2.8 Time Headway Concepts
2.8.1 Drivers’ behavior and time headway at roundabouts
Driver behaviour at roundabouts is mostly affected at the roundabout entry. This is because the circulating vehicles have continuous movement, while the entry vehicle must judge the gap to be accepted before entering the roundabout. The driver’s behaviour influences the number of vehicles that accept the same critical gap with the circulating vehicle, which is termed the follow-up headway. Follow-up headway is defined as the difference in time between a departure vehicle and the immediately following vehicle at the roundabout entry if the two vehicles accept the same gap in the circulating stream under queuing conditions (Tracz et al., 2004). The follow-up headway is influenced by many factors which include roundabout geometry, traffic movement at the roundabout, the circulating traffic volume, the pedestrian volume, vehicle type and speed, waiting time, driver age and gender (Lord-Attivor and Jha, 2012b, Zong et al., ND). Rain is a factor that affects the driver’s visibility at the entry, and when visibility is affected the entry driver may not be able to judge the gap to be accepted nor the speed of the circulating vehicle correctly. It can therefore be reasoned that it will have an effect on the entry vehicle’s waiting time and the follow- up time as the entry drivers will be more cautious before entering the roundabout. The effect of the waiting time might have a knock-on effect on the number of vehicles that enter the roundabout, affect the capacity, cause delay and decrease the quality of service of the roundabout. This may depend on the rain intensity, although the extent of rain intensity on follow-up headway is yet to be determined.
The follow-up time at roundabouts varies in places around the world. Rodegerdts et al. (2007) show that the average follow-up time in the US is 3.2 seconds, while Dahl and Lee (2012) found the average follow-up time in Canada to be 3.30 seconds. There are many factors that contribute to drivers’ behaviour at roundabouts. They include road infrastructure, vehicle type, traffic and ambient conditions as well as the ability to estimate the circulating vehicle speed (Lord-Attivor and Jha, 2012a, Johnson, 2013, Ben-Edigbe, 2016). The critical gap is the safe time heaway of the circulating vehicle for entrance of the entry vehicle. In general, critical gap is a parameter that depends on local conditions such as geometric layout, driver behaviour, vehicle characteristics, and traffic conditions. Follow-up time is the minimum time headway between two successive vehicles entering the roundabout if the available gap is big enough. Follow-up time is a time headway.
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The roundabout is an at-grade intersection that operates on the yield rule where the entry vehicles give priority to the circulating vehicles. The yield rule operates on the availability of a gap within the circulating traffic. Whenever a gap is available, the entry vehicle will look for a safe gap in the circulating traffic before accepting and entering the roundabout. Sometimes when safe gaps appear in the circulating traffic stream, they are not taken by drivers entering the roundabout. Others may select to enter the roundabout when it is deemed unsafe. After all, driver behaviour at roundabouts is what the driver actually does, not what the driver can or is expected to do at roundabouts. It raises the question of what exactly can be construed a safe gap. One thing is clear, the available gap determines the number of the vehicles that enters the roundabout.
Critical gap is one of the factors that determines the number of vehicles that enter the roundabout. HCM (2000) define “critical gap as the minimun time between successive major street vehicles where the street vehicles make a maneuver.” This is a generalised critical gap definition for intersection but the definition for the critical gap at a roundabout can be defined as the minimum acceptable time headway between sussessive circulating vehicles by the entry vehicle. Any changes in the critical gap will have an effect on the entry vehicles and could also affect the service provided by the roundabout. Critical gap is influenced by the drivers’ behaviour, entry angle, sight condition, pavement markings and vehicles’ performance (Xu and Tian, 2008, Lord-Attivor and Jha, 2012b, Raff, 1950, Kang et al., 2012).
