# Comparison of Single-Lane Roundabout Entry Degree of Saturation Estimations from Analytical and Regression Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{o}), the central island radius (R

_{i}), the circulatory roadway width (u), the number of legs (j), the angle between the legs (δ), entry width (e

_{j}), exit width (e

_{j}′), entry radius (R

_{j}), and exit radius (R

_{j+1}′). The selected R

_{o}is influenced by the location of the roundabout (urban, suburban, rural), roundabout task (e.g., traffic calming), spatial constraints, and the number of circulatory lanes. In this investigation, the roundabout geometric parameters, design vehicle used for swept path analysis, and traffic flow characteristics were defined based on the following assumptions: (1) the roundabouts were to be situated in a suburban environment and (2) the roundabouts were to act as single-lane traffic-calming devices along the transition path from the rural to the urban environment.

- R
_{o}applied in this investigation varied from 13.0 to 20.0 m, with a 0.5 m increment. According to previous research given in [22], these outer radii are commonly used for single-lane roundabouts worldwide. An increment of 0.5 m was chosen to capture the dispersity of the results and to create a sample that is representative, manageable, and easy to present at the same time; - The roundabout leg alignment was radial, as is standard in the suburban environment [22];
- The axes of legs 1 and 4 intersected at δ = 90°. The axes of legs 1 and 2 intersected at δ ranging from 75° to 90° with 5° increments. This range was defined after considering the condition given in [23], regarding the length of the arc (l) between the adjacent roundabout legs. According to these guidelines, the length of this arc should be longer than 20 m to ensure the efficiency of the roundabout. Namely, a shorter l makes it difficult for drivers to signal the exit of the intersection when turning right due to the very short time to turn on the turn signals;
- There were 15 m long triangular splitter islands designed at each leg with 0.5 m offset from the defined outer edge of the circulatory roadway;
- The initial alignment of 3.25 m wide entry and exit lanes was defined.

_{i}, R

_{j}, R

_{j+1}′, and u resulted from the design vehicle swept path analysis, which was performed in the AutoCAD software add-in Vehicle tracking 2022. The selected design vehicle was a 12 m long bus adopted from the German vehicle library FGSV 2001. The geometric parameters R

_{i}and u (Figure 1b) were defined based on the design vehicle swept path when driving in a full circle [7] while ensuring minimum lateral clearances of 0.5 m [24].

_{j}and R

_{j+1}′ were defined based on the design vehicle swept path when turning right [24], while ensuring minimum lateral clearances of 0.5 m. Where possible, to achieve wider exits, the roundabout’s right roadway edge was designed by considering the condition of R

_{j+1}′ ≥ R

_{j}+ 2 m (Figure 2a) [25]. Wider roundabout exits are favorable as they enable higher roundabout exit speeds, help minimize the likelihood of congestion and crashes at the exits, provide ease of navigation for long vehicles, and reduce the potential for trailers to track over the outside curb. At roundabouts where it was not possible to design the right roadway edge with radii R

_{j}and R

_{j+1}′ due to leg alignment, a different procedure was applied. Here, the right roadway edge was designed based on the trajectory of the vehicle’s right turn movement and lateral clearances of 0.5 m in cross-section a–a (Figure 2b). The geometric parameters e

_{j}and e

_{j}′ were defined as the shortest distance between the intersection point of the drawn line on the edge of the splitter island and the entry line and the right roadway edge on the roundabout entry and exit (Figure 2).

_{o}as an input for calculation. The regression Swiss Bovy model, given in [1], considers the influence of conflicting traffic on the circulatory roadway that is exiting the roundabout at the same leg as the observed entry. The influence of conflicting traffic on the circulatory roadway is defined as the distance between exiting and entering traffic flows along the center of the circulatory lane (b

_{j}) [8], i.e., it considers the joint influence of geometric parameters R

_{o}, R

_{i}, and δ.

_{j}was defined through the following procedure. First, the lines from the center of the outer radius R

_{o}to the center of the radii R

_{j}and R

_{j}′ were drawn (Figure 3a). At roundabouts where it was not possible to design the right roadway edge with radii R

_{j}and R

_{j+1}′, the line from the center of the outer radius R

_{o}perpendicular to the trajectory of the vehicle’s right turn movement was drawn (Figure 3b). Then, a circle of radius R

_{o}—u/2 was constructed from the R

_{o}center. Traffic stream conflicting points, exiting point (C) and entering point (C′), were defined as intersections of these entities. The distance between exiting and entering traffic streams along the center of the circulatory lane, i.e., the length of the circular arc between conflicting points b

_{j,}was then measured.

