Jim Hines says that I told him traffic is like an inventory
oscillator, so it must be true!
You can model the velocity control of individual cars as such an
oscillator. The distance between you and the car in front plays the
role of inventory. The distance between the cars accumulates the
difference between the velocity of the car in front and your
velocity. Your velocity is in turn your momentum divided by the mass
of your car. Momentum is a stock that integrates the net force
applied (accelerating force from the drive train less drag and
friction), or equivalently, your velocity integrates the net
acceleration of your car. Through the gas/brake pedals, you control
the change in acceleration (with a delay -- your reaction time plus
the reaction time of the drive train). Thus your momentum is
analogous to the labor force of the firm: it is a stock that
accumulates control actions. You determine acceleration via negative
feedbacks in which you try to control the distance between you and
the car in front, and also attempt to keep your speed at the desired
rate. You have a goal for the distance between you and the next car,
which, as you learned in drivers ed, should be about 2 seconds *
your velocity (except in Boston, where your goal is to maintain less
than one car length so no one can cut in front of you). Note that
the 2 second rule is exactly the same as the basic rule for desired
inventory:
Desired Distance between cars = Desired time between cars * velocity
(meters) = Seconds * meters/second
Desired Inventory = Desired Inventory Coverage * expected shipments
Widgets = days * widgets/day
The structure just described is isomorphic to the basic inventory
control structure, and both are essentially negative feedback systems
with lots of phase lag and some damping. Depending on the gains of
the distance control loop and velocity control loop, along with
parameters such as the mass of your car, the system can be over,
critically, or underdamped. If you are laid back and willing to
tolerate some variation in the distance between you and the car in
front, the gain is low and the system will be overdamped. If you
drive with a lead foot the gain is high and the system will be
underdamped and generate high amplification (do the mental simulation
of what happens when the car in front suddenly slows down by 5 m/sec).
A line of cars on the freeway is thus like a supply chain, and the
dynamics are much like those seen in the beer game, but with many,
many more links in the chain. If a car slows, the driver behind
cannot react instantly, so the distance closes up. That driver must
(temporarily) slow more than the car in front did, to restore the
desired distance. The car behind that must then slow even more, and
so on. There are important nonlinearities here: velocity is (almost
always) nonnegative (you can stop, but you usually dont back up on
the freeway), and the distance between you and the car in front of
you cant be less than zero (attempts to do so result in crushed
metal and much cursing). In off hours, traffic is sparse, and each
car runs at its desired velocity, since the distance between you and
the next car is irrelevant. As rush hour begins and traffic density
builds, the space between cars shrinks. As it does, drivers must
take more aggressive control actions to keep from rear-ending the
driver in front, increasing the gain of the negative loop with the
time delays. As this occurs, the system moves from over to
underdamped (the dominant eigenvalues of the linearized system become
complex conjugates), and then often into a regime of unstable
oscillations, where the eigenvalues of the linearized system cross
into the positive half plane (a Hopf bifurcation for fans of
nonlinear differential equations). At this point the system
generates oscillations in velocity that increase in amplitude until
constrained by the nonlinearities, principally the bounds that
velocity is constrained between zero and the speed limit (in Boston,
between zero and speed limit + 20 mph). The result is stop and go
traffic even on limited access highways with no stoplights. You find
yourself zipping along at 60 for a few seconds, then screeching to a
dead halt, then speeding up again, and so on. The period of the
cycle depends on parameters but is often on the order of 30-90
seconds (though it often seems longer while you sit there stopped).
The most interesting aspect of traffic jams is their spontaneous
character, often occurring on open freeways with no apparent cause
such as an accident. This occurs when the gain of the system is high
enough to move it into the underdamped or locally unstable regime.
In such a situation, even the slightest perturbation can trigger the
amplification leading to a large jam. Perturbations arise from the
heterogeneity of the driver population (drivers have different goals
for speed and distance-to-car-in-front), from lane changes, from
entry/exit ramps, and from driver inattention and variability (e.g.,
taking you eyes off the road to dial the phone -- which you can use
to report the accident that you are about to cause by phoning while
driving).
The model I just described is in the class of car following models,
in which traffic is seen from a microscopic view. There are also
many macroscopic models that abstract from the individual cars to
consider traffic as a fluid and the traffic problem as analogous to
understanding the transition from laminar to turbulent flow.
Formal models of traffic flow have a long and distinguished history,
going back at least to the traffic equation of the late physicist
Bob Herman, in whose honor the "Robert Herman Lifetime Achievement
Award in Transportation Science" is awarded by the INFORMS Section on
Transportation Science & Logistics. A good survey of the history of
traffic models, both macro and microscopic, is Denos C. Gazis (2002)
The origins of traffic theory. Operations Research, 50(1), Jan-Feb,
69-77, available online at
http://www.eng.tau.ac.il/~ami/cd/or50/1 ... 1-0069.pdf.
John Sterman
From: John Sterman <
jsterman@MIT.EDU>