Phase Type and Matrix Exponential Distributions in Stochastic Modeling

Chapter

Abstract

Since their introduction, properties of Phase Type (PH) distributions have been analyzed and many interesting theoretical results found. Thanks to these results, PH distributions have been profitably used in many modeling contexts where non-exponentially distributed behavior is present. Matrix Exponential (ME) distributions are distributions whose matrix representation is structurally similar to that of PH distributions but represent a larger class. For this reason, ME distributions can be usefully employed in modeling contexts in place of PH distributions using the same computational techniques and similar algorithms, giving rise to new opportunities. They are able to represent different dynamics, e.g., faster dynamics, or the same dynamics but at lower computational cost. In this chapter, we deal with the characteristics of PH and ME distributions, and their use in stochastic analysis of complex systems. Moreover, the techniques used in the analysis to take advantage of them are revised.

References

  1. 1.
    Asmussen S, Bladt M (1999) Point processes with finite-dimensional conditional probabilities. Stoch Process Appl 82:127–142MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bean NG, Nielsen BF (2010) Quasi-birth-and-death processes with rational arrival process components. Stoch Models 26(3):309–334MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Buchholz P, Horvath A, Telek M (2011) Stochastic Petri nets with low variation matrix exponentially distributed firing time. Int J Perform Eng 7:441–454, 2011 (Special issue on performance and dependability modeling of dynamic systems)Google Scholar
  4. 4.
    Buchholz P, Telek M (2010) Stochastic petri nets with matrix exponentially distributed firing times. Perform Eval 67:1373–1385CrossRefGoogle Scholar
  5. 5.
    Burch JR, Clarke EM, McMillan KL, Dill DL, Hwang LJ (1990) Symbolic model checking: 1020 states and beyond. In: Fifth annual IEEE symposium on logic in computer science, 1990. LICS ’90, Proceedings, pp 428–439Google Scholar
  6. 6.
    Ciardo G, Luttgen G, Siminiceanu R (2001) Saturation: an efficient iteration strategy for symbolic state space generation. In: Proceedings of tools and algorithms for the construction and analysis of systems (TACAS), LNCS 2031. Springer, pp 328–342Google Scholar
  7. 7.
    Ciardo G, Marmorstein R, Siminiceanu R (2003) Saturation unbound. In: Proceedings of TACAS. Springer, pp 379–393Google Scholar
  8. 8.
    Cox DR (1955) The analysis of non-markovian stochastic processes by the inclusion of supplementary variables. Proc Cambridge Philos Soc 51(3):433–441MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Distefano S, Longo F, Scarpa M, Trivedi KS (2014) Non-markovian modeling of a bladecenter chassis midplane. In: Computer performance engineering, vol 8721 of Lecture Notes in Computer Science. Springer International Publishing, pp 255–269Google Scholar
  10. 10.
    Kleinrock L (1975) Queuing systems, vol 1: theory. Wiley Interscience, New YorkGoogle Scholar
  11. 11.
    Kulkarni VG (1995) Modeling and analysis of stochastic systems. Chapman & HallGoogle Scholar
  12. 12.
    Lipsky L (2008) Queueing theory: a linear algebraic approach. SpringerGoogle Scholar
  13. 13.
    Longo F, Scarpa M (2015) Two-layer symbolic representation for stochastic models with phase-type distributed events. Int J Syst Sci 46(9):1540–1571MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Miner AS, Ciardo G (1999) Efficient reachability set generation and storage using decision diagrams. In: Application and Theory of Petri Nets 1999 (Proceedings 20th international conference on applications and theory of Petri Nets. Springer, pp 6–25)Google Scholar
  15. 15.
    Miner A, Parker D (2004) Symbolic representations and analysis of large state spaces. In: Validation of stochastic systems, LNCS 2925, Dagstuhl (Germany). Springer, pp 296–338Google Scholar
  16. 16.
    Neuts M (1975) Probability distributions of phase type. In: Amicorum L, Florin EH (eds) University of Louvain, pp 173–206Google Scholar
  17. 17.
    Scarpa M, Bobbio A (1998) Kronecker representation of stochastic petri nets with discrete ph distributions. In: Proceedings of IEEE international computer performance and dependability symposium, 1998. IPDS’98. pp 52–62Google Scholar
  18. 18.
    Srinivasan A, Ham T, Malik S, Brayton RK (1990) Algorithms for discrete function manipulation. In: IEEE international conference on computer-aided design, 1990. ICCAD-90. Digest of technical papers, pp 92–95Google Scholar
  19. 19.
    Trivedi K (1982) Probability and statistics with reliability, queueing and computer science applications. Prentice-Hall, Englewood CliffsGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità degli Studi di TorinoTurinItaly
  2. 2.Dipartimento di IngegneriaUniversità degli Studi di MessinaMessinaItaly
  3. 3.Department of Networked Systems and Services, MTA-BME Information Systems Research GroupBudapest University of Technology and EconomicsBudapestHungary

Personalised recommendations