_{i}, u, e

_{j}, e

_{j}′, and b

_{j}were systematized according to the R

_{o}and δ. When designing a roundabout according to the previously described procedure, it should be noted that the results depend on the experience and subjective approach of the designer. Therefore, to better present the influence of the chosen R

_{o}and δ on the designed parameters, regression analysis was performed, and best-fit curves with a coefficient of determination larger than 0.99 were created. Second-degree polynomial curves were used to describe R

_{i}and u as a function of R

_{o}, and third-degree polynomial curves were used to describe e

_{j}, e

_{j}′, and b

_{j}as a function of R

_{o}for different δ. Additionally, the average difference between b

_{j}(j = 1, 2, and 3) for δ = 90° and b

_{j}(j = 1, 2, and 3) for δ = 85°, 80°, and 75° was calculated.

_{mj}) was defined as the ratio of entering traffic flow and entry capacity. According to [14], sustainable values of x

_{mj}range from 0.0 to 1.0 (values above 1.0 indicate an excess of entering traffic demand over entry capacity). x

_{mj}was calculated for each designed scheme considering the following simplifications and assumptions on traffic flow volume, distribution, and composition:

- Three travel directions through the roundabout were considered at each roundabout entry (j = 1, …, 4): right turn, straight passage, and left turn (i = 1, …, 12).
- Traffic flow q
_{i}at each entry (j = 1, …, 4) in each travel direction (i = 1, …, 12) was 150 veh/h, adding up to a total of 1800 veh/h passing through the roundabout (Figure 4a). - The influence of pedestrian and bicycle traffic on roundabout capacity was not considered.
- The influence of heavy vehicles on the traffic flow quality was considered through the homogenization of traffic flows q
_{i}. Flat-rate conversions of each q_{i}from vehicles per hour (veh/h) to Q_{i}in passenger car units per hour (pcu/h) were made by using a conversion factor of f_{T}= 1.1, prescribed by [26] in case of lacking real data on flow composition:

_{i}= f

_{T}· q

_{i},

_{i}is homogenized traffic flow in the direction i (pcu/h), f

_{T}is conversion factor (set to 1.1), and q

_{i}is traffic flow in the direction i (veh/h).

_{i}for each travel direction (i = 1, …, 12) were calculated. Then, the entering traffic flows Q

_{Ej}, exiting traffic flows Q

_{Sj}, and the circulating traffic flow Q

_{Cj}(traffic flow in the circulatory roadway, i.e., the main flow, which has priority over the ones entering the circulatory roadway) were calculated for each leg (j = 1, …, 4) according to Figure 4b and Table 2.

_{mj}was calculated through the following analytical and regression model procedures.

_{Cj}was used as an input variable for determining the base capacity of the approach BC

_{j}. BC

_{j}values for the roundabout with one circulatory lane and the outer radius of roundabout R

_{o}were determined from the chart in Figure 5 [26]. As the chart gave data only for roundabouts with outer radii R

_{o}of 13.5, 15, 17.5, and 20 m, for other investigated R

_{o}values, BC

_{j}was defined through interpolation.

_{Ej}was calculated as

_{Ej}is roundabout entry capacity considering the impact of pedestrian crossings (pcu/h), BC

_{j}is the base entry capacity of the roundabout according to Figure 5 (pcu/h), and f

_{p}is the capacity reduction factor for pedestrian and cycling traffic flow at roundabouts. In this investigation, the influence of pedestrian and bicycle traffic on entering traffic flow was neglected and the f

_{p}factor was set to 1.0. The value of entry capacity C

_{Ej}was, therefore, equal to base capacity BC

_{j}.

_{mj}, it was necessary to back-calculate the entry capacity values C

_{Ej}from passenger car units to vehicles per hour. The entry capacity C

_{mj}for the mixed traffic flow was then calculated as

_{mj}is the entry capacity for mixed flow (veh/h), C

_{Ej}is the entry capacity (pcu/h), and f

_{T}is the conversion factor for traffic composition set to 1.1 because of the absence of real data on flow composition.

_{mj}in vehicles per hour was then calculated on each leg’s entry by summing up the three traffic flows q

_{i}with different directions of travel (right turn, straight passage, and left turn). x

_{mj,A}was calculated as the ratio of entering mixed traffic flow q

_{mj}in vehicles per hour and entry capacity C

_{mj}in vehicles per hour:

_{mj,A}is the entry degree of saturation according to the analytical model, q

_{mj}is mixed traffic flow on legs’ entry (veh/h), and C

_{mj}is entry capacity (veh/h).

