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Markov Reward Models and Markov Decision Processes in Discrete and Continuous Time: Performance Evaluation and Optimization

  • Alexander Gouberman
  • Markus Siegle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8453)

Abstract

State-based systems with discrete or continuous time are often modelled with the help of Markov chains. In order to specify performance measures for such systems, one can define a reward structure over the Markov chain, leading to the Markov Reward Model (MRM) formalism. Typical examples of performance measures that can be defined in this way are time-based measures (e.g. mean time to failure), average energy consumption, monetary cost (e.g. for repair, maintenance) or even combinations of such measures. These measures can also be regarded as target objects for system optimization. For that reason, an MRM can be enhanced with an additional control structure, leading to the formalism of Markov Decision Processes (MDP).

In this tutorial, we first introduce the MRM formalism with different types of reward structures and explain how these can be combined to a performance measure for the system model. We provide running examples which show how some of the above mentioned performance measures can be employed. Building on this, we extend to the MDP formalism and introduce the concept of a policy. The global optimization task (over the huge policy space) can be reduced to a greedy local optimization by exploiting the non-linear Bellman equations. We review several dynamic programming algorithms which can be used in order to solve the Bellman equations exactly. Moreover, we consider Markovian models in discrete and continuous time and study value-preserving transformations between them. We accompany the technical sections by applying the presented optimization algorithms to the example performance models.

