Team #5888 Page 1 of 16

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1 Team #5888 Page 1 of 16 Abstract Over the past couple decades, growing amounts of research in the United States and especially in Europe have been devoted to developing modern traffic circles. It has been clearly demonstrated in studies throughout the world that, compared to the performance of traditional intersections, traffic circles, or roundabouts, increase an intersection s traffic flow while decreasing the rate of both accidents and fatalities due to accidents. However, there has been much less research concerning the development of the most efficient way to manage traffic flow in and around the traffic circle. Our model analyzes the traffic capacity of traffic circles using several different methods of controlling traffic flow in and around the circle. The basic model is a traffic circle with a single lane that forms a perfect circle and four single lane roads entering the traffic circle ninety degrees apart. This model returns the capacity, in cars per hour, that the traffic circle can handle when using yield signs, stop signs, or traffic lights on the entryways or on the circle itself and predicts the optimal configuration. Finally, the basic model is expanded to find the capacity and optimal configuration of two lane and three lane traffic circles, while retaining the majority of our original assumptions. It can also be expanded further should a traffic circle be desired at an intersection that has more than three lanes. This model determines that the most efficient way to manage roundabout traffic, regardless of the number of lanes, is by placing yield signs on the incoming roads.

2 Team #5888 Page 2 of 16 Traffic Circle Optimization: Round and Round We Go Introduction Optimizing traffic flow in metropolitan areas is a challenging task for city planners. As different innovations are applied to this challenge, one rising star is the traffic circle. Instead of depending on conventional traffic lights to direct traffic, the traffic circle uses the very structure of the road to guide traffic. Traffic circles are becoming more and more popular overseas as they continue to prove themselves more efficient than traffic lights. Traffic circles also greatly reduce the number of traffic accidents and injuries at crowded intersections [1]. Research has led to the conclusion that roundabouts are the safest and most effective type of intersection traffic control available today" [1]. The goal of our research is to provide an effective tool of analysis to properly evaluate the most efficient way to control different traffic circles in different circumstances. Although research has proven that traffic roundabouts are a more efficient and safe way to direct traffic than traffic lights, the best way to assist traffic flow in these circles is still up for debate [2]. There are three visual symbols that can be used: yield signs, stop signs, and traffic lights. These can be positioned where vehicles enter the circle or inside the circle. Each of these positioning techniques produces different results in maximizing traffic flow through a circle. Using metering signals to control entering traffic is one popular option to direct traffic flow. These signals are on time controls that monitor the overall flow and queues of each input [3]. When flows and queues are at minimal values, the meters are put in a non-functioning state. The traffic simply yields as it enters the system. When the entering traffic flow increases beyond a given threshold, detectors located in the pavement activate the metering signals [4]. These signals then produce rotating red and green lights to control the traffic. This system makes sure to allow access to all roads in percentages that give precedence to inputs that have a higher flow rate of traffic [5]. Stop signs work much in the same manner, with the addition of having a constant rate of stopping vehicles at all intersections [6]. Yield signs are by far the most popular control for traffic circles. They are placed where vehicles are approaching the roundabout [7]. They depend on drivers using their own judgment to effectively merge into oncoming traffic. Because of reduced speeds within the circle, merges can happen with little difficulty [8]. Assumptions Perfect drivers-we assume no accidents occur and all timing is exact. There is a constant flow of traffic from all entrances. This allows the model to be modified for various situations. All drivers maintain optimal gap distance, or following distance, of one car length.

