On-line Optimal Control of a Class of Discrete Event Systems with Real-Time Constraints


We consider Discrete Event Systems (DES) involving tasks with real-time constraints and seek to control processing times so as to minimize a cost function subject to each task meeting its own constraint. It has been shown that the off-line version of this problem can be efficiently solved by the Critical Task Decomposition Algorithm (CTDA) (Mao et al., IEEE Trans Mobile Comput 6(6):678–688, 2007). In the on-line version, random task characteristics (e.g., arrival times) are not known in advance. To bypass this difficulty, worst-case analysis may be used. This, however, does not make use of probability distributions and results in an overly conservative solution. In this paper, we develop a new approach which does not rely on worst-case analysis but provides a “best solution in probability” efficiently obtained by estimating the probability distribution of sample-path-optimal solutions. We introduce a condition termed “non-singularity” under which the best solution in probability leads to the on-line optimal control. Numerical examples are included to illustrate our results and show substantial performance improvements over worst-case analysis.

This is a preview of subscription content, log in to check access.

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


  1. Aydin H, Melhem R, Mossé D, Mejia-Alvarez P (2004) Power-aware scheduling for periodic real-time tasks. IEEE Trans Comput 53(5):584–600, May

    Article  Google Scholar 

  2. Charnes A, Cooper WW (1959) Chance constrained programming. Manage Sci 6:73C79

    Article  MathSciNet  Google Scholar 

  3. Charnes A, Cooper WW, Symonds GH (1958) Cost horizons and certainty equivalents; an approach to stochastic programming of heating oil. Manage Sci 4:235C263

    Article  Google Scholar 

  4. Chong EK, Givan P, Chang HS (2000) A framework for simulation-based network control via hindsight optimization. In Proceedings of the 39th IEEE conference on decision and control, pp 1433–1438

  5. Gamal AE, Nair C, Prabhakar B, Uysal-Biyikoglu E, Zahedi S (2002) Energy-eff icient scheduling of packet transmissions over wireless networks. In Proceedings of IEEE INFOCOM, vol 3, 23–27, pp 1773–1782, New York City, USA

  6. Grant M, Boyd S, Ye Y (2006) Disciplined convex programming. In: Liberti L, Maculan N (eds) Chapter in global optimization: from theory to implementation. Springer, Berlin, pp 155–210

    Google Scholar 

  7. Hoeffding W (1963) Probability inequalities for sums of bounded random variables. J Am Stat Assoc 58(301):13C30, March

    Article  MathSciNet  Google Scholar 

  8. Jeffay K, Stanat DF, Martel CU (1991) On non-preemptive scheduling of periodic and sporadic tasks. In Proceedings of the IEEE real-time systems symposium, pp 129–139

  9. Jonsson J, Lonn H, Shin KG (1999) Non-preemptive scheduling of real-time threads on multi-level-context architectures. In Proceedings of the IEEE workshop on parallel and distributed real-time systems, vol 1586. Springer, Berlin, pp 363–374

    Google Scholar 

  10. Kall P, Wallace S (1994) Stochastic programming. Wiley, New York

    Google Scholar 

  11. Liu JWS (2000) Real-time systems. Prentice Hall, New Jersey

    Google Scholar 

  12. Mao J, Cassandras CG (2007) Optimal control of two-stage discrete event systems with real-time constraints. J Disc Event Dyn Syst 17(4):505–529

    MATH  Article  MathSciNet  Google Scholar 

  13. Mao J, Cassandras CG (2008) Optimal control of multi-layer discrete event system with real-time constraints. In Proceedings of the 17th international federation of automatic control(IFAC) world congress, pp 4120–4125, July

  14. Mao J, Cassandras CG (2009a) Optimal admission control of discrete event systems with real-time constraints. J Disc Event Dyn Syst. doi:10.1007/s10626-008-0052-5

  15. Mao J, Cassandras CG (2009b) Optimal control of multi-stage discrete event systems with real-time constraints. IEEE Trans Autom Control 54(1):108–123

    Article  MathSciNet  Google Scholar 

  16. Mao J, Cassandras CG, Zhao QC (2007) Optimal dynamic voltage scaling in power-limited systems with real-time constraints. IEEE Trans Mobile Comput 6(6):678–688, June

    Article  Google Scholar 

  17. Miao L, Cassandras CG (2005a) Optimality of static control policies in some discrete event systems. IEEE Trans Autom Control, 50(9):1427–1431, September

    Article  MathSciNet  Google Scholar 

  18. Miao L, Cassandras CG (2005b) Receding horizon control for a class of discrete event system with real-time constraints. In Proceedings of the 44th IEEE conference decision and control, pp 7714–7719

  19. Miao L, Cassandras CG (2006) Optimal transmission scheduling for energy-efficient wireless networks. In Proceedings of IEEE INFOCOM

  20. Pepyne DL, Cassandras CG (2000) Optimal control of hybrid systems in manufacturing. Proc IEEE, 88(7):1108–1123

    Article  Google Scholar 

  21. Ruszczynski A, Shapiro A (eds) (2003) Stochastic programming. Handbooks in operations research and management science, vol. 10. Elsevier, Amsterdam

    Google Scholar 

  22. Shapiro A, Ruszczynski A (2007) Lectures on stochastic programming. URL: http://www2.isye.gatech.edu/people/faculty/Alex_Shapiro/SPbook.pdf

  23. Uysal-Biyikoglu E, Prabhakar B, Gamal AE (2002) Energy-efficient packet transmission over a wireless link. IEEE/ACM Trans Network 10:487–499, August

    Article  Google Scholar 

  24. Wu G, Chong EKP, Givan RL (2002) Burst-level congestion control using hindsight optimization. IEEE Trans Autom Control, special issue on systems and control methods for communication networks 47(6):979–991, June

    MathSciNet  Google Scholar 

  25. Yao F, Demers A, Shenker S (1995) A scheduling model for reduced CPU energy. In Proceedings of the 36th annual symposium on foundations of computer science (FOCS’95), pp 374–382. IEEE Computer Society

Download references

Author information



Corresponding author

Correspondence to Jianfeng Mao.

Additional information

The authors’ work is supported in part by NSF under Grants DMI-0330171 and EFRI-0735974, by AFOSR under grants FA9550-04-1-0133 and FA9550-04-1-0208, and by DOE under grant DE-FG52-06NA27490.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mao, J., Cassandras, C.G. On-line Optimal Control of a Class of Discrete Event Systems with Real-Time Constraints. Discrete Event Dyn Syst 20, 187–213 (2010). https://doi.org/10.1007/s10626-008-0058-z

Download citation


  • On-line optimal control
  • Discrete event system
  • Real-time constraints