Abstract
We consider the problem of finding a separable solution for the equilibrium state probabilities in a Markovian process algebra model, in which the action rates may depend on the behaviour of other components. To do this we consider regular cycles in the underlying state space and show that a semi-product form solution exists when the functions describing the action rates have specific forms. The approach is illustrated with two examples, one a generalised version of a known state-dependent queueing network and the other in the domain of security protocols.
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References
Bonald, T., Proutiere, A.: Insensitivity in processor-sharing networks. Performance Evaluation 49, 193–209 (2002)
Boucherie, R., van Dijk, N.: Product forms for queueing networks with state-dependent multiple job transitions. Advances in Applied Probability 23, 152–187 (1991)
Chao, X., Miyazawa, M., Pinedo, M.: Queueing Networks: Customers, Signals and Product Form Solutions. Wiley (1999)
Derisavi, S., Hermanns, H., Sanders, W.: Optimal state-space lumping in Markov chains. Information Processing Letters 87, 309–315 (2003)
Harrison, P.G.: Turning back time in Markovian process algebra. Theoretical Computer Science 290(3), 1947–1986 (2003)
Harrison, P.G.: Reversed processes, product-forms and a non-product-form. Linear Algebra and Its Applications 386, 359–381 (2004)
Harrison, P.G., Lee, T.T.: Separable equilibrium state probabilities via time reversal in Markovian process algebra. Theoretical Comp. Sci. 346(1), 161–182 (2005)
Harrison, P.G.: Product-forms and functional rates. Performance Evaluation 66, 660–663 (2009)
Harrison, P.G.: Process algebraic non-product-forms. Electronic Notes in Theoretical Computer Science 151(3) (2006)
Harrison, P.G., Thomas, N.: Product form solution in PEPA via the reversed process. In: Kouvatsos, D. (ed.) Next Generation Internet. LNCS, vol. 5233, pp. 343–356. Springer, Heidelberg (2011)
Hayden, R., Bradley, J.: A fluid analysis framework for a Markovian process algebra. Theoretical Computer Science 411(22), 2260–2297 (2010)
Henderson, W., Taylor, P.: State-dependent Coupling of Quasireversible Nodes. Queueing Systems: Theory and Applications 37(1/3), 163–197 (2001)
Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press (1996)
Hillston, J.: Exploiting Structure in Solution: Decomposing Compositional Models. In: Brinksma, E., Hermanns, H., Katoen, J.-P. (eds.) FMPA 2000. LNCS, vol. 2090, pp. 278–314. Springer, Heidelberg (2001)
Hillston, J.: Fluid flow approximation of PEPA models. In: Proceedings of QEST 2005, pp. 33–43. IEEE Computer Society (2005)
Kelly, F.P.: Reversibility and stochastic networks. Wiley (1979)
Sevcik, K., Mitrani, I.: The distribution of queueing network states at input and output instants. JACM 28(2), 358–371 (1981)
Towsley, D.: Queuing Network Models with State-Dependent Routing. Journal of the ACM 27(2), 323–337 (1980)
Thomas, N.: Using ODEs from PEPA models to derive asymptotic solutions for a class of closed queueing networks. In: 8th Workshop on Process Algebra and Stochastically Timed Activities, PASTA (2008)
Thomas, N., Gilmore, S.: Applying quasi-separability to Markovian process algebra. In: Proceedings 6th International Workshop on Process Algebra and Performance Modelling (1998)
Thomas, N., Bradley, J., Thornley, D.: Approximate solution of PEPA models using component substitution. In: IEE Proceedings - Computers and Digital Techniques, vol. 150(2), pp. 67–74 (2003)
Thomas, N., Zhao, Y.: Fluid flow analysis of a model of a secure key distribution centre. In: Proceedings of the 24th Annual UK Performance Engineering Workshop, Imperial College London (2008)
Thomas, N., Zhao, Y.: Mean value analysis for a class of PEPA models. The Computer Journal 54(5), 643–652 (2011)
Tribastone, M.: Approximate Mean Value Analysis of Process Algebra Models. In: Proceedings of MASCOTS, pp. 369–378 (2011)
Zhao, Y., Thomas, N.: Comparing Methods for the Efficient Analysis of PEPA Models of Non-repudiation Protocols. In: Proceedings of 15th International Conference on Parallel and Distributed Systems, pp. 821–827 (2009)
Zhao, Y., Thomas, N.: Efficient solutions of a PEPA model of a key distribution centre. Performance Evaluation 67(8), 740–756 (2010)
Zhou, J., Gollmann, D.: An efficient non-repudiation protocol. In: Proceedings of the l0th Computer Security Foundations Workshop (CSFW 1997). IEEE Computer Society (1997)
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Thomas, N., Harrison, P.G. (2013). Semi-Product-Form Solution for PEPA Models with Functional Rates. In: Dudin, A., De Turck, K. (eds) Analytical and Stochastic Modeling Techniques and Applications. ASMTA 2013. Lecture Notes in Computer Science, vol 7984. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39408-9_29
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DOI: https://doi.org/10.1007/978-3-642-39408-9_29
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