  Existing methods for traffic signal design are either too simplistic to
capture realistic traffic characteristics or too complicated to be
mathematically tractable. In this study, we attempts to fill the gap by
presenting a new method based on the LWR model for performance analysis and
signal design in a stationary signalized ring road. We first solve the link
transmission model to obtain an equation for the boundary flow in stationary
states, which are defined to be time-periodic solutions in both flow-rate and
density with a period of the cycle length. We then derive an explicit
macroscopic fundamental diagram (MFD), in which the average flow-rate in
stationary states is a function of both traffic density and signal settings.
Finally we present simple formulas for optimal cycle lengths under five levels
of congestion with a start-up lost time. With numerical examples we verify our
analytical results and discuss the existence of near-optimal cycle lengths.
This study lays the foundation for future studies on performance analysis and
signal design for more general urban networks based on the kinematic wave
theory.
