Skip to main content

Computing the execution probability of jobs with replication in mixed-criticality schedules


Mixed-criticality scheduling addresses the problem of sharing common resources among jobs of different degrees of criticality and uncertain processing times. The processing time of jobs is observed during the online execution of the schedule with the prolongations of critical jobs being compensated by the rejection of less critical ones. One of the central questions in the field of mixed-criticality scheduling is ensuring the high reliability of the system with as few resources as possible. In this paper, we study the computation of the execution probability of jobs with uncertain processing times in a static mixed-criticality schedule. The aim is to compute the execution probability of jobs (i.e., the objective function of a schedule), which is a problem solvable by a closed-form formula when the jobs are not replicated. We introduce the job replication, i.e., scheduling a single job multiple times, as a new mechanism for increasing the execution probability of jobs. We show that the general problem with job replication becomes \(\#{\mathcal {P}}\)-hard, which is proven by the reduction from the counting variant of 3-sat problem. To compute the execution probability, we propose an algorithm utilizing the framework of Bayesian networks. Furthermore, we show that cases of practical interest admit a polynomial-time algorithm and are efficiently solvable. The proposed methodology demonstrates an interesting connection between schedules with uncertain execution and probabilistic graphical models and opens a new approach to the analysis of mixed-criticality schedules.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  • Agirre, I., Cazorla, F. J., Abella, J., Hernandez, C., Mezzetti, E., Azkarate-askatsua, M., & Vardanega, T. (2018). Fitting software execution-time exceedance into a residual random fault in ISO-26262. IEEE Transactions on Reliability 67(3), 1314–1327

  • Baruah, S. (2018). Predictability issues in mixed-criticality real-time systems (pp. 77–87). Cham: Springer.

    Book  Google Scholar 

  • Baruah, S., Bonifaci, V., D’angelo, G., Li, H., Marchetti-Spaccamela, A., Van Der Ster, S., & Stougie, L., (2015). Preemptive uniprocessor scheduling of mixed-criticality sporadic task systems. Journal of the ACM (JACM) 62(2), 14.

  • Baruah, S., Fohler, G., (2011). Certification-cognizant time-triggered scheduling of mixed-criticality systems. In 2011 IEEE 32nd real-time systems symposium (RTSS), (pp. 3–12). IEEE.

  • Behera, L., & Bhaduri, P. (2018). Time-triggered scheduling for multiprocessor mixed-criticality systems. In Distributed computing and internet technology (pp. 135–151). Springer International Publishing, Cham.

  • Bell, R. (2006). Introduction to IEC 61508. In Proceedings of the 10th Australian workshop on Safety critical systems and software (Vol. 55, pp. 3–12). Australian Computer Society, Inc.

  • Blazewicz, J., Ecker, K. H., Pesch, E., Schmidt, G., & Weglarz, J. (2007). Handbook on scheduling: From theory to applications. New York: Springer.

    Google Scholar 

  • Burns, A., Davis, R.I. (2017). A survey of research into mixed criticality systems. ACM Computing Surveys 50(6), 82:1–82:37

  • Burns, A., Davis, R. I., Baruah, S., & Bate, I. (2018). Robust mixed-criticality systems. IEEE Transactions on Computers, 67(10), 1478–1491.

    Article  Google Scholar 

  • Chang, Z., Ding, J. Y., & Song, S. (2019). Distributionally robust scheduling on parallel machines under moment uncertainty. European Journal of Operational Research, 272(3), 832–846.

    Article  Google Scholar 

  • Creignou, N., & Hermann, M. (1993). On P completeness of some counting problems. Ph.D. thesis, INRIA

  • Dagum, P., & Luby, M. (1997). An optimal approximation algorithm for Bayesian inference. Artificial Intelligence 93(1–2), 1–27

  • Daniels, R.L., & Carrillo, J.E. (1997). \(\beta \)-robust scheduling for single-machine systems with uncertain processing times. IIE Transactions 29(11), 977–985

  • Davis, R.I., Altmeyer, S., & Burns, A. (2018). Mixed criticality systems with varying context switch costs. In 2018 IEEE real-time and embedded technology and applications symposium (RTAS) (pp. 140–151).

  • Draskovic, S., Huang, P., & Thiele, L. (2016). On the safety of mixed-criticality scheduling. In Proceedings of the 4th international workshop on mixed criticality systems, RTSS (pp. 19 – 24). IEEE, Porto, Portugal

  • El-Hajj, R., Guibadj, R. N., Moukrim, A., & Serairi, M. (2020). A PSO based algorithm with an efficient optimal split procedure for the multiperiod vehicle routing problem with profit. Annals of Operations Research, 26, 1–36.

    Google Scholar 

  • Graham, R. L., Lawler, E. L., Lenstra, J. K., & Kan, A. R. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.

    Article  Google Scholar 

  • Guo, H., & Hsu, W. (2002). A survey of algorithms for real-time Bayesian network inference. In Join workshop on real time decision support and diagnosis systems.

  • Hamaz, I., Houssin, L., & Cafieri, S. (2018). A branch-and-bound procedure for the robust cyclic job shop problem. In International symposium on combinatorial optimization (pp. 228–240). Springer.

  • Hanzalek, Z., & Sucha, P. (2017). Time symmetry of resource constrained project scheduling with general temporal constraints and take-give resources. Annals of Operations Research, 248(1–2), 209–237.

