The Three-detector problem is a problem in traffic flow theory. Given is a homogeneous freeway and the vehicle counts at two detector stations. We seek the vehicle counts at some intermediate location. The method can be applied to incident detection and diagnosis by comparing the observed and predicted data, so a realistic solution to this problem is important. Newell G.F. proposed a simple method to solve this problem. In Newell's method, one gets the cumulative count curve (N-curve) of any intermediate location just by shifting the N-curves of the upstream and downstream detectors. Newell's method was developed before the variational theory of traffic flow was proposed to deal systematically with vehicle counts. This article shows how Newell's method fits in the context of variational theory.
Contents
A special case to demonstrate Newell's method
Assumption. In this special case, we use the Triangular Fundamental Diagram (TFD) with three parameters: free flow speed
The goal of three-detector problem is calculating the vehicle at a generic point (P) on the "world line" of detector M (See Figure 2). Upstream. Since the upstream state is uncongested, there must be a characteristic with slope
Downstream. Likewise, since the state over the downstream detector is queued, there will be a wave reaching P from a location
Actual count at M. In view of the Newell-Luke Minimum Principle, we see that the actual count at M should be the lower envelope of the U'- and D'-curves. This is the dark curves, M(t). The intersections of the U'- and D'- curves denote the shock's passages over the detector; i.e., the times when transitions between queued and unqueued states take place as the queue advances and recedes over the middle detector. The area between the U'- and M-curves is the delay experienced upstream of location M, trip times are the horizontal separation between curves U(t), M(t) and D(t), accumulation is given by vertical separations, etc.
Mathematical expression. In terms of the function N(t,x) and the detector location (
where
Basic principles of variational theory (VT)
Goal. Suppose we know the number of vehicles (N) along a boundary in a time-space region and we are looking for the number of vehicles at a generic point P (denoted as
Suppose, again, that an observer starts moving from the boundary to point P along path L. We know the vehicle number the observer sees,
So, if we now add the vehicle number on the boundary to the sum of all
Equations (1) and (2) are based on the relative capacity constraint which itself follows from the conservation law.
Maximum principle. It states that
Equation (4) is a shortest path(i.e., calculus of variations) problem with
Generalized solution
Three steps: 1. Find the minimum upstream count,Step 1
All possible observer straight lines between the upstream boundary and point P have to be constructed with observer speeds smaller than free flow speed:
where
Thus we need to minimize
Since
Thus,
Step 2
We have:
Since the FD is triangular,
Step 3
To get the solution we now choose the lower of
This is Newell's the recipe for the 3-detector problem.