## REVIEW article

Front. Built Environ., 16 November 2016
Sec. Transportation and Transit Systems
Volume 2 - 2016 | https://doi.org/10.3389/fbuil.2016.00028

# Methodological Frontier in Operational Analysis for Roundabouts: A Review

• Department of Civil, Environmental, Aerospace and Materials Engineering, University of Palermo, Palermo, Italy

## Introduction

### Calculation of Roundabout Operations

Gap acceptance models provide entry capacity estimates based on constant values of the critical headway and the follow-up headway which, in turn, represent average values for all the observed drivers. Considering constant values of the critical headway and the follow-up headway, the capacity of an entry always represents average conditions that are experienced by the users. Since actually variability and heterogeneity characterize drivers’ population, the assumptions on driver behavior above introduced can produce erroneous estimates of roundabout capacity. However, the critical headway and the follow-up headway being stochastically distributed cannot be considered as constant values, but each of them should be represented by a distribution of a set of values. Moreover, when capacity models based on the gap acceptance theory are used, analyst should specify the probability distribution of headways between vehicles in each major stream (Giuffrè et al., 2012a).

### The Aim of the Paper and Its Organization

Starting from these considerations and without claiming to be exhaustive, the article provides an overview of the key methodological issues in the operational analysis of the roundabouts. Focus is made on the derivation of the analytical-based models under steady-state (undersaturated) conditions at entries, the gap acceptance behavior, and the calculation of parameters included into the capacity equations, the issues of the stochastic nature of the traffic phenomena.

## Operational Analysis Issues at Roundabouts

Design and performance evaluation of a new (or an existing) roundabout is the core function of an operational analysis. In order to analyze operations of planned or existing roundabouts, the methods must allow a transportation analyst to assess the operational performance with regard to the use of the intersection and the elements of geometric design. However, modeling of real-world performances can result in a complex action especially when one has to evaluate: (1) the effect of exiting vehicles on entering driver’s decision (e.g., one can be uncertain of the intentions of the exiting or turning vehicles); (2) conditions of capacity constraint for one or more entries (with the consequent circulating flow downstream of the constrained entry less than the demand); (3) origin–destination patterns, which may influence the capacity of a given entry; (4) differences in vehicle fleet mixes, and so on. An operational analysis needs two kinds of estimates: the roundabout capacity and the level of service by using measures of effectiveness such as (control and geometric) delay and queues (Rodegerdts et al., 2010). When roundabout entry capacity must be calculated over a chosen observation period, steadiness and variability in traffic demand, as well as saturated or oversaturated conditions at entries, need to be specified. This requires the analysis of roundabouts with and without statistical equilibrium; based on the traffic conditions at entries, the use of probabilistic, deterministic, or time-dependent models is also needed (Mauro, 2010; Troutbeck and Brilon, 2016).

Roundabouts normally use gap acceptance rules. Since minor street drivers have to yield the right-of-way to circulating vehicles (that pass in front of the subject entry), entry capacities, just as service times, depend on the availability of major stream gaps, which should be large enough to enter into the intersection in a safe way. Thus, the operational performance of roundabouts can be influenced by the traffic volume desiring to enter a roundabout at a given time, the vehicle flow rate on the ring and the arrival headway distributions, as well as geometric design, vehicle and environment characteristics that affect each individual gap acceptance behavior. Geometry also plays a significant role in the evaluation of the operational performance at roundabouts: the angle at which a vehicle enters can affect the speed of circulating vehicles; the entry widths can determine the number of side-by-side vehicle streams at the yield line and can affect the rate at which the circulatory roadway may accommodate the vehicles; lane alignment can determine imbalanced lane flows on an entry and thus can influence entry capacity, etc. Thus, the geometric characteristics have an impact on the gap acceptance decision-making and then the capacity.

Entry capacity estimation is based on the critical headway and follow-up headway when the analytical-based (gap-acceptance) models are used to analyze roundabout performances. Thus, the accuracy of capacity estimation at roundabouts is dependent on the accurate estimation of these two parameters. Capacity calculation always provides average values, since it is based on constant values of critical headway and follow-up headway; however, the critical headway and follow-up headway are stochastically distributed and should be represented by a distribution of values. The analysis of this problem could be the starting point for assessing and trying to measure the uncertainty in roundabout capacity estimation.

The estimation of critical headway and follow-up headway cannot be end in itself, since the gap acceptance parameters are introduced into the capacity models for unsignalized intersections and roundabouts (Brilon et al., 1999). Most of methods published around the world for estimating the gap acceptance parameters are for unsignalized intersections; however, they can be extended to roundabouts, since they focus on the simple case of two one-way streets where only two movements are allowed: one minor stream, which gives priority to the major stream before entering the intersection. Today, for estimating the critical headway and the follow-up headway, and for calibrating the existing capacity models, the analysts necessarily have to distinguish data collected at single-lane sites from those surveyed at multilane sites to assess the effect of the number of lanes, the size of the diameter and the entry width, and to consider different traffic patterns with dominant and subdominant arrival flows (Rodegerdts et al., 2007, 2010).

