. 5, we fur-thermore notice a slight increase in the travel time at ap-proximately 18:45, for a short period of some 10 minutes.Investigation of the detector data, revealed that the flowremained constant at about 4500 vehicles per hour, butthe speed dropped to some 90 km/h (as opposed to 110km/h); we can conclude that all vehicles were probablysimultaneously slowing down during this period (perhapsa rubbernecking effect). Another possibility is a platoonof slower moving vehicles, but then it would seem to havedissipated rather quickly after 10 minutes.

Using ample historical data, we can analyse the traveltime over a period of many weeks, months, or even years.This would allow us to makeintuitivestatements such as:

“The typical travel time over this section ofthe road during a working Monday, lies ap-proximately between 4 and 6 minutes. Thereis however an 8% probability that the traveltime increases to some 22 minutes (e.g., dueto an occurring incident).”

Finally note that, besides the two previously mentionedtechniques for estimating travel times, an extensiveoverview can be found in theTravel Time Data CollectionHandbook[111]. Another concise but more theoretically-oriented overview is provided by Bovy and Thijs [9].

4. Reliability and robustness properties

As mentioned in the introduction of this section, peoplereason about their expected travel times based on a built-in safety margin. Central to this is the concept of theaverage travel time. Thereliability of such a travel timeis then characterised by itsstandard deviation. Driverstypically accept (and sometimes expect) a small delay intheir expected travel time. A traveller knows theexpectedtravel timebecause of the familiarity with the associatedtrip. To the traveller, this is personal historical informa-tion, for instance obtained by learning the trip’s details(e.g., the traffic conditions during a typical morning rushhour) [5].

Directly linked to the reliability of a certain expectedtravel time, is its variability. They are said to be unre-liable when both expected and experienced travel timesdiffer sufficiently. A typical characterisation of reliabilityinvolves the mean and standard deviation (i.e., the vari-ance, which is a measure of variability) of a travel timedistribution [19]. An example of such a travel time distri-bution for one day is shown in the histogram in the bot-tom part of Fig. 5.

Both first- and second-order measures of a distributionare by themselves insufficient to capture the completepicture. In order to grasp the notion of the previouslymentioned safety margin, another typical statistical mea-sure is considered, namely the90th percentile. The ra-tionale behind the use of this percentile is that travellersadopt a certain ‘safe’ threshold with respect to their ex-pected journey times. Considering the 90th percentile,this means that only one out of ten times the experi-enced travel time will differ significantly from the ex-pected travel time. Travel time reliability can thus beviewed upon as a measure of service quality (similar tothe concept of ‘quality of service’ (QoS) in telecommu-nications).

There has been some research into the analytic form oftravel time distributions (e.g., the work of Arroyo andKornhauser, concluding that a lognormal distributionseems the most appropriate [4]). There exist howeversignificant differences between travel time distributions:in general, a smaller standard deviation indicates a betterservice quality and reliability. In contrast to this, a largestandard deviation is indicative of chaotic behaviourof the traffic flow, the latter being totally unstable.Furthermore, travel time distributions can have a longtail; this signifies seldom events (e.g., incidents), that canhave significant repercussions on the quality of trafficoperations.

Let us finally note that there is an increased inter-est in thereliability of complete transportation net-works, and theirrobustnessagainst incidents. Tothis end, Immers et al. consider reliability as a user-oriented quality, whereas robustness is more a prop-erty of the system itself [51]. Among several charac-terising factors for robustness of transportation sys-tems, they also introduce the following practical no-tions in this context:redundancy, denoting a sparecapacity, andresilience, which is the ability to re-peatedly recover from a temporary overload. Theirconclusion is that the key element in securing trans-portation reliability lies in a good network design.

C. Level of service

Historically, one of the main performance indicators toassess the quality of traffic operations, was thelevel ofservice(LOS), introduced in the 1960s. It is representedas a grading system using one of six letters (A – F),whereby LOS A denotes the best operating conditionsand LOS F the worst. These LOS measures are basedon road characteristics such as speed, travel time, . . . ,and drivers’ perceptions of comfort, convenience, . . . [1].As is customary among traffic engineers, these represen-tative statistics of these characteristics are collectivelycalledmeasures of effectiveness(MOE).

Levels A through D are representative for free-flow con-ditions whereby LOS A corresponds to free flow, LOS Bto reasonable free flow, LOS C to stable traffic operations,and LOS D to bordering unstable traffic operations. LOSE is reminiscent of near-capacity flow conditions that areextremely unstable, whereas LOS F corresponds to con-gested flow conditions (caused by either structural or in-cidental congestion) [79].

As an example, we provide an overview of the differ-ent levels of service in Table I (based on [79], in sim-ilar form originally published in theHighway CapacityManual (HCM) of 1985 as theTransportation ResearchBoard’s(TRB)[127] special report #209.

LOS Density (veh/km)Occupancy (%)Speed (km/h)

A 0→ 7 0→ 5 ≥ 97B 7→ 12 5→ 8 ≥ 92C 12→ 19 8→ 12 ≥ 87D 19→ 26 12→ 17 ≥ 74E 26→ 42 17→ 28 ≥ 48F 42→ 62 28→ 42 < 48

> 62 > 42

TABLE I: Level of service (LOS) indicators for a motorway(adapted from [79], in similar form originally published inthe1985 HCM).

Calculating levels of service can be done using a mul-titude of methods; some examples include using thedensity (at motorways), using the space-mean speed(at arterial streets), using the delay (at signallised andunsignallised intersections), . . . [1]. The distinction

between different LOS is primarily based on the mea-sured average speed, and secondly on the density (oroccupancy). Furthermore, as traditional analyses onlyfocus on a select number of hours, a new trend is to con-duct whole year analyses(WYA) based on aggregatedmeasurements such as e.g., themonthly average dailytraffic (MADT) and the annual average daily traffic(AADT) [10]. The MADT is calculated as the averageamount of traffic recorded during each day of the week,averaged over all days within a month. Averaging theresulting twelve MADTs gives the AADT.

Regarding the use of the LOS, we note that it is arather old-fashioned method for evaluating the qual-ity of traffic operations. In general, it is difficultto calculate, mainly because the defined standardsat which the different levels are set, always dependon the specific type of traffic situation that is studied(e.g., type of road, . . . ). This makes the LOS more ofan engineering tool, used when assessing and plan-ning operational analyses. Instead of using the LOS,we therefore propose to adopt the more suited ap-proach based on oblique cumulative plots (we referthe reader to section III B 2). These allow for exam-ple to assess the differences between travel times un-der free-flow and congested conditions, thereby giv-ing a more meaningful and intuitive indication of thequality of traffic operations to the drivers.

D. Efficiency

In [18], Chen et al. state that the main reason for conges-tion is not demand exceeding capacity (i.e., the numberof travellers whowant to use a certain part of the trans-portation network, exceeds the available infrastructure’scapacity), but is in fact the inefficient operation of motor-ways during periods of high demand. In order to quan-tify this efficiency, they first look at what the prevailingspeed is when a motorway is operating at its maximumefficiency, i.e., the highest flow (corresponding to the ef-fective capacity, which is different from the HCM’s ca-pacity which is calculated from the road’s physical char-acteristics). Based on the distribution of 5-minute datasamples from some 3300 detectors, they investigate thespeed during periods of very high flows. This leads themto a so-called sustained speedvsust = 60 miles per hour(which corresponds to 60 mi/h× 1.609≈ 97 km/h).

The performance indicator they propose, is called theef-ficiencyη and it based on the ratio of thetotal vehiclemiles travelled(VMT), divided by thetotal vehicle hourstravelled(VHT). Note that as the units of VMT andvsust

should correspond to each other, we propose to use theterminology oftotal vehicle distance travelled(VDT) in-stead of the VMT, in order to eliminate possible confu-sion. Both VDT and VHT are defined as follows:

VDT = q K, (40)

VHT =VDTvs

, (41)

with, as before,q the flow,K the length of the road sec-tion, andvs the space-mean speed. Using the above defi-nitions, we can write the efficiency of a road section as:

η =VDT/VHT

vsust. (42)

The efficiency is expressed as a percentage, and it canrise above 100% when the recorded average speeds sur-pass the sustained speedvsust. In general, the discussedefficiency can also easily be calculated for a completeroad network and an arbitrary time period. It canfurthermore be seen as the ratio of the actual productivityof a road section (theoutput produced by this sectionduring one hour), to its maximum possible production(the input to the section) under high flow conditions.

Note that as a solution to their original claim (“con-gestion arises due to inefficient operation”), Chenet al. propose to increase the operational efficiency,mainly through the technique of suitable ramp meter-ing (using an idealised ramp metering control prac-tice that maintains the occupancy downstream of anon ramp to its critical level). But in our opinion,they neglect to take into account the entire situa-tion, i.e., they fail to consider the extra effects in-duced by holding vehicles back at some on ramps(e.g., the total time travelled byall the vehicles, in-cluding delays), thus rendering their statement prac-tically worthless by giving a feeble argument. Care-ful examination of their reasoning, reveals that theseextra effects are dealt with by shifting demand duringthe peak periods. . . but this just confirms our hypoth-esis that congestion occurs when demand exceedscapacity, even when this capacity is for example con-trolled through ramp metering !

In contrast to the work of Chen et al., Brilon proposesanother definition for theefficiency(now denoted asE):it is expressed as the number of vehicle kilometres thatare produced by a motorway section per unit of time [10]:

E = q vs Tmp, (43)

with now q the total flow recorded during the time in-tervalTmp. Brilon concludes that in order for motorwaysto operate at maximum efficiency, their hourly flows typi-cally have to remainbelowthe capacity flow (e.g., at 90%of qcap). Brilon also proposes to use this point of maxi-mum efficiency as the threshold when going from LOS Dto LOS E.

V. FUNDAMENTAL DIAGRAMS

Whereas the previous sections dealt with individual traf-fic flow characteristics, this section discusses some of therelations between them. We first give some characterisa-tions of different traffic flow conditions and the rudimen-tary transitions between them, followed by a discussionof the relations (which are expressed as fundamental di-agrams) between the traffic flow characteristics, givingspecial attention to the different points of view adoptedby traffic engineers.

A. Traffic flow regimes

Considering a stream of traffic flow, we can distin-guish different types of operational characteristics, calledregimes(two other commonly used terms are traffic flowphasesandstates). As each of these regimes is charac-terised by a certain set of unique properties, classificationof them is sometimes based on occupancy measurements(see for example the discussion about levels of service insection IV C), or it is based on combinations of differentmacroscopic traffic flow characteristics (e.g., the work ofKerner [55]).

In the following sections, we discuss the regimes knownasfree-flowtraffic, capacity-flowtraffic, congested, stop-and-go, and jammed traffic. Our discussion of theseregimes is in fact based on the commonly adopted wayof looking at traffic flows, as opposed to for exampleKerner’s three-phase traffic theory that includes a regimeknown assynchronisedtraffic (we refer the reader to sec-tion V D for more details). We conclude the section witha note on the transitions that occur from one regime toanother.

1. Free-flow traffic

Under light traffic conditions, vehicles are able to freelytravel at their desired speed. As they are largely unim-peded by other vehicles, drivers strive to attain their owncomfortable travelling speed (we assume that in case avehicle encounters a slower moving vehicle ahead, it caneasily change lanes in order to overtake the slower vehi-cle). Notwithstanding this ability for unconstrained trav-elling, drivers have to take into account the maximum al-lowed speed (denoted byvmax), as well as road-, engine-,and other vehicle characteristics. Note that in some cases,depending on the country under scrutiny, drivers performspeeding.

In essence, the previous description offree-flow trafficconsiders a traffic flow to be unrestricted, i.e., no sig-nificant delays are introduced due to possible overtakingmanoeuvres. As a consequence, thefree-flow speed(bysome called thenominal speed) is the mean speed of allvehicles, travelling at their own pace (e.g., 100 km/h); itis denoted byvff .

Free-flow traffic occurs exclusively at low densities, im-plying large average space headways according to equa-tion (10). As a result, small local disturbances in the tem-poral and spatial patterns of the traffic stream have nosignificant effects, hence traffic flow isstablein the free-flow regime.

2. Capacity-flow traffic

When the traffic density increases, vehicles are drivingcloser to each other. Considering the number of vehiclesthat pass a certain location alongside the road, anobserver will notice an increase in the flow. At a certainmoment, the flow will reach a maximum value (which isdetermined by the mean speed of the traffic stream andthe current density). This maximum flow is called thecapacity flow, denoted byqc, qcap, or evenqmax. A typicalvalue for the capacity flow on a three-lane Belgianmotorway with vmax equal to 120 km/h, can reach amaximum of some 7000 vehicles [113]. According toequation (17), the average time headway is minimal atcapacity-flow traffic, indicating the (local) formation oftightly packed clusters of vehicles (i.e., platoons), whichare moving at a certaincapacity-flow speedvc (or vcap)which is normally a bit lower than the free-flow speed.Note that some of these fast platoons are very unstablewhen they are composed of tail-gating vehicles: when-ever in such a string a vehicle slows down a little, it canhave a cascading effect, leading to exaggerate braking offollowing vehicles. Hence, these latter manoeuvres candestroy the local state of capacity-flow, and can in theworst case lead to multiple rear-end collisions. At thispoint, traffic becomesunstable.

The calculation of the capacity flow is a dauntingtask, holding traffic engineers occupied for the lastsix decades. The fact of the matter is that there ex-ists no rigourous definition for the concept of ‘capac-ity’. As a result, after many years of research, thisculminated in the publication of the fourth editionof the already previously mentionedHighway Ca-pacity Manual. It contains an impressive overview,spanning methodologies for assessing the capacity atspecific types of road infrastructures (motorway fa-cilities, weaving sections, on- and off-ramps, signal-lised and unsignallised urban intersections, . . . ) [1].

3. Congested, stop-and-go, and jammed traffic

Considering the regime of capacity-flow traffic, it is rea-sonable to assume that drivers are more mentally awareand alert in this regime, as they have to adapt their drivingstyle to the smaller space and time headways under highspeeds. However, when more vehicles are present, thedensity is increased even further, allowing a sufficientlylarge disturbance to take place. For example, a driverwith too small space and time headways, will have tobrake in order to avoid a collision with the leader directly

in front; this can lead to a local chain of reactions thatdisrupts the traffic stream and triggers a breakdown of theflow. The resulting state of saturated traffic conditions, iscalledcongested traffic. The moderately high density atwhich this breakdown occurs, is called thecritical den-sity, and is denoted bykc or kcrit (for a typical motorway,its value lies around 25 vehicles (PCUs) per kilometreper lane, [113]). From this knowledge, we can derivethe optimal driving speed for single-lane traffic flows asvs = qcap/kcrit = 2000÷ 25 =≈ 85 km/h.

Higher values for the density indicate almost always aworsening of the traffic conditions; congested traffic canresult instop-and-go traffic, whereby vehicles encounterso-calledstop-and-go waves. These waves require themto slow down severely, or even stop completely. Whentraffic becomes motionless, the space headway reaches aminimum as all vehicles are standing bumper-to-bumper;this extreme state is calledjammed traffic. Clearly, thereexists a maximum density at which the traffic seems toturn into a ‘parking lot’, called thejam densityand it isdenoted bykj , kjam, or kmax. For a typical motorway, itsvalue lies around 140 vehicles (PCUs) per kilometre perlane [113].

Note that the jam density is typically expressed in ve-hicles per kilometre. As already stated in the intro-duction of section III A, density ignores the effectsof traffic composition and vehicle lengths. For a typ-ical value of some 140 vehicles/km/lane for the jamdensity, this means that we express the density by us-ing passenger car units (see section III A 2 for moredetails). Suppose now for example that an averagetrailer truck equals 4.5 PCUs, then the jam densitywould decrease to some 140÷ 4.5≈ 31 trucks forthis class of vehicles.As a consequence, the value ofthe jam density is different for each vehicle class.

4. A note on the transitions between different regimes

Streams of traffic flows can be regarded as many-particlesystems (e.g., gasses, magnetic spin systems, . . . ); asthey have a large number of degrees of freedom, it isoften intractable when it comes to solving them exactly.However, from a physical point of view, these systemscan be described in the framework ofstatistical physics,whereby the collective behaviour of their constituents isapproximately treated using statistical techniques.

Within this context, the changeover from one trafficregime to another, can be looked upon as aphase tran-sition. Within thermodynamics and statistical physics, anorder parameteris often used to describe the phase tran-sition: when the system shifts from one phase to another(e.g., at acritical point for liquid-gas transitions), the or-der parameter expresses a different qualitative behaviour.Two examples of such an order parameter that is appli-cable to traffic flows, can be found in Schadschneider etal. who considered nearest neighbour correlations [104],and in Jost and Nagel who devised a measure of inhomo-geneity [53] (we refer the reader to our work in [78] for

an example in which they are used and compared whentracking phase transitions).

There exists a difference in which a phase transition canexpress itself. This difference is designated by the orderof the transition; generally speaking, the two most com-mon phase transitions arefirst-order and second-ordertransitions. According toEhrenfest’s classification, first-order transitions have an abrupt, discontinuous change inthe order parameter that characterises the transition. Incontrast to this, the changeover to the new phase occurssmoothly for second-order transitions [71, 119]. Notethat higher-order phase transitions also exist, e.g., in su-perconducting materials [23].

With respect to the description of regimes in traffic flows,it is commonly agreed that there exists a first-order phasetransition when going from the capacity-flow to the con-gested regime. The point at which this transition occurs,is the critical density. Studying the phase transitions en-countered in fluid dynamics, there exists a transition fromthelaminarflow (i.e., a fluid flowing in layers, each mov-ing at a different velocity) to theturbulentflow (i.e., thedisturbed random and unorganised state in which vorticesform). However, the transition here is triggered by an in-crease in the velocity of the fluid, as opposed to the tran-sition in traffic flows where a change in the density canlead to a cascading instability. In this respect, the anal-ogy for traffic flows holds better when comparing them togas-liquid transitions. Here free-flow traffic correspondsto a gaseous phase, in which particles are evenly spreadout in the system. At the point of the phase transition,liquid droplets will form, coagulating together into big-ger droplets. This leads to a state where both gaseous andliquid phases coexist, typically in the form of a big liq-uid droplet surrounded by gas particles. For even higherdensities, particles are so close to each other, and the onlyremaining state is the liquid phase [37, 52, 53, 54, 68, 93].

In conclusion, we refer the reader to the work of Tampere,where an excellent overview is given, detailing the differ-ent traffic flow regimes, their transitions, and mechanismswith respect to jamming behaviour [108].

B. Correlations between traffic flow characteristics

Whereas the previous sections all treated the macroscopictraffic flow characteristics on an individual basis, this sec-tion considers some of the relations between them. Westart our discussion with a look at the historic origin offundamental diagrams, after which we shed some light onthe different classical approaches. The section concludeswith some considerations with respect to empirical mea-surements.

1. The historic origin of the fundamental diagram

As in many scientific disciplines, the resulting statementsand theories are often preceded by an investigation of ob-tained experimental data, which serves as empirical ev-

idence for them. In this line of reasoning, Greenshieldswas among the first to provide — as far back as 1935 —a basis for most of the classic work on, what are called,empirical fundamental diagrams. In his seminal paper,he sketched alinear relation between the density and themean speed, based on empirically obtained data [44]:

v = vff

(1−

k

kj

). (44)

As can be seen from Greenshields’ relation, whenincreasing the density from zero to the jam densitykj ,the mean speed will monotonically decrease from thefree-flow speedvff to zero (note that we dropped the ‘s’or ‘t’ subscript from the mean speed, as it is not surewhether or not Greenshields used space- or time-meanspeed, respectively). The relation can be understoodintuitively, by assuming that drivers will tend to slowdown in crowded traffic, because this naturally givesthem more time to react to changes (e.g., sudden brakingof the lead vehicle). As it is reasonable to assume that themean speed remains unaffected for very low densities,Greenshields furthermore flattened the upper-left partof the regression line (corresponding to the free-flowspeed), although this effect is not incorporated in equa-tion (44).

0 20 40 60 80 100 120 140 160 180 2000

10

20

30

40

50

60

70

80

Density [vehicles/km]

Mea

n sp

eed

[km

/h]

FIG. 6: Greenshields’ original linear relation between theden-sity and the mean speed. Note that the regression line is basedon only seven measurements points, and that artificial flatteningof its upper-left part (figure based on [72] and [39]).

Although Greenshields’ derivation of the linear relationbetween density and space-mean speed appears elegantand simple, it should nevertheless be taken with a grainof salt. The fact of the matter is that his hypothesis is, ascan be seen in Fig. 6, based ononly seven measurementpoints, which comprise aerial observations taken onSeptember 3rd (Monday, Labor Day), 1934 [79]. One ofthe problems is that these observations arenot indepen-dent. An even more serious problem is that six of theseobservations were obtained for free-flow conditions,whereas the one single point that indicates congested

conditions, was obtained at an entirelydifferent road, ona different day[39] !

Some twenty years later, Lighthill and Whitham devel-oped a theory that describes the traffic flows on longcrowded roads using a first-order fluid-dynamic model[72]. As one of the main ingredients in their theory, theypostulated the following fundamental hypothesis:“at anypoint of the road, the flowq is a function of the den-sity k” . They called this function theflow-concentrationcurve(recall from section III C that density in the past gotsometimes referred to as concentration).

Continuing their reasoning, Lighthill and Whitham thenreferred to Greenshields’ earlier work, relating the space-mean speed to the density, and, by means of equation(31), thus relating the flow to the density. Theexistenceof the concept of the flow-concentration curve mentionedabove, was justified on the grounds that it describes traf-fic operating under steady-state conditions, i.e., homoge-neous and stationary traffic as explained in section III D 2.In this context, the flow-concentration curve therefore de-scribes theaverage characteristicsof a traffic flow. SoGreenshields first fitted a regression line to scarce data,after which his functional form seemed to be taken forgranted for the following seventy years. The key aspectin Lighthill and Whitham’s (and also Richards’ [102])approach, lay in the fact that they broadened the flow-concentration curve’s validity, including also conditionsof non-stationary traffic. They also stated that, becauseof e.g., changes in the traffic composition, the curve canvary from day to day, or even within a day (e.g., rushhours, . . . ). The same statement holds also true whenconsidering the flow-concentration curves of different ve-hicle classes (e.g., cars and trucks).

The term fundamental diagramitself, is historicallybased on Lighthill and Whitham’sfundamentalhy-pothesis of the existence of such aone-dimensionalflow-concentration curve. As traffic engineers grewaccustomed to the graphical representation of this curve,they started talking about thediagramthat represents it,i.e., the ’fundamental diagram’ [45].

In its original form, the fundamental diagram repre-sents anequilibrium relationbetween flow and den-sity, denoted byqe(k). But note that, because of thefundamental relation of traffic flow theory (see sec-tion III D 2), is it equally justified to talk about thevse

(k) or the vse(q) fundamental diagrams. Due

to this equilibrium property, the traffic states (i.e.,the density, flow, and space-mean speed) can bethought of as ‘moving’ over the fundamental dia-grams’ curves.

2. The general shape of a fundamental diagram

We now give an overview of some of the qualitativefeatures of the different possible fundamental diagrams,representing the equilibrium relations between density,

space-mean speed, and average space headway, and flow.Note that in each example, we consider apossiblefun-damental diagram, as they can take on many (functional)shapes.

Space-mean speed versus densityWe start our discussion based on the equilibrium relationbetween space-mean speed and density, i.e., thevse

(k)fundamental diagram. The main reason for starting here,is the fact that this diagram is the easiest to understand in-tuitively. Complementary to the example of Greenshieldsin Fig. 6, we give a small overview of its most prominentfeatures:

• the density is restricted between 0 and the maxi-mum density, i.e., the jam densitykj ,

• the space-mean speed is restricted between 0 andthe maximum average speed, i.e., the free-flowspeedvff ,

• as density increases, the space-mean speed mono-tonically decreases,

• there exists a small range of low densities, in whichthe space-mean speed remains unaffected and cor-responds more or less to the free-flow speed,

• and finally, the flow (equal to density times space-mean speed), can be derived as the area demarcatedby a rectangle who’s lower-left and upper-right cor-ners are the origin and a point on the fundamentaldiagram, respectively.

Space-mean speed versus average space headwayMicroscopic and macroscopic traffic flow characteristicsare related to each other by means of equations (10) and(17). According to the former, densityk is inversely pro-portional to the average space headwayhs. We can there-fore derive a fundamental diagram, similar to the pre-vious one, by substituting the density with the averagespace headway. As as result, the abscissa gets ‘inverted’,resulting in the fundamental diagram as shown in Fig. 7.

0

vs

vff

k−1∝ hsk−1

j k−1c

FIG. 7: A fundamental diagram relating the average spaceheadwayhs to the space-mean speedvs. Note that the aver-age space headway is proportional to the inverse of the density,i.e.,k−1.

The interesting features of this type of fundamental dia-gram, can be summed as follows:

• the curve starts not in the origin, but atk−1j , corre-

sponding to the average space headway when the

jam density is reached (i.e., all vehicles are stand-ing nearly bumper to bumper),

• as the average space headway increases, its inverse(the density) decreases, and the space-mean speedincreases,

• the space-mean speed continues to rise with an in-creasing average space headway, until it reachesthe maximum average speed, i.e., the free-flowspeedvff ; this happens at the inverse of the criti-cal densityk−1

c ,

• from then on, the space-mean speed remains con-stant with increasing average space headway.

The above features can be understood intuitively: atlarge average space headways, a driver experiences noinfluence from its direct frontal leader. However, thereexists a point at which the driver comes ‘close enough’ tothis leader (i.e., in crowded traffic), so that its speed willdecrease. This slowing down will continue to persist astraffic gets more dense (this the same reasoning behindGreenshields’ derivation in section V B 1).

Flow versus densityProbably the most encountered form of a fundamental di-agram, is that of flow versus density. Its origins date backto the seminal work of Lighthill and Whitham who, asdescribed earlier, referred to it as the flow-concentrationcurve. An example of theqe(k) fundamental diagram isdepicted in Fig. 8.

0

q

qcap

kkjkc

vs

w

free-flow congested

FIG. 8: A fundamental diagram relating the densityk to theflow q. The capacity flowqcap is reached at the critical densitykc. The space-mean speedvs for any point on the curve, isdefined as the slope of the line through that point and the origin.Taking the slope of the tangent to points on the curve, gives thecharacteristic wave speedw.

Noteworthy features of this type of fundamental diagramare:

• for moderately low densities (i.e., below the crit-ical densitykc), the flow increases more or lesslinearly (this is called thefree-flow branchof thefundamental diagram),

• near the critical densitykc, the fundamental dia-gram can bend slightly, due to faster vehicles beingobstructed by slower vehicles, thereby lowering thefree-flow speed [97],

• at the critical densitykc, the flow reaches a maxi-mum, called the capacity flow[128]qcap,

• in the congested regime (i.e., for densities higherthan the critical density), the flow starts to degradewith increasing density, until the jam densitykj isreached and traffic comes to a stand still, resultingin a zero flow (this is called thecongested branchof the fundamental diagram),

• the space-mean speedvs for any point on theqe(k)fundamental diagram, can be found as the slope ofthe line through that point and the origin,

There is one more piece of information revealed by theqe(k) fundamental diagram: When taking the slope ofthe tangent in any point of the diagram, we obtain whatis called thekinematic wave speed. These speedsw cor-respond toshock wavesencountered in traffic flows (e.g.,the stop-and-go waves). As can be seen from the fig-ure, the shock waves travel forwards, i.e.,downstream, infree-flow traffic (w ≥ 0), but backwards, i.e.,upstream,in congested traffic (w ≤ 0).

The above shape of theqe(k) fundamental diagram isjust one possibility. There exist many different flavours,originally derived by traffic engineers seeking a betterfit of these curves to empirical data. After the work ofGreenshields, another functional form — based on a log-arithm — was proposed by Greenberg [43]. Anotherpossible form was introduced by Underwood [112]. Allof the previous diagrams are calledsingle-regime mod-els, because they formulate only one relation betweenthe macroscopic traffic flow characteristics for the entirerange of densities (i.e., traffic flow regimes) [79]. In con-trast to this, Edie started developingmulti-regime mod-els, allowing for discontinuities and a better fit to empir-ical data coming from different traffic flow regimes [35].We refer the reader to the work of Drake et al. [34] andthe book of May [79] for an extensive comparison andoverview of these different modelling approaches (notethat Drake et al. used time-mean speed).

During the last two decades, other, sometimes more so-phisticated, functional relationships between density andflow have been proposed. Examples are the work ofSmulders who created a non-differentiable point at thecritical density in a two-regime fundamental diagram[106], the METANET model of Messmer and Papageor-giou who’s single-regime fundamental diagram containsan inflection point near the jam density [80], the workof De Romph who generalised Smulders’ functional de-scription of his two-regime fundamental diagram [103],the typical triangular shape of the fundamental diagramintroduced by Newell, resulting in only two possible val-ues for the kinematic wave speedw [97], . . . As canbe seen, these fundamental diagrams sometimes take onnon-concave forms, depending on the existence of in-flection points in the functional relation between flow

and density. In general, they can be convex, concave,(dis)continuous, piecewise-linear, everywhere differen-tiable, have inflection points, . . . Variations in shape willcontinu to be proposed, as it is for certain that there isno general consensus among traffic engineers regardingthe correct shape of this fundamental diagram. To illus-trate this, a more exotic approach is based oncatastrophetheory, which is, in a sense, a three-dimensional modelthat jointly treats density, flow, and space-mean speed.Acha-Daza and Hall applied the technique, resulting in asatisfactory fit with empirical data [2].

The most extreme argument with respect to the shapeof the fundamental diagram, came from Kerner whoquestioned its validity, and consequently rejected it alto-gether by replacing it with his fundamental hypothesis ofthree-phase traffic flow theory(refer to section V D formore details) [55].

Space-mean speed versus flowAn often spotted shape is that of thevse

(q) fundamentaldiagram, depicted in Fig. 9. As opposed to the earlierdiscussedvse

(k) fundamental diagram, the space-meanspeed versus flow curve no longer embodies a function inthe strict mathematical sense: for each value of the flow,there exists two different mean speeds, namely one in thefree-flow regime (upper branch) and one in the congestedregime (lower branch).

0 qqcap

vs

vff

FIG. 9: A fundamental diagram relating the flowq to the space-mean speedvs. The capacity flowqcap is located at the rightedge of the diagram, i.e., it is defined as the maximum averageflow. Note that there are two possible speeds associated witheach value of the flow.

Some people, e.g., economists who use the flow to rep-resent traffic demand, find this kind of fundamental dia-gram easy to cope with. But in our opinion, we are con-vinced however, that this diagram is rather difficult to un-derstand at first sight. We believe thevse

(k) fundamentaldiagram is a much better candidate, because density canintuitively be understood as a measure for how crowdedtraffic is, as opposed to some flow giving rise to two dif-ferent values for the space-mean speed.

As a final comment, we would like to point out that thepreviously discussed bivariate functional relationshipsbetween the traffic flow characteristics (e.g., density andflow), are based on observations. More importantly, this

means that there isno direct causal relationassumed be-tween any two variables. Fundamental diagrams sketchonly possible correlations, implying that thenature of thetransitionsbetween different traffic regimes thus remainsto be explored (see section V A 4 for a discussion).

3. Empirical measurements

As mentioned earlier, the fundamental diagrams dis-cussed in the previous section represent equilibrium re-lations between the macroscopic traffic flow characteris-tics of section III. In sharp contrast to this, real empiri-cal measurements from detector stations do not describesuch nice one-dimensional curves corresponding to thefunctional relationships.

As an illustrative example, we provide some scatter plotsin Fig. 10. The shown data comprises detector measure-ments (the sampling interval was one minute) during theentire year 2003; they were obtained by means of a videocamera [113] located at the E17 three-lane motorwaynear Linkeroever[129], Belgium. Because of the natureof this data, we only obtained flows, occupancies, andtime-mean speeds. After calculating the average vehiclelength, the occupancies were converted into densities us-ing equation (22). Using these recorded time series, wethen constructed scatter plots of the density, time-meanspeed, flow, and average space headway. Note that nosubstantial changes are introduced in these plots due toe.g., our using of densities calculated from occupancies,instead of using real measured densities.

0 50 100 150 2000

20

40

60

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FIG. 10: Illustrative scatter plots of the relations between trafficflow characteristics as measured by video camera CLO3 locatedat the E17 three-lane motorway near Linkeroever, Belgium. Themeasured occupancies were converted into densities, the time-mean speed remained unchanged. Shown are scatter plots of a(k,vt) diagram (top-left), a (hs,vt) diagram (top-right), a (k,q)diagram (bottom-left), and a (q,vt) diagram (bottom-right).

As the dimension of time is removed in these scatterplots, Daganzo calls themtime-independent models[27].It is important to understand thatthese scatter plots arenot fundamental diagrams, because the latter represent

one-dimensional equilibrium curves. According to Hel-bing, a better designation would beregression models[47]. In this dissertation, we introduce a terminologybased onphase spaces(or equivalentlystate spaces), re-sulting in e.g., the (k,q) diagram (note that we droppedthe adjective ‘fundamental’).

In reality, traffic is not homogeneous, nor is it stationary,thus having the effect of a large amount of scatter inthe presented diagrams. In free-flow traffic, interactionsbetween vehicles are rare, and their small local distur-bances have no significant effects on the traffic stream.As a result, all points are somewhat densely concentratedalong a line — representing the free-flow speed — inall four diagrams. However, in the congested regime,a wide range of scatter is visible due to the interactionsbetween vehicles. Furthermore, vehicle accelerationsand decelerations lead to large fluctuations in the trafficstream, as can be seen by the thin, but large, cloud ofdata points. The effect is especially pronounced forintermediate densities, leading to large fluctuations in thetime-mean speed and flow.

The occurrence of all this scatter in the data, leadssome traffic engineers to question the validity ofthe fundamental diagram. More specifically, thebehaviour in congested traffic seems ill-defined tosome. As stated earlier, Kerner is the most in-tense opponent in this debate, as he outright rejectsLighthill and Whitham’s hypothesis that remainedpopular over the last fifty years. Despite this criti-cism, the fundamental diagram remains, to the ma-jority of the community, a fairly accurate descriptionof the average behaviour of a traffic stream. Cas-sidy even provided quantifiable evidence of the ex-istence of well-defined bivariate relations betweentraffic flow characteristics. The key here was to sep-arate stationary periods from non-stationary ones inthe detector data (i.e., stratifying it) [14, 27]. Priorwork of Del Castillo and Benitez resulted in a moremathematically justified method, for fitting empiri-cal curves in data regions of stationary traffic, afterconstruction of a rigid set of properties that all fun-damental diagrams should satisfy [31, 32].

As a final note, we remark that the distribution of thecloud-like data points of the diagrams in Fig. 10, is a re-sult of various kinds of phenomena. First and foremost,there is the heterogeneity in the traffic composition (fastpassenger cars, slow trailer trucks, . . . ). Secondly, as al-ready mentioned, the non-stationary behaviour of trafficintroduces a significant amount of scatter in the congestedregime. Thirdly, each scatter plot is dependent on the typeof road, and the time of day at which the measurementswere collected. In this respect, the influence of (chang-ing) weather conditions is not to be underestimated (e.g.,rain fall results in different diagrams). In conclusion, itis clear that if we want these scatter plots to better fit thefundamental diagrams, all data points should be collectedunder similar conditions. Even more so, the relative lo-cation on the road at which the data points were recordedplays a significant role: e.g., a jam that propagates up-stream, passing an on-ramp will show different effects,

depending on where the observations were gathered (up-stream, right at, or downstream of the on-ramp) and onwhether or not the particular bottleneck was active [79].

C. Capacity drop and the hysteresis phenomenon

In the early sixties, traffic engineers frequently observedadiscontinuity in the measurements near the capacity flow.To this end, Edie proposed a two-regime model that in-cluded such a discontinuity at the critical density [35].Nowadays, this typical form of theqe(k) fundamental di-agram is known as areversed lambda shape(the namewas originally suggested by Koshi et al. [67]).

An example of such a reversedλ fundamental diagram,is shown in the left part of Fig. 11. Note however, thatthe depicted discontinuity apparently leads tooverlap-pingbranches of the free-flow and congested regimes, re-sulting in amulti-valuedfundamental diagram.

qqcap

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FIG. 11: Left: the typical invertedλ shape of the (k,q) funda-mental diagram, showing a capacity drop fromqcap to belowqout ≪ qcap (i.e., the queue discharge flow). The hysteresiseffect occurs when going from the congested to the free-flowbranch, as indicated by the three arrows (1) – (3).Right: a(k,q) diagram based on empirical data of one day, obtained byvideo camera CLO3, at the E17 three-lane motorway near Link-eroever, Belgium. The black dots denote minute measurements,whereas the thick solid line represents the time-traced evolutionof traffic conditions. The observed hysteresis loop was basedon consecutive 5-minute intervals covering a period that encom-passes the morning rush hour between 06:30 and 09:30.

Considering the left part of Fig. 11, it appears the flowcan take on two different values (hence the name ‘two-regime, two-capacity’ model) depending on the trafficconditions, i.e., whether traffic is moving from the free-flow to the congested regime on the equilibrium curve orvice versa. In order to comprehensively understand thishystereticbehaviour, we consider the following intuitivesequence of events:

(1) In the free-flow regime, the flow steadilyrises with increasing density, small perturba-tions in the traffic flow have no significanteffects (see section V A 1).

(2) At the critical densitykc, traffic is said tobe metastable: for small disturbances, traf-fic is stable, but when these disturbances aresufficiently large, they can lead to a cascad-ing effect (see section V A 2), resulting in abreakdown of traffic and kicking it onto thecongested branch. The state of capacity flow

atqcap is destroyed, due to a sudden decreaseof the flow, called thecapacity drop.

(3) In order to recover from the congestedto the free-flow regime, the traffic densityhas to bereduced substantially(in compari-son with the reverse transition), i.e., well be-low the critical densitykc. After this recov-ery, the flow will not be equal toqcap, but toqout ≪ qcap, which is called theoutflow froma jamor thequeue discharge capacity.

The above sequence signifies ahysteresis loopin the flowversus density fundamental diagram: going from the free-flow to the congested regime occurs via the capacity flow,but the reverse transition proceeds via another way. Thephenomenon was first observed by Treiterer and Meyers,who used aerial photography to calculate densities andspace-mean speeds, extracted from a platoon of movingvehicles [110]. Hall et al. later observed a similar phe-nomenon [46].

The right part of Fig. 11 shows a (k,q) diagram, obtainedwith empirical data collected at Monday September 10th,2001. The data was recorded by video camera CLO3,at the E17 three-lane motorway near Linkeroever, Bel-gium. The small dots represent minute-based measure-ments, whereas the thick solid line represents the time-traced evolution of traffic conditions. The observed hys-teresis loop was based on consecutive 5-minute inter-vals covering a period that encompasses the morning rushhour between 06:30 and 09:30.

Zhang is among the few who try to give a possiblerigourous mathematical explanationfor the occurrenceof this hysteresis phenomenon [120]. His exposition isbased on the behaviour of individual drivers during car-following: central to his interpretation is the existence ofanasymmetrybetween accelerating and decelerating ve-hicles (a related notion was already explored by Newellback in 1963 [94]). The former are associated with largerspace headways, whereas the latter typically have smallerspace headways. Both observations can be understoodwhen considering the characteristic ‘harmonica’ effect ofa string of consecutive vehicles: when the next stop-and-go wave is encountered, a driver is more alert as he typ-ically has to brake rather hard in order to avoid a colli-sion. But once this wave has passed, a driver gets morerelaxed, resulting in a larger response time when apply-ing the gas pedal. The deceleration reaction leads to asudden decrease of the space headway, whereas the ac-celeration reaction leads to a gradually developing largerspace headway. To this end, Zhang introduces three dis-tinct traffic phases, respectively called theaccelerationphase, thedeceleration phase, and astrong equilibrium(indicating a constant speed). Because the space head-way is thus treated differently under these qualitativelydifferent circumstances, the result is that there are nowdifferent functional relations for thevse

(hs) fundamentaldiagram. As a consequence, a hysteresis loop can appearin the (density,flow) state space. Note that Zhang’s workdescribes acontinuousloop in state space, whereas inmost cases hysteresis is assumed to follow a discontinu-ous fundamental diagram. Furthermore, as there are three

different ways for vehicles to reside in a traffic stream(i.e., Zhang’s traffic phases), there are now three differentcapacities related to these conditions; it is the capacityunder a stationary equilibrium flow that should be con-sidered as the ideal capacity of a roadway [121].

Note that depending on the location where the traf-fic stream measurements were performed, the transitionfrom the free-flow to the congested regime and vice versadoes not always have to pass via the capacity flow. In-stead, observations can indicate that the traffic state canjump abruptly from one branch to another in the diagram[39]. A possible explanation is that upstream of a jam,vehicles arrive with high speeds, resulting in strong de-celerations; a detector station located at this point wouldobserve traffic jumping from the uncongested branch im-mediately to the congested branch, without necessarilyhaving to pass via the capacity [95]. This has led Hall etal. to believe the reversed lambda shape is more correctlyreplaced by a continuous but non-differentiableinvertedV shape[46].

Continuing this latter train of thought, Daganzo believesthat many of these ‘extravagant’ phenomena (e.g., amulti-valued fundamental diagram) are uncalled for.Applying the stratification methodology of Cassidy [14],the scatter in the empirical data may vanish, restoring asmooth continuous equilibrium relation between densityand flow. One way of explaining the high tip of thelambda, is to assume that it is caused by statisticalfluctuations that comprise platoons of densely packedvehicles [27].

During the last seventy years, there has been acontinuing quest to find the ‘correct’ form of thefundamental diagrams. In this respect, we like tostress the fact that ‘only looking at the measure-ments’ is not sufficient: traffic engineers wantingto mine the gigabytes of empirical data, shouldalways look at the global picture. This means thatthe typical driving patterns,as well as the localgeometry/infrastructure, should also be taken intoaccount, so that the local measurements can beinterpreted with respect to the traffic flow dynamics.If this is neglected, the danger exists that traffic isonly sampled at discrete locations, giving a sortof ‘truncated’ view of the occurring dynamicalprocesses.

Finally, we like to agree with Zhang’s comments:the root cause of most of the differences in the con-struction of fundamental diagrams, is the erroneoustreatment of data (e.g., mixing data stemming fromdifferent traffic flow regimes) [120]. Because fun-damental diagrams imply the notion of an equilib-rium, care should be taken when using the data, i.e.,only considering stationary periods after removingthe transients.

D. Kerner’s three-phase theory

In the mid-nineties, Kerner and other fellow researchers,studied various traffic flow measurements stemming fromdetector stations along German motorways. Initially,they agreed with the classic notion of Lighthill andWhitham’s fundamental hypothesis of the existence ofone-dimensional equilibrium relation between the macro-scopic traffic flow characteristics (see section V B 1 formore details). However, upon discovery of a rich andcomplex set of empirical tempo-spatial patterns in con-gested traffic flow, Kerner decided to abolish this hypoth-esis, as it could not adequately capture all of these ob-served patterns. As a consequence, Kerner rejectsalltraffic flow theories and models that are based on this one-dimensional equilibrium relation [55].

In the search for a more correct theory that could accu-rately describe empirical traffic flow observations, Kernerdeveloped what is known as thethree-phase theoryoftraffic flow.

1. Free flow, synchronised flow, and wide-moving jam

In section V A, we elaborated on a classic approach totraffic flow, general assuming two qualitatively differentregimes, namely free-flow and congested traffic. Basedon empirical findings, Kerner and Rehborn in 1996 pro-posed three different regimes, separating the congestedregime into two other regimes. This led them to the in-troduction of the following regimes [61]:

• free flow,

• synchronised flow,

• and wide-moving jam.

The main difference betweensynchronised flowand thewide-moving jam, is that in the former low speeds buthigh flows (comparable to free-flow traffic) can be ob-served, whereas in the latter both low speeds and lowflows are observed. The description by the term ‘synchro-nised’ was based on the discovery that the time series offlows, densities, and mean speeds exhibited large degreesof correlation among neighbouring lanes. And althoughsynchronised flow is treated as a form of congestion, itnevertheless is characterised by a high continuous flow.Furthermore, a typical tempo-spatial region of synchro-nised flow has a fixed downstream front (that could belocated at a bottleneck’s position), whereas both the up-stream and downstream fronts of a wide-moving jam canpropagate undisturbed in the upstream direction of a traf-fic stream [60].

Kerner distinguishes several congestion patterns withrespect to traffic flows. A first typical pattern is asynchronised-flow pattern(SP), which can be furtherclassified as amoving SP(MSP), awidening SP(WSP),and alocalised SP(LSP). An SP can only contain syn-chronised flow; as we will shortly mention in section

V D 3, a moving jam can only occur inside such an SP.When such a jam transforms into a wide-moving jam,the resulting pattern is called ageneral pattern(GP); aGP therefore contains both synchronised flow and wide-moving jams. Just as with the SP, there exist differenttypes of GP. These are adissolving GP(DGP), aGP un-der weak congestion, and aGP under strong congestion.A final often encountered pattern occurs when two bottle-necks are spatially close to each other, resulting in whatis called anexpanded congested pattern(EP).

Taking the above considerations into account, the discov-ery and distinction between both types of congested traf-fic patterns should be made on the basis of tempo-spatialplots of the speed, rather than the flow (because the flowin synchronised traffic is difficult to differentiate fromthat of free-flow traffic) [55]. To this end, Kerner et al.developed two applications that are capable of accuratelyestimating, automatically tracking, and reliably predict-ing the above mentioned congested traffic patterns. Theirmodels are theForecasting of Traffic Objects(FOTO) andAutomatische StauDynamikAnalyse(ASDA) [63].

2. Fundamental hypothesis of three-phase traffic theory

Central to Kerner’s theory, is thefundamental hypothesisof three-phase traffic theory, which basically states thathypothetical steady states of synchronised flow, cover atwo-dimensional regionin a flow versus density diagram(as opposed to the classic notion of a one-dimensionalequilibrium relation). An example of such a diagram canbe seen in Fig. 12.

0

q

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kkjkout kc

F

S

J

FIG. 12: The flow versus density relation according to Kerner’sthree-phase traffic theory. The curve of free flow (denoted byF )is reminiscent of observations in the classic free-flow regime. Itlevels of a bit towards the capacity flowqcap at the critical den-sity kc. As a result of Kerner’s fundamental hypothesis, theregion of synchronised flow (denoted byS) covers a large two-dimensional part of the density-flow phase space. It is inter-sected by the lineJ , denoting the steady propagation of wide-moving jams. The lineJ also intersects the curve of free flowin the outflow from a jamqout ≪ qcap at the associated densitykout.

In the flow versus density diagram in Fig. 12, the threeregimes are depicted: the curve of free flow (denoted byF ), the region of synchronised flow (denoted byS) and

the wide-moving jam (denoted by the empirical lineJ).Just as in the classic fundamental diagrams, the observa-tions in free-flow traffic lie on a sharp line that linearlyincreases the flow with higher densities (note the level-ling of the curve near the capacity flowqcap associatedwith the critical densitykc). The region of synchronisedflow spans a large part of the density-flow phase space;an important remark here is that consecutive measure-ment points are scattered within this region, meaning thatan increase in the flow can happen with both higherandlower densities (as opposed to the free-flow regime) [61].

The characteristic lineJ denotes the steady, undisturbedpropagation of wide-moving jams. Its slope correspondsto the speed of a wide-moving jam’s downstream front,which typically lies aroundw ≈ -15 km/h [60]. Theupper-left point of the lineJ is located at a densitykout

corresponding to the outflowqout ≪ qcap from a wide-moving jam. This is an illustration of the capacity dropphenomenon, elucidated in section V C. The lineJ isdefined as follows:

q(k) =1T

(1−

k

kjam

), (45)

with T the time gap in congested traffic flows; it is usedto tune the outflow from a jam. Because wide-movingjams travel undisturbed, their outflow — caused by vehi-cles that leave the downstream front — can be either freeflow or synchronised flow. Typical values for this outflowrange from 1500 to 2000 vehicles/hour/lane [55]. The av-erage flow rate within such a wide-moving jam can be al-most zero, meaning that vehicles continuously encounterstop-and-go waves.

Related to the wild scatter in the (k,q) diagram of three-phase traffic theory, is the microscopic behaviour of in-dividual vehicles. The explanation given by Kerner andKlenov, is that vehicles in synchronised flow do not as-sume a fixed preferred distance to their direct frontalleader, but rather accept a certainrange of distances.Within this range, drivers have both the tendency to over-accelerate when they think there is the ability to overtake,and the tendency for drivers to adjust their speed to thatof their leader, when this overtaking can not be fulfilled[55, 58].

3. Transitions towards a wide-moving jam

The breakdown of traffic from the free-flow to the wide-moving jam state, is nearly always characterised by twosuccessiveF → S andS → J transitions, between freeflow and synchronised flow, and synchronised flow andwide-moving jam respectively. In the first stage, a stateof free flow changes to synchronised flow by theF → Stransition. Central to the idea of this phase transition, isthe fact that there is no explicit need for an external dis-turbance for its occurrence. A sufficiently large (i.e.,su-percritical) internal disturbance inside the traffic stream

(e.g., a lane change) causes anucleation effectthat in-stigates theF → S transition. Once it has set in, theonset of congestion is accompanied by a sharp drop inthe mean vehicle speed. During the second stage, a set ofnarrow-moving jams can grow inside the tempo-spatialregion of synchronised flow. A narrow-moving jam isdifferent from a wide-moving jam, in that vehicles typi-cally do not on average come to a full stop inside the jam.But, due to a compression of synchronised flow (an ef-fect termed thepinch effect), these narrow-moving jamscan coalesce into a wide-moving jam, thereby complet-ing the cascade of theF → S → J transition, resultingin stop-and-go traffic [57].

With respect to the flow versus density diagram inFig. 12, it can be seen that the lineJ actually dividesthe region of synchronised flow in two parts. Points thatlie underneath this line, characterise stable traffic stateswhere noS → J transition can occur. Points abovethe lineJ however, characterise metastable traffic states,meaning that sufficiently large disturbances can trigger aS → J transition [55, 66].

Note that the directF → J transition between free flowand wide-moving jam can also occur, but it has a verysmall probability, i.e., the critical perturbation needed, ismuch higher than that of the frequently occurringF → Stransition between free flow and synchronised flow. So ingeneral, wide-moving jams do not emerge spontaneouslyin free flow, but a situation where such a transition mayoccur, is when an off-ramp gets filled with slow-movingvehicles. This results in a local obstruction at the motor-way’s lane directly adjacent to the off-ramp, which cancause a local breakdown of the upstream traffic, resultingin a wide-moving jam. Finally, it is important to distin-guish the nature of this transition from that of theF → Stransition: the former is a transitioninducedby anexter-nal disturbanceof the local traffic flow, whereas the latteris considered as aspontaneoustransition due to aninter-nal disturbancewithin the local traffic flow (e.g., a lanechange) [55].

4. From descriptions to simulations

As Kerner himself describes his three-phase theory, itis a qualitative theory. In essence, it gives no expla-nation of why certain transitions occur, as it onlyde-scribesthem [55]. However, several exemplary micro-scopic traffic flow models have already been developed(i.e., treating all vehicles and their interactions individu-ally). These models can reproduce the different empiricaltempo-spatial patterns described by Kerner’s theory. Asexamples, we mention two models based on cellular au-tomata: a first attempt was made by Knospe et al., whodeveloped a model that takes into account a driver’s re-action to the brake-lights of his direct frontal leader [65].Kerner et al. refined this approach by extending it; theirwork resulted in a family of models based on the notionof asynchronisation distancefor individual vehicles; theyare commonly called the KKW-models (from its three au-thors, Kerner, Klenov, and Wolf) [56].

The theory can describe most of the encountered tempo-spatial features of congested traffic. And at the moment,successful microscopic models have been developed, butthe work is not yet over: an important challenge that re-mains for theoreticians, is the mathematical derivation ofa consistent macroscopic theory (i.e., one that treats traf-fic at a more aggregate level as a continuum) [55]. In pur-suit of such a model, Kim incorporated Kerner’s trafficregimes into a broader framework, encompassing six dif-ferent possible states: the transitions between these statesare tracked with a modified macroscopic model that usesconcepts from fuzzy logic theory [64].

E. Theories of traffic breakdown

A central question that is often asked in the field of traf-fic flow theory, is the following:“What causes conges-tion ?” Clearly, the answer to this question should bea bit more detailed than the obvious“Because there toomany vehicles on the road !”With respect to the phasetransitions that signal a breakdown of the traffic flow, var-ious — seemingly contradicting — theories exist. Arethey merely a matter of belief, or can they be rigourously‘proven’ ? Opinions are divided, but nowadays, two qual-itatively different mainstream theories exist, attributed todifferent schools of thought [12, 77, 108]:

The European (German) schoolIn the early seventies, Treiterer and Meyers performedsome aerial observations of a platoon of vehicles. As theyconstructed individual vehicle trajectories, they could ob-serve a growing instability in the stream of vehicles, lead-ing to an apparently emergingphantom jam(i.e., a jam‘out of nothing’) [110].

Some twenty years later, in the mid-nineties, Kerner andKonhauser made detailed studies of traffic flow measure-ments, obtained at various detector stations along Ger-man motorways. Their findings indicated that phantomjams seemed to emerge in regions of unstable traffic flow[59]. This stimulated Kerner and Rehborn to further re-search efforts directed towards the behaviour of propa-gating jams [60, 61]. They proposed a different set oftraffic flow regimes, culminating in what is now calledthree-phase traffic theory(see section V D for more de-tails) [55, 57, 62]. The main idea supported by followersof this school of researchers, is that traffic jamscanspon-taneously emerge, without necessarily having an infras-tructural reason (e.g., on-ramps, incidents, . . . )[130]. Indense enough traffic, phase transitions from the free-flowto the synchronised-flow regime can occur, after which alocal instability such as e.g., a lane change can grow (theso-calledpinch effect), triggering a stable jam leading tostop-and-go behaviour [57]. Kerner’s three-phase theorystands out as an archetypical example of these modernviews. But although his theory has, in our opinion, beenworked out well enough, he more than frequently encoun-ters harsh criticisms when conveying it to most audiences(perhaps the main cause for this human behaviour is thefact that Kerner always mentions the same view, i.e.,“allexisting traffic flow theories are wrong”).

Inspired by Kerner’s work, Helbing et al. gave in 1999an extended treatise on the different types of congestionpatterns that can be observed in the vicinity of spatial in-homogeneities (e.g., on-ramps). Their work resulted in auniversal phase diagram, containing a whole plethora ofpatterns of congested traffic states (calledhomogeneouslycongested traffic– HCT, oscillatory congested traffic–OCT, triggered stop-and-go traffic– TSG, pinned lo-calised cluster– PLC, andmoving localised cluster–MLC), each one having unique characteristics [48][131].In that same year, Lee et al. studied the patterns thatemerge at on-ramps, thereby agreeing with the findingsof Helbing et al. [70]. As the previous research intocongestion patterns was largely based on the use of ana-lytical traffic flow models and computer simulations, theneed for validation with empirical data grew. In 2000, thework of Treiber et al. among others, proved the existenceof the previously mentioned congestion patterns [109].

At this point, it is noteworthy to mention the seminalwork of Nagel and Schreckenberg [92], who in 1992developed a model that describes traffic flows in whichlocal jams can form spontaneously. As many variationson this model have been proposed, later work alsofocussed on the stability of traffic flows in these models,e.g., the work of Jost and Nagel [53].

The Berkeley schoolIncluding names such as the late Newell, Daganzo,Bertini, Cassidy, Munoz, . . . , the ‘Berkeley school’ (Uni-versity of California) supports the theory that all conges-tion is strictly induced by bottlenecks. The hypothesisholds for both recurrent and, in the case of an incident,non-recurrent congestion.

The main starting point states that there isalwaysa ‘ge-ometrical’ explanation for the breakdown. This expla-nation is based on the presence of road inhomogeneitiessuch as on- and off-ramps, tunnels, weaving areas, lanedrops, sharp bends, elevations, . . . Once a jam occurs dueto such a (temporary) bottleneck, it does not dissipate im-mediately; as a result, drivers can wonder why they enterand exit a congestion wave, without there being an ap-parent reason for its presence (since it happened earlierand the cause e.g., an incident, already got cleared). Da-ganzo uses this line of reasoning as an explanation for thedismissal of phantom jams [28].

The school uses a specific terminology with respect tobottlenecks (being road inhomogeneities). Two qualita-tively different regimes exist: thefree-flow regimeandthe queued regime. The latter occurs when a bottleneckbecomesactive, which will result in a queue growing up-stream of the bottleneck while a free-flow regime existsdownstream. Thebottleneck capacityis then defined asthe maximumsustainableflow downstream (which is dif-ferent from the maximum flow that can be observed priorto the bottleneck’s activation).

The location of these bottlenecks has some peculiaritiesinvolved: one of them is the concept of acapacity fun-nel [11]. It assumes that drivers are at times more alert,e.g., when they are driving on a motorway and nearing anon-ramp in rather dense traffic conditions [121]. This im-

pels them to accept shorter headways, so they are drivingclosely behind each other at a relatively high speed. Oncethey have passed the on-ramp’s location, they tend to re-lax, resulting in larger headways. The effect is that thebottleneck’sactualposition is located more downstream.

Shortly after the publication of Kerner and Rehborn’sfindings about the peculiar phase transitions that seemedto occur on German motorways, Daganzo et al. pro-vided a swift response where they stated that the occur-ring phase transitionscouldalso be caused by bottlenecksin a predictable way [29]. They implied that no spon-taneously emerging traffic jams are suggested, and thatthe observed traffic data from both German and NorthAmerican motorways did not contradict their own state-ments about the cause of the phase transitions [91]. Inshort, the subtle difference between their work and thatof Kerner and Rehborn, is that instabilities in the traf-fic stream are theresult and not the cause of the queuesthat emerge at active bottlenecks. With respect to a spon-taneous breakdown of traffic flow at on- and off-ramps(i.e., bottlenecks), Daganzo also states that this can beexplained using a simple traffic flow model operating un-der the assumption of a too high inflow from the on-rampor a caused by blocking of the off-ramp [26].

The studies undertaken by this school, are heavily basedon the researchers’ use of cumulative plots and elegantlysimple traffic flow models, as opposed to the classicmethodology that investigates time series of recordedcounts and speeds. As stated earlier (see section III B 2),some recent examples include the work of Munoz andDaganzo [85, 86, 87, 89] and Cassidy and Bertini [7, 15].

Recently, Tampere argued that both theories, as enunci-ated by the two schools, are not entirely contradictory.His statement is based on the fact that the mechanismsbehind the bottleneck-induced breakdown and spon-taneous breakdown are approximately the same, onlydiffering in theprobability of such a breakdown (whichis related to the instability of a traffic flow) [108].

In our view, both theories are sufficiently different,but compatible, in that the first school elaboratelydescribes traffic flow breakdown more or less as hav-ing an inherentlyprobabilistic nature, whereas thesecond school treats breakdown a strictdeterminis-tic process. The former introduces a complex vari-ety of congestion patterns, while the latter primarilyfocusses on an elegantly simple description of trafficflow breakdown. Even more characteristically, is theobservation that most adepts of the European school,inherently need stochasticity in the modelsin order toproduce their sought phantom traffic jams (note thatnotwithstanding the fact that stochastic models arein a strict sense also deterministic, we neverthelessadopt in this dissertation, the convention that deter-ministic means ‘non-stochastic’). Our argument is ina way also supported by Nagel and Nelson, who statethat the purpose of the traffic flow model (e.g., theeffect of moving bottlenecks versus predicting meantraffic behaviour) decide whether or not stochasticityin the model is required [91].

Furthermore, there might be some room for stochas-ticity in the Berkeley models after all, with thework of Laval which suggests that (disruptive) lanechanges form the main cause for instabilities in atraffic stream [69]. Deciding which school is right,is therefore in our opinion a matter of personal taste,but in the end, we agree with Daganzo when he statesthat research into bottleneck behaviour is the mostimportant in the context of traffic flow theory [30].

VI. CONCLUSIONS

In this paper, an extensive account was given, detailingseveral aspects related to the description of traffic flows.Most importantly, we have introduced a nomenclatureconvention, built upon a consistent set of notations. Ourdiscussion of traffic flow characteristics centred aroundthe space and time headways as microscopic characteris-tics, with densities and flows as their macroscopic coun-terparts. Several noteworthy highlights are the techniqueof oblique cumulative plots and the derivation of traveltimes based on these plots. A finally large part of thispaper reviewed some of the relations between traffic flowcharacteristics, i.e., the fundamental diagrams, and clari-fied some of the different points of view adopted by thetraffic engineering community.

APPENDIX A: GLOSSARY OF TERMS

1. Acronyms and abbreviations

4SM four step modelAADT annual average daily trafficABM activity-based modellingACC adaptive cruise controlACF average cost functionADAS advanced driver assistance systemsAIMSUN2 Advanced Interactive Microscopic

Simulator for Urban and Non-UrbanNetworks

AMICI Advanced Multi-agent Information andControl for Integrated multi-class trafficnetworks

AON all-or-nothingASDA Automatische StauDynamikAnalyseASEP asymmetric simple exclusion processATIS advanced traveller information systemsATMS advanced traffic management systemsBCA Burgers cellular automatonBJH Benjamin, Johnso, and HuiBJH-TCA Benjamin-Johnson-Hui traffic cellular

automatonBL-TCA brake-light traffic cellular automatonBML Biham, Middleton, and LevineBML-TCA Biham-Middleton-Levine traffic cellular

automatonBMW Beckmann, McGuire, and WinstenBPR Bureau of Public Roads

CA cellular automatonCA-184 Wolfram’s cellular automaton rule 184CAD computer aided designCBD central business districtCFD computational fluid dynamicsCFL Courant-Friedrichs-LewyChSch-TCA Chowdhury-Schadschneider traffic

cellular automatonCLO camera LinkeroeverCML coupled map latticeCONTRAM CONtinuous TRaffic Assignment

ModelCOMF car-oriented mean-field theoryCPM computational process modelsCTM cell transmission modelDDE delayed differential equationDFI-TCA deterministic Fukui-Ishibashi traffic

cellular automatonDGP dissolving general patternDLC discretionary lane changeDLD double inductive loop detectorDNL dynamic network loadingDRIP dynamic route information panelDTA dynamic traffic assignmentDTC dynamic traffic controlDTM dynamic traffic managementDUE deterministic user equilibriumDynaMIT Dynamic network assignment for the

Management of Information toTravellers

DYNASMART DYnamic Network Assignment-Simulation Model for AdvancedRoadway Telematics

ECA elementary cellular automatonEP expanded congested patternER-TCA Emmerich-Rank traffic cellular

automatonFCD floating car dataFDE finite difference equationFIFO first-in, first-outFOTO Forecasting of Traffic ObjectsGETRAM Generic Environment for TRaffic

Analysis and ModelingGHR Gazis-Herman-RotheryGIS geographical information systemsGNSS Global Navigation Satellite System

(e.g., Europe’s Galileo)GoE Garden of Eden stateGP general patternGPRS General Packet Radio ServiceGPS Global Positioning System

(e.g., USA’s NAVSTAR)GRP generalised Riemann problemGSM Groupe Speciale MobileGSMC Global System for Mobile

CommunicationsHAPP household activity pattern problemHCM Highway Capacity ManualHCT homogeneously congested trafficHDM human driver modelHKM human-kinetic modelHRB Highway Research Board

HS-TCA Helbing-Schreckenberg traffic cellularautomaton

ICC intelligent cruise controlIDM intelligent driver modelINDY INteractive DYnamic traffic assignmentITS intelligent transportation systemsIVP initial value problemJDK JavaTM Development KitKKT Karush-Kuhn-TuckerKKW-TCA Kerner-Klenov-Wolf traffic cellular

automatonKWM kinematic wave modelLGA lattice gas automatonLOD level of detailLOS level of serviceLSP localised synchronised-flow patternLTM link transmission modelLWR Lighthill, Whitham, and RichardsMADT monthly average daily trafficMC-STCA multi-cell stochastic traffic cellular

automatonMesoTS Mesoscopic Traffic SimulatorMFT mean-field theoryMITRASIM MIcroscopic TRAffic flow SIMulatorMITSIM MIcroscopic Traffic flow SIMulatorMIXIC Microscopic model for Simulation of

Intelligent Cruise ControlMLC mandatory lane change

moving localised clusterMOE measure of effectivenessMPA matrix-product ansatzMPCF marginal private cost functionMSA method of successive averagesMSCF marginal social cost functionMSP moving synchronised-flow patternMT movement timeMUC-PSD multi-class phase-space densityNaSch Nagel and SchreckenbergNAVSTAR Navigation Satellite Timing and RangingNCCA number conserving cellular automatonNSE Navier-Stokes equationsOCT oscillatory congested trafficOD origin-destinationODE ordinary differential equationOSS Open Source SoftwareOVF optimal velocity functionOVM optimal velocity modelParamics Parallel microscopic traffic simulatorPATH California Partners for Advanced Transit

and HighwaysProgram on Advanced Technology forthe Highway

PCE passenger car equivalentPCU passenger car unitPDE partial differential equationPELOPS Program for the dEvelopment of

Longitudinal micrOscopic trafficProcesses in a Systemrelevantenvironment

PeMS California Freeway PerformanceMeasurement System

PHF peak hour factor

PLC pinned localised clusterpMFT paradisiacal mean-field theoryPRT perception-reaction timePSD phase-space densityPW Payne-WhithamQoS quality of serviceSFI-TCA stochastic Fukui-Ishibashi traffic

cellular automatonSimone Simulation model of Motorways with

Next generation vehiclesSLD single inductive loop detectorSMARTEST Simulation Modelling Applied to Road

Transport European Scheme TestsSMS space-mean speedSOC self-organised criticalitySOMF site-oriented mean-field theorySP synchronised-flow patternSSEP symmetric simple exclusion processSTA static traffic assignmentSTCA stochastic traffic cellular automatonSTCA-CC stochastic traffic cellular automaton

with cruise controlSUE stochastic user equilibriumSUMO Simulation of Urban MObilityT2-TCA Takayasu-Takayasu traffic cellular

automatonTASEP totally asymmetric simple exclusion

processTCA traffic cellular automatonTDF travel demand functionTMC Traffic Message ChannelTMS time-mean speedTOCA time-oriented traffic cellular

automatonTRANSIMS TRansportation ANalysis and SIMulation

SystemTRB Transportation Research BoardTSG triggered stop-and-go trafficUDM ultra-discretisation methodUMTS Universal Mobile Telecommunications

SystemVDR-TCA velocity-dependent randomisation traffic

cellular automatonVDT total vehicle distance travelledVHT total vehicle hours travelledVMS variable message signVMT total vehicle miles travelledVOT value of timeWSP widening synchronised-flow patternWYA whole year analysis

2. List of symbols

ai the acceleration of vehicleiC the number of substreams in a traffic flowdx a single infinitesimal location in spacedt a single infinitesimal instant in timeη the efficiency of a road section

(according to Chen et al, [18])

E the efficiency of a road section(according to Brilon, [10])

F the free-flow curve in three-phase traffic theorygsi

the space gap of vehicleigl,b

sithe space gap at the left-back of vehiclei

gl,fsi

the space gap at the left-front of vehiclei

gr,bsi

the space gap at the right-back of vehiclei

gr,fsi

the space gap at the right-front of vehiclei

gtithe time gap of vehiclei

gl,bti

the time gap at the left-back of vehicleigl,f

tithe time gap at the left-front of vehiclei

gr,bti

the time gap at the right-back of vehiclei

gr,fti

the time gap at the right-front of vehicleihs the average space headwayhsi

the space headway of vehiclei

hl,bsi

the space headway at the left-back of vehiclei

hl,fsi

the space headway at the left-front of vehiclei

hr,bsi

the space headway at the right-back of vehiclei

hr,fsi

the space headway at the right-front of vehiclei

ht the average time headwayhti

the time headway of vehicleihl,b

tithe time headway at the left-back of vehiclei

hl,fti

the time headway at the left-front of vehiclei

hr,bti

the time headway at the right-back of vehiclei

hr,fti

the time headway at the right-front of vehiclei

J the wide moving jam lineJ in three-phasetraffic theory

k the densitykc the density of thec-th substream in a

traffic flowkc the critical densitykcrit the critical densitykj the jam densitykjam the jam densitykmax the jam densitykl the density in lanelkout the density associated with the queue discharge

capacityk(t) the density at timetK the length of a measurement region

(i.e., a certain road section)Kld the length of a detection zonel the average length of a vehicleli the length of vehicleiL the number of lanes on a roadN the number of vehicles in a measurement regionNl the number of vehicles in the measurement

region in lanelNl(t) the number of vehicles in the measurement

region in lanel at timet

N(t) a cumulative count function

N(t) a smooth approximation ofN(t)

ot the average on-time of a set of vehiclesoti

the on-time of vehicleioti,l

the on-time of vehiclei in lanel

q the flowq|15 the peak flow rate during one quarter hour

within an hourq|60 the average flow during the hour with the

maximum flow in one dayqb a background flowqc the flow of thec-th substream in a traffic flowqc the capacity flowqcap the capacity flowqe(k) an equilibrium relation between the flow and

the densityql the flow in lanelqmax the capacity flowqout the outflow from a (wide moving) jam,

the queue discharge capacityq(t) the flow at timetρ the occupancyρi the occupancy time of vehicleiρl the occupancy in lanelRs a spatial measurement region at a fixed time

instantRt a temporal measurement region at a fixed

locationRt,s a general measurement regionσ2

s the statistical sample variance of the space-mean speed

σ2t the statistical sample variance of the time-

mean speedS the synchronised-flow region in three-phase

traffic theoryτi the reaction time of vehiclei’s drivert a time instantTi the travel time of vehicleiTmp the duration of a measurement periodT (t0) the experienced dynamic travel time, starting at

time instantt0

T (t0) the experienced instantaneous travel time,starting at time instantt0

vc the capacity-flow speedvcap the capacity-flow speedvff the free-flow speedvi the speed of vehicleivi,l the speed of vehiclei in lanel

vi,l(t) the speed of vehiclei in lanel at timet

vmax the maximum allowed speed (e.g., by animposed speed limit)

vs the space-mean speed

vscthe space-mean speed of thec-th substream

vse(hs) an equilibrium relation between the

SMS and the average space headwayvse

(k) an equilibrium relation between theSMS and the density

vse(q) an equilibrium relation between the

SMS and the flowvsust the sustained speed during a period of high

flowvt the space-mean speedvtc

the time-mean speed of thec-th substreamv(t, x) the local instantaneous vehicle speed at time

instantt and locationxw the characteristic/kinematic wave speed (of a

wide moving jam)xi the longitudinal position of vehicleiXi the distance travelled by vehiclei

ACKNOWLEDGEMENTS

Dr. Bart De Moor is a full professor at the KatholiekeUniversiteit Leuven, Belgium. Our research is supported

by: Research Council KUL: GOA AMBioRICS, sev-eral PhD/postdoc & fellow grants,Flemish Govern-ment: FWO: PhD/postdoc grants, projects, G.0407.02(support vector machines), G.0197.02 (power islands),G.0141.03 (identification and cryptography), G.0491.03(control for intensive care glycemia), G.0120.03 (QIT),G.0452.04 (new quantum algorithms), G.0499.04 (statis-tics), G.0211.05 (Nonlinear), research communities (IC-CoS, ANMMM, MLDM), IWT : PhD Grants, GBOU(McKnow), Belgian Federal Science Policy Office:IUAP P5/22 (‘Dynamical Systems and Control: Com-putation, Identification and Modelling’, 2002-2006),PODO-II (CP/40: TMS and Sustainability),EU: FP5-Quprodis, ERNSI, Contract Research/agreements:ISMC/IPCOS, Data4s,TML, Elia, LMS, Mastercard.

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[122] Note that when calculating the total density using equa-tion (5), the partial densities can also correspond withoutloss of generality to different vehicle classes instead ofjust different lanes [27, 115].

[123] In most cases, volume denotes the number of vehiclescounted during a certain time period, as opposed to flowwhich is just the equivalent hourly rate.

[124] Note that the hypothesis also assumes that the variablesare spatially measured, e.g., space-mean speed.

[125] http://www.telematicscluster.be[126] Note that in a broader sense, queueing delays also en-

compass delays at signallised and unsignallised intersec-tions.

[127] The TRB was formerly known as theHighway ResearchBoard (HRB).

[128] Note that this capacity flow is not an extreme value, i.e.,it can be different from the maximum observed flow. Thereason is that, with respect to the nature of the fundamen-tal diagram, the capacity flow is taken to be anaveragevalue [44, 72].

[129] The detector station is called CLO3, which is an acronymfor ‘Camera Linkeroever’.

[130] But note that bottleneck-induced traffic flow breakdownsare not excluded by the theory of Kerner et al.

[131] In addition, they also provided a link with Kerner’s three-phase theory, whereby synchronised flow can correspondto HCT, OCT, or PLC, and moving jams can correspondto TSG or MLC states [49].