3 Team #5888 Page 3 of 16 All the cars are the same make and model being 10 feet long with a 2 second acceleration time to 15 mph. These parameters align with family car specifications [9]. No exit time. We assume that the entrances and exits are like on and off ramps with no deceleration times needed. The shape of the circle is perfectly round. The lane width of the roundabout is 12 feet with the middle circumference being 666 feet. Right turn overpass: all cars turning right do not enter the traffic circle. See point D on Figure 1 for an example. The number of cars entering the roundabout per second is constant and is divided equally among the entrances. The traffic circle runs at full Car Capacity at all times. Nomenclature Avg o =Average Circulation Avg stop =Stop Sign Circulation Avg light =Light Circulation Avg yieldinside =In Yield Circulation p n =Percentage Occurrence t=segmented Roundabout Time Period C=Roundabout Circumference ω=number of Inputs c=car Capacity c l =Car Length c s =Car Gap Cars yieldn =Yield Result Cars stopn =Stop Result Cars lightn =Light Result Cars yieldinsiden =In Yield Result n=number Occurrence ψ=average Speed α=rate of Entry per Input β=total Rate of Entry t o =Outer Segmented Roundabout Time Period t in =Inner Segmented Roundabout Time Period One-lane Model and Validation The formation of our model begins with the perfect world assumptions of a four-way perpendicular intersection converted into a roundabout. These assumptions, listed above, allow for ideal circumstances to be evaluated and basic analytical equations to be formed.

4 Team #5888 Page 4 of 16 Segmented Roundabout Time Period is the time it takes a car to travel one division of the circle relative to the number of entrance/exit opportunities. For the basic one-lane traffic circle with four perpendicular entrance/exits (inputs), 4t constitutes the time required for a vehicle to travel the entire circumference of the middle circle. The average time of each car within the circle is called Average Circulation. It is determined by the percentage of vehicles that chose each of the possible paths: right turn, straight, left turn, and u-turn. The cars always travel counterclockwise around the circle, but their path can result in a left turn by traveling three-quarters around the circle. Each percent, represented by p, is multiplied by the amount of time those cars are traveling within the circle. A car turning right travels 1t within the circle and then exits. Likewise a car proceeding straight is in the circle 2t. See Equation 1: Avg o = p 1 t + p 2 2t + p 3 3t + p 4 4t (1) However, in our model we assume that traffic turning right will completely divert to the rightturn overpass. This leaves 100% of the traffic entering the circle with three options: straight, left turn, and u-turn. To illustrate the model we choose the percentages of these choices as 75% straight (p 2 ), 24% left (p 3 ), and 1% u-turn (p 4 ). These numbers are reasonable estimates because most traffic going through an intersection usually passes straight through. Very little traffic chooses to u-turn. Because we do not have right turns passing through the circle, the remainder is a relatively small percentage for left turns. These values are represented in the Equation 2: Avg o = 0 t t t t (2) Next, the value of the Segmented Roundabout Time Period must be defined. The circumference, C, must be divided by the number of input locations then divided by the Average Speed of the traffic within the circle in miles per hour. Finally, the equation must be converted into seconds: t = C 1 1 mi 3600 sec ω ψ 5280 ft 1 r (3) t = 666ft 4 1 r 1 mi 3600 sec 15 mi 5280 ft 1 r With t defined, it is possible to solve for Average Circulation: = sec (4) Avg o = sec sec sec sec = 17.1 sec (5) To determine the number of cars that a circle can hold, the circumference of the circle driven must be divided by the sum of the length of the vehicle added to the space between it and the subsequent vehicle. In our sample model, the car is 10 feet long with one car length between vehicles. We calculate the resulting Car Capacity:

5 Team #5888 Page 5 of 16 In our basic model this equation results in: C c = (6) c l +c s c = 666 ft 10 ft +10 ft = 33.3 cars (7) Car Capacity must then be divided by the number of inputs to determine the number of cars resulting from each input at any time. The cars are then divided by Average Circulation to determine the Rate of Entry/Exit from each input: α = c sec ω Avg o 1 r (8) α = 33.3 cars sec 3600 sec 1 r 1753 cars r (9) The total number of cars per hour that the circle can handle is the Total Rate of Entry Cars yield β = 4α = c sec Avg o 1 r = 33.3 cars sec 3600 sec 1 r 7011 cars This model, with a 7010 cars/hr capacity in perfect conditions, assumes the use of yield signs on all roads entering the circle. As each car arrives, it immediately has a gap which it can fill which will keep the circle functioning constantly at full-capacity. r (10) (11)

