  This paper considers a single link with traffic light boundary conditions at
both ends, and investigates the traffic evolution over time with various signal
and system configurations. A hydrodynamic model and a modified stochastic
domain wall theory are proposed to describe the local density variation. The
Nagel-Schreckenberg model (NaSch), an agent based stochastic model, is used as
a benchmark. The hydrodynamic model provides good approximations over short
time scales. The domain wall model is found to reproduce the time evolution of
local densities, in good agreement with the NaSch simulations for both short
and long time scales. A systematic investigation of the impact of network
parameters, including system sizes, cycle lengths, phase splits and signal
offsets, on traffic flows suggests that the stationary flow is dominated by the
boundary with the smaller split. Nevertheless, the signal offset plays an
important role in determining the flow. Analytical expressions of the flow in
relation to those parameters are obtained for the deterministic domain wall
model and match the deterministic NaSch simulations. The analytic results agree
qualitatively with the general stochastic models. When the cycle is
sufficiently short, the stationary state is governed by effective inflow and
outflow rates, and the density profile is approximately linear and independent
of time.
