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This paper improves upon an approach utilizing Gaussian emission distributions within a Markov task execution model that analyzes bounds on deadline miss probabilities for tasks in a reservation-based server. Our method distinctly addresses the issue of runtime complexity, prevalent in existing methods, by employing a state merging technique. This not only maintains computational efficiency but also retains the accuracy of the deadline-miss probability estimations to a significant degree. The efficacy of this approach is demonstrated through the timing behavior analysis of a Kalman filter controlling a Furuta pendulum, comparing the derived deadline miss probability bounds against various benchmarks, including real-time Linux server metrics. Our results confirm that the proposed method effectively upper-bounds the actual deadline miss probabilities, showcasing a significant improvement in computational efficiency without significantly sacrificing accuracy.<\/jats:p>","DOI":"10.1007\/s11241-024-09431-7","type":"journal-article","created":{"date-parts":[[2024,10,10]],"date-time":"2024-10-10T14:03:15Z","timestamp":1728568995000},"page":"443-490","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Efficiently bounding deadline miss probabilities of Markov chain real-time tasks"],"prefix":"10.1007","volume":"60","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7431-5529","authenticated-orcid":false,"given":"Anna","family":"Friebe","sequence":"first","affiliation":[]},{"given":"Filip","family":"Markovi\u0107","sequence":"additional","affiliation":[]},{"given":"Alessandro V.","family":"Papadopoulos","sequence":"additional","affiliation":[]},{"given":"Thomas","family":"Nolte","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,10,10]]},"reference":[{"key":"9431_CR1","doi-asserted-by":"publisher","unstructured":"Abeni L, Buttazzo G (1998) Integrating multimedia applications in hard real-time systems. 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