The effect of rainfall on traffic flow rate and driver visibility as discussed shows that drivers’ visibility is affected, and it can therefore be reasoned that the circulating vehicles may be affected by the poor visibility due to rainfall. Impaired visibility might make drivers reduce speed and affects the time headway at the circulating stream, this might affect the availability of acceptable gap by the entry vehicles. The waiting time of the entry vehicles at the entry while taking time to cautiously judge correctly the circulating vehicle speed and the safe gap under rainfall might lead to a reduction in critical gap availability. It could be reasoned that rainfall might affect the critical gap, but the extent of the effect with the varying intensity is yet to be investigated. Given a rainfall scenario at roundabouts, it is necessary to know the interaction of vehicle entering and circulating the central island. The key question is whether established and probable critical gaps and follow- up time headway under dry weather are the same under rainy conditions? Should there be
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differentials, to what extent can be a concern. Would there be a significant differential along the rainfall precipitation line, it may be asked. The extent of driver behavioural changes and critical gap characteristics under rainy conditions has not been studied, hence the procedure adopted in this thesis is novel. There is a consensus among researchers that follow-up time and critical gap are two important roundabout performance measures, even though their values vary depending on the computation method used.
The best way to describe this stochastic process of vehicle arrivals is by using the time headways between vehicles. As the estimation of headways is important to several applications in traffic engineering, it has been researched and documented by (Van As, 1993, Akcelik and Chung, 1994a, Ashalatha and Chandra, 2011, Jenjiwattanakul and Sano, 2011), among others and will be discussed in this thesis. According to May (1990), time headways vary considerably between two boundary conditions. Under low traffic flow rate, time headways can be considered random, when flow rate is near capacity time headways are constant.
In previous studies, a maximum likelihood Raff (1950), Ashworth (1968), Siegloch (1973), and Wu (2012) methods among others were used to estimate critical gap. The Raff model is based on cumulative density function of the accepted and rejected headway used at the intercept as the critical headway. Rodegerdts (2007) show that critical gap in the US is in the range of 3.7 to 5.5 seconds, Dahl and Lee (2012) found the critical gap to be between 3.5 to 6.1 seconds in Canada, Manage et al. (2003) showed that the critical gap in Japan ranges between 3.26 to 4.90 seconds, while Qu et al. (2014)mentioned that the critical gap is in the range of 2.6 to 3.2 seconds in China. So it is postulated that time headway has no fixed value. It varies relative to prevailing conditions.
Raff’s model is widely used in many countries owing to its simplicity and practicality. Wu (2012) mentions that the critical headway based on the Raff model does not consider the average critical headway. The Raff method of estimating the critical gap uses the cumulative probability of the rejected and accepted gap (Fr and Fa). Raff’s threshold method is one of the earliest methods of
gap acceptance estimation and is adopted in many studies for its simplicity. It can be expressed as:
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1 – Fr (t) = Fa (t) [2.57]
Raff’s consideration that the number of rejected gaps larger than the critical gap is equal to the number of accepted gaps smaller than critical gap which can be expressed as;
𝑁𝑎𝐹𝑎(𝑡̂𝑐) = 𝑁𝑟[1 − 𝐹𝑟(𝑡̂𝑐)] [2.58]
Then;
𝐹𝑎(𝑡̂𝑐) = 𝛾𝑟
𝛾𝑎[1 − 𝐹𝑟(𝑡̂𝑐)] [2.59]
The proportion of rejected gaps larger than the critical gap is equal to the proportion of accepted gaps smaller than the critical gap because N is fixed. Two proportions can be counteracted, so that the total accepted coefficient is equal to the accumulated probability of the headway (t) larger than the critical gap as illustrated below;
𝛾𝑎= 𝑃(𝑇 ≥ 𝑡̂𝑐) + 𝛾𝑎𝐹𝑎(𝑡̂𝑐) − 𝛾𝑟[1 − 𝐹𝑟(𝑡̂𝑐)] [2.60] = 𝑃(𝑇 ≥ 𝑡̂𝑐) = ∫ 𝑓(𝑡) ∞ 𝑡̂𝑐 𝑑𝑡 = 1 − 𝐹𝑟(𝑡̂𝑐) = 𝛼𝑒 −𝜆(𝑡̂𝑐−𝑡𝑚) [2.61] Then, 𝑡̂𝑐= 𝑡𝑚− 1 𝜆𝑙𝑛 ( 𝛾𝑎 𝛼) [2.62] Where,
t denotes headway; Fr (t) is cumulative probability of the rejected gap: Fa (t) is the cumulative probability of the accepted gap; P (.) denotes the probability of gap interval; f(t) denotes the probability function of headway in a major stream; F (t) denotes the cumulative probability function of headway in a major stream; λ denotes decay constant, = αq/(1-qtm); tm is the minimum headway, and α denotes the proportion of free vehicles.
