Skip to main content

Stochastic Ordering of Semi-Markov Processes

  • Chapter
Network Performance Engineering

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 5233))

  • 2079 Accesses

Abstract

In this tutorial we address the stochastic ordering of semi-Markov processes in the usual and level-crossing stochastic ordering senses. We highlight the sample-path approach for the comparison of semi-Markov processes and for the simulation of processes with a given distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arnold, B.C.: Majorization and the Lorenz Order: A Brief Introduction. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  2. Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  3. Baccelli, F., Brémaud, P.: Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences, 2nd edn. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  4. Bawa, V.S.: Stochastic dominance: a research bibliography. Management Science 28(6), 698–712 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brockwell, P.J., Gani, J., Resnick, S.I.: Birth, immigration and catastrophe processes. Advances in Applied Probability 14(4), 709–731 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chao, X., Zheng, Y.: Transient analysis of immigration birth-death processes with total catastrophes. Probability in the Engineering and Informational Sciences 17(1), 83–106 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Çinlar, E.: Markov renewal theory. Advances in Applied Probability 1, 123–187 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  8. Çinlar, E.: On semi-Markov processes on arbitrary spaces. Proc. Cambridge Philos. Soc. 66, 381–392 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  9. Çinlar, E.: Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs (1975)

    MATH  Google Scholar 

  10. Di Crescenzo, A., Ricciardi, L.M.: Comparing first-passage times for semi-Markov skip-free processes. Statistics and Probability Letters 30(3), 247–256 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  11. Disney, R.L., Kiessler, P.C.: Traffic Processes in Queueing Networks: A Markov Renewal Approach. Johns Hopkins, Baltimore (1987)

    MATH  Google Scholar 

  12. Economou, A., Fakinos, D.: A continuous-time Markov chain under the influence of a regulating point process and applications in stochastic models with catastrophes. European Journal of Operational Research 149(3), 625–640 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. El-Taha, M., Stidham Jr., S.: Sample-Path Analysis of Queueing Systems. Kluwer, Boston (1999)

    Book  MATH  Google Scholar 

  14. Feller, W.: On semi-Markov processes. Proc. Nat. Acad. Sci. U.S.A. 51, 653–659 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ferreira, F.: Embedding, Uniformization and Stochastic Ordering in the Analysis of Level-Crossing Times and GIX/M(n)//c Systems. Ph.D thesis, Instituto Superior Técnico - Technical University of Lisbon, Lisbon, Portugal (2007)

    Google Scholar 

  16. Ferreira, F., Pacheco, A.: Level-crossing ordering of semi-Markov processes and Markov chains. Journal of Applied Probability 42(4), 989–1002 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ferreira, F., Pacheco, A.: Level-crossing ordering of skip-free to the right continuous time Markov chains. Journal of Applied Probability 42(1), 52–60 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ferreira, F., Pacheco, A.: Comparison of level crossing times for Markov and semi-Markov processes. Statistics and Probability Letters 77(2), 151–157 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gross, D., Harris, C.M.: Fundamentals of Queueing Theory, 3rd edn. Wiley, Chichester (1998)

    MATH  Google Scholar 

  20. Irle, A.: Stochastic ordering for continuous-time processes. Journal of Applied Probability 40(2), 361–375 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Irle, A., Gani, J.: The detection of words and an ordering for Markov chains. Journal of Applied Probability 38A, 66–77 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Janssen, J., Limnios, N. (eds.): Semi-Markov Models and Applications. Kluwer, Dordrecht (1999)

    MATH  Google Scholar 

  23. Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997)

    Book  MATH  Google Scholar 

  24. Kalbfleisch, J.D., Prentice, R.L.: The Statistical Analysis of Failure Time Data. Wiley, New York (1980)

    MATH  Google Scholar 

  25. Kamae, T., Krengel, U., O’Brien, G.L.: Stochastic inequalities on partially ordered spaces. The Annals of Probability 5(6), 899–912 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kijima, M.: Markov Processes for Stochastic Modeling. Chapman and Hall, London (1997)

