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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The case of different Jordan blocks with identical eigenvalue is not considered here, because it cannot occur in non-redundant PH representations.
- 2.
Failure is more like an event than an activity but, in order to keep the discussion clearer, we refer to it as failure activity.
References
Asmussen S, Bladt M (1999) Point processes with finite-dimensional conditional probabilities. Stoch Process Appl 82:127–142
Bean NG, Nielsen BF (2010) Quasi-birth-and-death processes with rational arrival process components. Stoch Models 26(3):309–334
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)
Buchholz P, Telek M (2010) Stochastic petri nets with matrix exponentially distributed firing times. Perform Eval 67:1373–1385
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 428–439
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 328–342
Ciardo G, Marmorstein R, Siminiceanu R (2003) Saturation unbound. In: Proceedings of TACAS. Springer, pp 379–393
Cox DR (1955) The analysis of non-markovian stochastic processes by the inclusion of supplementary variables. Proc Cambridge Philos Soc 51(3):433–441
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 255–269
Kleinrock L (1975) Queuing systems, vol 1: theory. Wiley Interscience, New York
Kulkarni VG (1995) Modeling and analysis of stochastic systems. Chapman & Hall
Lipsky L (2008) Queueing theory: a linear algebraic approach. Springer
Longo F, Scarpa M (2015) Two-layer symbolic representation for stochastic models with phase-type distributed events. Int J Syst Sci 46(9):1540–1571
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)
Miner A, Parker D (2004) Symbolic representations and analysis of large state spaces. In: Validation of stochastic systems, LNCS 2925, Dagstuhl (Germany). Springer, pp 296–338
Neuts M (1975) Probability distributions of phase type. In: Amicorum L, Florin EH (eds) University of Louvain, pp 173–206
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 52–62
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 92–95
Trivedi K (1982) Probability and statistics with reliability, queueing and computer science applications. Prentice-Hall, Englewood Cliffs
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Horvath, A., Scarpa, M., Telek, M. (2016). Phase Type and Matrix Exponential Distributions in Stochastic Modeling. In: Fiondella, L., Puliafito, A. (eds) Principles of Performance and Reliability Modeling and Evaluation. Springer Series in Reliability Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30599-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-30599-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30597-4
Online ISBN: 978-3-319-30599-8
eBook Packages: EngineeringEngineering (R0)