Keyword

1 Introduction

It is adopted that the critical headway for drivers of vehicles moving from the roundabout entry (tg) is the value of headway between the vehicles in the main stream such that each headway with the value equal or greater will be used for performing a manoeuvre of entering the roundabout lane by the respective driver from the subordinated entry (average in statistic terms) whereas each distance with value lower (that prevents performing the intended manoeuver) cannot be used. Critical headway is not a constant value. It adopts different values for different drivers and for each driver at different times. Critical headway is a random variable, with its value depending on the characteristics of human and vehicle and geometrical and movement conditions of the intersection, which can be characterized by probability distribution.

In the models used for determination of traffic capacity of the entries to intersections with right of way and roundabouts based fully or partially on gap acceptance theory, the critical headway and follow-up headway (t f) are the basis parameters of the decision-making process in organization of vehicle traffic. These parameters usually represent the average behaviours in a population of drivers. In more extended models used in the literature, based on complex probability distributions, the number of model parameters increases. For example, in models based on the Cowan’s M3 distribution, apart from parameters t g and t f, determination of two other parameters is also needed. These are values of minimal headways between vehicles in the main stream and the share of vehicles moving freely. This group of models includes the models presented in a study [1].

There are a number of studies concerning various problems connected with traffic on roundabouts [2–16]. The paper presents an analysis in terms of studies connected with determination of critical headways for drivers of vehicles at the entries of roundabouts. Due to the very rich set of studies published so far, the focus was on the most important and most popular models and the most recent studies in this field.

2 Methods of Determination of Numerical Values of Critical Headways

Values of critical headway are not measured directly. They can be determined directly based on the headways rejected and headway accepted by individual drivers from the entries. Therefore, apart from typical errors connected with performing measurements, evaluation of this parameter also involves the error that results from indirect method of determination. This fact caused the development of many different methodologies and techniques to determine the consistent and unbiased estimator of the critical headway. There were over thirty various techniques and methods used for evaluation of critical headway at the intersections without traffic lights. Individual methods of estimation often yield very different values of critical headway. The most popular and the most frequently used techniques and methodologies for determination of critical headways are [17–20]:

  • Method by M. Raff and Hart based on cumulative curves. It is one of the first methods of estimation, where cumulative curves of rejected and accepted headways are construed. The value t g is read at the location of intersection of the curves. Therefore, t g is the value of the headway for which the number of headways accepted shorter than this value is equal to the number of headways rejected longer than this value. According to this method, critical headway corresponds to median (the second quartile, medium value).

  • Graphical techniques that use the gap acceptance curve (e.g. Harders method, Blunden method, Ashworth method with correction that depends on the variance of distribution of individual critical headways). It is adopted that t g corresponds to the value of 50 % of the acceptance curve. The acceptance curve is construed through determination of quotients of the headways accepted and the total of headways rejected and accepted for each class range of headways.

  • The method of estimation for parameters of critical headway at the assumed type of probability distribution that uses the tools of mathematical statistics. These include: probit method, logit method and the method of highest credibility [22]. As explained in studies [18, 23], the method of highest credibility has a very high practical importance since it allows for obtaining estimators which are consistent, unbiased, best asymptotically efficient and linear with respect to random variables. The random distribution of the critical headway for the population of drivers is logarithmic-normal distribution which is characterized by the fact that it is rightward skewed and does not adopt negative values although it accepts occurrence of a short headway t g.

There is a probability in the estimation procedure that t g should occur between values of logarithms: the highest headway rejected by the driver and the headway accepted. This method assumes that the sample obtained from measurements represents the event with the highest probability possible, which is identical with the condition that credibility reaches maximum. The probability that t g will be between the values of logarithms of the highest headway rejected by a driver and the headway accepted is evaluated from the equation [18, 22]:

$$L = \prod\limits_{i = 1}^{n} {\left[ {F\left( {a_{i} } \right) - F\left( {r_{i} } \right)} \right]}$$
(1)

where:

F(a i):

cumulative distribution function for headways accepted,

a i :

logarithm of the i headway accepted,

F(r i):

cumulative distribution function for headways rejected,

r i :

logarithm of the i headway rejected.

