An Analytical Solution for Probabilistic Guarantees of Reservation Based Soft Real--Time Systems

We show a methodology for the computation of the probability of deadline miss for a periodic real–time task scheduled by a resource reservation algorithm. We propose a modelling technique for the system that reduces the computation of such a probability to that of the steady state probability of an infinite state Discrete Time Markov Chain with a periodic struc- ture. This structure is exploited to develop an efficient numeric solution where different accuracy/computation time trade–offs can be obtained by operating on the granularity of the model. More importantly we offer a closed form conservative bound for the probability of a deadline miss. Our experiments reveal that the bound remains reasonably close to the experimental probability in one real–time application of practical interest. When this bound is used for the optimisation of the overall Quality of Service for a set of tasks sharing the CPU, it produces a good sub–optimal solution in a small amount of time.