Before introducing the key concepts and the methods to perform the roundabout capacity analyzes, the techniques actually used to estimate the critical headways and follow-up headways will be described in the following section.

### Estimation of the Critical Headway and the Follow-up Headway

The critical headway can be estimated from on-field observations by employing several techniques which, in general, fall into two classes: the first class of techniques is based on a regression analysis between the number of users, which can enter into a major stream gap and the time duration of this gap; in this case, saturated conditions are required and the queue must have at least one vehicle in it over the observation period. The second class of techniques, in turn, estimates the distribution of the critical headways and the distribution of follow-up headways independently.

Probabilistic approaches must be used to estimate the critical headway when the minor stream does not continuously queue. Thus, most of these methods require the appropriate observation of a minor street driver under unsaturated traffic conditions and his/her gap acceptance decisions at an entry of unsignalized intersections or roundabouts.

With reference to regression techniques, Siegloch (1973) proposed to observe a condition of continuous queuing on the minor street; thus, one can observe n realizations (that are always integer numbers) for the function n(τ) by counting the number of the minor stream vehicles that enter the roundabout using major stream headways of size τ: In order to represent the observed data, we can use the linear regression on the average headway size values (i.e. the dependent variable) against the number of vehicles that enter during this average headway size, $n¯$, or:

$τ=a+b⋅n¯$

in which the coefficients a and b have to be estimated. However, the average headway size from the observed τ values, for each realization n, should be computed before starting the regression; otherwise the more numerous observations for the smaller n would govern the whole result (Brilon et al., 1999).

The linear regression function in Eq. 1 would be correct if the critical headway and the follow-up headway were constant values; then it can be written as follows:

$n(τ)=0forτ<τ0τ−τ0τfforτ≥τ0$

where τ0 = τc τf/2 represents the intercept of the headway size axis and τf the slope of the linear regression above introduced. In this way, the critical headway τc can be calculated from the regression technique directly.

Under unsaturated traffic conditions, the regression techniques cannot be applied (see above); thus, the critical headway can be calculated through probabilistic approaches. On this regard, the most commonly used methods – but not limited to these – are Raff’s method (Raff and Hart, 1950), Ashworth’ method (Ashworth, 1970), the maximum likelihood technique (Troutbeck, 1992); subsequently, a brief description will be given.

According to Raff and Hart (1950), the critical headway (τc) represents that value of τ at which 1 − Fr(τ) = Fa(τ), that is the cross point of the cumulative distribution function Fr(τ) of the rejected headways (>τ) and the cumulative distribution function Fa(τ) of the accepted headways (<τ). The critical headway represents the median value (not the mean value) of the distribution.

Differently from Raff and Hart (1950); Ashworth (1970) found that the critical headway (τc) can be estimated from the mean (μa) of the accepted headways (τa) and the SD of the accepted headways (σa): τc = μaqc ⋅ σa2, where qc stands for the major stream traffic volume (in volume per second). It is noteworthy that the equation above is valid under the assumption of exponentially distributed major stream headways (with statistical independence between consecutive headways) and normal distribution for τc and τa. Thus, for the critical headway estimation, Ashworth’s method uses only the accepted headways, neglecting the rejected headways. In turn, the maximum likelihood method requires information about the accepted gap and the largest rejected gap for each driver.

Troutbeck (1992) described in detail this method based on the assumption that a driver’s critical headway is larger than the largest rejected headway (τr) and smaller than the accepted headway (τa). Thus the method calculates the probability of the critical headway being between the largest rejected headway (τr) and the accepted headway (τa). In order to estimate this probability, the driver’s behavior is assumed to be consistent. The likelihood that the driver’s critical headway τc will be between τr and τa is given by the difference between the two corresponding cumulative distribution functions Faa) − Frr).

Based on the two vectors of the observed {τr} and {τa}, the likelihood L* for a sample of n observed entering drivers is given by:

$L∗=∏a,r=1nFaτa−Frτrs$

whereas the logarithm L of the likelihood L* is given by:

$L=∑a,r=1nlnFaτa−Frτrs$

The probabilistic distribution for the critical headways is usually assumed to be log-normal. The likelihood estimators μ and σ2 (the mean and the variance of the critical headway distribution), which maximize L are the solutions to the two equations $∂L∂μ=0$ and $∂L∂σ2=0$. This leads to a set of two equations, which are depending on the vectors of the observed {τr} and {τa} and must be solved iteratively by using numerical methods. Troutbeck (1992) proposed a solution by using iterative numerical solution techniques; thus the mean critical headway and its variance could be computed by: $τ¯c=e(μ+0.5σ2)$ and $s2=τ¯c2⋅(eσ2−1)$. According to Tian et al. (1999), the mean critical headway could be calculated and used in various gap acceptance capacity and delay models, since it was an acceptable quantity for representing the average driver behavior.

