On the Convolution Efficiency for Probabilistic Analysis of Real-Time Systems

Authors Filip Marković , Alessandro Vittorio Papadopoulos , Thomas Nolte

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Filip Marković
  • Mälardalen University, Västerås, Sweden
Alessandro Vittorio Papadopoulos
  • Mälardalen University, Västerås, Sweden
Thomas Nolte
  • Mälardalen University, Västerås, Sweden


We want to thank Davor Čirkinagić who borrowed his computing system for performing the evaluation. Also, we are very grateful to the anonymous reviewers for their comments.

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Filip Marković, Alessandro Vittorio Papadopoulos, and Thomas Nolte. On the Convolution Efficiency for Probabilistic Analysis of Real-Time Systems. In 33rd Euromicro Conference on Real-Time Systems (ECRTS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 196, pp. 16:1-16:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


This paper addresses two major problems in probabilistic analysis of real-time systems: space and time complexity of convolution of discrete random variables. For years, these two problems have limited the applicability of many methods for the probabilistic analysis of real-time systems, that rely on convolution as the main operation. Convolution in probabilistic analysis leads to a substantial space explosion and therefore space reductions may be necessary to make the problem tractable. However, the reductions lead to pessimism in the obtained probabilistic distributions, affecting the accuracy of the timing analysis. In this paper, we propose an optimal algorithm for down-sampling, which minimises the probabilistic expectation (i.e., the pessimism) in polynomial time. The second problem relates to the time complexity of the convolution between discrete random variables. It has been shown that quadratic time complexity of a single linear convolution, together with the space explosion of probabilistic analysis, limits its applicability for systems with a large number of tasks, jobs, and other analysed entities. In this paper, we show that the problem can be solved with a complexity of 𝒪(n log(n)), by proposing an algorithm that utilises circular convolution and vector space reductions. Evaluation results show several important improvements with respect to other state-of-the-art techniques.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
  • Computer systems organization → Real-time system specification
  • Probabilistic analysis
  • Random variables
  • Algorithm Complexity


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