_{Ej}in passenger car units per hour was defined as [1]

_{Ej}is entry capacity (pcu/h), Q

_{Cj}is circulating traffic flow in front of the leg being considered (pcu/h), Q

_{Sj}is exiting traffic flow on the same leg as the entry (pcu/h), α is a factor reflecting the impact of exiting traffic on entry capacity by distance b

_{j}, β is a factor for adjusting circulating flow depending on the number of circulatory lanes, and γ is a factor for adjusting entry capacity depending on the number of circulatory lanes.

_{j}and middle curve. The factors β and γ were set to 1.0, as single-lane circulatory roadways were investigated [1].

_{mj,R}is the entry degree of saturation according to the regression model, γ is a factor for adjusting entry capacity depending on the number of circulatory lanes (set to 1.0), Q

_{Ej}is entering traffic flow (pcu/h), and C

_{Ej}is entry capacity (pcu/h).

_{o}and δ on calculated x

_{mj}, regression analysis was performed, and best-fit curves for x

_{mj}values were created. Second-degree polynomial curves with a coefficient of determination larger than 0.99 were used to visualize the trend.

_{mj,A}). With this reversed calculation, the potential traffic flow volume in veh/h was determined for each roundabout, summarized, and compared with the input total flow rate of 1800 veh/h.

## 3. Results

_{o}for different δ should be (1) R

_{o}≥ 13.5 m for δ = 85°, (2) R

_{o}≥ 16.5 m for δ = 80°, and (3) R

_{o}≥ 19.0 m for δ = 75°.

_{mj}were performed for all 60 designed roundabouts, based on the calculated traffic flow volumes shown in Figure 7.

_{i}, e

_{j}, e

_{j}′, and b

_{j}were proportional to R

_{o}, and u values were inversely proportional. The change in e

_{1}was (1) proportional to δ, (2) identical for δ = 90 and 85°, (3) identical for δ = 80 and 75° at R

_{o}= 13.0 m, and (4) identical for δ = 90, 85, and 80° at R

_{o}≥ 19.0 m. The change in e

_{2}was inversely proportional to δ at R

_{o}≥ 14.0 m. The change in e

_{2}′ and e

_{3}′ at R

_{o}≥ 16.0 m showed an uneven trend for different δ. As expected, δ did not affect the entrance and exit widths e

_{3}, e

_{4}, e

_{1}′, and e

_{4}′.

_{j}increasing for different δ corresponded to those of e

_{j}and e

_{j}′. Thus, the change in b

_{1}(1) was proportional to δ and (2) was identical for δ = 90 and 85°. The change in b

_{2}was inversely proportional to δ at R

_{o}≥ 14.0 m. δ had a negligible effect on the values of b

_{3}at R

_{o}< 16 m. The change in δ had no effect on the values of b

_{4}. Extreme values of b

_{j}were observed for δ = 75° and R

_{o}= 13.0 m (b

_{1}= 12.5 m) at leg 1, and R

_{o}= 20.0 m (b

_{2}= 25.7 m) at leg 2.

_{o}of roundabouts and δ according to the analytical model (x

_{mj,A}) and regression model (x

_{mj,R}) showed that x

_{mj}values were inversely proportional to R

_{o}. x

_{mj,R}values followed the observed trend of b

_{j}for all analyzed δ at each roundabout leg (Figure 9b, Figure 10, Figure 11 and Figure 12b). For R

_{o}≤ 16.5 m, an established trend showed a more rapid decrease in b

_{j}and, consequently, in x

_{mj,R}. On the other hand, for R

_{o}≥ 19.0 m, it can be stated that the differences in the right roadway edge design, regardless of δ, do not affect x

_{mj,R}. The values of x

_{mj,A}decreased at a lower and uniform rate as R

_{o}increased. x

_{mj,A}values were higher than x

_{mj,R}by, on average, 16%. The average difference between x

_{mj,A}and x

_{mj,R}varied between 0.088 and 0.100. These extreme differences were observed for δ = 75°, at leg 1 and leg 2, respectively. At each leg, the average observed difference between the calculated x

_{mj,R}for δ = 90° and x

_{mj,R}for the other δ values amounted to (1) 0.3% at leg 2 for δ = 85°, (2) 1.1% at leg 1, and 0.4% at leg 2 for δ = 80°, (3) 1.7% at leg 1, 0.5% at leg 2, and 0.2% at leg 3 for δ = 75°.

