## Specialized technology for solving scheduling problems with timed automata

The zone-based technology has its roots in verification where the temporal uncertainty is viewed as coming from the external environment and the system should be correct with respect to all environment choices.

Around the beginning of the AMETIST project it was observed that when uncertainty is associated with the scheduler decisions, for example in problems of scheduling under certainty, sometimes there is a unique successor among the uncountably many which gives the optimum (non-lazy schedules).

Consequently the problem can be solved without using zones at all but rather using vectors of clock variables. This way certain problems can be formulated as shortest paths in discrete weighted graphs and solved much more efficiently. They can also benefit from existing search algorithms for on game graphs in order to find sub-optimal schedules for scheduling with discrete uncertainties.

Even in the case of dense uncertainty coming from the environment/adversary side, there might be some clever ways to avoid zones. When the adversary has a choice in some interval I = [a, b] we may still relax the problem by assuming only a finite subset I0 of the interval.

Solving the problem of synthesizing an optimal scheduling strategy with respect to I0 may have the following consequences:

- The actual value of the chosen strategy with respect to I may be worse than the value computed based on I0. This is more problematic for qualitative criteria where the system may be correct with respect to I0 but not with respect to I. For quantitative criteria this is less critical because we already accept sub optimal solutions when the problem is large.

- During execution the adversary can make a choice in I-I0 and the system will find itself in a state for which the optimal action has not been computed. However in many problems some default rules can be used to determine the action at such a state based on the optimal computed action in its neighbourhood.

Around the beginning of the AMETIST project it was observed that when uncertainty is associated with the scheduler decisions, for example in problems of scheduling under certainty, sometimes there is a unique successor among the uncountably many which gives the optimum (non-lazy schedules).

Consequently the problem can be solved without using zones at all but rather using vectors of clock variables. This way certain problems can be formulated as shortest paths in discrete weighted graphs and solved much more efficiently. They can also benefit from existing search algorithms for on game graphs in order to find sub-optimal schedules for scheduling with discrete uncertainties.

Even in the case of dense uncertainty coming from the environment/adversary side, there might be some clever ways to avoid zones. When the adversary has a choice in some interval I = [a, b] we may still relax the problem by assuming only a finite subset I0 of the interval.

Solving the problem of synthesizing an optimal scheduling strategy with respect to I0 may have the following consequences:

- The actual value of the chosen strategy with respect to I may be worse than the value computed based on I0. This is more problematic for qualitative criteria where the system may be correct with respect to I0 but not with respect to I. For quantitative criteria this is less critical because we already accept sub optimal solutions when the problem is large.

- During execution the adversary can make a choice in I-I0 and the system will find itself in a state for which the optimal action has not been computed. However in many problems some default rules can be used to determine the action at such a state based on the optimal computed action in its neighbourhood.