  In this paper we present a stable day-to-day dynamical system for drivers'
departure time choice at a single bottleneck. We first define within-day
traffic dynamics with the point queue model, costs, the departure time user
equilibrium (DTUE), and the arrival time user equilibrium (ATUE). We then
identify three behavioral principles: (i) Drivers choose their departure and
arrival times in a backward fashion (backward choice principle); (ii) After
choosing the arrival times, they update their departure times to balance the
total costs (cost balancing principle); (iii) They choose their arrival times
to reduce their scheduling costs or gain their scheduling payoffs (scheduling
cost reducing or scheduling payoff gaining principle). In this sense, drivers'
departure and arrival time choices are driven by their scheduling payoff
choice. With a single tube or imaginary road model, we convert the nonlocal
day-to-day arrival time shifting problem to a local scheduling payoff shifting
problem. After introducing a new variable for the imaginary density, we apply
the Lighthill-Whitham-Richards (LWR) model to describe the day-to-day dynamics
of scheduling payoff choice and present splitting and cost balancing schemes to
determine arrival and departure flow-rates accordingly. We also develop the
corresponding discrete models for numerical solutions. We theoretically prove
that the day-to-day stationary state of the LWR model leads to the scheduling
payoff user equilibrium (SPUE), which is equivalent to both DTUE and ATUE and
stable. We use one numerical example to demonstrate the effectiveness and
stability of the new day-to-day dynamical model.
