A linear complementarity system approach to macroscopic freeway traffic modeling

In this paper, the (modified) cell transmission model (CTM/MCTM) is formulated as a linear complementarity system (LCS). The LCS formulation of the CTM/MCTM presented here is a discrete time linear system with a complementarity condition. The time evolution of such kind of LCS consists of a series of “events” which cause changes in dynamics and possibly jumps in the state vector. The occurrence of events is governed by certain complementarity conditions which are similar to those in the linear complementarity problem of mathematical programming. The discrete time linear system corresponds to the flow conservation equation of the CTM/MCTM while the complementarity condition governs the sending and receiving function defined by a series of “min” operations in the original CTM/MCTM. Technical difficulties encountered in application of the CTM and its extensions such as the hard nonlinearity caused by the “min” operator can be avoided by the LCS reformulation. On the other hand, by this formulation, the theory of LCS developed in control theory and mathematical programming communities can be applied to the qualitative analysis of the CTM and its modifications. The new formulation makes the CTM convenient for the design of traffic state estimators, ramp metering controllers and for the dynamic traffic assignment purposes. For example, the new model contributes to the dynamic user equilibrium (DUE) problem with physical queueing models as network loading model by converting the DUE problem into a uniform complementarity system. This may benefit the existence issue of DUE with CTM as network loading model which is regarded as an important yet difficult problem. The new formulation can benefit freeway control strategies design by adopting the results from control and mathematical programming communities.