6 Team #5888 Page 6 of 16 Figure 1.1 In Figure 1.1 each entrance s traffic is represented by a different color. All of the right-turning traffic exits through the right bypass. The remaining traffic then goes primarily to the opposite exit with some continuing around the circle to result in a left turn or u-turn. The yield sign is placed at position A in the diagram above to allow cars to gauge the position of other vehicles before entry. If a stop sign or traffic light were utilized, it would be placed at position B. This position allows for a 2 second acceleration time for the stopped traffic to get up to 15 mph before entering the traffic circle. Position C is the theoretical position of any traffic control devices within the circle. We then compare this result to the use of stop signs at all entering intersections. The stop sign is placed back away from the circle at the distance that it will take our accelerating family car to accelerate from 0-15 mph in 2 seconds. Because all cars will experience this 2 second additional time, the Average Circulation goes from 17.1 seconds to 19.1 seconds for the Stop Sign Circulation. Running our model s calculations with this new average time results in a maximum capacity of 6276 cars/hr:

7 Team #5888 Page 7 of 16 Cars stop = 33.3 cars s 6276 cars 19.1 s 1 r r (12) Using stop lights instead of yield signs has a similar negative result. One full rotation of the traffic circle is 4t which is approximately seconds. Continuing under our assumption of equal traffic flow coming from all directions, all four traffic lights will function with equal amounts of green and red. Yellow accumulates with the red and green because we assume that during the first half of the yellow light, the driver will continue as if the light were green. During the second half of the yellow light, the driver will stop, treating the signal as a red light. Therefore, the seconds is divided in half to give periods of seconds for both the red and green cycles. At any point in time, there are two green traffic lights and two red traffic lights. The green cycles are offset so that when the traffic light at point A is in the second 7.56 seconds of its green cycle, the traffic light at point B is also green (See Figure 1.2). However, after a time lapse of 7.56 seconds, when the traffic light at point A reaches the end of its green cycle and turns red, after seconds of being green, the traffic light at point B will enter the second half of its green cycle and the traffic light at point C will turn green. The traffic lights will continue to turn green in a clockwise fashion as traffic flows in a counterclockwise fashion. Because the time it takes a car traveling at 15 mph to travel from its point of entry to the next entry is 7.56 seconds, cars traveling on the roundabout itself will never pass by an entry whose traffic light is green. In Figure 1.2, the first car at the traffic light at point B enters the roundabout. After the time lapse of 7.56 seconds, this car reaches point X in Figure 1.2, while the traffic light at point A is now red. Therefore the flow of cars on the roundabout continues uninterrupted by the traffic coming onto the circle. Figure 1.2

8 Team #5888 Page 8 of 16 For all of those waiting at a red light there are wait times from 0 seconds to 15 seconds, and assuming constant traffic flow, the average wait time at the red light will be 7.5 seconds. Those cars which arrive during the green light will have a 0 second wait time. This results in an overall average of 3.75 seconds wait time for traffic passing through the traffic lights. Added to the 17.1 second Average Circulation, the Light Circulation is seconds: Cars ligt = 33.3 cars sec 3600 sec 1 r 5750 cars r (13) One other alternative which we consider is the placement of yield signs within the traffic circle itself. These signs award the right-of-way to the entering traffic instead of the traffic already within the circle. The traffic in the circle has to take a 1 second deceleration/acceleration period to account for looking for incoming traffic at each of the inputs it passes. This process is illustrated in Equation 14: Avg yieldinside = sec + 1 sec sec + 1 sec sec + 1 sec sec + 1 = 19.4 sec (14) Cars yieldinside = 33.3 cars sec 3600 sec 1 r 6179 cars r (15) In this scenario, the best traffic flow apparatus is the yield signs outside the circle. This solution is followed in decreasing efficiency by stop signs, yields inside the circle and then traffic lights. These results are summarized in Table 1. For a 1-Lane Roundabout # Cars per Hour # Cars per Hour per Entrance Yield Sign on the Roundabout 6,179 1,545 Yield Sign on the On Ramp 7,011 1,753 Stop Sign on the On Ramp 6,276 1,569 Stop Lights on the On Ramp 5,750 1,437 Table 1 Two-lane Model and Verification After calculating a basic model for a one-lane roundabout, we add a second lane to all inputs and to the roundabout. This change results in more cars arriving at the system and a higher capacity for the roundabout. The time around the circle is now calculated by t being the time around the inner circle plus the time around the outer circle averaged. The inner circle retains its diameter of 212 ft. The outer circle has a diameter or 236 ft. These give inner circumference (C in ) of 666ft and outer circumference (C o ) of 741 ft. We also increase the space between cars to two car lengths to accommodate merging between the inner and outer loops. This results in a 47 car fullcapacity model as shown in Equation 16: c = C o +C in c l +c s = 741 ft +666 ft 10 ft +20 ft = 1407 ft 30 ft 47 cars (16)

