Using Markov’s Inequality with Power-Of-k Function for Probabilistic WCET Estimation

Authors Sergi Vilardell , Isabel Serra , Enrico Mezzetti , Jaume Abella , Francisco J. Cazorla , Joan del Castillo

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Sergi Vilardell
  • Polytechnic University of Catalonia, Barcelona, Spain
  • Barcelona Supercomputing Center (BSC), Spain
Isabel Serra
  • Barcelona Supercomputing Center (BSC), Spain
  • Centre de Recerca Matemàtica, Barcelona, Spain
Enrico Mezzetti
  • Barcelona Supercomputing Center (BSC), Spain
  • Maspatechnologies S.L, Barcelona, Spain
Jaume Abella
  • Barcelona Supercomputing Center (BSC), Spain
Francisco J. Cazorla
  • Barcelona Supercomputing Center (BSC), Spain
  • Maspatechnologies S.L, Barcelona, Spain
Joan del Castillo
  • Autonomous University of Barcelona, Spain

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Sergi Vilardell, Isabel Serra, Enrico Mezzetti, Jaume Abella, Francisco J. Cazorla, and Joan del Castillo. Using Markov’s Inequality with Power-Of-k Function for Probabilistic WCET Estimation. In 34th Euromicro Conference on Real-Time Systems (ECRTS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 231, pp. 20:1-20:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Deriving WCET estimates for software programs with probabilistic means (a.k.a. pWCET estimation) has received significant attention during last years as a way to deal with the increased complexity of the processors used in real-time systems. Many works build on Extreme Value Theory (EVT) that is fed with a sample of the collected data (execution times). In its application, EVT carries two sources of uncertainty: the first one that is intrinsic to the EVT model and relates to determining the subset of the sample that belongs to the (upper) tail, and hence, is actually used by EVT for prediction; and the second one that is induced by the sampling process and hence is inherent to all sample-based methods. In this work, we show that Markov’s inequality can be used to obtain provable trustworthy probabilistic bounds to the tail of a distribution without incurring any model-intrinsic uncertainty. Yet, it produces pessimistic estimates that we shave substantially by proposing the use of a power-of-k function instead of the default identity function used by Markov’s inequality. Lastly, we propose a method to deal with sampling uncertainty for Markov’s inequality that consistently improves EVT estimates on synthetic and real data obtained from a railway application.

Subject Classification

ACM Subject Classification
  • Computer systems organization → Real-time system architecture
  • Markov’s inequality
  • probabilistic time estimates
  • probabilistic WCET
  • Extreme Value Theory


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