Fitting Methods Based on Distance Measures of Marked Markov Arrival Processes

  • Gábor HorváthEmail author
  • Miklós Telek
Part of the Simulation Foundations, Methods and Applications book series (SFMA)


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.


Traffic 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.


  1. 1.
    Asmussen S, Bladt M (1999) Point processes with finite-dimensional conditional probabilities. Stochast Process Appl 82:127–142MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asmussen S, Koole G (1993) Marked point processes as limits of Markovian arrival streams. J Appl Probab 365–372Google Scholar
  3. 3.
    Bean NG, Nielsen BF (2010) Quasi-birth-and-death processes with rational arrival process components. Stochast Models 26(3):309–334MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buchholz Peter, Kemper Peter, Kriege Jan (2010) Multi-class Markovian arrival processes and their parameter fitting. Perform Eval 67(11):1092–1106CrossRefGoogle Scholar
  5. 5.
    Buchholz Peter, Telek Miklós (2013) On minimal representations of rational arrival processes. Ann Oper Res 202(1):35–58MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Casale G, Zhang EZ, Smirni E (2010) Trace data characterization and fitting for Markov modeling. Perform Eval 67(2):61–79Google Scholar
  7. 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
  8. 8.
    He Qi-Ming, Neuts Marcel (1998) Markov arrival processes with marked transitions. Stochast Process Appl 74:37–52MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Horváth András, Horváth Gábor, Telek Miklós (2010) A joint moments based analysis of networks of MAP/MAP/1 queues. Perform Eval 67(9):759–778CrossRefGoogle Scholar
  10. 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
  11. 11.
    Horváth Gábor, Van Houdt Benny (2013) Departure process analysis of the multi-type MMAP[K]/PH[K]/1 FCFS queue. Perform Eval 70(6):423–439CrossRefGoogle Scholar
  12. 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. 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. 14.
    Latouche G, Ramaswami V (1987) Introduction to matrix analytic methods in stochastic modeling, volume 5. Soc Ind Appl MathGoogle Scholar
  15. 15.
    Laub AJ (2005) Matrix analysis for scientists and engineers. SIAMGoogle Scholar
  16. 16.
    Lipsky L (2008) Queueing theory: A linear algebraic approach. SpringerGoogle Scholar
  17. 17.
    Neuts M (1975) Probability distributions of phase type. In: Liber Amicorum Prof. Emeritus H. Florin, pp 173–206. University of LouvainGoogle Scholar
  18. 18.
    Neuts MF (1979) A versatile Markovian point process. J Appl Probab 16:764–779MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic processes for finance and insurance. Willey, New YorkCrossRefzbMATHGoogle Scholar
  20. 20.
    Steeb WH (1997) Matrix calculus and Kronecker product with applications and C++ programs. World ScientificGoogle Scholar
  21. 21.
    Telek Miklós, Horváth Gábor (2007) A minimal representation of Markov arrival processes and a moments matching method. Perform Eval 64(9):1153–1168CrossRefGoogle Scholar
  22. 22.
    van de Liefvoort A (1990) The moment problem for continuous distributions. Technical report, University of Missouri, WP-CM-1990-02, Kansas CityGoogle Scholar
  23. 23.
    Zhang Qi, Heindl Armin, Smirni Evgenia (2005) Characterizing the BMAP/MAP/1 departure process via the ETAQA truncation. Stochast Models 21(2–3):821–846MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Networked Systems and ServicesBudapest University of Technology and EconomicsBudapestHungary
  2. 2.MTA-BME Information Systems Research GroupBudapestHungary

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