    Book  MATH  Google Scholar 

  27. Kijima, M., Ohnishi, M.: Stochastic orders and their applications in financial optimization. Mathematical Methods of Operations Research 50(2), 351–372 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kirstein, B.M.: Monotonicity and comparability of time-homogeneous Markov processes with discrete state space. Mathematische Operations Forschung und Statistik 7, 151–168 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kulkarni, V.G.: Modeling and Analysis of Stochastic Systems. Chapman and Hall, London (1995)

    MATH  Google Scholar 

  30. Last, G., Brandt, A.: Marked Point Processes on the Real Line: The Dynamic Approach. Springer, New York (1995)

    MATH  Google Scholar 

  31. Levy, H.: Stochastic dominance and expected utility: survey and analysis. Management Science 38(5), 555–593 (1992)

    Article  MATH  Google Scholar 

  32. Levy, P.: Processus semi-markoviens. In: Proceedings of the International Congress of Mathematicians 1954, Amsterdam, vol. III, pp. 416–426. Erven P. Noordhoff N.V., Groningen (1956)

    Google Scholar 

  33. Limnios, N., Oprişan, G.: Semi-Markov Processes and Reliability. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  34. Lindvall, T.: Comparisons between certain queuing systems. Probability in the Engineering and Informational Sciences 16(1), 1–17 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lindvall, T.: Lectures on the Coupling Method. Dover Publications, Mineola (2002)

    MATH  Google Scholar 

  36. Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and its Applications. Academic Press, New York (1979)

    MATH  Google Scholar 

  37. Massey, W.A.: Stochastic orderings for Markov processes on partially ordered spaces. Mathematics of Operations Research 12(2), 350–367 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  38. Morais, M.J.C.: Stochastic Ordering in the Performance Analysis of Quality Control Schemes. PhD thesis, Instituto Superior Técnico, Technical University of Lisbon, Portugal (2002)

    Google Scholar 

  39. Mosler, K., Scarsini, M.: Stochastic Orders and Applications: A Classified Bibliography. Springer, Heidelberg (1993)

    Book  MATH  Google Scholar 

  40. Müller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)

    MATH  Google Scholar 

  41. Pyke, R.: Markov renewal processes: definitions and preliminary properties. The Annals of Mathematical Statistics 32, 1231–1242 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pyke, R.: Markov renewal processes with finitely many states. The Annals of Mathematical Statistics 32, 1243–1259 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  43. Ross, S.M.: Stochastic Processes, 2nd edn. Wiley, New York (1996)

    MATH  Google Scholar 

  44. Shaked, M., Shanthikumar, J.G.: Stochastic Orders and Their Applications. Academic Press, San Diego (1994)

    MATH  Google Scholar 

  45. Smith, W.L.: Regenerative stochastic processes. Proc. Roy. Soc. London. Ser. A. 232, 6–31 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  46. Smith, W.L.: Renewal theory and its ramifications. J. Roy. Statist. Soc. Ser. B 20, 243–302 (1958)

    MathSciNet  MATH  Google Scholar 

  47. Sonderman, D.: Comparing semi-Markov processes. Mathematics of Operations Research 5(1), 110–119 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983)

    MATH  Google Scholar 

  49. Szekli, R.: Stochastic Ordering and Dependence in Applied Probability. Springer, New York (1995)

    Book  MATH  Google Scholar 

  50. Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000)

    Book  MATH  Google Scholar 

  51. Tong, Y.L.: Probability Inequalities in Multivariate Distributions. Academic Press, New York (1980)

    MATH  Google Scholar 

  52. van Doorn, E.A.: Stochastic Monotonicity and Queueing Applications of Birth-Death Processes. Springer, New York (1981)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Ferreira, F., Pacheco, A. (2011). Stochastic Ordering of Semi-Markov Processes. In: Kouvatsos, D.D. (eds) Network Performance Engineering. Lecture Notes in Computer Science, vol 5233. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02742-0_18

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-02742-0_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02741-3

  • Online ISBN: 978-3-642-02742-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics