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Accident Analysis and Prevention 35 (2003) 427433

A comparison of headway and time to collision as safety indicatorsKatja Vogel

Swedish National Road and Transport Research Institute, S-581 95 Linkping, Sweden

Received 13 September 2001; received in revised form 21 January 2002; accepted 13 February 2002

Abstract

The two safety indicators headway and time to collision (TTC) are discussed and compared with respect to their usefulness indetermining the safety of different traffic situations, like different locations in a junction. Over a 6-day-period traffic flow measures weretaken in a four-way junction with stop signs on the minor road. It was found that for vehicles in a car following situation headway and TTC

are independent of each other. The percentage of small headways is relatively constant across different locations in the junction, while thepercentage of small TTC values varies between different locations. It is recommended to use headway for enforcement purposes, becausesmall headways generate potentially dangerous situations. TTC, on the other hand, should be used when a certain traffic environment is tobe evaluated in terms of safety, because it indicates the actual occurrences of dangerous situations. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Safety assessment; Safety indicator; Time to collision; Headway

1. Introduction

Time headway (H) is one of the indicators that is usedto estimate the criticality of a certain traffic situation. It has

been defined as the elapsed time between the front of thelead vehicle passing a point on the roadway and the front ofthe following vehicle passing the same point (Evans, 1991,p. 313). In some countries, this indicator is also used bythe authorities to impose fines for close following. Anotherwidely used safety indicator is time to collision (TTC), aconcept introduced by Hayward (1972). It indicates the timespan left before two vehicles collide, if nobody takes evasiveaction. In this study, these two indicators will be compared,both theoretically and in relation to empirical data.

1.1. Time headway

Time headway is measured by taking the time that passesbetween two vehicles reaching the same location (seeEq. (1)).

H = ti ti1 (1)

with ti denoting the time at which the vehicle i passes acertain location and ti1 the time at which the vehicle aheadof vehicle i passes the same location.

Tel.: +46-13-20-43-06; fax: +46-13-14-14-36.E-mail address: [email protected] (K. Vogel).

Different countries have slightly different rules with re-gard to the legal or recommended safety distance. In the US,e.g. several driver training programs (Michael et al., 2000)state that it is impossible to follow a vehicle safely with

a headway of less than 2 s. In Germany, the recommendedminimum distance is half the speedometer, which means,a car traveling at 80 km/h should keep a distance of at least40 m. This rule translates to a recommended time headwayof 1.8 s. Fines are imposed when the time headway is smallerthan 0.9 s. In Sweden the National Road Administration rec-ommends a time headway of 3 s in rural areas, and the po-lice use a time headway of 1 s as orientation for imposingfines.

Researchers investigated whether any connection betweenpreferred time headway, accident involvement, and drivercharacteristics existed, but the results are not consistent.Evans and Wasielewski (1982), e.g. claimed that drivers whokeep longer time headways tend to have a history of feweraccidents and violations. On the other hand, the same authorsstated 1 year later (Evans and Wasielewski, 1983), that noreliable relation between preferred time headway and acci-dent involvement could be detected. Van Winsum and Heino(1996) investigated in a simulator study whether a closerfollowing distance was connected to more expertise in ac-curately estimating TTC, but the relationship they foundwas not significant. Michael et al. (2000) found that a sub-stantial percentage of drivers in several urban locations didnot observe the 2 s rule, but compliance increased moder-ately when hand-held signs urged drivers to heed the rule.

0001-4575/02/$ see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S 0001-457 5(02)000 22-2

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428 K. Vogel/ Accident Analysis and Prevention 35 (2003) 427433

However, no direct relation to the occurrence of traffic con-flicts was provided.

1.2. Time to collision

TTC is computed according to Eq. (2).

TTCi =Xi1(t) Xi(t) li

Xi(t) Xi1(t) Xi(t) > Xi1(t) (2)

with Xi denoting the speed of vehicle i, Xi the position ofvehicle i, li the length of vehicle i and i1 the vehicle aheadof vehicle i.

The TTC is the time that is left until a collision occurs ifboth vehicles continue on the same course and at the samespeed. Therefore it is the time that is needed to cover thedistance between the lead and the following vehicle withthe relative speed between the lead and the following ve-hicle. Hayward (1972) first introduced this concept, and it

is discussed extensively in Hydn (1987), for example. Forcalculation of TTC to be possible, the vehicles in ques-tion have to be on collision course, but not necessarily ina car-following situation. TTC in car-following situationsis only defined when the speed of the following vehicle ishigher than the speed of the lead vehicle. In research TTChas often been used as a safety indicator for certain ma-neuvers by determining the minimum TTC measured dur-ing the maneuver (e.g. van Winsum and Heino, 1996; Hirstand Graham, 1997; Janssen and Nilsson, 1991). Accordingto Svensson (1998), TTC is an indicator for a traffic con-flict and is, thus, inversely related to accident risk (smallerTTC values indicate higher accident risks and vice versa).In a recent article, Minderhoud and Bovy (2001) suggest amethod, which allows using TTC to compare the safety ofdifferent drivers, road environments, or situations in gen-eral. The basic idea is to sample TTC values over time, andto examine how often a certain driver undershoots a givenlower safety limit, or how often this limit is breached on aparticular road stretch or under particular conditions. In theliterature different opinions can be found as to which valueshould be used as safety limitsuggestions range from 1.5sin urban areas (Svensson, 1998) to 5 s (Maretzke and Jacob,1992). Minderhouds and Bovys (2001) method will be ap-plied on empirical data in this study, and different threshold

values will be compared.

1.3. Relation and comparison

A comparison of the two equations shows that more vari-ables have to be known to determine TTC than to determine

H. The relationship between these two measures is presentedin Eq. (3).

TTC =Xi

Xi Xi1H, where H = H

li1

Xi= gap

(3)

with Xi1 denoting the speed of the lead vehicle, Xi thespeed of the following vehicle and li1 the length of the leadvehicle.

To obtain TTC, the speed of both the involved vehicleshas to be known in addition to the time gap. An interestingdifference between the two measures exists with respect to

traffic safety. It could be formulated such that time headwayis a step further away from a crash than TTC. This claimis based on the reasoning described later.

Let us consider a vehicle in the following mode. Sucha vehicle can have a relatively small headway, but a largeor even undefined TTC value. This situation occurs, whenXi1 is equal to or larger than Xi (cf. Eq. (3)). The situationbecomes critical only when something in the constellationchanges, like when the lead vehicle brakes such that Xi1becomes smaller than Xi. Thus, under stable circumstances,a small time headway can be maintained over extended pe-riods of time without resulting in an immediately danger-ous situation. If, on the other hand, the TTC value of the

following vehicle is small, something has to change in theconstellation if a crash is to be avoided. In a car-followingsituation, the average relative speed between following andlead vehicle cannot be larger than 0 if a collision is to beavoided in the long run.

To summarize, vehicles with small time headways can(and frequently do) have large or undefined TTC values,while small TTC values (in car following) are impossiblefor vehicles with long time headways. In a car-followingsituation, TTC can, in fact, never be smaller than H, becausethe term Xi/(Xi Xi1) (cf. Eq. (3)) can never be smallerthan 1. This is due to the fact that Xi1 is never negative,

which would mean that the lead vehicle was reversing. Inthe special case of a stopped lead vehicle (Xi1 = 0), theactual time gap (H) equals TTC.

As mentioned earlier, in a car-following situation TTCcan never be smaller than the time gap between the lead andthe following vehicle (H). Thus, if the two values are to becompared, it seems reasonable to exclude those cases thatare not safety critical with respect to any of the two measures(Table 1). In order to determine the threshold between safetycritical small and safe long headways, the existing liter-ature on the concept of free and following vehicles wasconsulted. A free vehicle is by definition not in interactionwith any vehicle ahead of it. For this reason, the analysesin the present study were limited to following vehicles. Thedefinition for a free vehicle that was adopted here is basedon an empirical analysis by Vogel (2002), which shows that

Table 1Relationship between TTC, headway and safety

Headway

Small Large

TTCSmall Danger imminent ImpossibleLarge Potential danger Safe

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vehicles with a time headway of more than 6 s choose theirtraveling speed independent of the vehicle ahead. The choiceof this threshold value is supported by the fact that no authorconsiders a TTC that is larger than 6s to be dangerous. Thismeans that even a stopped lead vehicle will not force thefollower to perform dangerous avoidance maneuvers if the

headway is larger than 6s (and the following driver can seethe vehicle ahead). Only for speeds higher than 130 km/h alarger time headway is needed to stop a vehicle safely be-hind a stopped vehicle.

An additional reason for limiting the analysis to follow-ing vehicles only is the possibility to compare different sit-uations with each other. It allows, e.g. comparing the safetylevel for situations with different traffic densities, eitheracross locations, or in the same location at different times.If it were of interest whether the safety level is influencedby time of day (daytime versus night-time), a simple com-parison of the percentage of small TTCs across all vehi-cles would mostly reflect the lower traffic volume during

night-time. This might lead to the assumption that safety in-creases during the night. If, on the other hand, only thosevehicles are considered that actually are in a car-followingsituation, a relative increase of small TTC values might befound during the night, which could for instance be ex-plained with driver fatigue and extended reaction times.

The last reason for restricting the analysis to followingvehicles is of methodological nature. A consequence of themeasurement technique used in this study is that TTC val-ues become less reliable the longer the time headway forthe vehicle in question. This is explained in more detail inSection 3. By excluding vehicles with a long headway from

Fig. 1. Schematic overview of the junction with indication of the location of the measurement sites.

the analysis, those unreliable TTC values are excluded aswell.

2. Method

The location of the study was a four-way junction in amid-sized town in Sweden. The junction is one of the mostaccident-prone locations in town. Stop signs were placedon the subordinate road and right-of-way signs were placedon the main road. The posted speed limit was 50km/h onall arms of the junction. All arms had one lane in eachdirection, except for one arm on the main road, which had aseparate lane for left-turning vehicles (Fig. 1). The junctionwas located in the outskirts of the town, the two streetsforming it were major roads leading to residential areas. Thecontinuation of the main road after measurement site Main1(indicated in Fig. 1) led out of town, the continuation aftermeasurement site Main5 led to the city center. Lampposts

were installed at the junction on all four arms.For 6 days in spring 2000, traffic flow point measure-

ments were taken 24 h a day at seven locations around thejunction. The weather during the measurement week waseither sunny or cloudy, but there was no precipitation. Themeasurement devices (described in detail in Anund, 1992)record the speed of each passing vehicle, its direction, thetime interval between two passing vehicles, and the axle dis-tance of each vehicle, from which the vehicle type can bededuced to a certain extent (Srensen, 1996).

The locations of the measurement devices are indicatedin Fig. 1. The measuring sites Main1 and Main4, as well as

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Sub1 and Sub2 were located at 115m distance from the cen-ter of the junction, the sites Main2 and Main3 were locatedat 17.5m distance from the center of the junction. Main5was located on the main road, approximately 600 m awayfrom Main4. There was no junction in the vicinity, exceptfor one small and not very frequently used by-road lead-

ing to a block of houses, at a distance of about 20m fromSub2. On site Main3 (direction out of town), left-turningvehicles could be registered separately due to the dividedlane.

Measurement sites were defined to be corresponding,when they were installed on equivalent locations with re-spect to the junction. In this sense, the locations Sub1 (direc-tion towards junction) and Sub2 (direction towards junction)are corresponding, as well as the locations Main2 (directionaway from junction) and Main3 (direction away from junc-tion).

3. Results

Between 20,000 and 40,000 vehicles were registered ateach location, depending on traffic volume. For each vehi-cle the driving speed, driving direction and the passing timewere recorded. Each driving direction was analyzed sepa-rately for each site. Only vehicles with time headways of atmost 6 s were considered in the analysis. Time headway andTTC were calculated as follows.

Time headway was calculated precisely as in Eq. (1). Ithas to be noted that time headway usually is defined as theelapsed time between the front of the lead vehicles and the

front of the following vehicles reaching the same location.In the present study, the vehicles were recorded as soon asthe front axle passed the measurement site. Thus, the elapsedtime between the two front axles reaching the same locationis taken as approximation of time headway.

For the calculation of TTC another approximation hadto be made. According to Eq. (2), the numerator should beequal to the distance headway minus the length of the leadvehicle, which is the distance gap between the two vehicles.As distance headway was not directly available from the datain this study, it was derived from time headway and travelingspeed of the following vehicle, based on the assumptionthat the vehicles traveled at a constant speed during themeasurement period. Vehicle length was approximated byadding 1.80 m to the axle distance of the lead vehicle. 1.80 mis equal to the average difference between vehicle lengthand axle distance for passenger cars. It was felt that thisapproximation was acceptable, because 95% of all recordedvehicles were passenger cars. However, no vehicle typeswere excluded from the analysis.

The denominator should be the relative speed between thetwo vehicles at measurement time ti. It was approximatedby the difference between the speed of the following vehicleat measurement time ti and the speed of the lead vehicleat measurement time ti1, with ti being the time when the

following vehicle was recorded at the measurement site, andti1 being the time when the lead vehicle was recorded atthe measurement site.

3.1. Traffic density

Neither the subordinate nor the main road operated at ca-pacity level any time. For both roads the number of vehi-cles per hour was very similar across weekdays, the patternlooked different for Saturday and Sunday, though, as therewere no rush hour peaks. The increase in traffic volume dur-ing the morning and the afternoon rush hours on weekdayswas more pronounced on the main road.

The percentage of following vehicles lay on averageslightly above 30% on the main road and somewhat below30% on the subordinate roads. The percentage of followingvehicles increased with increasing traffic volume, but at anytime of day there were at least 50% free vehicles on bothroads.

3.2. Relationship between time to collision and headway

For each site and each direction correlations between timeheadway and TTC were calculated for cases with valid TTCvalues (speed of following vehicle larger than speed of leadvehicle). At each site, the correlation was substantial if allvehicles were considered (average correlation r = 0.423;S.D. = 0.169), but close to 0 when only following vehicleswere considered (average correlation r = 0.077; S.D. =0.047). If only following vehicles are considered, TTC and

H can be regarded as practically independent of each other

and can be investigated separately.

3.3. Headway

For each measurement site and each direction, the per-centage of measured time headways that were below 1, re-spectively 2 s, given all following vehicles, was calculated.The results are presented in Fig. 2. The length of the blackfield shows the percentage of time headways below 1 s, andthe length of the gray field shows the percentage of timeheadways between 1 and 2s. The sum of the lengths of bothshows the percentage of time headways below 2 s (percent-age scale in lower left-hand corner), always given all fol-lowing vehicles at the site in question.

As can be seen in Fig. 2, the percentage of vehicles thatdrive with a time headway below 2s is relatively evenly dis-tributed on the main road (30.3% on average, S.D. = 5.99).The percentage of following vehicles traveling with a timeheadway below 1 s is on average 1.5% with a S.D. of 0.95.There are no big differences between the direction towardsthe junction and the direction away from the junction. Onthe subordinate road, on the other hand, more vehicles drivewith a time headway below 2 s when they are on their waytowards the junction. In general, the percentage of smallheadways is relatively similar on the corresponding sites. A

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Fig. 2. Percentage of vehicles with time headways smaller than 2 and 1 s, respectively, given all following vehicles, for each measurement site and eachdirection. The percentage scale is indicated in the lower left-hand corner.

comparison of the sites close to the junction (both directionson Main2 and Main3) with the sites further away from thejunction (both directions on all other sites) does not showany noteworthy differences with respect to percentage ofsmall time headway values.

