1. Early History of Roundabout Capacity Estimation
  2. Development of the Modern Roundabout Capacity Equation
  3. Geometry and Safety
  4. Methods of Capacity Measurement
  5. U.S. Roundabout Capacity Estimation Methods
  6. Calibration

Early History of Roundabout Capacity Estimation

Roundabout capacity has been debated since the first projects in Paris, New York, and London more than a century ago. Lacking the yield sign, early circles performed very differently from modern roundabouts, but the principle was similar: traffic entered and exited a one-way circulating roadway. By the 1940s the US Bureau of Public Roads stated that the practical capacity of the US' 'rotary' was 3000 vehicles per hour. Meanwhile, the UK had a 'roundabout' carrying 10,000 vehicles per hour, and in Paris, the Place de l’Etoile moved 20,000 vehicles per hour with frequent crashes.

Clearly, capacity assumptions were wrong and safety factors weren't understood.

In 1950, Tulsa police officer Clinton Riggs invented the YIELD sign, and starting in 1956, UK road agencies tested the sign at roundabout entries: creating the first yield- controlled roundabouts. The change was wildly successful: eliminating lock-ups and significantly improving capacity and safety. With encouragement from Frank Blackmore and the Transport and Road Research Laboratory (TRRL), Britain made Yield-at-Entry (offside priority) the national standard in 1966, and the 'modern' roundabout became a circular roadway with entering traffic yielding to circulating vehicles. Unfortunately, because yield-at-entry virtually eliminated weaving on the circulating roadway, the previous weaving theories of capacity became obsolete, and there was no suitable capacity formula!1

As traffic continued to increase through the 1960's, and without adequate capacity and design guidance on this new intersection type, UK road agencies began widening roundabout entries and altering geometries in a variety of intuitive attempts to increase capacity and reduce queues and delay. Results were haphazard, and in some cases unsafe, but local road agencies had inadvertently created a broad variety of designs for valuable empirical research.

Given national imperatives to improve road capacity and safety in the late 60's and early 70's, the TRRL under the leadership of Frank Blackmore carried out experiments on the TRRL test track and at sites in the field. Experimenters modified intersection geometries, and police blocked traffic to create approach queues so that saturation capacity could be measured. (One valuable product of these experiments was the modern "mini- roundabout.") The experiments revealed that subtle changes in geometry had powerful capacity effects, but they still lacked equations to describe the effects of geometry on entry capacity.

The TRRL and UK universities tested behavioral (gap) models of vehicle-vehicle interactions during the 1970's, but these proved statistically unwieldy, because driver behavior is not constant and roundabout geometry is complex. As intersection volumes approach capacity saturation, field studies showed drivers accept increasingly smaller gaps, and with low speeds, entering drivers can safely nose forward and force gaps where no gap exists. In addition, subsequent drivers follow on in brief periods of 'priority reversal,' creating additional capacity where gap theory predicted no capacity should exist! Gap based theoretical models assumed that circulating flow doesn't react to entering vehicles, but this was found not to be the case. Other observed behaviors includes follow on time, shy distance, merging, and residual weaving.

Because of the daunting complexity, it proved statistically impossible to collect enough sample data to correlate the many geometric variables with the multiple complex human behaviors, and arrive at aggregate capacity/geometry relationships with any accuracy or confidence. Since geometric design guidance was the main objective, the consensus was reached that a new capacity approach was needed to derive roundabout geometries optimized for capacity and safety, and that theoretical (gap) model mechanisms were not as accurate . Given these issues, TRRL abandoned the gap acceptance model of roundabout capacity in the mid 1970's, and focused future efforts on direct regression of geometry vs. observed aggregate capacity and safety.

Development of the Modern Roundabout Capacity Equation

In the late 1970's, Professor Rod Kimber, then Head of Junction Design Section at TRL, spearheaded a series of controlled experiments on the TRL test track, accompanied by extensive field measurements at sites on the road network. On the TRL test track (a wide paved area), staff sequentially set up and tested 35 different roundabout geometries operating under traffic, while a control roundabout operated nearby with no changes to its geometry. Drivers, including housewives with children at school, drove the experimental roundabouts as flaggers directed which way to turn. The track experiment allowed researchers to develop the basic structure of the equation.

