Compositional approximate markov chain aggregation for PEPA models
Pages 96 - 110
Abstract
Approximate Markov chain aggregation involves the construction of a smaller Markov chain that approximates the behaviour of a given chain. We discuss two different approaches to obtain a nearly optimal partition of the state-space, based on different notions of approximate state equivalence.
Both approximate aggregation methods require an explicit representation of the transition matrix, a fact that renders them inefficient for large models. The main objective of this work is to investigate the possibility of compositionally applying such an approximate aggregation technique. We make use of the Kronecker representation of PEPA models, in order to aggregate the state-space of components rather than of the entire model.
References
[1]
Buchholz, P.: Exact and ordinary lumpability in finite Markov chains. Journal of Applied Probability 31(1), 59-75 (1994)
[2]
Bušić, A., Fourneau, J.: Bounds based on lumpable matrices for partially ordered state space. In: ICST Workshop on Tools for Solving Markov Chains. ACM (2006)
[3]
Courtois, P.: Decomposability, instabilities, and saturation in multiprogramming systems. Communications of the ACM 18(7), 371-377 (1975)
[4]
Daly, D., Buchholz, P., Sanders, W. H.: Bound-preserving composition for Markov reward models. In: Quantitative Evaluation of Systems, pp. 243-252. IEEE Computer Society (2006)
[5]
Dayar, T., Stewart, W.: Quasi lumpability, lower-bounding coupling matrices, and nearly completely decomposable Markov chains. SIAM Journal on Matrix Analysis and Applications 18(2), 482-498 (1997)
[6]
Deng, K., Sun, Y., Mehta, P., Meyn, S.: An information-theoretic framework to aggregate a Markov chain. In: American Control Conference, pp. 731-736. IEEE Press (2009)
[7]
Deuflhard, P., Huisinga, W., Fischer, A., Schütte, C.: Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains. Linear Algebra and its Applications 315(1-3), 39-59 (2000)
[8]
Fourneau, J., Lecoz, M., Quessette, F.: Algorithms for an irreducible and lumpable strong stochastic bound. Linear Algebra and its Applications 386, 167-185 (2004)
[9]
Franceschinis, G., Muntz, R.: Bounds for quasi-lumpable Markov chains. Performance Evaluation 20(1-3), 223-243 (1994)
[10]
Franceschinis, G., Muntz, R.: Computing bounds for the performance indices of quasi-lumpable stochastic well-formed nets. IEEE Transactions on Software Engineering 20(7), 516-525 (1994)
[11]
Gilmore, S., Hillston, J., Ribaudo, M.: An efficient algorithm for aggregating PEPA models. IEEE Transactions on Software Engineering 27(5), 449-464 (2001)
[12]
Hartfiel, D.: On the structure of stochastic matrices with a subdominant eigenvalue near 1. Linear Algebra and its Applications 272(1-3), 193-203 (1998)
[13]
Hillston, J.: A compositional approach to performance modelling. Cambridge University Press (1996)
[14]
Hillston, J.: Fluid flow approximation of PEPA models. In: Quantitative Evaluation of Systems, pp. 33-42. IEEE Computer Society (2005)
[15]
Hillston, J., Kloul, L.: An Efficient Kronecker Representation for PEPA Models. In: de Luca, L., Gilmore, S. (eds.) PAPM-PROBMIV 2001. LNCS, vol. 2165, pp. 120-135. Springer, Heidelberg (2001)
[16]
Jensen, A.: Markoff chains as an aid in the study of Markoff processes. Skandinavisk Aktuarietidskrift 36, 87-91 (1953)
[17]
Kemeny, J., Snell, J.: Finite Markov Chains. Springer (1976)
[18]
Kwiatkowska, M., Norman, G., Parker, D.: PRISM: probabilistic model checking for performance and reliability analysis. ACM SIGMETRICS Performance Evaluation Review 36(4), 40-45 (2009)
[19]
Malik, J., Shi, J.: Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(8), 888-905 (2000)
[20]
Meyer, C. D.: Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Review 31(2), 240-272 (1989)
[21]
Ng, A., Jordan, M., Weiss, Y.: On spectral clustering: Analysis and an algorithm. Advances in Neural Information Processing Systems 14(1), 849-856 (2001)
[22]
Pokarowski, P.: Uncoupling measures and eigenvalues of stochastic matrices. Journal of Applied Analysis 4(2), 259-267 (1998)
[23]
Runolfsson, T., Ma, Y.: Model reduction of nonreversible Markov chains. In: IEEE Conference on Decision and Control, pp. 3739-3744. IEEE (2008)
[24]
Smith, M.: Compositional abstractions for long-run properties of stochastic systems. In: Quantitative Evaluation of Systems, pp. 223-232. IEEE Computer Society (2011)
[25]
Thomas, N., Bradley, J.: Analysis of Non-product Form Parallel Queues Using Markovian Process Algebra. In: Kouvatsos, D. D. (ed.) Next Generation Internet. LNCS, vol. 5233, pp. 331-342. Springer, Heidelberg (2011)
Recommendations
Pseudo-marginal Markov Chain Monte Carlo for Nonnegative Matrix Factorization
A pseudo-marginal Markov chain Monte Carlo (PMCMC) method is proposed for nonnegative matrix factorization (NMF). The sampler jointly simulates the joint posterior distribution for the nonnegative matrices and the matrix dimensions which indicate the ...
Analyzing Markov chain Monte Carlo output
AbstractMarkov chain Monte Carlo (MCMC) is a sampling‐based method for estimating features of probability distributions. MCMC methods produce a serially correlated, yet representative, sample from the desired distribution. As such it can be difficult to ...
Visual representation of the multivariate correlation structure in a Markov chain. Output analysis that accounts for this multivariate structure is able to more accurately summarize variability in the simulation process. We review the output analysis ...
Comments
Information & Contributors
Information
Published In
Sponsors
- SICSA: The Scottish Informatics and Computer Science Alliance
Publisher
Springer-Verlag
Berlin, Heidelberg
Publication History
Published: 30 July 2012
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 26 Jan 2025