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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

Specialty section: This article was submitted to Transportation and Transit Systems, a section of the journal Frontiers in Built Environment

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.

Several studies and researches have shown that modern roundabouts are safe and effective as engineering countermeasures for traffic calming, and they are now widely used worldwide. The increasing use of roundabouts and, more recently, turbo and flower roundabouts, has induced a great variety of experiences in the field of intersection design, traffic safety, and capacity modeling. As for unsignalized intersections, which represent the starting point to extend knowledge about the operational analysis to roundabouts, the general situation in capacity estimation is still characterized by the discussion between gap acceptance models and empirical regression models. However, capacity modeling must contain both the analytical construction and then solution of the model, and the implementation of driver behavior. Thus, issues on a realistic modeling of driver behavior by the parameters that are included into the models are always of interest for practitioners and analysts in transportation and road infrastructure engineering. Based on these considerations, this paper presents a literature review about the key methodological issues in the operational analysis of modern roundabouts. Focus is made on the aspects associated with the gap acceptance behavior, the derivation of the analytical-based models, and the calculation of parameters included into the capacity equations, as well as steady-state and non-steady-state conditions and uncertainty in entry capacity estimation. At last, insights on future developments of the research in this field of investigation will be also outlined.

The roundabout technologies continue to grow across the world mostly because of the long-lasting benefits associated with planning, geometric design, safety, operations, environmental, and landscaping issues. Indeed, roundabouts often represent safer and more efficient solutions than other at-grade intersections; with regard to the installation context, the shape, and the dimensions, roundabouts can be also esthetically appealing design alternatives compared to other unsignalized intersections (Rodegerdts et al.,

Over time, a large number of studies have been developed to analyze operations of existing or planned roundabouts. However, the topical discussion between gap acceptance theory and empirical regression models still characterizes the general situation about capacity calculation of unsignalized intersections and roundabouts (Brilon et al.,

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.,

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.

According to these objectives, the paper is organized as follows: Section “

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.,

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.,

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.

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 (

The linear regression function in Eq. _{0} = τ_{c} −_{f}_{f}_{c}

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,

According to Raff and Hart (_{c}_{r}_{a}_{r}_{a}

Differently from Raff and Hart (_{c}_{a}_{a}_{a}_{c}_{a}_{c}_{a}^{2}, where _{c}_{c}_{a}

Troutbeck (_{r}_{a}_{r}_{a}_{c}_{r}_{a}_{a}_{a}_{r}_{r}

Based on the two vectors of the observed {τ_{r}_{a}

The probabilistic distribution for the critical headways is usually assumed to be log-normal. The likelihood estimators μ and σ^{2}_{r}_{a}

Other methods for estimating critical headways have been also recommended for practical applications: Harder’s method (Harders,

More recently, microscopic approaches have been also used to estimate the critical headway at roundabouts. For instance, Vasconcelos et al. (

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. (

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. (

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 _{c}

Usually,

When one divides this mean value

Equation _{c} ⋅ f_{c}

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 “

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

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,

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.

According to Mannering and Washburn (

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,

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,

The headway _{c}

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

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

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 “_{c}/3600 and the other symbols have the meaning already explained. This model was also adopted in the National Research Council and Transportation Research Board (_{c}_{0} = τ_{c} −_{f}

The above capacity model is an exponential regression model based on a gap acceptance theory (Akçelik,

Troutbeck (

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:
_{c}_{c}

Fisk (

The Hagring model (Hagring, _{c,i}_{c,e}_{c}_{f}_{c,e}_{c,i}

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

Reference | Country | Model applicability | Model input | Note |
---|---|---|---|---|

Arem and Kneepkens ( |
The Netherlands | Single-lane roundabouts | Circulating flow; exiting flow on leg; critical and follow-up headways; minimum gap | The formula is based on Tanner’s equation (Tanner, |

National Research Council and Transportation Research Board ( |
USA | Single-lane roundabouts | Circulating flow; critical and follow-up headways | The formula is based on Harder’s model (Harders, |

CAPCAL2 ( |
Sweden | Single-lane and two-lane roundabouts | Percentage of heavy vehicles; critical and follow-up headways; minimum gap; proportion of random arrivals; length and width of weaving area | CAPCAL2 is the new version of the Swedish software for estimating capacity and performance measures in roundabouts and intersections both with and without traffic signals; it was introduced in 1996. The calculation procedure for roundabouts and intersections (without traffic signals) is based on gap acceptance |