Then, measured saturation capacity data from 86 public road roundabouts was used to refine and calibrate the formula to field observed capacities. Fortunately, this data had available some of the unusual designs that road agencies had built before design guidance, offering a very wide range of variation in the geometric parameters. In all, the field data sample for the multiple regression included 11,000 minutes of at-capacity observation of about one million vehicles. (In today's funds, the twenty years of studies cost roughly $13 million.)

Researchers measured and tested an array of geometric variables, of which six were found to be logically independent and statistically significant in the unified capacity formula. When you consider them, the variables are also readily intuitive. Significant predictive variables were found to be:

Independent Variable Symbol Data Range Unit
Entry Width3 e 3.6 - 16.5 Meters
Approach Half Width v 1.9 - 12.5 Meters
Effective Flare Length l1 1.0 - Inf. Meters
Inscribed Circle Diameter D 13.5 - 171.6 Meters
Entry Radius r 3.4 - Inf. Meters
Entry Angle Phi 0.0 - 77 Degrees
Circulating Flow Qc 0 - 4,700 PCUs/hr

Geometric Definitions

The final version of the TRL regression equation achieved a known r2 of .78. (I.E. The independent variables explain 78% of variation in entry capacity.) Between-site variation leaves a statistically irreducible standard error of about 200 vehicles per hour, due to variation in network location, local conditions, driver behavior, weather, and other random factors. This statistic is critically important, because it's essential for designers to know not just the mean capacity they should expect, but also the probability that actual capacity will be more or less than predicted.

RODEL includes between site variation as the 'Confidence Level' function. Engineering judgment is necessary to consider whether a specific site is likely to be high or low on the capacity distribution, and RODEL allows the designer to quickly test and correct for potential effects. Among factors to consider are urban vs. rural location, daily commuter vs. shopping or tourist traffic, commercial percentages, and roadway functional class.

In 1992, follow-on field studies were conducted to re-confirm the accuracy of the empirical capacity equations. The results revealed that all subject intersections were operating in the predicted capacity range, so the capacity model was not modified. Consequently, Kimber's capacity equation has now been in continuous use, unmodified, for more than thirty years, worldwide, under the full range of roundabout sizes, driver behaviors, and road conditions. This track record is consistent and unmatched.

Geometry and Safety

With completion of the unified roundabout capacity formula in 1980, researchers turned their attention to safety, with results published by Mr. Geoff Maycock of TRL and Dr. Richard Hall of Southampton University in 1984.2 The analysis included injury crash data from eighty-four roundabouts, with a total of 431 junction years of crash data: an average of five years of crash data per roundabout. Fatal and injury crashes were regressed against geometry and traffic flows to find statistically significant relationships for design safety. The findings are critically important for roundabout geometric designers.

A Key Finding: Some geometric variables that increase capacity also increase certain types of injury crashes. For this reason, it is critical NOT to under-predict capacity and overdesign the roundabout, as this can unnecessarily increase injury crash risk.

Findings revealed that different types of injury crashes are related to different geometry and flow variables. Here are some of the significant safety findings for different causal variables:

