Cost-flow curves for static assignment

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, $U=RI$. With constant $R$, cost is proportional to flow, but $R$ can also depend on $I$, 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 $v(\rho )$ and $q(\rho )$, see Fig. 27.15 left and top, where dashed lines are used in the congested regimes. One can construct or calculate $v(q)$ from that (center right in Fig. 27.15). Link travel time is $T(q) = L/v(q)$; a sketch of this is shown at the bottom of Fig. 27.15.

Figure 27.14: Illustration of steady-state network flow.

Figure 27.15: Construction of $v(q)$ and thus $T(q)$ for link dynamics. Starting points are the $v(\rho )$ diagram at the left and the $q(\rho )$ diagram at the top.

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 $v(q)$, which corresponds to the lower branch of $T_{link}(q)$. This results in functions $T(q)$ which in general start at the free speed travel time for zero flow, and which increase with increasing flow, which is plausible.

Figure 27.16: Construction of speed and link travel time as function of flow, now for a link with a bottleneck at the end. Inputs are the speed-density relation on the left and the flow-density relation on the bottom.

However, what happens if the assignment model assigns more flow to a link than capacity $cap$? 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 $q > cap$. This is in fact what static assignment models essentially do, except that they use a smooth function (i.e. no jump at $q=cap$). 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.

Figure 27.17: A freeway ending in a bottleneck.

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.