Fig. 3. Percentage of vehicles with TTC values smaller than 5, 4, 3, 2 and 1 s, given all following vehicles, for each measurement site and each direction.The percentage scale is indicated in the lower left-hand corner.

3.4. Time to collision

TTC values are presented in a similar fashion in Fig. 3.The length of the black rectangle indicates the percentageof vehicles that had a TTC value of less than 1s at this

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measurement site, based on all following vehicles. Thelength of the dark gray field indicates the percentage ofvehicles with TTC values between 1 and 2 s, and so on.

Large differences with respect to the percentage of smallTTC values can be found between the different measurementsites. Generally, the sites closer to the junction have a higher

percentage of small TTC values, especially values below 1 soccur almost only close to the junction.Visual inspection shows that most corresponding mea-

surement sites have relatively similar percentages of smallTTC values. The corresponding measurement sites Main3(towards the junction, split into two lanes) and Main2 (to-wards the junction) are an exception. Main2 has a muchlarger percentage of small TTC values than Main3. More-over, the percentage of small TTC values on Main2 is con-siderably larger than at any other site.

3.5. Comment on the use of inferential statistics

Due to the following reasons, no inferential tests wereperformed to examine whether any of the observed differ-ences were statistically significant or not: (a) the number ofmeasurement sites was small (15), (b) the percentage valuesfor the different time boundaries (

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traffic. A left-turning lane releases both the pressure on theleft-turning vehicles that arises when a queue develops be-hind them, and the potential anger of those who have to waitbehind a blocked left-turner. If the number of left-turnersis approximately equally high at site Main2 as on Main3,adding a left-turning lane could lead to an increase in traf-

fic safety. It seems less likely that driving direction per sehas an effect (going into town vs. leaving town), becauseno such effect can be observed on the other correspondingsites. Additionally, the junction was located well away fromboth the city center and the city limits.

5. Conclusion

It was found that headway and TTC are independent ofeach other for following vehicles. Due to the fact that TTCvalues cannot be smaller than headway values, a short head-way can be interpreted as potential danger, because onlyvehicles that travel with short headways have the possibil-ity to produce small TTC values. A small TTC value, onthe other hand, represents actual danger, because an acci-dent can only be avoided by changing the situation actively.The two values are suitable for different purposes, becausethey provide different information. It is recommended thatauthorities use headway as criterion for tailgating, becauseit is easy to measure, it is easily understandable and inter-pretable, and most important of all, it is directed againstpotential danger, which effectively prevents dangerous TTCvalues from occurring at all.

TTC values, on the other hand, should be used if the ac-

tual safety of a situation has to be evaluated. A particularroad design or driver can be evaluated with respect to safetyby examining the actual percentage of dangerously smallTTC values within a given time frame. Similarly, it is pos-sible to evaluate the safety of in-car systems like intelligenttransport systems (ITSs) by comparing the same driver withand without the system. Traffic environments can be com-pared with respect to safety, and the same environment canbe analyzed at different times of day. Possible re-design ofa road stretch or recommendations for ITS devices can bebased on empirical grounds. The method could prove to bevery useful within the field of traffic simulation, as long asthe traffic model is based on accurate distributions of speedand headway.

Acknowledgements

I would like to thank VINNOVA, Sweden and the SwedishNational Road Administration for providing the financialsupport that made this study possible. I also thank AlbertKircher for helpful comments on the manuscript.

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Vogel 2003