9 Team #5888 Page 9 of 16 Applying this new data to Equation 16 for the Segmented Roundabout Time Period result is shown in Equation 17 which represents one half of the circumference of the circle: t = 1407 ft 4 1 r 1 mi 3600 sec 15 mi 5280 ft 1 r = sec (17) Figure 2 has the same labeling system as Figure 1. This diagram emphasizes the movement of the drivers moving in tandem upon entry in a two lane roundabout. Figure 2 The original equation has to be slightly altered to find the average time that a vehicle spends on the roundabout. We work under the assumption that traffic enters the roundabout at a constant flow from both lanes in tandem entering from any particular direction. The traffic entering in the right lane is required to take the opposite exit, resulting in passing straight through the roundabout. The other 50% of the traffic enters the inner circle. 25% of this traffic passes straight through the roundabout as well, resulting in 75% of the vehicles taking the straight route. The other 25% will exit with a resulting left turn or u-turn. Because the system is now functioning at two car-length spacing, there is enough room for cars to safely exit directly from the inner circle. These cars will exit without merging into the outer circle. The percentages of people taking the straight, left turn, and u-turn options remain the same from the previous model. Outer Segmented Roundabout Time Period represents the time to travel one division of the outer loop relative to the number of entrance/exit opportunities, while Inner Segmented Roundabout Time Period represents the same for the inner loop. We also add 2 seconds to each of the inner circle times to allow for passing through the outer circle upon entrance and exit, which results in the following model:

10 Team #5888 Page 10 of 16 Avg yield 2 = 0.5(2t o ) t in + 2 sec t in + 2 sec (4t in + 2 sec) (18) In practice, with our example, this equation will result in: t o = t in = 741 ft ft 4 1 r 1 mi 3600 sec 15 mi 5280 ft 1 r 1 r 1 mi 3600 sec 15 mi 5280 ft 1 r = sec (19) = sec (20) Avg yield 2 = sec sec sec sec + 2 = sec (21) With this adjustment to the model we run the calculations for number of cars per hour for yield sign, stop sign, and stop light again: Cars yield 2 = 47 cars sec 3600 sec 1 r 8926 cars r (22) Once again for the stop sign we added 2 seconds to get the cars up to speed. We also add an additional.5 seconds to account for the additional time needed for the cars to accelerate into their respective lanes. This results in a second average time: Cars stop 2 = 47 cars sec 3600 sec 1 r 7886 cars r (23) Because traveling half the average circumference of the roundabout takes seconds, or the Segmented Roundabout Time Period from above, the stop lights will be on rotations of 16 seconds green and 16 seconds red. Cars waiting at red lights will wait an average of 8 seconds because they are arriving at a constant rate and waiting from 0 to 16 seconds. Cars at green lights will not have to wait at all. This yields a total average wait time of 4 seconds, which results in second average time in the circle: Cars ligt2 = 47 cars sec 3600 sec 1 r 7371 cars r (24) The yield signs within the traffic circle add one second for each intersection in which the vehicle has to yield: Avg yield 2 = sec sec sec sec = sec (25) Cars yieldinside 2 = 47 cars sec 3600 sec 1 r 8370 cars r (26)

