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

Abstract

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 structure. 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

The PROSIT tool: Toward the optimal design of probabilistic soft real‐time systems

“…This result holds both for fixed‐priority scheduling and for resource reservations, and it allowed us to use some of the most efficient numeric and analytic algorithms as a basis for the implementation of the system analysis. Moreover, the resulting DTMC is a conservative approximation of the system, leading to a pessimistic result in the probabilistic analysis, as proved in the works of Diaz et al and Palopoli et al…”

“…The analysis is significantly simpler if we adopt a resource reservation scheduler, in which each task executes as if on a dedicated computer and can be analyzed independently of the other tasks (the temporal isolation property). This radical simplification makes the model easy to construct and to analyze and brought different authors to develop efficient numerical methods or analytic bounds …”

“…To address this limitation, Villalba et al have recently proposed an extension of the analysis techniques first proposed for i.i.d. computation times to the case of Markov modulated computation times (MMP), a correlated process that is apparently applicable to a large class of robotic applications. The price to pay for this extension is, understandably, an additional complication of the model.…”

“…In particular, as mentioned before, in the PROSIT tool, we have implemented four different solvers for the probabilistic guarantees: the Cyclic Reduction algorithm, the Logarithmic Reduction algorithm, the Companion Form algorithm, and the approximated Analytic Bound …”

“…Additionally, PROSIT was used in the work of Palopoli et al for the experimental validation and the probabilistic quality optimization, in the work of Frías et al for the stochastic analysis, and the works of Frías et al and Abeni et al for the estimation of the parameters of the model. Moreover, PROSIT is freely available online for the real‐time systems research community and the industrial practitioners.…”