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Ashworth (1968) assumes that the headway of a major stream follows the negative exponential distribution and critical gap and the accepted gap follows normal distribution, Ashworth gives the calculation formula of critical gap as follows:
𝑡̅𝑐 = 𝑡̅𝑎− 𝑞𝑎𝜎
2
[2.63]
Where; q is the flow rate rate of the major stream (veh/s), 𝑡̅𝑐 is the average critical gap (s), 𝑡̅𝑎 is
the average accepted gap (s), 𝑎𝑛𝑑 𝑎𝜎
2
is the variance of accepted gaps (s2). The standard deviation of the accepted gaps (s) is shown below in equation 2.64 where s is the standard deviation of the accepted gap (s), x is the accepted gap (s), 𝑥̅ is th mean of the accepted gap (s) and n is the total number of accepted gap.
[2.64]
Ashworth (1970) estimates critical gap using standard deviation of the accepted gaps within the circulating traffic stream, the circulating traffic flow rate and the mean accepted gap. However, Miller (1974) modified Ashworth’s equation on the hypothesis that critical gap follows distribution. 𝑡̅𝑐 = 𝑡̅𝑎− 𝑣𝑝𝑐𝜎 2 [2.65] 𝜎𝑐 = 𝜎𝑎 𝑡̅𝑐 𝑡̅𝑎 [2.66] Where, 𝑐𝜎 2
is the variance of critical gap (s2).
Wu (2012) did not require any assumptions concerning the distribution function of critical gaps and the driver behaviour, rather probability density function was used to estimate critical headways as illustrated below in equation 2.67.
Ftc(t) =
Fa(t)
Fa(t)+1−Fr(t) [2.67]
Where: Ftc(t) = PDF of the critical headway; Fta(t) = PDF of an accepted gap t, and
Fr(t) = PDF of a maximum rejected gap t.
1
)
(
2n
x
x
s
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If a time gap is sorted in ascending order, critical headway is estimated with the equation 2.68.
𝑡𝑐= ∑ [𝑃𝑡𝑐(𝑡𝑗). 𝑡𝑗+𝑡𝑗−1 2 ] 𝑁 𝑗=1 [2.68] Where 𝑃𝑡𝑐(𝑡𝑗) is the frequency of the calculated critical headways between j and j-1.
Logit’s method is a weighted regression model to estimate the critical gap as shown in equation 2.69.
[2.69]
Where:
= Probability of gap acceptance by the entry vehicle; x = gap duration. = Regression coefficient; = Regression coefficient
Solving for x by assuming that the probability of accepting a gap is 50 percent, then substitute 0.5 for P to obtain the critical gap. This method is adopted for critical gap estimation in many studies (Gattis and Low, 1999, Ashalatha and Chandra, 2011, Vasconcelos et al., 2013). Brilon et al. (1999) and Vasconcelos et al. (2013) discovered that this method underestimates critical gap when compared to other methods of estimating critical gap. The Logit method was not recommended for critical gap estimation by Brilon et al. (1999) because it was found to be dependent on conflicting traffic volume. The Probit method of critical gap estimation uses a best-line fit to a weighted linear regression of gap data. The interval is divided into suitable portions, and the proportion of accepted gap is determined. The process of gap acceptance is a binomial response and is dependent on the size of the gap. On the assumption that the critical gap is normally distributed, the probit of the proportion accepting a gap is shown in equation 2.70.
