---

Figure 1 reprises a simplified intersection with only straight-through traffic.

Vehicles on Wi and Ei are being given green signals as indicated, and they can pass through the intersection without slowing.  Meanwhile vehicles on Si and Ni are being given red signals and must stop without entering.

Let us assume that the traffic 'aspects' reverse on one-minute cycles.  In 'congested flow', theoretically a maximum of 2,050 / 60 = 34 vehicles will pass through the intersection during each green interval. 

Meanwhile, in congested flow, 34 vehicles will be stopped in a queue (platoon) to wait out each red signal.  These are oversimplified estimates, of course.  As analyzed in the solution to the Density Lock puzzle, additional time must be allowed to bring vehicles to a stop and to clear the intersection of vehicles in a platoon, operating at vC and then accelerating to vL.

Many relevant cases can be represented by non-congested flow.  Let us suppose, for example, that density = 12 veh/mi (roughly one-fifth of the maximum for 35 mi/hr).  Fewer than two vehicles per minute show up for either green or red signals.  The latter must remain stopped a full minute, during which fewer than two vehicles will pass through the intersection.  Red signals can keep the intersection almost empty, reducing its traffic flux to near zero.  

---

Figure 2 reprises the same simplified intersection, now depicting turns, along with straight-through traffic.

Here one can see a performance shortcoming in the left-turning procedure.  Whereas it is forbidden to enter an intersection on a red signal,  getting caught in the middle of an intersection when the signal changes to red is a common experience. 

Such is often the case for a left-turning vehicle -- plus its consequent queue -- which must wait for an opening in on-coming traffic.  The result is delay in the release of cross-traffic under its green aspect. 

---

Performance of an Intersection can be judged by the average speed of vehicles that pass through it.  Solvers of several puzzles, for example Train Speed, have discovered the need to put 'time in the numerator' for calculating average speeds.  That is what we shall do here.

Let us make the following assumptions, intentionally generous to intersection performance:

• that each individual approaching vehicle is traveling at vL = 35 mi/hr = 51 ft/sec;
• that all incoming lanes Si, Ei, Ni, Wi are accommodating equal traffic flux;
• that equal durations of one minute have been programed for signal aspects. 

Thus an approaching vehicle will be equally likely to face either a green or red signal.  The first vehicle stopped as its aspect changes to red is called the platoon 'leader'...  Assume...

• that the 'leader' decelerates at g/10 = 3.2 ft/sec/sec, which requires 16 seconds;
• that after stopping the 'leader' waits 60 - 16 seconds for the aspect to change to green;
• that the 'leader' then accelerates to vL at g/10 = 3.2 ft/sec/sec, which also requires 16 seconds.

Vehicles in the platoon (other than the 'leader') will be constrained to lower average  performance in speed by their need to travel at creep speed vC before being released to accelerate to vL.  We shall ignore those vehicles in the platoon for this comparison.

The total distance covered by the 'leader' during the red sequence described above can be calculated as xS = 3.2 x 16² = 819 ft.  That distance is accompanied by a total elapsed time of tS = 60 + 16 = 76 seconds.  Meanwhile, vehicles with the green signal will cover that same distance, xS = 819 ft, at vL = 51 ft/sec in 819 / 51 = 16 seconds. 

Accordingly, vehicles pass through the intersection in either 76 seconds or 16 seconds.  With time in the numerator, we calculate the average elapsed time as (76 + 16) / 2 = 46 seconds, for an average speed given by (819 / 46)(3600 / 5280) =  12 mi/hr. 

---

Figure 3 reprises the Rond-Point 
which was postulated in the puzzle to replace the intersection shown in Figures 1 and 2 above.

In the absence of traffic signals, vehicles can flow continuously into and out of the circle, limited by the flux that can be accommodated in the circle itself. Vehicular speed is also limited by the diameter of the circle.
 
Merging of vehicular flows are shown at four points.  We observe that for straight-through traffic, each incoming lane must yield to only one other incoming lane, which is already established in the circle: Si yields to Wi, Ei yields to Si, Ni yields to Ei, and Wi yeilds to Ni.

---

Performance of a Rond-Point can be judged by the average speed of vehicles that pass through it.  For comparison with the performance of the intersection above, let us make the following assumptions, which are intentionally ungenerous to the rond-point:

• that each approaching vehicle is traveling at vL = 35 mi/hr = 51 ft/sec;
• that vehicular speed within the circle is limited to vP = 20 mi/hr = 29 ft/sec;
• that on entry each vehicle must yield to one vehicle already in the circle.

