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Section: Research Program

Probabilistic Worst Case Reasoning for Real-Time Systems

Participants : Liliana Cucu, Robert Davis, Yves Sorel.

The arrival of modern hardware responding to the increasing demand for new functionalities exacerbates the limitations of the current worst-case real-time reasoning, mainly to the rarity of worst-case scenarios. Several solutions exist to overcome this important pessimism and our solution takes into account the extremely low probability of appearance of a worst-case scenario within one hour of functioning (${10}^{-45}$), compared to the certification requirements for instance (${10}^{-9}$ for the highest level of certification in avionics). Thus we model and analyze real-time systems with time parameters described by using probabilistic models. Our results for such models address both schedulability analyses as well as timing analyses. Both such analyses are impacted by existing misunderstanding. The independence between tasks is a property of real-time systems that is often used for its basic results. Any complex model takes into account different dependences caused by sharing resources other than the processor. On another hand, the probabilistic operations require, generally, the (probabilistic) independence between the random variables describing some parameters of a probabilistic real-time system. The main (original) criticism to probabilistic is based on this hypothesis of independence judged too restrictive to model real-time systems. In reality the two notions of independence are different. Providing arguments to underline this confusion is at the center of our dissemination effort in the last years.

We provide below the bases driving our current research as follows:

• Optimality of scheduling algorithms stays an important aspect of the probabilistic real-time systems, especially that the introduction of probabilistic time parameters has a direct impact on the optimality of the existing scheduling algorithms. For instance Rate Monotonic scheduling policy is no longer optimal in the case of one processor when a preemptive fixed-priority solution exists. We expect other classes of algorithms to lose their optimality and we concentrate our efforts to propose new scheduling solutions in this context [22].

• Increased complexity of schedulability analysis due to the introduction of probabilistic parameters requires appropriate complexity reasoning, especially with the emergence of probabilistic schedulability analyses for mixed-criticality real-time systems [23]. Moreover the real-time applications are rarely independent and precedence constraint using graph-based models are appropriate in this context. Precedence constraints do decrease the number of possible schedulers, but they also imposes an "heritage" of probabilistic description from execution times to release times for instance.

• Proving feasibility intervals is crucial for these approaches that are often used in industry on top of simulation. As worst-case situations are rare events, then observing them or at least observe those events that do provoke later the appearance of worst-case situations is difficult. By proposing an iterative process of composition between different statistical models [17], we provide the basis to build a solution to this essential problem to prove any probabilistic real-time reasoning based on measurements.

• Providing representativeness of a measurement-based estimator is the final proof that a probabilistic worst-case reasoning may receive. Our first negative results [24] indicate that the measurement protocol is tighly connected to the statistical estimator and that both must verified properties of reproducibility in order to contribute to a convergence proof.