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On a bilateral birth-death process with alternating rates

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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.

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References

  1. Anderson W.J., McDunnough P.M.: On the representation of symmetric transition functions. Adv. Appl. Prob. 22, 548–563 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  3. Conolly B.W.: On randomized random walks. SIAM Rev. 13, 81–99 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Conolly B.W., Parthasarathy P.R., Dharmaraja S.: A chemical queue. Math. Sci. 22, 83–91 (1997)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  9. Di Crescenzo A., Martinucci B.: On a symmetric, nonlinear birth-death process with bimodal transition probabilities. Symmetry. 1, 201–214 (2009)

    Article  MathSciNet  Google Scholar 

  10. Giorno V., Nobile A.G.: On the distribution of the range of an asymmetric random walk. Ricerche Mat. 37, 315–324 (1988)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. Lente G.: The role of stochastic models in interpreting the origins of biological chirality. Symmetry 2, 767–798 (2010)

    Article  Google Scholar 

  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)

  16. Pollett P.K.: Similar Markov chains. Probability, statistics and seismology. J. Appl. Prob. 38A, 53–65 (2001)

    MathSciNet  MATH  Google Scholar 

  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)

  18. Ricciardi L.M., Sato S.: On the range of a one-dimensional asymmetric random walk. Ricerche Mat. 36, 153–160 (1987)

    MathSciNet  MATH  Google Scholar 

  19. Stockmayer W.H., Gobush W., Norvich R.: Local-jump models for chain dynamics. Pure Appl. Chem. 26, 555–561 (1971)

    Article  Google Scholar 

  20. Tarabia A.M.K., El-Baz A.H.: A new explicit solution for a chemical queue. Math. Sci. 27, 16–24 (2002)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Salerno, 84084, Fisciano, SA, Italy

    Antonio Di Crescenzo, Antonella Iuliano & Barbara Martinucci

Authors
  1. Antonio Di Crescenzo
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  2. Antonella Iuliano
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  3. Barbara Martinucci
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Corresponding author

Correspondence to Antonio Di Crescenzo.

Additional information

Communicated by Editor in Chief.

This paper is dedicated to the memory of Professor Luigi Maria Ricciardi, who passed away in Naples on May 7, 2011.

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Di Crescenzo, A., Iuliano, A. & Martinucci, B. On a bilateral birth-death process with alternating rates. Ricerche mat. 61, 157–169 (2012). https://doi.org/10.1007/s11587-011-0122-0

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  • DOI: https://doi.org/10.1007/s11587-011-0122-0

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