Transient analysis of piecewise homogeneous Markov fluid models
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Transient analysis of piecewise homogeneous Markov fluid models Salah Al-Deen Almousa1 · Gábor Horváth1 · Miklós Telek1 Accepted: 13 October 2020 © The Author(s) 2020
Abstract Piecewise homogeneous Markov fluid models are composed by homogeneous intervals where the model is governed by an interval dependent pair of generators and the model behaviour changes at the boundaries. The main difficulty of the transient analysis of piecewise homogeneous Markov fluid models is the appropriate description of the various boundary cases. The paper proposes an analytical approach to handle the wide variety of the possible boundary cases in a relatively simple to describe and implement manner. Keywords Piecewise homogeneous Markov fluid models · Transient solution · Laplace transform · Numerical analysis
1 Introduction Markov fluid models (MFMs) gained significant popularity in modeling telecommunication systems in the 1980’s (Anick et al. 1982). The first methodology to analyze the behaviour of such systems was based on spectral decomposition (Kulkarni 1997). In 1999, Ramaswami initiated a research line to analyze stochastic fluid models via matrix analytic methods (Ramaswami 1999), while Akar and Sohraby recommended the use of purely numerical matrix iterative methods (Akar and Sohraby 2004). Both methods provide numerically stable analysis, e.g., for the stationary distribution of the fluid level, but the approach based on matrix analytic methods gained more popularity due to the fact that it provides a stochastic interpretation of the considered performance measures.
This work is partially supported by the OTKA K-123914 and the TUDFO/51757/2019-ITM grants.
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Miklós Telek [email protected] Salah Al-Deen Almousa [email protected] Gábor Horváth [email protected]
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Department of Networked Systems and Services, Budapest University of Technology and Economics, Budapest, Hungary
123
Annals of Operations Research
In a series of consecutive papers the stationary (Ahn et al. 2005; da Silva Soares and Latouche 2006) and the transient (Ahn and Ramaswami 2004, 2005, 2006) analysis of homogeneous (finite and infinite) MFMs has been investigated. At the same time, the ingredients of the computational methods for various performance measures of fluid models has also been enhanced (Remiche 2005; Bean et al. 2009). Especially, the combination of two matrix exponential terms for describing the behaviour of finite buffer homogeneous fluid models got established. Motivated by several practical examples, e.g. (Mandjes et al. 2003), the analysis of homogeneous MFMs has been extended to the analysis of piecewise homogeneous models, where the characterizing matrices of the model are constant in a region of the fluid level, but they might differ region by region. The terminology used to describe this set of models is rather diverse: “level dependent evolution” (da Silva Soares and Latouche 2009), “multi-layer” (Bean and O’Reilly 2008), “multi-regime” (Kankaya and Akar 2008), etc. We refer to such modes as piecewise homogeneous Markov flu
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