Traditional models for transportation planning, called ``static assignment'', do not use any representation of link dynamics at all. The purpose of this section is to explain the traffic dynamics representation of static assignment, and how that relates to the traffic dynamics we have seen so far.
Quite in general, any assignment method needs to be able to calculate link travel times from demand for traffic on a link. Intuitively, travel times increase with demand. The problem seems to be to find a good equation for that - it will however turn out that there is no simple solution.
Static assignment generates steady state solutions. So from a dynamic point of view, steady state assignment would be a better name. This means that continuous streams of traffic are fed into the system at the origins, and they move via their routes to their destinations, where they are removed. In consequence, demand for a link comes as a flow. So for a simple demand-cost relation we need to find link delay as a function of link flow.
This is actually similar to electricity, where steady-state currents
follow an equilibrium pattern through a network according to
Kirchhoff's laws. The cost function is Ohm's law, . With
constant
, cost is proportional to flow, but
can also depend on
, making this non-linear. The main difference to steady state
assignment is that in traffic the particles have fixed destinations
which cannot be interchanged.
Now let us construct link travel time as a function of steady state
flow for link dynamics. We start from simplified link fundamental
diagrams and
, see Fig. 27.15 left and
top, where dashed lines are used in the congested regimes. One can
construct or calculate
from that (center right in
Fig. 27.15). Link travel time is
; a sketch
of this is shown at the bottom of Fig. 27.15.
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A problem with this is that there is in general either more than one
or no velocity/time value for every given flow value. Looking at the
case where the node capacity is the restricting quantity
(Fig. 27.16), we see that the problem remains similar for that
case. The normal simplification for static assignment has been to
only use the upper branch of , which corresponds to the lower
branch of
. This results in functions
which in
general start at the free speed travel time for zero flow, and which
increase with increasing flow, which is plausible.
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However, what happens if the assignment model assigns more flow to a
link than capacity ? We know that this is dynamically impossible
under steady state conditions. So the only consistent choice for this
situation is to set the link travel time to infinity for
.
This is in fact what static assignment models essentially do, except
that they use a smooth function (i.e. no jump at
). The main
difference between different cost-flow-curves is which cost they give
to assigned flows above capacity.
In that sense, it is more reasonable to think about capacity for static assignment as just a free parameter of a cost-flow curve. The calibration of a cost-flow curve is quite difficult, and given the fact that there is no dynamical basis for such a curve, it is clear that it has to be more an art than a science. Nevertheless, the resulting models work quite well, and in spite of knowing better from a theoretical perspective, it is difficult to come up with models that work better in practice.
So far, we have described steady state traffic dynamics and how they are mapped on cost-flow curves for steady state assignment. We have described that one aspect that such models do not pick up are queues upstream of bottlenecks. Note that such queues can well exist under steady state conditions; they violate however the condition that there should only be one velocity/travel time value for each flow value.
There are dynamic aspects of traffic that steady state models cannot pick up at all. A typical scenario is that we have a wide freeway eventually ending in a bottleneck. During rush-hour build-up, the freeway may be used at capacity, resulting in a growing queue at the bottleneck, which will not vanish until the end of the rush period (Fig. 27.17). The steady-state solution would not allow that amount of traffic for the freeway. So here lies one of the reasons why assigment models that are used in practice allow flows above capacity.
There have been attempts to make static assignment models dynamic by solving separate models for several time slices. It is clear that from a dynamical perspective this is not a realistic solution - e.g., the above example with the freeway being used above the bottleneck capacity could still not be picked up.