Probabilistic timing analysis: an approach using copulas

Timing analysis and scheduling for complex systems now requires tools to deal with the fact that systems behaviour is stochastic and contains dependencies which are difficult to determine analytically. This paper introduces copulas, a general statistical tool for constructing multivariate distributions and describing dependence structures of random variables. Copulas are used to solve the problem of determining the probability distribution of the worst-case execution time of a real-time program. The common assumption of statistical independence is generally incorrect for real-time systems and results in severe underestimation of the probability of the worst case. Copulas allow the description of the dependence structure between blocks of a program, and whenever that dependence can not be determined it is possible to provide lower bounds for the distribution of the worst case execution time for any possible dependence between them. The method is illustrated through a case study.