Density Lock

The expression density lock was coined by a certain research assistant during  landmark studies at UCLA's Institute of Transportation and Traffic Engineering in 1954.  Traffic flow analyses using primitive Monte Carlo simulation identified
various causes of traffic viscosity, including marginal friction and critical absorption volumes.
Fifteen years later, in the 1970s, density lock, which pertains to highways not surface streets, became known by a misnomer: gridlock.  Hey, where's the "grid"?
Everybody in the world has suffered from the adverse effects of density lock on roadway capacity and speed. The work at UCLA resulted in the earliest deployment of ramp metering, which is now widely applied to relieve highway congestion.

 Let us explore another concept, first proposed in 1954, for preventing density lock?

We begin with a generalized representation in Figure 1 of a vehicular platoon operating in steady state, which means traveling at a constant, limited speed vL with a constant following-distance between vehicles, measured from bumper-to-bumper xF.  The instantaneous location of each vehicle along the x-axis will be referenced herein to its front-bumper

In 'congested flow', the speed of each vehicle is constrained by the speed of the vehicle ahead, such that vN v(N-1).  The following-distance xF is controlled by the driver of the following vehicle.  Inasmuch as 40% of highway accidents involve rear-end collisions, a 'safe' following distance, often called a 'space cushion', needs to be maintained, taking into consideration roadway conditions and visibility.  The vernacular expression 'tail-gating' is used to describe the unsafe practice of following too closely.

In Figure 1, the symbol xH represents 'spatial headway', which is the sum of the vehicle length xL and the following-distance xF.  For a typical automobile, 12 ft < xL < 15 ft.  In the Density Lock puzzle, we shall assume that xL = 13 ft.  Officially, the term headway refers to the time tH required for the passage of successive vehicles, and tH = xH / vL.  Modern driver-handbooks recommend the "two-second rule," such that tF 2 sec, which would mean that tH varies with vL according as tH = 2 sec + xL / vL.  However,...
Your puzzle-master lives in France, where he has observed stripes painted alongside highways that mark one-second intervals corresponding to the posted speed limit vL.  Pictorial signboards on the wayside provide guidance to drivers for xF by indicating that vehicle N should leave an empty stripe behind any stripe ahead that is occupied by vehicle N-1.  Observations of driving behaviors on two continents suggest that an estimate of tF = 1.5 seconds would be more realistic for use  in the Density Lock puzzle.
As an illustration, consider a roadway in the U.S. with a speed-limit vL = 35 mph (~50 ft/sec).  To assure a safe following-distance using tF = 1.5 sec, xF = 75 ft, which calls for xH = 88 ft, and headway tH = 1.75 sec.  It may be worth noting that 'recommendations' in mid-twentieth century driver-manuals often said, "allow one car length for each 10 mph of speed."  Back then cars were longer, say, 15 feet, which suggests that xF = 53 ft for 35 mph (22 ft closer), and, since xH = 68 ft, the headway tH = 1.4 sec.

Speed = Flux / Density

Individual users of transportation systems care mostly about Speed.  They want to get from where they are to where they want to be.  Faster the better.  Speed is measured by distance traveled per unit of time -- miles per hour.   In vehicular transportation, limits are imposed by roadway conditions, by neighborhoods such as residential areas and business districts and school zones, by weather and visibility -- oh, and by traffic.
Managers of transportation systems care mostly about traffic flow -- Flux. More the better.  The obvious objective of 'traffic managers' is to maximize the number of vehicles moving through the system per unit of time -- vehicles per hour.  Flux is limited to various capacities within the infrastructure: by extra lanes on roadways and ramps on highways, by cross-streets and signals and stops -- oh, and by traffic.
Traffic means congestion -- Density.  Congestion comes from demands on the system exceeding  'free-flow' capacities.  Density is measured as the number of vehicles per unit of distance -- vehicles per mile.  Ironically, the objective for traffic managers is not to minimize density.  Here's why.  When roads and highways are relatively empty, congestion may indeed be low, but so is flow.  Individual users will benefit with low flow wherein speeds may be faster.  For sure, speed will be limited by something other than traffic.  Ah, but then there are fewer users being served by the transportation system.  Do traffic managers actually worry when their capital-intensive investments are under-utilized?  Some do, maybe, but not a lot.

The simple expression v mi/hr = q veh/hr / k veh/mi offers us a rich set of relationships.  Solvers will benefit from studying the fundamental diagram of traffic flow, where they will learn that for a given value for q, there are two values for k and thus two values for v, one in 'free-flow', the other in 'congested flow'.  To test remedies for Density Lock we must focus on congested flow.

 Speed = Flux / Density Management in Congested Flow Speed of each vehicle N is limited by the speed of vehicle N-1 immediately ahead. Flux through a system is framed by traffic management via designs and regulations. Density is absolutely controlled by the individual drivers in vehicles!  Sophisticated solvers will take note of the exclamation point.

Below is a graph of the three traffic variables...

speed = vL ft/sec, flux = 3,600 / tH veh/hrdensity = 5,280 / xH veh/mi.
...on a section of highway. The green dots in the graph show that in congested flow, a platoon of vehicles traveling at 60 mph will be limited to a maximum density of 33 veh/mi which imposes a limit of 2,113 veh/hr per lane maximum flux in that section.

