A formal solution for the stationary distribution of queue-length at a fixed-cycle traffic light is found for a fairly general distribution of arrivals and for a single stream of vehicles which either all turn left or else all go straight on or turn right. (We assume that the vehicles are driving on the right of the road.) Some inequalities are derived for the expected queue-length and for the expected delay per vehicle.
"On the Traffic-light Queue." Ann. Math. Statist. 35 (1) 380 - 388, March, 1964. https://doi.org/10.1214/aoms/1177703761