Other methods for estimating critical headways have been also recommended for practical applications: Harder’s method (Harders, 1968) discussed in more detail by Brilon et al. (1999), the logit procedures, which provide many similarities to the classical logit models of transportation planning [see, e.g., Polus et al. (2005)], the probit procedures, having formulations similar to the logit models, and used to estimate the probability that a gap will be accepted; however, in the last flows should be managed a lot more carefully (Solberg and Oppenlander, 1964). Wu (2012) proposed a method for estimating the distribution function of critical headways at unsignalized intersections based on equilibrium of probabilities; in turn, Hewitt’s method (Hewitt, 1983) enabled the calculation of the probability distribution of the critical headways for entering drivers, which reject the initial lag; the method is based on observations of the time duration of the headways refused and eventually accepted by drivers. Hewitt (1985) also performed a comparison between some methods for measuring critical headway.

Differently from the critical headway, the follow-up headway can be estimated directly from on-field observations by measuring the difference between the entry departure times of the minor-street queued vehicles using the same gap in the major stream. Rodegerdts et al. (2007) observed that vehicles using the same gap usually have the same opposing vehicle time, which may be calculated based on the accepted lag or the accepted gap. By using the accepted lag, the opposing vehicle time can be calculated by adding the entry arrival time to the accepted lag; by using the accepted gap, in turn, the opposing vehicle time can be calculated by adding the entry arrival time to the total rejected gaps and lag. It is noteworthy that, at multilane sites, the follow-up headway may be also influenced by the dominant and subdominant arrival flows. However, further sites with dominant left-lane arrival flows should be examined to validate the concept; what is more, interdependencies between entering and circulating vehicles at multilane roundabouts can be observed, because of the priority reversal between entering and circulating vehicles. In these cases, specific analytical capacity models should be derived from observations of the driver behavior [see, e.g., Giuffrè et al. (2012b)]. The complexity of these models may lie behind the difficulty of observing the behavioral parameters, which are required to implement the model. Giuffrè et al. (2014) proposed a procedure to get the unknown behavioral parameters from traffic surveys; these parameters concerned the saturation headways, which often elude the direct observations, since traffic conditions in which they can be observed rarely occur. Thus, the unknown parameters were estimated through a regression model based on-field data surveyed at a multilane roundabout in Palermo, Italy; for further details, the reader is referred to the original sources abovementioned.

Based on a systematic literature review of empirical studies and researches developed in different countries with the objective to measure the major gap-acceptance parameters at existing roundabouts, Giuffrè et al. (2016) noted that the effect size – i.e., the statistical mean values of the critical headway and the follow-up headway, which each (primary) study presents – varied from study to study; hence, they performed the meta-analysis of effect sizes as part of the literature review through the random-effects model. The meta-analysis, indeed, was developed in order to work directly with the effect size from each study; thus, a summary effect was computed (and tested) for each gap-acceptance parameter – at different types of roundabouts – by synthesizing the site-specific estimates from prior studies. Compared to the results of each study, the single quantitative meta-analytic estimate, both for the critical headway and the follow-up headway, represented an appropriate and reliable quantity for describing an average driver behavior. At last, the meta-analytic estimate gave, with greater power than each effect size, a comprehensive measure for the parameters of interest; it could be used to recalibrate existing capacity models for single-lane roundabouts and double-lane roundabouts.

## Modeling Methods for Roundabout Capacity Analysis

Starting from the simple queuing model – in which a single minor traffic stream crosses a single major traffic stream – the capacity calculation for a roundabout in steady-state condition can be addressed by specifying the arrival headway distribution in the major stream of volume Qc (veh/h) and the gap-acceptance function which expresses the number of minor stream vehicles that can depart during an acceptable headway of size τ (Siegloch, 1973). However, an understanding of the interaction of two traffic streams can represent the basic sources of knowledge about capacity estimation for unsignalized intersections and roundabouts with more than two traffic streams.