Keywords

Markov Decision Process Reward Function Laurent Series Policy Iteration Average Reward 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Altman, E.: Constrained Markov Decision Processes. Chapman & Hall (1999)Google Scholar
  2. 2.
    Altman, E.: Applications of Markov Decision Processes in Communication Networks. In: Feinberg, E.A., Shwartz, A. (eds.) Handbook of Markov Decision Processes. International Series in Operations Research & Management Science, vol. 40, pp. 489–536. Springer, US (2002)CrossRefGoogle Scholar
  3. 3.
    Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P.: Model-Checking Algorithms for Continuous-Time Markov Chains. IEEE Transactions on Software Engineering 29(6), 524–541 (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bäuerle, N., Rieder, U.: Markov Decision Processes with Applications to Finance. Springer, Heidelberg (2011)Google Scholar
  5. 5.
    Bellman, R.: Dynamic Programming. Princeton University Press, Princeton (1957)zbMATHGoogle Scholar
  6. 6.
    Benini, L., Bogliolo, A., Paleologo, G.A., De Micheli, G.: Policy Optimization for Dynamic Power Management. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 18, 813–833 (1998)CrossRefGoogle Scholar
  7. 7.
    Bertsekas, D.: Dynamic Programming and Optimal Control, 3rd edn., vol. I. Athena Scientific (1995) (revised in 2005)Google Scholar
  8. 8.
    Bertsekas, D.: Dynamic Programming and Optimal Control, 4th edn., vol. II. Athena Scientific (1995) (revised in 2012)Google Scholar
  9. 9.
    Bertsekas, D., Tsitsiklis, J.: An analysis of stochastic shortest path problems. Mathematics of Operations Research 16(3), 580–595 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bertsekas, D., Tsitsiklis, J.: Neuro-Dynamic Programming, 1st edn. Athena Scientific (1996)Google Scholar
  11. 11.
    Beynier, A., Mouaddib, A.I.: Decentralized Markov decision processes for handling temporal and resource constraints in a multiple robot system. In: Proceedings of the 7th International Symposium on Distributed Autonomous Robotic System, DARS (2004)Google Scholar
  12. 12.
    Bolch, G., Greiner, S., de Meer, H., Trivedi, K.S.: Queueing Networks and Markov Chains - Modelling and Performance Evaluation with Computer Science Applications, 2nd edn. Wiley (2006)Google Scholar
  13. 13.
    Cassandra, A.R.: A survey of POMDP applications. In: Working Notes of AAAI 1998 Fall Symposium on Planning with Partially Observable Markov Decision Processes, pp. 17–24 (1998)Google Scholar
  14. 14.
    Diz, F.J., Palacios, M.A., Arias, M.: MDPs in medicine: opportunities and challenges. In: Decision Making in Partially Observable, Uncertain Worlds: Exploring Insights from Multiple Communities, IJCAI Workshop (2011)Google Scholar
  15. 15.
    Fox, B.L., Landi, D.M.: An algorithm for identifying the ergodic subchains and transient states of a stochastic matrix. Communications of the ACM 11(9), 619–621 (1968)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gouberman, A., Siegle, M.: On Lifetime Optimization of Boolean Parallel Systems with Erlang Repair Distributions. In: Operations Research Proceedings 2010 - Selected Papers of the Annual International Conference of the German Operations Research Society, pp. 187–192. Springer (January 2011)Google Scholar
  17. 17.
    Guo, X., Hernandez-Lerma, O.: Continuous-Time Markov Decision Processes - Theory and Applications. Springer (2009)Google Scholar
  18. 18.
    Heidergott, B., Hordijk, A., Van Uitert, M.: Series Expansions For Finite-State Markov Chains. Probability in the Engineering and Informational Sciences 21(3), 381–400 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hou, Z., Filar, J.A., Chen, A. (eds.): Markov Processes and Controlled Markov Chains. Springer (2002)Google Scholar
  20. 20.
    Howard, R.A.: Dynamic Programming and Markov Processes. John Wiley & Sons, New York (1960)zbMATHGoogle Scholar
  21. 21.
    Hu, Q., Yue, W.: Markov Decision Processes with their Applications. Springer (2008)Google Scholar
  22. 22.
    Janssen, J., Manca, R.: Markov and Semi-Markov Reward Processes. In: Applied Semi-Markov Processes, pp. 247–293. Springer, US (2006)Google Scholar
  23. 23.
    Janssen, J., Manca, R.: Semi-Markov Risk Models for Finance, Insurance and Reliability. Springer (2007)Google Scholar
  24. 24.
    Jensen, A.: Markoff chains as an aid in the study of Markoff processes. Skandinavisk Aktuarietidskrift 36, 87–91 (1953)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Stidham Jr., S., Weber, R.: A survey of Markov decision models for control of networks of queues. Queueing Systems 13(1-3), 291–314 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mahadevan, S.: Learning Representation and Control in Markov Decision Processes: New Frontiers. Foundations and Trends in Machine Learning 1(4), 403–565 (2009)CrossRefzbMATHGoogle Scholar
  27. 27.
    Mahadevan, S., Maggioni, M.: Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes. Journal of Machine Learning Research 8, 2169–2231 (2007)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mausam, Kolobov, A.: Planning with Markov Decision Processes: An AI Perspective. Synthesis Lectures on Artificial Intelligence and Machine Learning. Morgan & Claypool Publishers (2012)Google Scholar
  29. 29.
    Momtazi, S., Kafi, S., Beigy, H.: Solving Stochastic Path Problem: Particle Swarm Optimization Approach. In: Nguyen, N.T., Borzemski, L., Grzech, A., Ali, M. (eds.) IEA/AIE 2008. LNCS (LNAI), vol. 5027, pp. 590–600. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  30. 30.
    Obal, W.D., Sanders, W.H.: State-space support for path-based reward variables. In: Proceedings of the Third IEEE International Performance and Dependability Symposium on International Performance and Dependability Symposium, IPDS 1998, pp. 233–251. Elsevier Science Publishers B. V. (1999)Google Scholar
  31. 31.
    Ott, J.T.: A Markov Decision Model for a Surveillance Application and Risk-Sensitive Markov Decision Processes. PhD thesis, Karlsruhe Institute of Technology (2010)Google Scholar
  32. 32.
    Powell, W.B.: Approximate Dynamic Programming - Solving the Curses of Dimensionality. Wiley (2007)Google Scholar
  33. 33.
    Puterman, M.L.: Markov Decision Processes - Discrete Stochastic Dynamic Programming. John Wiley & Sons INC. (1994)Google Scholar
  34. 34.
    Qiu, Q., Pedram, M.: Dynamic power management based on continuous-time Markov decision processes. In: Proceedings of the 36th Annual ACM/IEEE Design Automation Conference, DAC 1999, pp. 555–561. ACM (1999)Google Scholar
  35. 35.
    Sanders, W.H., Meyer, J.F.: A Unified Approach for Specifying Measures of Performance, Dependability, and Performability. Dependable Computing for Critical Applications 4, 215–238 (1991)CrossRefGoogle Scholar
  36. 36.
    Schaefer, A.J., Bailey, M.D., Shechter, S.M., Roberts, M.S.: Modeling medical treatment using Markov decision processes. In: Brandeau, M.L., Sainfort, F., Pierskalla, W.P. (eds.) Operations Research and Health Care. International Series in Operations Research & Management Science, vol. 70, pp. 593–612. Kluwer Academic Publishers (2005)Google Scholar
  37. 37.
    Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. A Bradford Book. MIT Press (March 1998)Google Scholar
  38. 38.
    Trivedi, K.S., Malhotra, M.: Reliability and Performability Techniques and Tools: A Survey. In: Messung, Modellierung und Bewertung von Rechen- und Kommunikationssystemen. Informatik aktuell, pp. 27–48. Springer, Heidelberg (1993)Google Scholar
  39. 39.
    Tsitsiklis, J.N.: NP-Hardness of checking the unichain condition in average cost MDPs. Operations Research Letters 35(3), 319–323 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    White, D.J.: A Survey of Applications of Markov Decision Processes. The Journal of the Operational Research Society 44(11), 1073–1096 (1993)CrossRefzbMATHGoogle Scholar
  41. 41.
    Wolff, R.W.: Poisson Arrivals See Time Averages. Operations Research 30(2), 223–231 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Alexander Gouberman
    • 1
  • Markus Siegle
    • 1
  1. 1.Department of Computer ScienceUniversität der BundeswehrMünchenGermany

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