First, let us consider phases B14 and B20 where two traffic flows cross and exclude mutually. For each traffic light we distinguish the usual three phases, resp. states: Red (Stop!), Green (Go!), and Intermediate (amber phase). Unlike the phases R and G, the duration of the phase I is fixed.
We introduce the notion of time (Riedel, 1991) and we distinguish between states and sets of states (a sequence of consecutive states). Thus the following four elements of a timed state diagram are defined: A Bar represents a state at a discrete point of time t = t1. A Circle represents a set of states at consecutive points of time t1 < t < t2, where t2 is free. An Arc represents an instantaneous transition from one state or set of states to another at a given time t = t1. A Double arc represents a time consuming transition from one state or set of states to another with t1 < t < t1 + Dt, for a given Dt.
In a second step, the timed state diagram is further simplified by neglecting the uncontrollable states, as shown in Fig. 4.
Fig. 4. Timed state diagrams
We now consider the approach roads at time t = 0. The situation is represented by the diagram shown in Fig. 5.
Fig. 5. Diagram of customers and their locations at time t = 0
Fig. 6. Delayed traffic of approach B14 at t = tr
On the horizontal axis denoted by the distance s, the arrival-pattern of customers at t = 0 is shown. For simplicity we assume here that the customers can instantaneously stop and re-accelerate. Suppose that the traffic light of access road B14 has been red for a time tr and the arriving customers have stopped. The corresponding waiting time diagram is shown in Fig. 6. As time is going on, customers are being scrolled left with constant speed according to a default speed. When a customer cannot be scrolled farther along the horizontal axis because of red light or a stopped customer in front of him, he gets "scrolled" along the vertical axis. Thus the accumulated waiting time of a customer is displayed on his vertical axis.
The crossing spans a two-dimensional grid (see the upper part of Fig. 7). Due to such a projection the information about the arrival time of a customer is lost. In Fig. 7, when we give green to a phase, we move through the graph along the corresponding axis according to the number of passing customers. Moving from one vertex (state) to the next produces some costs (depending on the choice of the objective) given by the weight of the edges. The goal is to reach the vertex which is most far from the origin.
Fig. 7. State space representation