Usually, n(τ) denotes the number of the minor stream vehicles, which can enter the roundabout using a time headway of size τ and f(τ) denotes the probability function of all headways in the major circulating stream. Based on the assumptions about user behavior at unsignalized intersections and roundabouts as introduced in the previous sections, a biunique correspondence exists between n(τ) and τ. This is why f(τ) can be also viewed as the probability distribution of n(τ) (Mauro, 2010); then the mean value $n¯$ can be calculated as follows:

$n¯=∫τ=0∞nτ⋅fτdτ$

When one divides this mean value $n¯$ to the average size of headways $τ¯$, clearly equal to $τ¯=1Qc$, one can get:

$C=n¯τ¯=Qc∫0∞nτ⋅fτdτ$

Equation 6 gives the mean number of minor vehicles, which perform their maneuvering in the time unit, i.e., the entry capacity. This equation for the entry capacity of unsignalized intersections and roundabouts forms the foundation of the gap acceptance theory; indeed, most of the analytical capacity models found in literature are based on this concept [see, e.g., Brilon et al. (1999) and Mauro (2010)]. The capacity provided by τ headways per hour is then Qc ⋅ f(τ) ⋅ n(τ); thus, the capacity is a function of the circulating flow (Qc, veh/h), which is synthesized by f(τ); in turn, n(τ) takes into account the users’ psycho-technical attitudes, which are synthesized by the critical headway and the follow-up headway (Mauro, 2010).

As a consequence of the equation above, the capacity of the simple two-stream situation can be calculated by methods based on the elementary probability theory when the following assumptions are met: (1) constant values for the critical headway and the follow-up headway; (2) exponential distribution for major stream headways (see “Counting Distributions”); (3) each traffic stream is characterized by constant values of the traffic volumes.

Considering constant values for the critical headway and the follow-up headway, two different types of capacity equations can be distinguished based on two different formulations for n(τ): the first type of capacity equations assumes a stepwise (constant) function for n(τ) (Harders, 1968), whereas the second type of capacity equations assumes a continuous linear function for n(τ) (Siegloch, 1973; Rodegerdts et al., 2010). It is noteworthy that when one models the capacity of entries conflicted by two (or more) circulating lanes, the conflicting flow rate is the total of all major streams (National Research Council and Transportation Research Board, 2010); thus, all major streams are combined as one traffic stream and examined by using proper multilane stream parameters. The headway distribution in the major traffic stream can be dealt by using appropriate values of the minimum headway and the bunching parameters.

In general, modeling arrivals of vehicles at a road cross-section is a fundamental step in traffic flow theory. An important application concerns traffic flow simulation in which vehicle generation has to represent vehicles arrivals. However, the vehicle arrival is a random process since several vehicles can come together, or vehicle arrivals can be rare events. Modeling vehicle arrivals means modeling how many vehicles arrive in a given interval of time, or modeling what is the time interval between two arrivals of successive vehicles. In the first process, the random variable is the number of vehicle arrivals observed in a given interval of time; it takes some integer values. Thus, the process can be modeled by a discrete distribution. In the second process, the random variable is represented by the time interval between successive arrival of vehicles and it can be any positive real values; thus, some continuous distributions can be considered to model the vehicle arrivals. It is noteworthy that, being these processes correlated, the distributions that describe them should be also inter-related for better explaining this traffic phenomenon (Kadiyali, 1987; May, 1990).

Bearing in mind the objective to estimate entry capacity at roundabouts, in the following, we will refer about some discrete distributions, which account for traffic counts and are used to model the vehicle arrivals; then, we will present some continuous distributions used for (time) headway modeling.

### Counting Distributions

According to Mannering and Washburn (2013), the derivation of models that take into account the non-uniformity in traffic flow is based on the assumption that vehicle arrivals, at a specified cross-section, correspond to some random process. Thus, one should select a probability distribution suitable to represent the observed patterns of traffic arrivals. Among the counting distributions, we can remember the Poisson distribution; the corresponding probability mass function is given as follows:

$Pn=λ⋅tn⋅e−λtn!$

where P(n) is the probability of having n vehicle arrivals in the time interval t, λ is the average vehicle flow (i.e., the arrival rate in vehicles per unit time), and t stands for the duration of the time interval over which vehicles are counted. The Poisson distribution (also known as the law of rare events) was introduced by Kinzer (1933); until now, several applications have been carried out in transportation engineering.

Based on the statistical assumptions concerning the derivation of Poisson distribution, the model lends itself well as arrival model in a single lane (or two or more adjacent lanes) when steady-state conditions persist over the analysis time period, and the arrival of one vehicle is independent of the arrival of another vehicle (i.e., no interaction is experienced between the arrivals of two successive vehicles). Empirical observations have shown that the assumption of Poisson-distributed traffic arrivals is most realistic in lightly congested traffic conditions; thus, the model can be consistent with experimental data when the flow is rare and, hence, it can be used when flow rates up to 400–500 veh/h are accommodated. The Poisson distribution cannot be used without a steady-state condition or when traffic flows reach heavily congested conditions; in these cases, other traffic flow distributions can be considered more appropriate (Mauro, 2010). Another limitation of Poisson model is that the mean of the observations equals the variance (Mannering and Washburn, 2013). However, many real count data do not adhere to the assumption that the mean and the variance are equal, and another distribution should be used. When the variance exceeds the mean of the counts, the negative binomial distribution can be used; it captures overdispersion, which can take place in various contexts [see, e.g., Hilbe (2008)]. When, in turn, the mean of the counts exceeds the variance the choice of the probability distribution can fall on the binomial distribution; however, it should be particularized with the measured data [see, e.g., Devroye (1986)]. Such distributions are discussed in more specialized sources [such as, for instance, May (1990), Lord and Mannering (2010), and Mauro and Branco (2012)]. The criterion for choosing alternative traffic counting models has been exposed by Mauro and Branco (2013); the same source shows the theoretical probability distributions of arrivals (namely, the binomial, Poisson, and negative binomial distribution) and their expressions as a function of the sample statistics.

