Vortex-Based Zero-Conflict Design of Urban Road Networks David Eichler 1, Hillel Bar-Gera 2, Meir Blachman 1.Physics Department, Ben-Gurion University of the Negev 2. Department of Industrial Engineering and Management, Ben-Gurion University of the Negev

Part I: Motivation Conflicting (intersecting) traffic flow is a liability and a drag. Green lights are tolerable, red lights can be extremely annoying.

Traffic conflict is dangerous. Replacing conflicting flow with merging (e.g. roundabouts) saves lives. 60% fewer fatalities at roundabouts than at traffic intersections, including signaled ones. Roundabouts are slow (capacity per lane = 1200 vs vph ) and cause much traffic congestion during rush hours Traffic signals also cause congestion…obviously IS THERE A BETTER WAY?

Saturated traffic flow from a resting state is a rarefaction wave. Flow rate (vehicles passing per unit time) = vehicular density (vehicles per unit length) x “ sound speed ” (length traveled per unit time) = one vehicle per human reaction time Saturated traffic flow rate [velocity/distance between cars] through unsignalled, unconflicted intersection is also one vehicle per human “ reaction time ” [because distance ~ velocity x reaction time].

But reaction time to acceleration from rest is slower than reaction to braking. So in saturated traffic flow vehicles can flow into the back end of a traffic jam faster than they can flow out the front. This is why the jam persists long after the cause has disappeared (phantom jams). Saturated traffic flow at much more than 2200 vph is observed to be unstable to jamming. Jamming observed even on conflict free race tracks!

So there is a cost for stopping and starting. Eliminating traffic conflict at road intersections would avoid stopping, and this increases intersection capacity.

II. Conflict – free intersections (or zero traffic conflict (ZTC) intersections) Definition: a turning movement is an ordered pair of directions (legs stemming from an intersection) e.g. NS, SN, SE, etc. Definition: A maximal ZTC road intersection is one in which no additional turning movement can be added while keeping the other with zero conflict. A “sidewalk” turn can never conflict with any other turning movement.

We assume two lanes per leg, with merging, but in reality, more lanes could be added. This assumption is equivalent to insisting that there are no disconnected lanes in the same direction, e.g. We do not assume the driving must be all on right or all on left.

We wish to classify all conflict-free intersections. Classification via number of sidewalk turns proves useful.

M MM 3-legged conflict free intersections

The familiar four conflict-free intersections permitted by Israeli traffic signals. MM MM All maximal conflict-free 4 way intersections with 4 right sidewalk turns

a. Figure 2: Four-leg zero-traffic-conflict intersection designs, all legs are two-way-driving-on-the-right. a. through & through ; b. left & left; c. through & left from one leg; d. through & left to one leg. a. 3b. 3c.3d. 3e. 3a. M Note that 3a,c,d,e are not maximal conflict-free intersections because additional turning movements can be added without blocking existing ones. Lemma: In fact, any adjacent legs that are both 1-way would allow additional turning movement with one change, so it cannot be maximal ZTC intersection. (HBG)

a. Figure 2: Four-leg zero-traffic-conflict intersection designs, all legs are two-way-driving-on-the-right. a. through & through ; b. left & left; c. through & left from one leg; d. through & left to one leg. a. 3b. 3c.3d. 3e. 3a. M Note that 3a,c,d,e are not maximal conflict free intersection because additional turning movements can be added without blocking existing ones. Lemma: In fact, any adjacent legs that are both 1-way legs would allow additional turning movement with one directional change, so it cannot be maximal ZTC intersection. (HbG)

lemma: There are no 4-way ZTC intersections with 3 sidewalk turns in the same direction and 1 in the opposite direction [e.g. 3 right and 1 left sidewalk turn] (HbG ). The legs on both sides of the opposite direction sidewalk turn are one-way, and therefore by Lemma the resulting designs cannot be maximal ZTC. (See Fig. 3c, 3d & 3e for illustration).

a. b. 12c. 12d. f. Figure 12: additional maximal connected zero-traffic-conflict four-leg intersection designs. 12b. b. 12a. c. e.f. 12e. 12f additional maximal connected zero-traffic-conflict four-leg intersection designs. 3 STMs 2R +2L STMs M M MM 12d

Theorem: There are no other maximal 4-way conflict-free intersections (than the above 9) to within obvious symmetries (HbG). Proof: There are no others with 4 right sidewalk turns … with 2 right and 2 left sidewalk turns ….with 2 of one and only 1 of the other ….with only 2 or 1 total sidewalk turns

III. Can we use this set of conflict-free intersections to build an efficient conflict- free traffic network? Note Braess ’ s paradox: Reducing freedom to pursue individual interests is sometimes in everyone ’ s interest.

Low Conflict 1 (LC1)

33% increase in travel distance

22% increase in travel distance

Low Conflict 2 (LC2)

Square “ target ” design

LC3

LC4 10 x 20 16x 6 12 x 2 vorticies

How do these compare with simply eliminating left turns (which would lessen, but not eliminate, the need for traffic signals)? Note: the calculations below are for uniformly distributed origins and destinations.

