Ricerche di Matematica

, Volume 61, Issue 1, pp 157–169 | Cite as

On a bilateral birth-death process with alternating rates

  • Antonio Di Crescenzo
  • Antonella Iuliano
  • Barbara Martinucci
Article
First Online:
Received:
Revised:

Abstract

We consider a bilateral birth-death process characterized by a constant transition rate λ from even states and a possibly different transition rate μ from odd states. We determine the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state. Some features of the birth-death process confined to the non-negative integers by a reflecting boundary in the zero-state are also analyzed. In particular, making use of a Laplace transform approach we obtain a series form of the transition probability from state 1 to the zero-state.

Keywords

Birth-death processes Alternating rates Probability generating functions Transition probabilities Symmetry 

Mathematics Subject Classification (2010)

60J80 60J85 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderson W.J., McDunnough P.M.: On the representation of symmetric transition functions. Adv. Appl. Prob. 22, 548–563 (1990)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Böhm, W., Hornik, K.: On two-periodic random walks with boundaries. Research Report Series/Department of Statistics and Mathematics, 75. Department of Statistics and Mathematics, WU Vienna University of Economics and Business, Vienna (2008)Google Scholar
  3. 3.
    Conolly B.W.: On randomized random walks. SIAM Rev. 13, 81–99 (1971)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Conolly B.W., Parthasarathy P.R., Dharmaraja S.: A chemical queue. Math. Sci. 22, 83–91 (1997)MathSciNetMATHGoogle Scholar
  5. 5.
    Di Crescenzo, A.: On some transformations of bilateral birth-and-death processes with applications to first passage time evaluations. In Sita ’94–Proceedings of 17th Symposium on Information Theory Appl, pp. 739–742. Hiroshima. Available at http://arXiv.org/pdf/0803.1413 (1994)
  6. 6.
    Di Crescenzo A.: First-passage-time densities and avoiding probabilities for birth and death processes with symmetric simple paths. J. Appl. Prob. 35, 383–394 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Di Crescenzo A., Giorno V., Nobile A.G., Ricciardi L.M.: On a symmetry-based constructive approach to probability densities for two-dimensional diffusion processes. J. Appl. Prob. 32, 316–336 (1995)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Di Crescenzo A., Giorno V., Nobile A.G., Ricciardi L.M.: On first-passage-time and transition densities for strongly symmetric diffusion processes. Nagoya Math. J. 145, 143–161 (1997)MathSciNetMATHGoogle Scholar
  9. 9.
    Di Crescenzo A., Martinucci B.: On a symmetric, nonlinear birth-death process with bimodal transition probabilities. Symmetry. 1, 201–214 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Giorno V., Nobile A.G.: On the distribution of the range of an asymmetric random walk. Ricerche Mat. 37, 315–324 (1988)MathSciNetMATHGoogle Scholar
  11. 11.
    Giorno V., Nobile A.G., Ricciardi L.M.: A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes. J. Appl. Prob. 26, 707–721 (1989)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Giorno, V., Nobile, A.G., Ricciardi, L.M.: On the densities of certain bounded diffusion processes. Ricerche Mat. (2010). doi: 10.1007/s11587-010-0097-2
  13. 13.
    Iuliano A., Martinucci B.: Transient analysis of a birth-death process with alternating rates. In: Trappl, R. (eds) Cybernetics and Systems 2010, pp. 187–191. Austrian Society for Cybernetic Studies, Vienna (2010)Google Scholar
  14. 14.
    Lente G.: The role of stochastic models in interpreting the origins of biological chirality. Symmetry 2, 767–798 (2010)CrossRefGoogle Scholar
  15. 15.
    Parthasarathy, P.R., Lenin, R.B.: Birth and death process (BDP) models with applications–queueing, communication systems, chemical models, biological models: the state-of-the-art with a time-dependent perspective. American Series in Mathematical and Management Sciences, vol. 51, American Sciences Press, Columbus (2004)Google Scholar
  16. 16.
    Pollett P.K.: Similar Markov chains. Probability, statistics and seismology. J. Appl. Prob. 38A, 53–65 (2001)MathSciNetMATHGoogle 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, Berlin (1986)Google Scholar
  18. 18.
    Ricciardi L.M., Sato S.: On the range of a one-dimensional asymmetric random walk. Ricerche Mat. 36, 153–160 (1987)MathSciNetMATHGoogle Scholar
  19. 19.
    Stockmayer W.H., Gobush W., Norvich R.: Local-jump models for chain dynamics. Pure Appl. Chem. 26, 555–561 (1971)CrossRefGoogle Scholar
  20. 20.
    Tarabia A.M.K., El-Baz A.H.: A new explicit solution for a chemical queue. Math. Sci. 27, 16–24 (2002)MathSciNetMATHGoogle Scholar
  21. 21.
    Tarabia A.M.K., El-Baz A.H.: Analysis of the busy period of the chemical queue: a series approach. Math. Sci. 27, 108–116 (2002)MathSciNetMATHGoogle Scholar
  22. 22.
    Tarabia A.M.K., Takagi H., El-Baz A.H.: Transient solution of a non-empty chemical queueing system. Math. Meth. Oper. Res. 70, 77–98 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2011

Authors and Affiliations

  • Antonio Di Crescenzo
    • 1
  • Antonella Iuliano
    • 1
  • Barbara Martinucci
    • 1
  1. 1.Dipartimento di MatematicaUniversità di SalernoFiscianoItaly

Personalised recommendations