    Article  Google Scholar 

  • Hanzalek, Z., Tunys, T., & Sucha, P. (2016). An analysis of the non-preemptive mixed-criticality match-up scheduling problem. Journal of Scheduling, 19(5), 601–607.

    Article  Google Scholar 

  • Herroelen, W., & Leus, R. (2005). Project scheduling under uncertainty: Survey and research potentials. European Journal of Operational Research, 165(2), 289–306.

    Article  Google Scholar 

  • Ahmad, I., & Kwok, Y.-K. (1998). On exploiting task duplication in parallel program scheduling. IEEE Transactions on Parallel and Distributed Systems, 9(9), 872–892.

    Article  Google Scholar 

  • Jaramillo, F., Keles, B., & Erkoc, M. (2020). Modeling single machine preemptive scheduling problems for computational efficiency. Annals of Operations Research, 285(1), 197–222.

    Article  Google Scholar 

  • Kopetz, H. (1991). Event-triggered versus time-triggered real-time systems (pp. 86–101). Berlin: Springer.

    Book  Google Scholar 

  • Kwisthout, J., Bodlaender, H. L., & van der Gaag, L. C. (2010). The necessity of bounded treewidth for efficient inference in Bayesian networks. In ECAI (vol. 215, pp. 237–242).

  • Li, Y. F., Huang, H. Z., Mi, J., Peng, W., & Han, X. (2019). Reliability analysis of multi-state systems with common cause failures based on Bayesian network and fuzzy probability. Annals of Operations Research, 2, 1–15.

    Google Scholar 

  • Murphy, K. P., Weiss, Y., & Jordan, M. I. (1999). Loopy belief propagation for approximate inference: An empirical study. In Proceedings of the Fifteenth conference on Uncertainty in artificial intelligence (pp. 467–475). Morgan Kaufmann Publishers Inc.

  • Novak, A., Sucha, P., & Hanzalek, Z. (2019). Scheduling with uncertain processing times in mixed-criticality systems. European Journal of Operational Research, 279(3), 687–703.

    Article  Google Scholar 

  • Obermaisser, R., Kopetz, H., El Salloum, C., & Huber, B. (2007). Error containment in the time-triggered system-on-a-chip architecture. In Embedded system design: Topics, techniques and trends (pp. 339–352). Springer.

  • Papadimitriou, C. H., & Yannakakis, M. (1990). Towards an architecture-independent analysis of parallel algorithms. SIAM Journal on Computing, 19(2), 322–328.

    Article  Google Scholar 

  • Paredes, R., Dueñas-Osorio, L., Meel, K., & Vardi, M. (2019). Principled network reliability approximation: A counting-based approach. Reliability Engineering & System Safety, 191, 106472.

    Article  Google Scholar 

  • Ranjbar, M., Davari, M., & Leus, R. (2012). Two branch-and-bound algorithms for the robust parallel machine scheduling problem. Computers & Operations Research, 39(7), 1652–1660.

    Article  Google Scholar 

  • Russell, S.J., & Norvig, P. (2016). Artificial intelligence: A modern approach. Pearson Education Limited:Malaysia.

  • Sang, T., Bearne, P., & Kautz, H. (2005). Performing bayesian inference by weighted model counting. In Proceedings of the 20th national conference on artificial intelligence (Vol. 1, pp. 475–481), AAAI’05. AAAI Press

  • Santiváñez, J.A., & Melachrinoudis, E. (2020). Reliable maximin-maxisum locations for maximum service availability on tree networks vulnerable to disruptions. Annals of Operations Research 286(1), 669–701

  • Seddik, Y., & Hanzalek, Z. (2017). Match-up scheduling of mixed-criticality jobs: Maximizing the probability of jobs execution. European Journal of Operational Research, 262(1), 46–59.

    Article  Google Scholar 

  • Theis, J., Fohler, G., & Baruah, S. (2013). Schedule table generation for time-triggered mixed criticality systems. In Proceedings of the 1st international workshop on mixed criticality systems (pp. 79–84), RTSS .

  • Valiant, L. (1979). The complexity of computing the permanent. Theoretical Computer Science, 8(2), 189–201.

    Article  Google Scholar 

  • Valiant, L. (1979). The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3), 410–421.

    Article  Google Scholar 

  • Vestal, S. (2007). Preemptive scheduling of multi-criticality systems with varying degrees of execution time assurance. In 28th IEEE international real-time systems symposium (pp. 239–243), RTSS 2007. IEEE.

  • Yeh, C. T. (2020). Binary-state line assignment optimization to maximize the reliability of an information network under time and budget constraints. Annals of Operations Research, 287(1), 439–463.

    Article  Google Scholar 

Download references


This work was supported by the EU and the Ministry of Industry and Trade of the Czech Republic under the Project OP PIK CZ.01.1.02/0.0/0.0/20_321/0024399, and by the European Regional Development Fund under the project AI&Reasoning (reg. no. CZ.02.1.01/0.0/0.0/15_003/0000466). Next, we want to thank Daniel Slunecko for his early work on the problem and inspiring discussions.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Antonin Novak.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Novak, A., Hanzalek, Z. Computing the execution probability of jobs with replication in mixed-criticality schedules. Ann Oper Res 309, 209–232 (2022).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Mixed-criticality
  • Job replication
  • Scheduling
  • Bayesian networks
  • Computational complexity
  • Uncertain processing time