  • 1. Traffic Causes Crashes. The key predictors of crashes are vehicle and pedestrian volumes. The designer cannot usually influence flow variables, but the pattern of traffic flows will indicate the safer geometric design solution(s).
  • 2. Deflection (Ce = 1/Re) (aka Entry Path Curvature) exhibits a powerful and continuous relationship to crashes, with no break point at any specific curvature. Increased entry path deflection sharply reduces the risk of entering-circulating crashes, but simultaneously increases risks of approach (rear-end) and single vehicle (loss of control) crashes to a lesser degree. Optimal safe deflection is therefore not a single number, but depends on the entering and circulating flows at each roundabout approach.
  • 3. Entry Width (e) increases risk of entering-circulating crashes, and slightly decreases the risk of approach crashes.
  • 4. Sight Distance to the Left (VL) (VR to the right in the UK) was unexpectedly found to be strongly - and inversely - related to approach (rear end) and single vehicle loss-of- control crashes. Entering circulating crashes are not affected. Reason: The earlier drivers can see the next upstream approach, the earlier drivers look that direction and don't watch the road ahead of them: resulting in rear-end and loss-of-control crashes. For this reason, it is important to focus driver attention in the direction of relevant conflicts, by restricting visibility upstream until about 15 meters from the yield line.
  • 5. Visibility Around the Central Island was also found to increase crashes. Here again, blocking irrelevant views helps direct driver attention where they should be looking: at oncoming traffic and at the road and conflicts ahead of them.
  • 6. Angle Between Arms Increasing the angle to the next downstream entry changes the entering-circulating conflict from crossing to merge. The wider the angle between arms, the less risk of entering-circulating injury crashes. (Recent experience shows however, that an increased angle between arms, and associated merge movements, can lead to exit fender bender crashes at the downstream exit!)
  • 7. Approach Visibility (Distance at first sight of the roundabout) was significant for pedestrian crashes in 30-40 MPH approach category. No other geometric variable exhibited a significant correlation with pedestrian crashes. The key variables related to pedestrian crashes are vehicle and pedestrian crossing volumes.
  • 8. Approach Curvature or curvature of the approach roadway, was found to be strongly related to single vehicle crashes, however this is generally a function of road alignment and typically outside the scope of the roundabout design.
  • 9. Approach Width (v) was positively correlated with increased single vehicle crashes.

  • Disclaimer: These few brief points by no means exhaust the numerous issues related to roundabout design safety, and the reader is encouraged to review available design guidance and research materials on the topic.

    Methods of Capacity Measurement

    The 2010 Highway Capacity Manual defines capacity as: “the maximum sustainable hourly flow rate at which persons or vehicles can be expected to traverse a point or a uniform section of a lane or roadway during a given time period under prevailing roadway, environmental, traffic, and control conditions."3

    When referring to capacity estimation for a roundabout entry, we’re actually talking about determining the mean capacity line that relates maximum hourly entry flow rate (Qe) to prevailing circulating flow (Qc). To identify the mean capacity line, we need to observe and measure the maximum sustainable flow at not just one roundabout entry, but a range of observed capacities across many roundabout entries and prevailing conditions.

    There are two common methods for capacity measurement:

    • Direct Capacity Measurement

      Capacity is measured directly by repeatedly counting entering traffic and circulating traffic under "at capacity" conditions, typically for 1 to 5 minute intervals, with the observations plotted on a graph. The line / curve of capacity against circulating flow is then derived by regression. This method avoids any theoretical assumptions about driver behavior, by directly observing fully motivated drivers in saturated conditions.

      Direct capacity measurement will yield the most accurate results for a given intersection, but only if the entry is operating at genuinely “at-capacity” conditions. A large sample of data is necessary in order to determine the slopes and distribution of the regression lines.

    • Indirect Capacity Measurement

      The entry capacity / circulating flow graph can also be estimated indirectly by measuring gaps and using them to calculate capacity using theoretical equations. Indirect capacity estimation depends on the assumption that capacity is purely a gap acceptance mechanism. This allows for capacity measurements to be made without saturated conditions. However, it has been extensively observed that as the Volume/Capacity Ratio increases, the gap acceptance mechanism fades and is replaced by several other capacity mechanisms, including gap forcing, priority reversal, weaving, and interaction with drivers in the circulating flow. Thus, gap acceptance tends to underestimate entry capacity at high circulating flows.

    U.S. Capacity Estimation Methods

    Since modern roundabouts began gaining popularity in the U.S. in the 1990s, US practitioners have debated what capacity estimation method(s) are most suited to US conditions. Some assume that UK field capacity measurements may not apply to the U.S., because UK drivers are more experienced with roundabouts and employ different driver behaviors.

    Are UK drivers supermen? We'd like to think so, but like American drivers, ours are merely human, and geometry is geometry, humans drive by what they see, that is the foundation of much of our roadway design principles.

