Julianna Bor
Abstract
Matrix exponential (ME) distributions generalize phase-type distributions; however, their use in queueing theory is hampered by the difficulty of checking their feasibility. We propose a novel ME fitting algorithm that produces a valid distribution by construction. The ME distribution used during the fitting is a product of independent random variables that are easy to control in isolation. Consequently, the calculation of the CDF and the Mellin transform factorizes, making it possible to use these measures for the fitting without significant restriction on the distribution order. Trace-driven queueing simulations indicate that the resulting distributions yield highly accurate results.
- A. Andersen et al. A Markovian approach for modeling packet traffic with long-range dependence. IEEE JSAC, 16(5):719--732, 1998.Google Scholar
- N. G. Bean and B. F. Nielsen. Analysis of queues with rational arrival process components: A general approach. Perform. Eval. Rev., 39(4):31, 2012.Google ScholarDigital Library
- M. Bladt and B. F. Nielsen. Matrix-Exponential Distributions in Applied Probability. Springer, 2017.Google ScholarCross Ref
- P. Buchholz. An EM-algorithm for MAP fitting from real traffic data. In Comp. Perf. Eval. Modelling Tech. and Tools, pages 218--236. Springer, 2003.Google ScholarCross Ref
- P. Buchholz et al. Stochastic Petri nets with low variation matrix exponentially distributed firing time. Int. Journal of Perf. Eng., 7:441--454, 2011.Google Scholar
- P. Buchholz and J. Kriege. A heuristic approach for fitting MAPs to moments and joint moments. In Proc. of QEST, pages 53--62, 2009.Google ScholarDigital Library
- G. Casale et al. Characterization of moments and autocorrelation in MAPs. Perform. Eval. Rev., 35(2):27--29, sep 2007.Google ScholarDigital Library
- G. Casale, N. Mi, and E. Smirni. Model-driven system capacity planning under workload burstiness. IEEE Trans. Comp., 59(1):66--80, 2010.Google ScholarDigital Library
- M. Fackrell. Fitting with matrix-exponential distributions. Stoch. Models, 21(2--3):377--400, 2005.Google Scholar
- C. M. Harris and W. G. Marchal. Distribution estimation using Laplace transforms. INFORMS J. Comput., 10:448--458, 1998.Google ScholarCross Ref
- A. Horv´ath et al. Markovian modeling of real data traffic: Heuristic phase type and MAP fitting of heavy tailed and fractal like samples. In Perf. Eval. of Comp. Sys.: Tech. and Tools, pages 405--434. Springer, 2002.Google ScholarCross Ref
- G. Horv´ath and M. Telek. Butools 2: a rich toolbox for Markovian perf. eval. In Proc. of VALUETOOLS. ACM, 2017.Google ScholarDigital Library
- I. Horv´ath et al. Concentrated matrix exponential distributions. In Comp. Perf. Eng., pages 18--31. Springer, 2016.Google ScholarCross Ref
- G. Latouche and V. Ramaswami. Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, 1999.Google ScholarCross Ref
- R. S. Maier and C. A. O'Cinneide. A closure characterisation of phase-type distributions. J. App. Prob., 29(1):92--103, 1992.Google ScholarCross Ref
- T. Osogami and M. Harchol-Balter. Closed form solutions for mapping general distributions to quasi-minimal PH distributions. Perf. Eval., 63(6):524--552, 2006.Google ScholarDigital Library
- A. Riska, V. Diev, and E. Smirni. An EM-based technique for approximating long-tailed data sets with PH distributions. Perf. Eval., 55(1):147--164, 2004.Google ScholarDigital Library
- M. Telek et al. Matching moments for acyclic discrete and continuous phase-type distributions of second order. Int. Jour. of Sim. Systems, Sci. and Tech., 3, 04 2003.Google Scholar
- M. Telek and G. Horv´ath. A minimal representation of Markov arrival processes and a moments matching method. Perf. Eval., 64(9):1153--1168, 2007.Google ScholarDigital Library
- B. Van Houdt and J. S. van Leeuwaarden. Triangular M/G/1-type and tree-like quasi-birth-death Markov chains. INFORMS J. Computing, 23(1):165--171, 2011.Google ScholarDigital Library
Recommendations
Bilateral matrix-exponential distributions (abstract only)
In this article we define the classes of bilateral and multivariate bilateral matrix-exponential distributions. These distributions have support on the entire real space and have rational moment-generating functions. These distributions extend the class ...
Computing discrete mixtures of continuous distributions: noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient
In this article, we address the problem of computing the distribution functions that can be expressed as discrete mixtures of continuous distributions. Examples include noncentral chisquare, noncentral beta, noncentral F, noncentral t, and the ...
Fitting Mixtures of Exponentials to Long-Tail Distributions to Analyze Network Performance Models
INFOCOM '97: Proceedings of the INFOCOM '97. Sixteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Driving the Information RevolutionTraffic measurements from communication networks have shown that many quantities characterizing network performance have long-tail probability distributions, i.e., with tails that decays more slowly than exponentially. Long-tail distributions can have a ...
Comments