𝑌 = 5 + (𝑥− µ)σ [2.70]
Where: x = Accepted gaps proportion; and = normal distribution parameters. Y = Probit of x; 5 is added to the equation to keep the probit value.
) ( 0 1 1 1 x e p p 0
1
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Plotting probit versus gap size logarithm with a best-fit line, the critical gap can be obtained from the best-fit line as the x corresponding value of probit is 5. Siegloch (1973) uses the average number of entry vehicles and the accepted gaps to determine the critical gap and follow-up headway. He developed a linear relationship with the number of vehicles that accept gaps. He determined the accepted gap by plotting the number of vehicles accepting the gap as the dependent variable, and average accepted gap as the independent variable. The reciprocal of the gradient of the positive linear relationship gives the follow-up headway. He used the x-intercept and the estimated follow-up time to estimate the critical gap as:
𝑡𝑐 = 𝑡0+ 0.5𝑡𝑓 [2.71]
Where:
tc= Critical gap (s);
t0= X-axis intercept (s) = 2.51s (HCM 2000) tf = Estimated follow-up headway (s).
According to HCM (2010) follow-up time (tf) and critical gap (tc) can be estimated with the following equations: 𝑄𝐸= 𝐴𝑒(−𝐵𝑞𝑐) [2.72] Where; A = 1130 = 3600 𝑡𝑓 → tf = 3.19s [2.73] 𝐵 = 0.0007 = 𝑡𝑜 3600= 𝑡𝑐−0.5𝑡𝑓 3600 → tc = 4.11s [2.74]
Note that in equations 2.73 and 2.74, the parameters have fixed values for A and B. By implication, if the values of A and B can be computed by any valid method, follow-up time and critical gap can be estimated along the HCM (2010) line, it can be argued. If Kimber’s equation is modified to include a dummy variable as shown in equation 2.75, and simple substitution of F for A and fc for B, then the prevailing follow-up time and critical gap can be computed thus;
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𝑄𝐸= 𝑘 {(𝐹 − 𝑓𝑐𝑄𝑐)+ ∈} → fcQc ≤ F [2.75] QE = 0 when fcQc > F
𝑘 = 1 − 0.00347(∅ − 30) − 0.978( 1r− 0.05) [2.76]
Where;
Qe = entry flow rate (pcu/h) and
Qc = circulating flow rate (pcu/h)
Consider equation 2.75 again, when it rains, a dummy variable (ε) is introduced to depict that condition, hence one (1), otherwise zero (0) for dry weather.
𝑡 = 𝑌 𝑒
−𝛽𝑄 [2.77] t(s) Dry Rainfall∫ (
𝑌𝐷 𝑒−𝛽𝑄𝑐)
𝑒 𝑎𝜕𝑥 − ∫ (
𝑌𝑅 𝑒 −𝛽𝑄𝑐)
𝑒 𝑎𝜕𝑥 [2.78]
tc a b c d e =1.0 xa is the volume capacity ratio of 0, b, c and d are the volume capacity ratio increase towards 1.
Figure 2.11: Hypothetical time headway changes caused by rainfall
The key parameter F allows the influence of geometrical parameters like entry width, flare length and approach width to be determined. By adjusting F, the slope of the linear equation that also
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contains the major capacity geometrical relationships is preserved. Average time headway for vehicles circulating the roundabout can be estimated and adjusted to critical gap by considering the average vehicle length given that the difference between headway and gap is the lead vehicle length. When converting time headway to gap, the average circulating vehicle speed is also needed. For example, assuming F is 1000 veh/h, the average follow-up time headway will be 3600/1000 = 3.6 seconds. Assuming the average travel speed is 10m/s and the average vehicle length is 5m then the average gap time = 3.6 – [5/10] = 3.1seconds. So, there is no need to build a new model. What is needed is a modification of the relevant existing methods to accommodate rainy conditions. Shown below in figure 2.11 is a hypotheical time headway (t) relationship with degree of saturation. It is postulated that once the optimum flow rate is reached, the influence of rainfall is nullified. Thereafter, the peak traffic conditions set in and control time headway (t).