This paper identifies a family of linear transformations where conservation laws are invariant. In the case of a triangular fundamental diagram, it is shown that for a subset of these transformations, flow, total distance traveled and total delay are invariant. This means that for capacity or delay computations one may choose the transformation-i.e., the shape of the triangular diagram-that simplifies the problem the most, which does not require knowing the actual fundamental diagram. This is appealing also for delay-optimizing control problems since they may be solved using an isosceles fundamental diagram, which provides the most efficient numerical methods. Examples are given.
This paper shows that a wide range of stochastic extensions of the kinematic wave model tend to the same parameter-free expression for the probability of congestion at a given time-space point. This is shown for white noise initial density with deterministic and stochastic fundamental diagram in the case of Riemann problems and the bottleneck problem. It is also found that the stochastic solution (i) preserves the structure of the deterministic solution and (ii) tends to the deterministic solution with time at a given location.
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