Abstract
We consider a class of Markov chains known for its closed form transient and steady-state distributions. We show that some absorbing chains can be also seen as members of this class and we provide the closed form solution for their absorption time distributions. By constructing upper and lower bounding chains that belong to this particular class one can easily compute both lower and upper bounds for absorption time distribution of an arbitrary absorbing Markov chain. We provide a new algorithm for the construction of bounding chains from this class and we show a possible application of these bounds.
This work was supported by SMS, ANR-05-BLAN-009-02 and Checkbound, ANR-06-SETI-002.
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References
Abu-Amsha, O., Vincent, J.-M.: An algorithm to bound functionals of markov chains with large state space. In: 4th INFORMS Conference on Telecommunications, Boca Raton, FL (1998)
Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.-P.: Synchronization and Linearity: An Algebra for Discrete Event Systems. Willey, New York (1992)
Ben Mamoun, M., Busic, A., Fourneau, J.M., Pekergin, N.: Increasing convex monotone markov chains: Theory, algorithm and applications. In: MAM 2006: Markov Anniversary Meeting, Raleigh, North Carolina, USA, pp. 189–210 (2006)
Ben Mamoun, M., Busic, A., Pekergin, N.: Generalized class C Markov chains and computation of closed-form bounding distributions. Probability in the Engineering and Informational Sciences 21(2), 235–260 (2007)
Ben Mamoun, M., Pekergin, N.: Computing closed-form stochastic bounds on the stationary distribution of Markov chains. In: Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, Basel, pp. 197–209. Birkhauser (2000)
Ben Mamoun, M., Pekergin, N.: Closed-form stochastic bounds on the stationary distribution of Markov chains. Probability in the Engineering and Informational Sciences 16(4), 403–426 (2002)
Ben Mamoun, M., Pekergin, N.: Computing closed-form stochastic bounds on transient distributions of Markov chains. In: SAINT-W 2005: Proceedings of the 2005 Symposium on Applications and the Internet Workshops (SAINT 2005 Workshops), pp. 260–263. IEEE Computer Society Press, Washington, DC, USA (2005)
Ben Mamoun, M., Pekergin, N., Younès, S.: Model checking of continuous-time Markov chains by closed-form bounding distributions. In: QEST, pp. 189–198. IEEE Computer Society Press, Los Alamitos (2006)
Fourneau, J.-M., Kloul, L.: A precedence pepa model for performance and reliability analysis. In: Horváth, A., Telek, M. (eds.) EPEW 2006. LNCS, vol. 4054, pp. 1–15. Springer, Heidelberg (2006)
Fourneau, J.-M., Pekergin, N.: An algorithmic approach to stochastic bounds. In: Performance Evaluation of Complex Systems: Techniques and Tools, Performance 2002, Tutorial Lectures, pp. 64–88. Springer, London (2002)
Mahevas, S., Rubino, G.: Bound computation of dependability and performance measures. IEEE Trans. Comput. 50(5), 399–413 (2001)
Muller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002)
Muntz, R.R., de Souza e Silva, E., Goyal, A.: Bounding availability of repairable computer systems. IEEE Trans. on Computers 38(12), 1714–1723 (1989)
Pekergin, N., Vincent, J.-M.: Stochastic bounds on execution times of parallel programs. IEEE Trans. Softw. Eng. 17(10), 1005–1012 (1991)
Truffet, L.: Reduction technique for discrete time Markov chains on totally ordered state space using stochastic comparisons. Journal of Applied Probability 37(3), 795–806 (2000)
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Bušić, A., Pekergin, N. (2007). Closed Form Absorption Time Bounds . In: Wolter, K. (eds) Formal Methods and Stochastic Models for Performance Evaluation. EPEW 2007. Lecture Notes in Computer Science, vol 4748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75211-0_4
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DOI: https://doi.org/10.1007/978-3-540-75211-0_4
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