Copyright ©2015 by Paul Niquette. All rights reserved.
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.
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.
'congested flow', the speed of each vehicle is
constrained by the speed of the vehicle ahead, such that
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
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.
Below is a graph of the three traffic variables...
speed = vL ft/sec, flux = 3,600 / tH veh/hr, density = 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...
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.
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...
...thus in stopping from 60 mph, density goes from 33 veh/mi to 352 veh/mi, a ratio of ~11:1.speed vN → 0 ft/sec , flux → 0 veh/hr, density → 5,280 / xH veh/mi
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.
Using the symbols in Figure 1 for a given value of tF = xF / vL, we write xH = xF + xL, and...
xH = vL tF + xL
tH = xH / vL = (vL tF + xL) / vL