_{1}, b

_{2}, and b

_{3}for δ = 90° and b

_{1}, b

_{2}, and b

_{3}for δ = 85°, 80°, and 75°, respectively. The change in these differences in b

_{j}values is linear. As expected, b

_{1}shortened as δ decreased. The opposite is true for b

_{2}and b

_{3}. The main reason for this is the design of the right roadway edge based on the design vehicle swept path analysis. Namely, on the roundabout exit at legs 2 and 3, the body of the design vehicle swept a wider surface than on the roundabout entry at leg 1, and this surface was ever wider as δ decreased. Therefore, the designed exit radii R

_{2}′ and R

_{3}′ were significantly larger than the recommended ones and, consequently, the distances b

_{2}and b

_{3}were inversely proportional to δ.

_{mj}results as the analytical one for 6% to 15% higher traffic flows q

_{i}. The difference in traffic flow values obtained through this calculation is shown as a percentage concerning the initial total value of 1800 veh/h (Figure 14).

## 4. Discussion and Conclusions

_{m,R}= 0.85, which is defined as the maximum entrance degree of saturation [28]. This will allow the investigation of how this distance affects other roundabout geometric parameters and traffic flows. The described approach to roundabout capacity model comparison and appropriateness evaluation could be applied to more complex roundabout setups, i.e., to roundabouts with two-lane approaches, two-lane circulatory roadways, different leg numbers and alignments, and the presence of pedestrian and cyclist flow. This investigation could be also conducted for urban areas where spatial and territorial constraints are more stringent, i.e., for roundabouts with minimal outer radii of 6.5 m and with leg alignments defining the minimum distance between exiting and entering traffic flows along the center of the circulatory lane of 9 m.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Design of initial geometric parameters: (

**a**) the outer radius, legs, and splitter islands; (

**b**) the central island radius and the circulatory roadway width defined based on the design vehicle swept path when driving in a full circle.

**Figure 2.**Roadway right edge design based on the trajectory of the vehicle’s right-turn movement: (

**a**) defined by entry radius and exit radius; (

**b**) defined by the trajectory of the design vehicle.

**Figure 3.**Defining the distance b along the center of the circulatory lane, for roadway right edge design defined by the: (

**a**) entry and exit radius; (

**b**) design vehicle trajectory.

**Figure 4.**Traffic flows at a four-legged roundabout: (

**a**) each entry in each travel direction, in veh/h; (

**b**) entering, exiting, and circulating the roundabout.

**Figure 6.**Conflict factor reflecting the impact of exiting traffic on entry capacity by the distance between exiting and entering traffic streams along the center of the circulatory lane.

**Figure 9.**Investigated parameters at leg 1 as a function of R

_{o}for different δ, considering the application of design conditions 1 and 2: (

**a**) geometric parameter b

_{1}; (

**b**) traffic parameter x

_{m1}.

**Figure 10.**Investigated parameters at leg 2 as a function of R

_{o}for different δ, considering the application of design conditions 1 and 2: (

**a**) geometric parameter b

_{2}; (

**b**) traffic parameter x

_{m2}.

**Figure 11.**Investigated parameters at leg 3 as a function of R

_{o}for different δ, considering the application of design conditions 1 and 2: (

**a**) geometric parameter b

_{3}; (

**b**) traffic parameter x

_{m3}.

**Figure 12.**Investigated parameters at leg 4 as a function of R

_{o}for different δ, considering the application of design conditions 1 and 2: (

**a**) geometric parameter b

_{4}; (

**b**) traffic parameter x

_{m4}.

**Figure 13.**The average difference between b

_{1}, b

_{2}, and b

_{3}for δ = 90° and b

_{1}, b

_{2}, and b

_{3}for δ = 85°, 80°, and 75°.

**Figure 14.**Results of the reversed calculation of traffic flow—the difference in traffic flow values shown as the percentage of the initial 1800 veh/h.