11 Team #5888 Page 11 of 16 In this scenario the best traffic flow apparatus is the yield signs outside of the circle. This solution is followed by yield signs located inside the circle, stop signs, and then stop lights. These results are summarized in Table 2: For a 2-lane Roundabout # Cars per Hour # Cars per Hour per Entrance Yield Sign on the Roundabout 8,367 2,091 Yield Sign on the On Ramp 8,924 2,231 Stop Sign on the On Ramp 7,884 1,961 Stop Lights on the On Ramp 7,369 1,842 Table 2 Model Convertibility These formulas can be used for more than just the two sample roundabouts above. The information needed is the average diameter of each circle, number of lanes, speed of vehicles, acceleration times, length of vehicles, space between vehicles, and percentage of vehicles taking each route. With this model, any traffic circle can have its data analyzed for its most efficient traffic flow apparatus. A specific example is adding a third lane to the model. The third lane would make a larger circumference with three vehicles now entering the circle from each side at any one time. With lane number being the only variable that changes in this scenario, the Cars yield3 would be 12,907 cars. Summary Both Table 1 and Table 2 show similar results. The yield sign outside of the roundabout produces significantly more efficient traffic flow than do the stop sign, traffic light, and yield inside the roundabout. Traffic engineers can analyze the data that this model produces and decide between the top two options produced. We recommend using the yield signs outside of the circle under these conditions. Recognizing that in real life, drivers are not perfect, one reason that another method might be used is if there were a high number of collisions in a particular location. In this situation using a more controlled method such as the timed stoplights might work more efficiently. For a 1-Lane Roundabout # Cars per Hour # Cars per Hour per Entrance Yield Sign on the Roundabout 6,179 1,545 Yield Sign on the On Ramp 7,011 1,753 Stop Sign on the On Ramp 6,276 1,569 Stop Lights on the On Ramp 5,750 1,437 Table 1

12 Team #5888 Page 12 of 16 For a 2-lane Roundabout # Cars per Hour # Cars per Hour per Entrance Yield Sign on the Roundabout 8,367 2,091 Yield Sign on the On Ramp 8,924 2,231 Stop Sign on the On Ramp 7,884 1,961 Stop Lights on the On Ramp 7,369 1,842 Table 2 Further Research There are several interesting features that could be explored in more detail to enhance the viability of our model. One example is testing the model with more lanes. Our basic two lane model could be easily modified for larger quantities of lanes. The test would be if the numbers remain semi-realistic using the model in this circumstance. Another area for future research is varying flow rates for opposing directions. Currently, our model shows the efficiency of the roundabout with constant traffic flow from all directions. Analyzing the traffic flow at different rates from different inputs would make the model more complex.

13 Team #5888 Page 13 of 16 References [1] Russel, Eugene R. Ph.D. P.E., Margaret J. Rys, Ph.D., Srinivas Mandavilli. Operational Efficiency of Roundabouts. 2 nd Urban Street Symposium, Anaheim July [2] US Department of Transportation. Planning. [3] Natalizio, Emmanuel. Roundabouts with Metering Signals. Institute of Transportation Engineers 2005 Annual Meeting, Melbourne, Australia. August [4] Akcelik, Rahmi. Capacity and Performance Analysis of Roundabout Metering Signals. TRB National Roundabout Conference, Vail, Colorado, USA May [5] Akcelik, Rahmi. Roundabout Case Study Comparing Capacity Estimates from Alternative Analytical Models, A. 2 nd Urban Street Symposium, Anaheim July ments/akcelik_uss2roundaboutpaper_revised.pdf+akcelik++a+roundabout+case+stu dy [6] Akcelik, Rahmi. Roundabout Model Calibration Issues and a Case Study. TRB National Roundabout Conference, Vail, Colorado, USA May [7] Weber, Philip P.E.Internationally Recognized Roundabout Signs. TRB National Roundabout Conference, Vail, Colorado, USA May [8] US Department of Transportation. Traffic Design and Landscaping. [9] Riches, Erin. Civic at Fever Pitch. 17 Nov

