Acta Applicandae Mathematica

, Volume 34, Issue 1–2, pp 213–223 | Cite as

Further results for semiregenerative phenomena

  • N. U. Prabhu
Part III: Regeneration


A theory of semiregenerative phenomena was developed by the author. The set of points at which such a phenomenon occurs is called a semi regenerative set. There is a correspondence between a semiregenerative set and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete time case). Prabhu, Tang, and Zhu showed that the properties of semiregenerative sets associated with Markov random walks completely characterize the fluctuation behaviour of these processes in the nondegenerate case and also established a Wiener-Hopf factorization based on these sets. These results are surveyed in this paper.

Mathematics Subject Classifications (1991)

60K05 60K15 60J15 

Key words

Borel measures fluctuation behaviour linked regenerative phenomena Markov-additive process Markov random walk Markov renewal process quasi-Markov chain recurrent and regenerative phenomena semiregenerative phenomena semiregenerative sets subordinator Wiener-Hopf factorization 


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  1. 1.
    Asmussen, S.: Aspects of matrix Wiener-Hopf factorization in applied probability,The Mathematical Scientist 14 (1989), 101–116.Google Scholar
  2. 2.
    Asmussen, S.: Ladder heights and the Markov-modulated M/G/1 queue,Stochast. Processes Appl. 37 (1991), 313–326.Google Scholar
  3. 3.
    Barlow, M. T., Rogers, L. C. G. and Williams, D.: Wiener-Hopf factorization for matrices,Seminaire de Probabilities XIV, Lecture Notes in Math. 784, Springer-Verlag, New York, 1980, pp. 324–331.Google Scholar
  4. 4.
    Feller, W.:An Introduction to Probability Theory and its Applications, vol. 1, 3rd edn, Wiley, New York, 1967.Google Scholar
  5. 5.
    Gut, A. and Prabhu, N. U.: Renewal theory, regenerative phenomena and random walks, Dept. of Math., Uppsala University, June 1984.Google Scholar
  6. 6.
    Kaspi, H.: On the symmetric Wiener-Hopf factorization for Markov additive processes, Z.Wahrsch. verw. Gebiete 59 (1982), 179–196.Google Scholar
  7. 7.
    Kennedy, J. and Williams, D.: Probabilistic factorization of a quadratic matrix polynomial,Math. Proc. Cambridge Philos. Soc. 107 (1990), 591–600.Google Scholar
  8. 8.
    Kingman, J. F. C.: Linked systems of regenerative events,Proc. London Math. Soc. 15 (1965), 125–150.Google Scholar
  9. 9.
    Kingman, J. F. C.:Regenerative Phenomena, Wiley, New York, 1972.Google Scholar
  10. 10.
    Neveu, J.: Une généralisation des processus à acroissements positifs indépendents,Abh. Math. Sem. Univ. Hamburg 25 (1961), 36–61.Google Scholar
  11. 11.
    Prabhu, N. U.: Theory of semiregenerative phenomena,J. Appl. Probab. 25A (1988), 257–274.Google Scholar
  12. 12.
    Prabhu, N. U.: Markov renewal and Markov-additive processes —A review and some new results, in B. D. Choi and J.-W. Yim (eds),Proceedings of KAIST Mathematics Workshop, vol. 6, Korea Advanced Institute of Science and Technology, Taejon, 1991, pp. 57–94.Google Scholar
  13. 13.
    Prabhu, N. U. and Tang, L. C.: Markov-modulated single server queueing systems, to appear inJ. Appl. Probab. Google Scholar
  14. 14.
    Prabhu, N. U., Tang, L.-C. and Zhu, Y.: Some new results for the Markov random walk,J. Math. Phys. Sci. 25 (1991), 635–663.Google Scholar
  15. 15.
    Prabhu, N. U. and Zhu, Y.: Markov-modulated queueing systems,Queueing Systems Theory Appl. 5 (1989), 215–245.Google Scholar
  16. 16.
    Presman, E. L.: Factorization methods and boundary problems for sums of random variables given on Markov chains,Math, USSR Izv. 3 (1969), 815–852.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • N. U. Prabhu
    • 1
  1. 1.School of Operations Research and Industrial EngineeringCornell UniversityIthacaUSA

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