The interaction between the modules can lead to logical deadlocks. For example, plans depend on congestion, but congestion depends on plans. As said before, widely accepted method to resolve this is systematic relaxation (e.g. [3]) - that is, make preliminary plans, run the traffic micro-simulation, adjust the plans, run the traffic micro-simulation again, etc., until consistency between modules is reached. The method is somewhat similar to a standard relaxation technique in numerical analysis.
Fig. 1.2 shows an example of the effect of this. The scenario here is that 50000 travelers, distributed randomly throughout Switzerland, all want to travel to Lugano, which is indicated by the circle. The scenario is used as a test case, but it has some resemblance with vacation traffic in Switzerland,
The left figure shows traffic when every driver selects the route which would be fastest on an empty network. The micro-simulation here uses the so-called queue model [7], which is a queuing model with an added link storage constraint. That is, links are characterized by a service rate (capacity), and a maximum number of cars on the link. If the link is full, no more vehicles can enter, causing spill-back. Compared to the original version of Ref [8], our model has an improved intersection dynamics [7]. After the initial routing and the initial micro-simulation, iterations are run as follows:
Fig. 1.2 right is the result after 49 such iterations. Quite visibly traffic has spread out over many more different routes.
Fig. 1.3 shows the relaxation behavior for a scenario in
Dallas [9,10]. The plot shows the
sum of all travel times as a function of the iteration number. From
this plot and from other observations it seems that here, broken
ergodicity is not a problem, and all relaxation methods go to the same
state, although with different convergence speeds.
The result is in fact similar to a fixed strategy Nash Equilibrium: for a single run in the relaxed state, it is approximately true that no traveler could improve by changing routes. The players follow a ``best reply'' dynamics (i.e. find the best answer to yesterday's traffic), and for some (non-traffic) systems it can even be proven that this converges to a Nash equilibrium [11].
As mentioned above, traffic has a tendency to spread out in reaction
to congestion. That is, people shift from congested to uncongested
links, even if under uncongested conditions the new route would take
more time. One would expect that one could note this in the strategy
landscape, i.e. that the performance difference between the first
fastest path is smaller under congested than under uncongested
conditions. This conjecture is in fact
correct [12]. Fig. 1.4 (left) shows
performance decrease incurred by selecting the
-best route instead
of the best one, once for an uncongested network and once for a
congested one. Clearly, the strategy landscape is much flatter in the
congested situation. It is however equally flat in the relaxed and in
the unrelaxed situation. Fig. 1.4 (right) shows that the
additional options under relaxed congested conditions are truly
different from the best option, which is not true for the unrelaxed
congested situation. This, in summary: Under uncongested conditions,
deviating from the fastest path incurs a high penalty. Under
unrelaxed congested conditions, there are many good options, but they
are all similar to each other. Only under relaxed congested
conditions do we find many truly different options with similar
performance.
The above results were obtained with a Dallas/Fort Worth scenario within the TRANSIMS project. ``Uncongested'' refers to the empty network, where free speeds were used to compute link travel times. ``Unrelaxed congested'' refers to the zeroth iteration traffic in the middle of the rush hour. ``Relaxed congested'' refers to a situation after many iterations; in fact, link travel times were averaged over 10 iterations. All curves in Fig. 1.4 refer to an average over 955 randomly selected OD (origin-destination) pairs in the study area which are between 7 km and 7.01 km apart.
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