During 'rush hour', the density of vehicles in the section of highway will necessarily increase as vehicles enter the lane on ramps from side-roads and other highways.  The graph shows that when the traffic density in that section of highway doubles to 66 veh/mi, the maximum flux, which is the carrying capacity of the lane, will be slightly reduced to 2,003 veh/hr, but the maximum speed of individual vehicles will be decreased 50% -- implying twice the trip-time.  Exclamatory punctuation is invited.

We consider now two transient-states in congested flow, beginning with the inevitable need for vehicles to slow down or to stop.  Stops occur in various forms.  Three come readily to mind...
1. Emergency Stops can occur in any lane on any roadway.  Whereas in congested flow, a sudden slow-down can produce a traffic 'jam', depending on density, but in 'free-flow', not so much.  A 'platoon' can form behind the stopped vehicle, its length determined by the duration of the stop.  Release of the jam is generally an ad hoc proposition.
2. Boulevard Stops require all vehicles in each lane to come to rest or to queue up at an intersection and wait for one vehicle of cross-traffic, if any.  Subsequent vehicles in the lane, if any, creep forward and wait their turn, first-come-first-served, then start up one-by-one, with releases being self-regulated and acceleration generally in 'free-flow'.
3. Signalized Intersections, wherein vehicles are regulated for stopping and releasing by automatic controls, often synchronized among groups of intersections, apportioned by traffic detection that make 'calls' for lane-control and for coordination with pedestrians.  In congested flow, signals intentionally create traffic 'jams' along with 'platooning'.
For exemplifying transient-states in congested flow, we shall use the Signalized Intersection
Figure 2 applies a tool known as a phase-plane, which plots the first derivative of a variable against the variable.  In this case, vehicle speed v = dx/dt is plotted on the vertical axis against linear dimensions on the horizontal axis x.  Events referenced to time cannot be depicted on a phase-plane plot.  Instead, two 'snap-shots' are superimposed in the diagram.  They represent [a] the point in space where the driver of the first vehicle N=0 has observed a yellow aspect signal and has decided that the vehicle can be brought to a safe stop at the intersection before the red aspect appears and [b] any time after all the vehicles in the platoon have been brought to a stop.

 Vehicles are shown here being braked to a stop at a red signal.  This begins the formation of a platoon.  The first vehicle N=0 is assumed to be slowing at a constant braking deceleration b0 reaching v0 = 0 at location x = 0.  A constant deceleration appears as a parabola on a phase-plane plot, inasmuch as braking distance x0 = vL2 / (2 b0).  A 'reaction' distance xR is shown as the same for all vehicles in the platoon.  It is an estimate for the distance traveled at vL during tR, commonly called 'reaction time' in most jurisdictions.  Signals in the U.S. are typically located on the far side of the intersection, such that drivers in some vehicles behind N=0 will be able to see the light and plan their deceleration.  That interval tR allows for [a] the 'perception' by each driver of an existential need to decelerate based on conditions ahead and [b] the 'mode-change delay' from propulsion to braking. For the base case in the Density Lock puzzle, we will let tR = 0.75 sec.  The total stopping distance can be generalized for each vehicle in the platoon in the following expression vL tR + vL2 / (2 bN).   Figure 2 shows common following-distance xF between vehicles moving at vL, which was introduced in Figure 1 above.  Introduced in Figure 2 is a typical separation distance xS between stopped vehicles.  Most drivers allow 1 ft < xS < 3 ft.  For the base case in the Density Lock puzzle, we shall use xS = 2 ft.  An emergency stop on a highway might let vL = 60 mph (88 fps), and xF = 132 ft.  Nota bene, each 'space cushion' will get 'squeezed' from xF = 132 ft to xS = 2 ft.  With xF > xS, the  allowable braking distances cumulatively increase for successive vehicles in the platoon, and xN > x(N-1), assuring the safety of bN < b(N-1).  With elementary algebra, one finds that xN = x0 + N (xH - xS - xL).  Notice that xN does not vary with tR, inasmuch as xR is nested within both xF and xH.  To ascertain the required deceleration for each vehicle, we write vL2 / 2bN = vL2 / 2b0 + N (xH - xS - xL) and solve for bN as a function of b0. Thus bN = b0 vL2 / [vL2 + 2N (xF - xS) b0].  As we see here, traffic density sure does increase when vehicles in a platoon slow down... speed vN → 0 ft/sec , flux → 0 veh/hr,  density → 5,280 / xH veh/mi ...thus in stopping from 60 mph, density goes from 33 veh/mi to 352 veh/mi, a ratio of ~11:1.  The most extreme congestion is called a 'jam' by traffic managers. Vehicles within a jam are prohibited by density lock from moving much.  Solvers will be invited to evaluate a counter-intuitive proposal for relieving traffic jams.  But first we need to consider the realities that arise during another transient-state in vehicular flow.  Figure 3 applies a phase-plane to help us understand what takes place in a platoon of vehicles recovering from a traffic jam.