Troutbeck ( |
Australia | Single-lane and multilane roundabouts | Circulating flow; turning flow; entry flow; number of lanes; entry width; diameter; critical and follow-up headways | Separate equations for left and right lanes have been proposed |

Wu ( |
Germany | Single-lane and multilane roundabouts | Circulating flow, number of lanes; critical and follow-up headways; minimum gap | Based on Tanner’s equation (Tanner, |

_{c}

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 (

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 (

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 (_{i}_{ei}_{ei}

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 (

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 (_{e}_{e}

Based on these considerations, in the operational analysis of roundabouts, one of the primary objectives should be to derive an entry capacity distribution, which accounts for the variations in the contributing random variables. In order to match this goal, Monte Carlo simulation can be useful to obtain the entry capacity distribution at roundabouts. Thus, one has to perform the random sampling from the probability density functions, which have to be chosen for the contributing parameters, according to an adopted capacity formulation. Here, for a first exploration, Crystal Ball software was used for performing many runs based on the Hagring model; the values of the critical headways and follow-up headways were randomly drawn from their corresponding (selected) distributions in each run. By way of example, for single-lane roundabouts, Figure

In some countries (and among these in Italy), a specific formula for the determination of the roundabout capacity has not yet been defined with any precision. Thus, several positive experiences made worldwide can be considered the starting point for the prediction of the roundabout capacity. It should be noted, first of all, that it is essential to compare the types of roundabouts recommended in each national standard (if available) with the roundabout schemes proposed by the foreign standards that address the same types of intersections. Indeed, different countries around the world use different classifications, and their standards can result not fully interchangeable with respect to the conformity between the geometric standards and the context of the roundabout installation. The same consideration may interest the calculation procedure especially regarding the entry capacity. We have introduced that, in capacity calculation, the general situation is (even today) characterized by the discussion between gap acceptance and empirical regression models. We also have focused attention on the capacity analysis procedures, which are based on gap acceptance theory to some extent and/or have understood that this theory can be considered the basis for the roundabout operations even if they have not used the same theory explicitly. Thus, attention must be placed on the precise evaluation of critical headways and follow-up headways from traffic observation, since these behavioral parameters have an important role in determining the entry capacity at roundabouts. The behavioral parameters, indeed, are random variables and they should be characterized by their probability distributions; thus, the resulting capacity value based on the mean values of the major gap acceptance parameters may underestimate or overestimate the real value of entry capacity. Moreover, based on a systematic literature review of studies and researches developed worldwide to measure critical headways and follow-up headways at existing roundabouts, the authors noted that the effect size for each parameter varied from study to study. Thus the meta-analysis of effect sizes performed as part of the literature review through the random-effects model can represent a useful approach to provide a single quantitative meta-analytic estimate, both for the critical headway and the follow-up headway; in turn, the quantitative meta-analytic estimate can be considered more appropriate for representing the gap acceptance behavior of drivers on roundabouts. It was also pointed out that traffic demand and entry capacity are also random variables and they should be characterized by their probability functions. This could increase the uncertainty in estimation procedures. In order to detect these kinds of effects, the managing of uncertainty in roundabout entry capacity evaluation, as well as the evaluation of the roundabout reliability could become fields requiring increasing attention in the next future.

Since many random factors affect roundabout operations (e.g., vehicles arriving and gap acceptance), and the application of the statistical methods can result expensive and time consuming (since reliable results can require lots of well-chosen field observations on different geometric and traffic conditions), microscopic traffic simulation can result the favorite method for the study of the roundabout operations. Recent advances in research and applications to road and highway engineering outline the great potential for useful application of microscopic traffic simulation models to accurately account for road operations and performances, since they capture the interactions of road traffic through a series of complex algorithms describing car-following, lane-changing, gap-acceptance as they happen in the real world. Differently to traditional analytical models and techniques that provide a simplified (aggregated) representation of road traffic, microscopic traffic simulation models represent for researchers and practitioners the favorite methods for analyzing operations of road networks or single road infrastructures, and taking decisions on their geometric design development, since they allow the accurate modeling of some road planning and design problems. Microsimulation, indeed, enables to create increasing levels of complexity and uncertainty in the operations of road traffic networks and single road installations, but concerns are often expressed by practitioners regarding its possible misuse. In simulation studies, moreover, model calibration is a crucial task when reliable results have to be obtained from the analysis that we made.

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.

OG is full professor

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.