Fitting Methods Based on Distance Measures of Marked Markov Arrival Processes
Approximating various real-world observations with stochastic processes is an essential modelling step in several fields of applied sciences. In this chapter, we focus on the family of Markov-modulated point processes, and propose some fitting methods. The core of these methods is the computation of the distance between elements of the model family. First, we introduce a methodology for computing the squared distance between the density functions of two phase-type (PH) distributions. Later, we generalize this methodology for computing the distance between the joint density functions of \( k \) successive inter-arrival times of Markovian arrival processes (MAPs) and marked Markovian arrival processes (MMAPs). We also discuss the distance between the autocorrelation functions of such processes. Based on these computable distances, various versions of simple fitting procedures are introduced to approximate real-world observations with the mentioned Markov modulated point processes.
KeywordsTraffic modelling Marked Markovian arrival process Squared distance
This work was supported by the Hungarian research project OTKA K101150 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.
- 2.Asmussen S, Koole G (1993) Marked point processes as limits of Markovian arrival streams. J Appl Probab 365–372Google Scholar
- 6.Casale G, Zhang EZ, Smirni E (2010) Trace data characterization and fitting for Markov modeling. Perform Eval 67(2):61–79Google Scholar
- 7.Golub GH, Nash S, Van Loan C (1979) A Hessenberg-Schur method for the problem AX + XB = C. IEEE Trans Autom Control 24(6):909–913Google Scholar
- 10.Horváth G (2013) Moment matching-based distribution fitting with generalized hyper-Erlang distributions. In: Analytical and stochastic modeling techniques and applications, pp 232–246. SpringerGoogle Scholar
- 12.Horváth G (2015) Measuring the distance between maps and some applications. In: Gribaudo M, Manini D, Remke A (eds) Analytical and Stochastic Modelling Techniques and Applications, Lecture Notes in Computer Science, vol 9081, pp 100–114. SpringerGoogle Scholar
- 13.Johnson MA, Taaffe MR (1989) Matching moments to phase distributions: mixtures of Erlang distributions of common order. Stochast Models 5(4):711–743Google Scholar
- 14.Latouche G, Ramaswami V (1987) Introduction to matrix analytic methods in stochastic modeling, volume 5. Soc Ind Appl MathGoogle Scholar
- 15.Laub AJ (2005) Matrix analysis for scientists and engineers. SIAMGoogle Scholar
- 16.Lipsky L (2008) Queueing theory: A linear algebraic approach. SpringerGoogle Scholar
- 17.Neuts M (1975) Probability distributions of phase type. In: Liber Amicorum Prof. Emeritus H. Florin, pp 173–206. University of LouvainGoogle Scholar
- 20.Steeb WH (1997) Matrix calculus and Kronecker product with applications and C++ programs. World ScientificGoogle Scholar
- 22.van de Liefvoort A (1990) The moment problem for continuous distributions. Technical report, University of Missouri, WP-CM-1990-02, Kansas CityGoogle Scholar