Models of random arrivals are widely discussed in the technical literature and used since they are fundamental to the gap acceptance modeling. Besides counting distributions, suitable for describing counts of discrete units, such as cars, under various conditions of occurrence, another class of distributions is that of interval distributions, which describe the probability of intervals (headways) of different sizes between events and need to be characterized statistically. However, counts of cars deal with discrete events, whereas headways can be measured on a continuous scale. For purely random events, arrival headways are described by the negative exponential distribution; when drivers are forced into non-random behavior as during congested traffic conditions, other distributions can result more appropriate.

In detail, for populations whose counts are described by the Poisson distribution, the headways between counts can be described by the negative exponential distribution (M1). This distribution has been extensively used in literature; it is based on the assumption that each vehicle arrives at random without dependence between successive vehicle arrivals (Troutbeck and Brilon, 2016).

The headway h must be then greater than t and the probability density function may be stated as:

$Ph>t=λ⋅e−λ⋅t$

where h stands for the headway between events and λ = Qc/3600 is the average arrival rate in the opposing stream (in vehicle per second). The cumulative probability function of headways is given by

$Fh≤t=1−e−λ⋅t$

However, the M1 distribution allows unrealistic short headways and does not describe platooning. When traffic volume is so high that each car tends to follow the car ahead, M1 distribution may be unsuitable to describe the headways between cars and can be considered realistic for a very low traffic flow rate (about less than 150 veh/h). Thus, the shifted negative exponential distribution (M2) can result more suitable. Indeed, M2 distribution represents the probability that the headway h is less than t with a prohibition of headways less than Δ, that is the shifted exponential distribution assumes that there is a minimum headway between vehicles. The cumulative probability distribution of headways may be stated:

$Fh≤t=1−e−λ⋅t−Δ$

where Δ is the amount of the shift since short headways are prohibited, τ ≥ Δ and λ is a model parameter calculated as $λ=q1−Δq$. However, the M2 distribution is used for single-lane traffic only. Although, the negative and the shifted negative exponential distribution (M1 and M2) are widely used as headway distribution models, the bunched exponential distribution of arrival headways (M3) improves the representation of inter-vehicular time intervals in the (major) circulating stream, and gives a more accurate prediction of arrival headways about up to 12 s, that is particularly useful for analyzing urban roads and streets. The M3 distribution was proposed by Cowan (1975, 1987) and then extensively used for estimating entry capacity of unsignalized intersections and roundabouts [see, e.g., Akçelik and Chung (1994)]. Cowan (1987) discussed the value of the above models for their use as arrival processes in stochastic model building and described some traffic situations where the models could be appropriate. The cumulative distribution function for the bunched exponential distribution (M3 distribution) represents the probability of a headway less than t seconds and may be stated as follows:

$Fh≤t=1−φ⋅e−λ⋅t−Δ,t≥Δ0,t<Δ$

where Δ in this case is the average intrabunch (minimum) arrival headway, is the proportion of unbunched (free) vehicles and λ is a model parameter calculated as $λ=φ⋅q1−Δq$ with q ≤ 0.98/Δ (note that the arrival flow rate is q in vehicle per second). The intrabunch headway (or the headway within each bunch equal to the minimum arrival headway Δ) and the proportion of unbunched (free) vehicles (with randomly distributed headways) are related to the distribution of the circulating stream headways. The average intrabunch headway corresponds to the average headway at capacity (Δ = 3600/C, where C is the capacity in veh/h).

The M3 distribution explicitly takes into account the number of bunched vehicles through the φ parameter representing the proportion of free vehicles. Application of the M3 parameters to each circulating lane of the roundabouts allows to use capacity formulas for n-lanes, each having different Cowan M3 parameters. The M1 and the M2 distributions can be derived from the M3 distribution by assuming Δ = 0 and φ = 1 (and therefore λ = q) for the M1 distribution and φ = 1 and therefore λ = q/(1 − Δq), with q ≤ 0.98/Δ, for the M2 distribution. One can observe that both distributions assume no bunching, whereas the M3 distribution model can be applied by estimating φ or using a bunching model, which estimates φ as a function of the opposing flow. In practical application, indeed, only the traffic flow is known, not the headway distribution; thus, it is necessary to relate φ or Δ with the opposing flow. Further discussions on the M3 model and gap acceptance models can be found in more specialized sources [see, e.g., Luttinen (1999) and Akçelik (2007)].