The condition of the highest credibility is used for determination of the estimators of parameters of critical headway distribution, i.e. mean value E(t g) and variance D 2(t g). These estimators are a function of the parameters of logarithmic-normal distribution m and \(\sigma^{2}\) and are evaluated using the following relationships:

$$E\left( {t_{g} } \right) = \mathop {t_{g} }\limits^{ - } = e^{{m + \frac{{\sigma^{2} }}{2}}}$$
(2)
$$D^{2} \left( {t_{g} } \right) = E\left( {t_{g} } \right)^{2} \cdot \left( {e^{{\sigma^{2} }} - 1} \right)$$
(3)

  • The algebraic method that uses the relationship proposed by Drew [21] (this method is similar to the method by M. Raff and Hart):

    $$t_{g} = t + \frac{{\left( {t_{3} - t_{1} } \right)\Delta t}}{{\left( {t_{2} + t_{3} } \right) - \left( {t_{1} + t_{4} } \right)}}\begin{array}{*{20}c} {} & {[s]} \\ \end{array}$$
    (4)

    where:

    t 1 :

    t 2, t 3, t 4, are values of headways which are searched in the sample so that they meet the condition that the number of headways accepted (t 1 , t 2), lower than the specific value t is similar (or equal in ideal case) to the number of headways rejected (t 3, t 4) greater than the specific value t [s],

    T :

    time that corresponds to the beginning of the range where the values t 1, t 2, t 3, t 4 [s] were located,

    Δt :

    span of the time range [s].

  • The method based on headway histograms. It is assumed that critical headway corresponds to the modal value in the test.

  • Method by Weiss and Maradudin [24] that takes into consideration time loss incurred by drivers at the entries and drivers impatience. According to the assumptions of this method, the value t g decreases with the increase in time loss incurred by drivers at entries. With the increase in time loss, drivers are willing to accept shorter headways, i.e. probability of acceptance of a headway is increasing and the following inequalities are true:

    $$\eta_{1} \left( {t_{g} } \right) \le \eta_{2} \left( {t_{g} } \right) \le \ldots \le \eta_{n} \left( {t_{g} } \right)$$
    (5)

    where:

    \(\eta_{i} \left( {t_{g} } \right)\) :

    probability that the headway t g will be accepted by the driver i at the entry.

  • Method by Hagring [25] where t g is obtained as an optimal value between the values of critical headway that ensures entering to the main roadway of the intersection with maximal safety and the critical headway accepted by the driver in the case of high values of time loss incurred at the entry (in this case, t g is connected with risky and dangerous entering the circular roadway).

  • Method by Siegloch for conditions of saturation with traffic [18] where the value t g is evaluated from the function of using headways in the main streams.

In many cases, values of critical headways evaluated based on the above methods differ from each other.

3 Comparison of Models for Estimate Critical Headways at Roundabouts

Roundabouts are one of the more interesting road traffic arrangements, so they are frequently elements of multifaceted analysis (f. eg. [26–33]). Due to the importance of gap-acceptance and its impact on roundabouts capacity, a large number of critical headway researches and studies have been conducted. The selected studies characterizing the critical headway for drivers at one-lane roundabouts entries are presented in Table 1. Hagring [34] state that gap acceptance models can be classified as macro and micro approach. Macro approach applies to interactions between traffic streams and geometric considerations and micro approach applies to the driver level. The approach adopted in this article is based on micro analysis using detailed data of drivers behavior and vehicles trajectories. The main challenge of critical headway estimating is that the parameters of the distribution of the critical headways, can be estimated as a function of various explanatory variables like speeds of vehicles, traffic conditions, roundabout location, waiting time for a appear gap, type of vehicle and much more other factors. Depending on the structure of model, for estimation of parameters of the models a number of methods have been employed. For many modeling approaches the maximum likelihood method provides a lot of advantages and gave superior results. Taking into account estimating the critical headway at roundabouts, a number of studies have been conducted but focused mainly on one-lane roundabouts.

Table 1 Comparison of critical headway models for one-lane roundabouts
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It can be concluded from Fig. 1 that mean value of the parameter t g for all models is 4.40 s. The values t g below the mean can be observed in such countries as Israel, Portugal, Spain, Italy, Australia and the USA (data from HCM 2000 [38]) and Germany (studies by Brilon and Wu [37]). Furthermore, the values of the parameter t g which are higher than the average were found for such countries as Slovenia, Denmark, Poland and Germany (studies by Baier et all. [36]) and the USA (with is shown in the most recent studies cited in NCHRP 572 [39] and HCM 2010 [40].