    Conversely, proponents argue that gap acceptance models have proven flawed at high capacity, will consistently yield poor results in the face of saturated intersections, and require too much data collection on a per-project basis, making the design process less efficient.

    Debate continues, and acceptable methods for determining the best estimation method for the U.S..

    Although the differences between drivers in different countries have never been quantified, it's reasonable to imagine that some environmental variables (physical restraints, cultural expectations, weather, etc.) might result in different behaviors and therefore different approaches to traffic control. We don't know of any such differences, but it's possible. UK signal capacity, for example, is identical to US signal capacity.

    The uncertainty is exacerbated by the fact that US roundabouts are designed for future traffic. Lacking a time machine, it's impossible to know how US drivers will drive roundabouts after decades of daily practice. Still, it's worth noting that the UK capacities were observed only ten-to-twelve years after modern yield-at-entry roundabouts were implemented, and observed capacity hasn't changed in thirty years. UK drivers actually had half as much experience as US drivers will have by the project design year.

    The relatively recent US roundabout movement has so far produced only a few roundabouts operating at capacity. Seventeen years after it opened to traffic, one roundabout interchange in Vail, Colorado now exhibits saturated conditions during peak ski season. Data collected from these roundabouts appear to confirm the accuracy of Kimber’s equations in the US. But, data is still sparse and conclusions aren't viable until a far wider array of data is collected from multiple sites. As more data becomes available Rodel can adjust to fine tune (if necessary) US capacity estimation through its calibration inputs.


    Calibration is advocated to compensate for regional variation in capacity, but it can be a very slippery slope, as misapplication of calibration can result in wildly inaccurate capacity estimates. Measuring gaps for calibration can completely negate capacity estimation due to design geometry and differing V/C ratios, as gaps will be significantly different for different geometric designs and V/C ratios.

    Roundabouts that are too large are less feasible and less safe. Roundabouts without adequate capacity will create unexpected congestion early in the design’s life. Being wary of these pitfalls is important when approaching calibration.

    Capacity underestimation results in roundabouts that are may be larger , less feasible, and less safe. Underestimation will lead to false conclusions and substandard planning and engineering decisions, including overdesign, increased cost and land take, unnecessary socioeconomic and environmental impacts, incorrect choice of intersection type, and sub- optimal public safety.

    Capacity overestimation results in roundabouts with inadequate capacity, leading to unexpected congestion early in the design’s life. Practitioners need to be wary of both these pitfalls when approaching calibration.

    Calibration requires a very large sample of “at-capacity” data over many sites to derive the mean of the capacity measurements. If these Qe/Qc lines are produced from data collected from an intersection that isn’t "at-capacity," the calibrated line will not represent true capacity. Measuring gaps is challenging by itself, with wide variation and potential error in measuring gaps. This aspect of calibration introduces a major obstacle since data collection requires very specific conditions and varies greatly from site to site.

    RODEL accounts for the inherent variation in driver behavior for a given geometry and V/C ratio by utilizing a confidence level check.

    All models estimate the mean capacity of a population of identical roundabouts. The standard error about the population mean is about ± 200 cars per hour. RODEL allows the analyst or designer to test a design with pessimistic capacity lines from the population of capacity lines using the Confidence Level test. This checks the robustness of a design and warns the designer when large queues and delay will occur if the actual capacity is less than the population mean. This encourages subtle changes in geometry to increase capacity sufficiently to resolve latent problems that other software cannot detect.

    The recently released Rodel V1-Win has many new features added to facilitate the Planning, Geometric Design and Traffic Analysis of roundabouts.

    1. Ashworth R, and Field J.C.: The Capacity of Rotary Intersections, The Highway Engineer, 1973, 20(3), pp. 14-21

    2. Maycock, G and Hall, R, Accidents at 4-Arm Roundabouts, TRRL, Crowthorne, 1984

    3. U.S. Highway Capacity Manual 2010, U.S. DOT, Washington, D.C., April 2011, pg. 8-2

    4. R. M. Kimber, "The Traffic Capacity of Roundabouts", TRL, 1980