### Gap Acceptance Capacity Models

The arrival headway distribution models can be used together with gap acceptance parameters to derive the capacity models. As above introduced, gap acceptance models are (macroscopic) analytical models, which express the capacity in an exponential function of the circulating flow; thus the rate of reduction in capacity decreases as the circulating flow increases.

Based on the gap acceptance process, for the simple two-stream situation entry capacity can be estimated by elementary probability theory methods if the assumptions introduced in Section “Modeling Methods for Roundabout Capacity Analysis” are met. Harders (1968), for instance, used Eq. 6 and combined a stepwise constant function for the number of minor stream vehicles n(τ) (which can enter the roundabout using a major stream headway of size τ) and the M1 distribution for headways in the major circulating stream. Thus, he obtained the well-known capacity model as follows:

$Ce=λ⋅e−λ⋅τc1−e−λ⋅τf$

where λ = Qc/3600 and the other symbols have the meaning already explained. This model was also adopted in the National Research Council and Transportation Research Board (2000). Siegloch (1973) first derived another capacity formula resulting in a relation of capacity versus conflicting flow Qc = λ⋅3600, veh/h, by applying a continuous linear function for n(τ) as follows:

$Ce=3600τfe−λ⋅τ0$

where τ0 = τc τf/2, s. More recently, this capacity model was revised in the National Research Council, and Transportation Research Board (2010) as follows:

$Ce=3600τfe−Qc⋅τc−0.5τf3600$

The above capacity model is an exponential regression model based on a gap acceptance theory (Akçelik, 2011); this model can be calibrated by using site-specific values for the critical headway and the follow-up headway. Geometry is classified in terms of the numbers of circulating lanes and entry lanes. In this model, shorter critical headways were used for a multilane roundabout than a single-lane roundabout.

Troutbeck (1986), as reported by Brilon et al. (1999), highlighted that the assumptions introduced above (see “Modeling Methods for Roundabout Capacity Analysis”) for the capacity of the simple two-stream situation, could lead to an unrealistic representation of the phenomenon under examination and different efforts to drop one or the other assumption were made also by using traffic simulations technique [see, e.g., among the first researches, Grossmann (1988)]. However, Troutbeck and Brilon (2016), also based on the ongoing experience, observed that Harder’s model (Harders, 1968) and Siegloch’s model (Siegloch, 1973) can give quite realistic results in practice.

More general solutions for the capacity models have been obtained by replacing the M1 distribution with the more realistic M2 and M3 distributions. For instance, a more general capacity formula is derived by using a dichotomized distribution as follows:

$Ce=φ⋅Qc⋅e−λ⋅τc−Δ1−e−λ⋅τf$

where λ = (φ⋅Qc)/(3600 − Δ⋅Qc). If φ = 1 and Δ = 0, then Harders’ equation is obtained. If φ = 1 − λ⋅Δ the capacity equation derived by Tanner (1962) is obtained with a step function for n(τ), whereas, by using the linear relationship for n(τ), the capacity equation derived by Jacobs (1979) is given:

$Ce=φ⋅Qc⋅e−λ⋅τ0−Δλ⋅τf$

Fisk (1989) extended Tanner’s model to multilane roundabouts by assuming that drivers had a different critical gap when crossing different streams, whereas Hagring (1998) presented a generalization of the earlier gap acceptance model by extending Troutbeck’s model (Troutbeck, 1986). Hagring (1998) indeed derived the capacity of a minor stream crossing or merging n (independent) major streams, by using for each stream a Cowan’s M3 headway distribution. This is why Cowan’s M3 headway distribution explicitly accounts for the number of bunched vehicles through the proportion of free vehicles expressed by the φ parameter. Some bunching models to estimate φ formulated by several authors have been collected by Giuffrè et al. (2012a) and Akçelik (2007).

The Hagring model (Hagring, 1998) has been rewritten and applied several times. For instance, Giuffrè et al. (2012a) used this model to compare performances of turbo roundabouts and double-lane roundabouts; so they specified the model in relation to values of the conflicting traffic flow (moving on the inner circulating lane Qc,i and/or the outer circulating lane Qc,e), and τc, τf, and Δvalues and obtained:

$Ce=Qc,e⋅1−Δ⋅Qc,e3600⋅e−Qc,e3600⋅τc−Δ1−e−Qc,e3600⋅τcf$

when the subject entry drivers face one antagonist traffic flow; this is the case of right-lane capacity estimation. They also estimated left-lane capacity when the subject entry drivers face both the circulating traffic flow in the outer lane (Qc,e) and the circulating traffic flow in the inner lane (Qc,i) on the circulatory roadway as follows:

$Ce=Qc,e+Qc,i⋅1−Δ⋅Qc,e3600⋅1−Δ⋅Qc,i3600×e−Qc,e3600⋅τc−Δ−Qc,i3600⋅τc−Δ1−e−Qc,e+Qc,i3600⋅τf$

It should be noted that the capacity models above were built for unsignalized intersections. Since their understanding is based on the operation of the interacting streams, these models can be extended to the roundabout operation with one circulating stream or more circulating streams. Table 1 shows, in turn, a summary of gap acceptance capacity models specifically developed for roundabouts at steady-state conditions. However, the reader is invited to consult Rodegerdts et al. (2007) for a wider summary of roundabout operational models, also including details about linear and exponential regression models for capacity calculation. Figure 1 shows, in turn, the comparison among some capacity models for single-lane roundabouts; two models (Hagring, 1998; National Research Council and Transportation Research Board, 2010, recalibrated model) were also calibrated on the values of the critical headway and the follow-up headway as calculated by the meta-analysis previously performed by the authors (Giuffrè et al., 2016).

TABLE 1

Table 1. A summary of gap acceptance capacity models for roundabouts (references in the first column).

FIGURE 1

Figure 1. Comparison of capacity models for single-lane roundabouts [note that C stands for capacity and Qc for circulating flow; the Brilon and Wu formula is recommended in the German Highway Capacity Manual (FGSV, 2001)].

It should be noted that, for calculating roundabout capacity, as well as queue lengths and waiting times, steadiness and variability of traffic demand must be specified for the time period chosen; the presence of one or more saturated (or oversaturated) entries must be also highlighted. This requires the analysis of the roundabout with and without statistical equilibrium; moreover, based on the state at entries, probabilistic and/or deterministic models not only will be applied but also the time-dependent models. Statistical equilibrium and steady-state conditions will be briefly discussed in the following section. However, capacity calculation at saturation or oversaturation conditions of entries have been widely described by Mauro (2010) to which the interested reader is referred.

## Some Remarks on Steady-State and Non-Steady-State Conditions

In addition to capacity, the indices that are taken into account for the assessment of traffic flow performance at roundabouts are the queue lengths, measured by the number of vehicles in terms of means and percentiles, the waiting times due to the queuing up, and the average delay for vehicles entering the intersection. To evaluate these indices, two tools can be used to solve the problems of gap acceptance: queuing theory and simulation.

Each solution based on the conventional queuing theory is a steady-state solution. Indeed, this kind of solutions are usually expected for non-time dependent traffic volumes, which are subsequent to infinitely long time, and when the demand that is experiencing compared to capacity gives a degree of saturation less than 1.

It should be noted that, calculation of roundabout capacity, queue lengths, and waiting times require that steadiness and variability of traffic demand are specified for the time period chosen; the presence of one or more saturated (or oversaturated) entries must be also highlighted. According to Mauro (2010), the analysis of the roundabout with and without statistical equilibrium is required; moreover, based on the state at entries, probabilistic and/or deterministic models not only can be applied but also the time-dependent models can be used (Troutbeck and Brilon, 2016).

In general, the operational conditions of a roundabout may be studied through the succession of states, whose evolution requires that the probability associated with each state of the system is known. However, this probability for the same state may vary any time. Thus, the system exists in a transient condition. In turn, the system reaches a statistical equilibrium (i.e., the system is in a steady-state condition) when the probabilities of the states remain constant over time. According to Mauro (2010), rather than evaluating the time-invariant state probability distribution, the finding that a steady-state condition exists, relies on the evaluation of the time invariance of some appropriate statistical values for one or more variables, that evolve randomly, and which are deemed to be related to the operating conditions of the system. Thus, a roundabout can be considered at a steady-state condition when entering traffic demand does not change over time and the roundabout entries are characterized by undersaturated conditions. In practical terms, a steady-state condition is reached, being the entries undersaturated, when the traffic demand is constant for a finite time interval T which must be:

$T>max1Ci3600−Qei36002$

known as Morse’s inequality (Morse, 1982). This condition can only be applied if Ci (capacity of entry i) and Qei (demand volume of entry i) – both expressed in hourly volumes – may be assumed constant during T, the entries are undersaturated in T and the degree of saturation is suitably smaller than unity. Morse’s inequality (Morse, 1982), indeed, should allow to stabilize the traffic conditions of the roundabout around constant mean values of the state variables (as queue lengths and waiting times). If Morse’s inequality (Morse, 1982) is not fulfilled, time-dependent solutions should be used. In other words, the non-steady-state situations (characterized by the variability of Qei and/or the oversaturation of the entry under examination having the degree of saturation not sufficiently greater than one) cannot be evaluated through probabilistic and deterministic approaches. Mauro (2010) reviewed some results of stochastic and deterministic theory about waiting phenomena. He also presented a general heuristic criterion for evaluating the system state variables at any operational conditions at roundabout entries, namely undersaturated, saturated, and oversaturated conditions, as well as the methods for the study of roundabouts with time-dependent traffic demands (especially roundabouts with traffic peaks, which occur between two steady-state periods of flows). The coordinate transformation method is one technique, which provides estimates between these two states (Troutbeck and Brilon, 2016). Newell (1982) developed the first mathematical solutions for the time-dependent problem, which now needs to be made more accessible to practitioners. However, a heuristic (approximate) solution for the case of the peak hour effect was proposed by Kimber and Hollis (1979).

In general, the derivation of time-dependent relationships is based on the assumption that the statistical equilibrium solutions (which allow to reach deterministic solutions) are relative to Poissonian arrivals and exponential service time. Also time-dependent solutions for traffic peaks, which occur between two periods at steady-state conditions were based on the assumption that statistical equilibrium conditions, both before and after the peaks, are the same. Mauro (2010) again highlighted that the result obtained by using time-dependent formulas (of heuristic nature) does not match the results given by the queuing theory without statistical equilibrium; however, differences can be neglected for practical interest. The approximate solutions for studying the transient states of waiting phenomena are now thoroughly explored [see, e.g., Troutbeck and Brilon (2016)]. Indeed, the modern tools of stochastic simulation allow to overcome almost all problems easily, increasing (to any desired level) the same degree of reality of the model. However, restrictions are due to the efforts to be done and the available computer time.

## Managing Uncertainty in Roundabout Entry Capacity Evaluation

Most of the technical literature agrees that, in gap acceptance process, the critical headway and the follow-up headway have a significant role in determining the roundabout entry capacity as a function of the major stream flow rate with a specified arrival headway distribution. In the calculation process, current practices replace these random variables by single mean values, neglecting their changes, and providing a single-value of entry capacity. In order to manage uncertainty in capacity estimation at roundabouts, entry capacity distributions should be estimated, once the probability distributions of the critical headways and follow-up headways have been assumed. Thus, the results of the calculation should be expressed probabilistically, meaning that the probability distributions of entry capacity rather than the simple point estimates of the performance measure have to be obtained. In this view, random variables are not just the flows of the various legs (or the traffic demand), but also the entry capacities that are depending on them; furthermore, it must be said that these variables are non-statistically independent. Since traffic demand and entry capacity are random variables, they should be characterized by their probability functions. This is necessary for the evaluation of reliability in each leg, that is to say the probability that the system does not fail and, in the specific case of the roundabouts, that traffic demand does not exceed the single entry capacity.

Mauro (2010) presented general criteria for the evaluation of reliability at roundabout legs based on the study of the performance function represented by the reserve capacity (C Qe) or the reciprocal of the rate of capacity (C/Qe); he provided the analytical relations when capacity and demand result normally distributed (having means and variances known) and he also proved that the two indices to which we usually resort (i.e., the reserve capacity or the rate of capacity) are not enough to say that the system does not fail. It is noteworthy, however, that when the mean values of the two indices in question are constant, the reliability is depending on the dispersion around the mean values of the flows, i.e., the level of uncertainty that affects the values above introduced.

FIGURE 2

Figure 2. Uncertainty in capacity estimation at single-lane roundabouts {note that the capacity equation in figure is calculated by using the mean critical and follow-up headways provided by the meta-analysis [see Giuffrè et al. (2016)]}.

## Conclusion and Future Developments

In the recent past, the authors have published the first results of a research in which the calibration of the microscopic traffic simulation model was formulated as an optimization problem based on a genetic algorithm; the objective function was defined in order to minimize the differences between the simulated and real data set in the speed–density graphs for a freeway segment. The application of the proposed methodology to roundabouts can represent an interesting starting point for future research activities. A comparison could be performed between the capacity functions based on the critical headways and the follow-up headways derived from meta-analysis and simulation outputs for a roundabout built in Aimsun microscopic simulator.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Citation: Giuffrè O, Granà A and Tumminello ML (2016) Methodological Frontier in Operational Analysis for Roundabouts: A Review. Front. Built Environ. 2:28. doi: 10.3389/fbuil.2016.00028

Received: 17 May 2016; Accepted: 26 October 2016;
Published: 16 November 2016

Edited by:

Cholachat Rujikiatkamjorn, University of Wollongong, Australia

Reviewed by:

Pabitra Rajbongshi, National Institute of Technology Silchar, India
Rasa Ušpalytė-Vitkūnienė, Vilnius Gediminas Technical University, Lithuania

Copyright: © 2016 Giuffrè, Granà and Tumminello. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Anna Granà, anna.grana@unipa.it