J. Haddad, B. De Schutter, D. Mahalel, I. Ioslovich, and P.-O. Gutman, "Optimal steady-state control for isolated traffic intersections," IEEE Transactions on Automatic Control, vol. 55, no. 11, pp. 2612-2617, Nov. 2010.
A simplified isolated controlled vehicular traffic intersection with two movements is considered. A discrete-event max-plus model is proposed to formulate an optimization problem for the green-red switching sequence. In the case when the criterion is a strictly increasing, linear function of the queue lengths, the problem becomes a linear programming (LP) problem. Also, in this case, the steady-state control problem can be solved analytically. A sufficient and necessary condition for steady-state control is derived, and the structure of optimal steady-state traffic control is revealed. Our condition is the same as the necessary condition in the paper "The oversaturated intersection," by D. Gazis and R. Potts (Proceedings of the 2nd International Symposium on the Theory of Traffic Flow, London, UK, 1963, pp. 221-237) for both queue lengths to be non-increasing at an isolated intersection.