Fig. 1
figure 1

Critical headways for one-lane roundabouts by different authors (Source Own)

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Moreover, the potential impact one of psychotechnical parameter t g on one-lane roundabout entry capacity estimate is examined by varying the parameter t g from minimum to maximum limits under different circulating flows (Q nwl). Critical gap value change between 2.2 and 5.2 s. This t g values were adopted on the basis of various authors research presented in Table 1 (see Fig. 2). Follow-up parameter is assumed to be constants. As can be seen from Fig. 2 one-lane roundabout entry capacity is higher if smaller critical gap values are accepted by drivers. Therefore, more accurate determination of the critical headway value is very important because improves the accuracy of roundabout entry capacity estimation.

Fig. 2
figure 2

The change in one-lane roundabout entry capacity estimated with different values of parameter t g (Source Own)

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Fewer models and results of evaluation of the parameter tg compared to single-lane roundabout can be found in the literature for entries at two-lane roundabouts. Selected models of the parameter t g for two-lane roundabouts were presented in Table 2. Table 2 shows that part of models were defined with accuracy of a single traffic lane at the entry and part of them concern only the general value of the parameter t g for the whole entry.

Table 2 Comparison of critical headway models for two-lane roundabouts entry
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It should be noted that for two-lane roundabouts the average values of critical headways are different by each entry lane and they are distinguished for the inner and outer circulating lanes. Figure 3 shows that mean value of the parameter tg for drivers from the right and left traffic lane at the entry of the two-lane roundabout amounts to 3.83 and 4.05 s. This means that the drivers from the right lane accept lower headways between the vehicles on the circular roadway than the drivers of vehicles from the left lane at the entry. The values t g which are over the mean value can be observed in such countries as Sweden, the USA and Poland (study by Macioszek [1]).

Fig. 3
figure 3

Estimates of critical headway for two-lane roundabouts by different authors (Source Own)

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Such type of roundabouts like turbo roundabouts are fairly recent development and projects connected with critical headways have only been implemented in a few works (Table 3; Fig. 4). Parameter t g depending on the scheme of conflict area with one or two circulating streams. So, the t g values are calculated separately for left and right lane on turbo roundabouts entry. The entering vehicles are faced by one or two circulating streams depending on scheme of conflict area.

Table 3 Comparison of critical headway models for turbo roundabouts entry
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Fig. 4
figure 4

Estimates of critical headway for turbo roundabouts by different authors (Source Own)

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The study [20] presented the empirical examinations aimed at determination of the value of the parameter t g on single-lane roundabouts. For the collected samples, the critical headway t g was determined graphically based on cumulative curves of the accepted and rejected headways and acceptance gaps as well as by means of the D. Drew’s method. Mean values and medians were also evaluated for individual samples of the accepted headways. Values of critical headways estimated using various calculation methods were presented in Table 4 and in Fig. 5.

Table 4 One-lane roundabouts results comparison for different estimation method
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Fig. 5
figure 5

One-lane roundabouts results comparison for each estimation method (Source Own)

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Values of critical headways evaluated based on the above methods differ from each other. Critical headways estimated from cumulative curves for headways accepted and headways rejected, acceptance curves and using the D. Drew’s method adopt the same values. However, these values are slightly lower than the values estimated as mean value and median of headways accepted (maximal difference of ±1.0 s).

4 Conclusions

The comparative analysis presented in the paper in terms of studies connected with determination of critical headways for drivers of vehicles at the entries to single-lane roundabouts, two-lane roundabouts and turbo roundabouts reveals that the values of the parameter tg obtained by individual authors differ from each other. These differences are in particular caused by the fact that values of this parameter cannot be measured in an indirect manner, which causes that, apart from the error connected with measurement, there are also errors that result from direct determination of the parameter tg. The values of the parameter tg presented in Tables 1, 2 and 3 were obtained by different methods such as regression, maximum likelihood method, logit method, Raff method, Troutbeck’s and other methods.

Differences in the values of the parameter tg for individual countries are also caused by cultural diversity, differences in behaviours of drivers, their habits and customs. The consequence of this fact is difference in traffic capacity of roundabouts in individual countries. Therefore, the attempt to adapt the model used for determination of traffic capacity at entries of roundabouts built in a country to the conditions of another country requires previous calibration of the model, which should in particular concern psychotechnical parameters of drivers’ vehicles at the entries of roundabouts i.e. critical headway and the follow-up headway.