Group | Symbol | Parameter |
---|---|---|

Geometric parameters | R_{o} | outer radius (m) |

R_{i} | central island radius (m) | |

u | circulatory roadway width (m) | |

j | number of roundabout legs, j = 1, …, 4 | |

δ | angle between the roundabout legs (°) | |

l | length of the arc between the adjacent roundabout legs (m) | |

e_{j} | entry width (m) | |

e_{j}′ | exit width (m) | |

R_{j} | entry radius (m) | |

R_{j+1}′ | exit radius (m) | |

b_{j} | distance between exiting and entering traffic flows along the center of the circulatory lane (m) | |

Traffic parameters | i | direction of travel through roundabout, i = 1, …, 12 |

q_{i} | traffic flow in the direction i (veh/h) | |

Q_{i} | traffic flow in passenger car units in the direction i (pcu/h) | |

f_{T} | conversion factor for traffic composition (pcu/veh) | |

Q_{Cj} | circulating traffic flow (pcu/h) | |

Q_{Sj} | exiting traffic flows at leg j (pcu/h) | |

Q_{Ej} | entering traffic flows at leg j (pcu/h) | |

BC_{j} | base entry capacity of the roundabout at leg j (pcu/h) | |

f_{P} | capacity reduction factor for pedestrian and cycling traffic flow at roundabouts (-) | |

C_{Ej} | entry capacity at leg j (pcu/h) | |

C_{mj} | entry capacity for mixed traffic flow at leg j (veh/h) | |

q_{mj} | mixed traffic flow at leg j (veh/h) | |

x_{mj,A} | degree of saturation of entry at leg j according to analytical model (-) | |

x_{mj,R} | degree of saturation of entry at leg j according to regression model (-) | |

α | factor reflecting the impact of exiting traffic on entry capacity by distance b (-) | |

β | factor for adjusting circulating flow depending on the number of circulating lanes (-) | |

γ | factor for adjusting entry capacity depending on the number of circulating lanes (-) |

Leg j | Q_{Ej} (pcu/h) | Q_{Sj} (pcu/h) | Q_{Cj} (pcu/h) |
---|---|---|---|

1 | Q_{E1} = Q_{1} + Q_{2} + Q_{3} | Q_{S1} = Q_{4} + Q_{8} + Q_{12} | Q_{C1} = Q_{7} + Q_{10} + Q_{11} |

2 | Q_{E2} = Q_{4} + Q_{5} + Q_{6} | Q_{S2} = Q_{3} + Q_{7} + Q_{11} | Q_{C2} = Q_{1} + Q_{2} + Q_{10} |

3 | Q_{E3} = Q_{7} + Q_{8} + Q_{9} | Q_{S3} = Q_{2} + Q_{6} + Q_{10} | Q_{C3} = Q_{1} + Q_{4} + Q_{5} |

4 | Q_{E4} = Q_{10} + Q_{11} + Q_{12} | Q_{S4} = Q_{1} + Q_{5} + Q_{9} | Q_{C4} = Q_{4} + Q_{7} + Q_{8} |

**Table 3.**Overview of two roundabout design conditions’ fulfillment (marked with +): Condition 1 (l ≥ 20 m)/Condition 2 (R

_{j+1}≥ R

_{j}+ 2 m).

R_{o} (m) | 13.0 | 13.5 | 14.0 | 14.5 | 15.0 | 15.5 | 16.0 | 16.5 | 17.0 | 17.5 | 18.0 | 18.5 | 19.0 | 19.5 | 20.0 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

δ = 90° | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ |

δ = 85° | −/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ |

δ = 80° | −/− | −/− | −/− | +/− | +/− | +/− | +/− | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ | +/+ |

δ = 75° | −/− | −/− | −/− | −/− | −/− | +/− | +/− | +/− | +/− | +/− | +/− | +/− | +/+ | +/+ | +/+ |

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Čudina Ivančev, A.; Ahac, M.; Ahac, S.; Dragčević, V.
Comparison of Single-Lane Roundabout Entry Degree of Saturation Estimations from Analytical and Regression Models. *Algorithms* **2023**, *16*, 164.
https://doi.org/10.3390/a16030164

**AMA Style**

Čudina Ivančev A, Ahac M, Ahac S, Dragčević V.
Comparison of Single-Lane Roundabout Entry Degree of Saturation Estimations from Analytical and Regression Models. *Algorithms*. 2023; 16(3):164.
https://doi.org/10.3390/a16030164

**Chicago/Turabian Style**

Čudina Ivančev, Ana, Maja Ahac, Saša Ahac, and Vesna Dragčević.
2023. "Comparison of Single-Lane Roundabout Entry Degree of Saturation Estimations from Analytical and Regression Models" *Algorithms* 16, no. 3: 164.
https://doi.org/10.3390/a16030164