Table 1: average additional distance for 10 by 10 nodes grid designs. (Average rectilinear distances are 6.00 blocks under CBP (center-of-block parking) and 6.04 under SP (street parking).) LC2LC1TargetOneWayNoLeftUnrestrictedAccPark U turn NLADSPPU FADSPPU NLADCBPPU FADCBPPU NLADSPAU FADSPAU NLADCBPAU FADCBPAU Street parking Central block Prohibited U turn Allowed U turn

Table 2: average additional distance for 10 by 20 nodes grid designs. (Average rectilinear distances are 9.33 blocks under CBP (center-of-block parking) and 9.37 under SP (street parking.) LC4LC3OneWayNoLeftUnrestrictedAccParkU turn NLADSPPU FADSPPU NLADCBPPU FADCBPPU NLADSPAU FADSPAU NLADCBPAU FADCBPAU Street Parking Central block parking LC4 always beats LC3

Table 3: average additional distance for 20 by 20 nodes grid designs. (Average rectilinear distances are blocks under CBP and under SP.) TargetOneWayNoLeftUnrestrictedAccPark U turn NLADSPPU FADSPPU NLADCBPPU FADCBPPU NLADSPAU FADSPAU NLADCBPAU FADCBPAU

Features in data: The most important factor in increased distance cost of conflict elimination is getting a bad start. Street parking vs. center-of-block parking makes a larger difference than U- turn option, lane access or even network design. Relative cost of conflict elimination should decline with length of trip.

Figure 11: Average additional distance as a function of the rectilinear OD distance for 10 by 10 nodes network assuming street-parking nearest-lane-access-direction with prohibited-U-turns (SP/NLAD/PU). Reference designs are: unrestricted, one-way and no-left turn; proposed low-conflict designs are: Target, LC1 and LC2. 10 x 10 SP NLAD Prohibited U turns

10 x 20 NLAD CBP Allowed U turns

20 x 20 NLAD CBP

Relative cost (percentage of increased trip length) of conflict elimination decreases with size of city, whereas cost of traffic signals does not. Conjecture: for large cities, conflicting traffic flow and attendant traffic signals increase travel time.

What about pedestrians? Vortexes should allow much easier design of green waves. Pedestrian crossing time much less than typical waiting time. Green wave can be ~85% of vortex length. This should make it much easier to stay on one. 3D infrastructure for pedestrians – e.g. bridges, tunnels – far cheaper than for vehicles.

A Tale of Two Cities Green waves for all Vortex based green waves

Green waves for all

Time: 08:10:01

Time: 08:20:01

Time: 08:30:01

Time: 08:40:01

Time: 08:50:01

Time: 09:00:01

Time: 09:10:01

Time: 09:20:01

Time: 09:30:01

Time: 09:40:01

Time: 09:50:01

Time: 10:00:01

Time: 10:10:01

Time: 10:20:01

Time: 10:30:01

Time: 10:40:01

Time: 10:50:01

Time: 11:00:01

Time: 11:10:01

Time: 11:20:01

Time: 11:30:01

Time: 11:40:01

Time: 11:50:01

Time: 12:00:01

Time: 12:10:01

Time: 12:20:01

Time: 12:30:01

Time: 12:40:01

Time: 12:50:01

Time: 13:00:01

Time: 13:10:01

Time: 13:20:01

Time: 13:30:01

Time: 13:40:01

Time: 13:50:01

Time: 14:00:01

Square “ target ” design with pedestrian crossing

Square “ target ” design

Time: 08:10:01

Time: 08:20:01

Time: 08:30:01

Time: 08:40:01

Time: 08:50:01

Time: 09:00:01

Time: 09:10:01

Time: 09:20:01

Time: 09:30:01

Time: 09:40:01

Time: 09:50:01

Time: 10:00:01

Time: 10:10:01

Time: 10:20:01

Time: 10:30:01

Time: 10:40:01

Time: 10:50:01

Time: 11:00:01

Time: 11:10:01

Time: 11:20:01

Time: 11:30:01

Time: 11:40:01

Time: 11:50:01

Time: 12:00:01

Time: 12:10:01

Time: 12:20:01

Time: 12:30:01

Time: 12:40:01

Time: 12:50:01

Time: 13:00:01

Time: 13:10:01

Time: 13:20:01

Time: 13:30:01

Time: 13:40:01

Time: 13:50:01

Time: 14:00:01

210K

270K

Is there an algorithm for determining the best conflict-free routing scheme for a given town? So far, the algorithms we try are not as good as our imagination.

Note any game where the “ players ” are intersections, each trying to reduce its own waiting time, encourages equality, and this is not what is desired. e.g. signaled intersections are typically designed to favor the longer queue - i.e. equalize waiting time between crossing options - on the grounds that total waiting time at that intersection is thereby reduced. (Let the few wait for the many, don ’ t have the many waiting for the few.) But, by encouraging equality between conflicting traffic streams, it encourages traffic conflict, because each traveler has less of a reason to favor one route over another. Need strong central authority that reduces personal options.

So the following alternative strategy was tried: Discourage equality, reduce options. Eliminate the crossing option with the lower demand. But it doesn ’ t work as well as guessing. Brute force would require at least O(9 n 2 ) But invoking 4-fold symmetry, on 9 x 9 town (=64 blocks), choosing vortex sign for each block, reduces total number of choices to ~ =

A City of No Conflict

Concluding remarks: Cities don ’ t need traffic signals, except possibly for pedestrians. Travel time would probably be reduced with conflict-free routing. Optimizing solution an unsolved problem, but not hopeless.