14 Team #5888 Page 14 of 16 Technical Summary Our basic model deals with four single-lane inputs and a one-lane on the roundabout, where the roundabout has an inner diameter of 200 feet, which gives us a middle diameter of 212 feet and an outer diameter of 224 feet. All of our calculations for time averages are proportional to the radius, making our model easily modifiable for any size roundabout. Our model has a right lane offshoot, allowing us to ignore all traffic turning right; this removes almost a fourth of the cars on the circle and would be an optimal choice regardless of traffic flow. Our basic model has four equally spaced entrances and exits (see Figure 1), and has a mid-lane circumference of 666 ft. We begin by assuming that 75% of cars not turning right go straight, that 24% go left, and 1% go around in a U-Turn. We calculate the average time that it takes for a car at 15 mph to travel between consecutive entrances as t = 7.57 sec. Using this information we calculate t avg = 17.1 sec for the basic model with a yield sign on the on-ramp. Then we can calculate that the maximum number of cars that can fit on the main circle of the roundabout, with a one car-length gap between them, is 33. We use this number and t avg to calculate the total number of cars per hour, for the number of cars per entrance per hour multiply that by ¼ (see Table 1). Using this format, we can quickly modify our calculations for yield signs on the roundabout (at point C on Figure 1). We simply add 1 second to t and recalculate everything. For a stop sign at point B on Figure 1, we add 2 seconds to our t avg result.

15 Team #5888 Page 15 of 16 Figure 1 Figure 2 For a stop light at point B in Figure 1, we set the green lights at seconds and the red/yellow at seconds. Because we calculate that it takes a car seconds to go completely around the circle at 15 mph, the lights are arranged to be offset by half the time of the total green light cycle. This means that when a car gets the green light at point B in Figure 2, by the time it gets to point X, the light at point A has gone red; thus, there is no traffic conflict. Using this method we calculated the flow rates for four different situations and obtain the results in Table 1. These show that a yield sign on the on-ramp is the optimal configuration for an intersection with a constant flow rate from all four directions. Table 1 For a 1-Lane Roundabout # Cars per Hour # Cars per Hour per Entrance Yield Sign on the Roundabout 6,179 1,545 Yield Sign on the On-ramp 7,011 1,753 Stop Sign on the On-ramp 6,276 1,569 Stop Lights on the On-ramp 5,750 1,437 For a two lane model, we change our basic t avg equation using the assumptions that the right lane of the on-ramp is used only by people going straight, that it takes an extra two seconds for the inner lane to load and exit, that cars are two car-lengths apart, and that a car in the inner lane exits straight off instead of merging into the outer lane. This gives us the two lane average time t avg 2l = 19 sec. We calculate t out and t in using the outer and inner middle diameters respectively. Using the same methods as above, but using longer times for the stop sign and lights, we get the results in Table 2. These results still show that an on-ramp yield is the best.

16 Team #5888 Page 16 of 16 Table 2 For a 2-lane Roundabout # Cars per Hour # Cars per Hour per Entrance Yield Sign on the Roundabout 8,367 2,091 Yield Sign on the On-ramp 8,924 2,231 Stop Sign on the On-ramp 7,884 1,961 Stop Lights on the On-ramp 7,369 1,842 To convert to a three-lane format, we can evenly distribute the weighting factors between the lanes and add one second to the required time for each lane the cars have to cross to enter and exit, and one second of slowdown time. To modify our model for different flow rates, we recommend using lights on all of the on-ramps. If there are two opposite ramps with a significantly higher flow rate than the other two, we would have the higher-rated ramps with green lights that run a cycle of seconds and then have all four lights go red for 7.57 seconds, and continue in an appropriate sequence. This light cycle would keep the main ramps moving rather quickly and would give them higher priority than the light flow ramps. Our roundabout would not be full at all times, but would never be empty, thus maintaining a somewhat stable flow rate. If you have one lane with a significantly higher flow rate, we would use lights as well but run the green light on that lane for seconds. When this light reaches the last 7.57 seconds of its cycle we would resume the green light cycle from the original model. When the quarterly cycle reaches the main light again, it would run for the allotted time. This method keeps the traffic circle full, but gives priority to traffic on the high flow ramp.