Advertisement

On a Bilateral Linear Birth and Death Process in the Presence of Catastrophes

  • Virginia Giorno
  • Amelia G. Nobile
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8111)

Abstract

A bilateral linear birth-death process with disasters in zero is considered. The Laplace transforms of the transition probabilities are determined and the steady-state distribution is analyzed. The first-visit time to zero state is also studied.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Buonocore, A., Di Crescenzo, A., Giorno, V., Nobile, A.G., Ricciardi, L.M.: A Markov chain-based model for actomyosin dynamics. Sci. Math. Jpn. 70, 159–174 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Conolly, B.: On Randomized Random Walks. SIAM Review 13(1), 81–99 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Conolly, B.: Lecture Notes on Queueing Systems. Ellis Horwood Ltd., Halsted (John Wiley & Sons), Chichester, New York (1975)Google Scholar
  4. 4.
    Di Crescenzo, A., Nastro, A.: On first-passage-time densities for certain symmetric Markov chains. Sci. Math. Jpn. 60(2), 381–390 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Di Crescenzo, A., Giorno, V., Nobile, A.G., Ricciardi, L.M.: A note on birth-death processes with catastrophes. Statistics and Probability Letters 78, 2248–2257 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Di Crescenzo, A., Martinucci, B.: On a symmetric, nonlinear birth-death process with bimodal transition probabilities. Symmetry 1, 201–214 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Di Crescenzo, A., Giorno, V., Nobile, A.G., Ricciardi, L.M.: On time non-homogeneous stochastic processes with catastrophes. In: Trappl, R. (ed.) Cybernetics and Systems 2010, pp. 169–174. Austrian Society for Cybernetics Studies, Vienna (2010)Google Scholar
  8. 8.
    Di Crescenzo, A., Giorno, V., Krishna Kumar, B., Nobile, A.G.: A double-ended queue with catastrophes and repairs, and a jump-diffusion approximation. Method. Comput. Appl. Probab. 14, 937–954 (2012)CrossRefzbMATHGoogle Scholar
  9. 9.
    Di Crescenzo, A., Iuliano, A., Martinucci, B.: On a bilateral birth-death process with alternating rates. Ricerche di Matematica 61(1), 157–169 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dimou, S., Economou, A.: The single server queue with catastrophes and geometric reneging. Method. Comput. Appl. Probab. (2011), doi:10.1007/s11009-011-9271-6Google Scholar
  11. 11.
    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 J. Oper. Res. (Stochastics and Statistics) 149, 625–640 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hongler, M.-O., Parthasarathy, P.R.: On a super-diffusive, non linear birth and death process. Physics Letters A 372, 3360–3362 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Karlin, S., McGregor, J.: Linear growth, birth and death processes. Journal of Mathematics and Mechanics 7(4), 643–662 (1958)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Medhi, J.: Stochastic Models in Queueing Theory. Academic Press, Amsterdam (2003)zbMATHGoogle Scholar
  15. 15.
    Pollett, P.K.: Similar Markov chain. J. Appl. Probab. 38A, 53–65 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Pruitt, W.E.: Bilateral birth and death processes. Trans. Amer. Math. Soc. 107, 508–525 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ricciardi, L.M.: Stochastic population theory: birth and death processes. In: Hallam, T.G., Levin, S.A. (eds.) Mathematical Ecology, Biomathematics, vol. 17, pp. 155–190. Springer, Heidelberg (1986)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Virginia Giorno
    • 1
  • Amelia G. Nobile
    • 1
  1. 1.Dipartimento di Studi e Ricerche Aziendali (Management & Information Technology)Università di SalernoFiscianoItaly

Personalised recommendations