Resource automata

Following the above observations, the process model can be formulated by TA. For formal definitions of the syntax and semantics of TA, see [15]. TA are used to model the individual resources by resource automata and to describe timing constraints by place automata. The former are used to start and to finish tasks which are uniquely assigned to resources, the latter establish timing constraints of places. 

Special clock variables called docks a, i = 1,..., NC are used to measure the time between events. The values of the clocks increase permanently with the rate t = 1. Clocks are used to measure the durations of tasks for the resource automata, and to measure the waiting between two tasks for place automata. 

The above model consists of two main parts a scheduler and the plant to schedule. Since the scheduler defines exactly when to start and finish the tasks, the behavior of the entire system is deterministic. Running both parts, the scheduler and the plant, corresponds to a simulation in which one single behavior of the composed system is obtained. Obviously, this situation requires the presence of a scheduler which knows the (optimal) schedule. If such a scheduler is absent, then the resource automata in the plant receive no signals. Scheduling of a plant can be understood as the task of finding a scheduler automaton which provides the optimal schedule with respect to an optimization criterion. In the sequel it is assumed that a scheduler does not exist and the automata model is designed as shown in Figure 10.4. 

Consider the situation when sufficient quantities of raw and intermediate materials are present and the resource automata in Figure 10.4 are waiting in the idle locations. Without a scheduler which exactly determines the next production step, either Ri or f 2 or Rj can start processing a batch. The possible actions are s(Ti), s(T2), s(T3) or w(e) with e e R-°. Hence, the scheduler has to choose from a set of possible decisions. This set is infinite because the waiting time e is a real number. 

Each valid evolution starts in a predefined initial state in which sufficient quantities of the raw materials are given. Furthermore, it must meet the requirements from Section 10.1.3. A schedule will be accepted only if the market demand has been satisfied. Hence, at the end of the evolution, the demanded quantities of final products must be present in the storages. Each evolution which does not satisfy the above requirements leads to an invalid schedule and thus must be rejected. In order to synthesize a valid and optimal schedule, the scheduler has to fix all degrees of freedom by choosing appropriate signals for the resource automata. A valid and optimal schedule corresponds to evolutions which minimize a given optimality criterion. 

Each resource automaton has a cyclic structure with two locations, idle and busy, and two transitions connecting these two locations. When the signal to start a task is received, the corresponding resource automaton changes its location from idle to busy by taking the first transition. When a task is finished, the resource automaton returns to the idle location by taking the second transition. 

Fig. 10.4 The timed automata models of the resources Ri, R2, R3 and the place automaton Pi.

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