Typically vehicles entering a rond-point merely slow down to give right-of-way to vehicles that are already established in the circle; however, for this comparison, we shall assume...

• that every incoming vehicle must stop for five seconds outside the circle;
• that after waiting, vehicles accelerate from zero to vP at g/10 = 3.2 ft/sec/sec;
• that the circumferential distance from incoming to outgoing is 700 ft;
• that outgoing vehicles accelerate from vP to vL at g/10 = 3.2 ft/sec/sec.
  

The time increments and associated distances can be calculated as follows:

• time decelerating from vL to stop = 16 sec, requiring 410 ft outside the rond-point;
• time waiting for passing vehicle = 5 sec with no movement (distance = 0 ft);
• time accelerating from stop to vP = 9 sec, requiring 130 ft;
• time in circle at vP = 20 sec for distance of 700 - 130 = 570 ft;
  
• time accelerating from vP to vL = 7 sec requiring 281 ft.

Total = 16 + 5 + 9 + 20 + 7 = 57 sec over a distance of 410 + 130 + 570 + 281 = 1,391 ft, for a speed = (1,391 / 57)( 3600 / 5280) = 16.6 mi/hr -- suggesting that, under the assumptions above, the Rond-Point is as much as 40% faster than a convention intersection!

---

Epilog:

Comments received from solvers near and far include queries about the applicability of rond-points in locations where there are multi-lane intersections.  For example: Richard Alexander wrote, "I suspect one reason rond-point intersections are not popular in the U.S. is because U.S. streets tend to be wider (multiple lanes) and rond-points work best with narrow streets feeding vehicles that circle single-file."

This animation depicts a rond-point that serves six-lane roadways and may give us some insights.  Each incoming roadway comprises three separately marked lanes, and for our observations here, they will be numbered by convention (left-to-right).  Each outgoing roadway offers space for three lanes, but they are unmarked.  Generally, one vehicle at a time seems to be exiting there, and, the driver is at liberty to position his or her vehicle laterally as appropriate for roadway conditions ahead.  The circumferential roadway is wide enough to accommodate three unmarked lanes, which will be identified here as C1, C2, C3.

---

Just for fun, let us trace the eight vehicle pathways as shown in the animation…

    Right Turning: 

• grey vehicle begins right signaling at Si3, yields to aqua vehicle in C3, and exits at Wo3
• orange vehicle begins right signaling at Ni3, takes lane C3, and exits at E

    Straight Through:

• blue vehicle enters at Ni2 not signaling, takes lane C2, begins right signaling, and exits at No2
• violet vehicle enters at Ei2 not signaling, takes lane C2, begins right signaling, and exits at Eo3
• aqua vehicle enters at Wi2 not signaling, takes lane C2, begins right signaling, and exits at Wo2
  

    Left Turning:

• red vehicle begins left signaling at Si1, takes lane C1, begins right signaling, and exits Eo1
• yellow vehicle begins left signaling at Wi1, talks lane C, begins right signaling, and exits at So1
  

    U-Turning

• green vehicle begins left signaling at Ei1, yields to yellow vehicle in C1, begins right signaling, and exits at Wo3

Admittedly the traffic density in the animation is low; however, we do observe vehicles operating safely side by side within the rond-point and changing 'lanes' as well.  Based on first-hand experiences here in France, driving in the circle is much like driving slowly on a limited-access expressway, especially where the diameter is large. 

The success of the rond-point here in France may be attributable to one salient observation: In six years, I have never seen any vehicle pass another on the right.  Not even once.

Sophisticated solvers may want to confirm for themselves that the multi-lane conventional intersection suffers a worse performance comparison than the single-lane analysis evidenced above.  Hint: To accommodate left turns, a green aspect must be periodically given to one pair of lanes, which necessitates prolonged red aspects for all other lanes in the intersection.

Finally, there is the matter of right-signaling approaching outgoing lanes, which is offered mostly as a courtesy to vehicles waiting at an incoming lane.  An optional courtesy, it seems.  There is a bit of hazard to consider.  If right-signaling for an upcoming exit is inadvertently offered too early the driver of an entering vehicle might stop yielding right-of-way and try to enter the circle.  In such a case, it would have been safer not to signal at all, which will merely result in a few seconds of stop-time for the entering vehicle.