The leading three vehicles in a platoon are shown stopped at a red light,  with the first vehicle, N=0, at location x = 0.  They are separated by bumper-to-bumper distances xS.  As the light turns green, the first vehicle N=0 is the only vehicle in the whole platoon that is not jammed but free to begin its acceleration a0 straight away, limited only by roadway conditions ahead.  All the rest of the vehicles in the platoon are operating in congested flow, with separations constrained to increase gradually from xS toward xF.  For modeling the typical profile for vehicles N = 1, 2,... three alternative concepts come to mind...
[1] Postulate a waiting time tW, and each vehicle N is assumed to remain stopped after the start-up of vehicle N-1 before beginning its own acceleration; however...
[2] An interesting model for driving behavior would
allow each vehicle to start-up at will and accelerate gradually one-by-one to facilitate the build-up of space between vehicles.  Such is the driving behavior modeled in Figure 3 above; however...
[3] That research assistant at UCLA, now an octogenarian living in France, has come to realized that the most reasonable model would have each vehicle N creep forward to the entrance of the intersection as allowed by the instantaneous location of vehicle N-1 ahead.  The creep-to-the-stop-line concept will be brought into consideration on the solution page of the Density Lock puzzle.
The first vehicle is shown accelerating at a0 in 'free-flow' to the limiting speed vL over a distance of x0 = vL2 / (2 a0), thence to continue at vL.  Vehicle N=1 begins accelerating at about the same time, but its acceleration a1 < a0The two profiles appear to cross each other at a point P beyond location x = 0.  No reason to get alarmed by that, though...
Keeping in mind that the phase-plane plot is not correlated in time, we realize that the common point P is reached at different times for the two vehicle.  Same for other vehicles in the platoon as they start up and follow along behind.  They each reach some intermediate speed vP at about the same location xP. Sophisticated solvers may enjoy the challenge of finding the location of xP and the value of vP.  Or not.
Acceleration a1 will be modeled as chosen intuitively by the driver of vehicle N=1 -- chosen so as to reach vL at a point along the roadway behind vehicle N=0 by the headway distance xH, where xH = xF + xL.  Kind of astonishing, when you think about it, that such an outcome can be assured by instuition.  Likewise, vehicle N=2 accelerates at a2 so as to reach vL at a distance xH behind vehicle N=1, and so forth.  One can generalize the acceleration distances as xN = vL2 / (2 aN).  Each vehicle reaches vL at the location which satisfies the following equation:  xN = x0 + N (xH + xS + xL), which can be expanded to read vL2 / 2aN)  = vL22a0 + N (xH + xS + xL) and solved for aN as a function of a0
Thus, aN = a0 vL2 / [vL2 +2 N (xF - xS) a0].
Sophisticated solvers will not be surprised to see that the expressions for aN and bN are identical.  It is merely a consequence of symmetry in the model formulation.

Here is a graph showing how aN and bN decrease with N in the illustration above with vL = 35 mph.   The value of b0 was selected to represent a case of an extreme deceleration -- 1 g, the limit that can be achieved with four-wheel braking and unity coefficient of friction between rubber and roadway surface.   The chosen value of a0 = b0 / 2, and produces a a curve that might reasonably characterizes the behavior of vehicles when the light changing from red to green.

Dean's Award for 1955

As mentioned in the introduction to the puzzle, 60 years ago a certain research assistant at UCLA's Institute of Transportation and Traffic Engineering made a radical proposal to the world in the form of a question...

 Would mandating that vehicles stop at prescribed separations prevent density lock?

He obtained sponsorship from renowned traffic expert Dan Gerlough and conducted an experiment to find out.  Ten fellow undergraduates in UCLA's College of Engineering gave up a Saturday morning to drive their cars around the campus on a blocked off street, where a temporary traffic signal had been set up.  Each driver was assigned a number and given a 'script'.  They assembled in a parking lot to be randomized.  They then drove as a platoon onto the street and stopped under the control of the signal.  Every car had a prescribed stopping location.  Ten runs were conducted for each of two configurations.
The base case prescribed xS = 2 ft.  The experiment case prescribed xS = xF / 2.
The stop-locations were marked with numbered traffic cones.  As the light was switched to green, the drivers accelerated to vL = 35 mi/hr.   At several places along the roadway were pairs of vehicle detectors, each pair separated by 3 ft to ascertain speeds.  Signals from the detectors were charted on a roll of paper, to be analyzed in timing measurements.  The resulting thesis won the dean's award for 1955.  Solvers can decide if that honor was deserved.

Using the symbols in Figure 1
for a given value of  tF = xF / vL, we write xH = xF + xL, and...
xH = vL tF + xL
kL = 1 / xH = 1 / (vL tF + xL)......density as a function of speed
vL = 1 / kL tF - xL / tF
..............speed
as a function of density
tH = xH / vL = (vL tF + xL) / vL
qL = 1 / tH = vL / (vL tF + xL)......flux as a function of speed
vL = qL xL / (1 - qL tF) ...............speed as a function of flux

kL = (1 - qL tF)  / xL.....................density as a function of flux
qL =
(1 - kL xL) /  tF.....................flux as a function of density