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Constructing transient birth–death processes by means of suitable transformations

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  • Di Crescenzo, Antonio
  • Giorno, Virginia
  • Nobile, Amelia G.

Abstract

For a birth–death process N(t) with a reflecting state at 0 we propose a method able to construct a new birth–death process M(t) defined on the same state-space. The birth and death rates of M(t) depend on the rates of N(t) and on the probability law of the process N(t) evaluated at an exponentially distributed random time. Under a suitable assumption we obtain the conditional probabilities, the mean of the process, and the Laplace transforms of the downward first-passage-time densities of M(t). We also discuss the connection between the proposed method and the notion of ν-similarity, as well as a relation between the distribution of M(t) and the steady-state probabilities of N(t) subject to catastrophes governed by a Poisson process. We investigate new processes constructed from (i) a birth–death process with constant rates, and (ii) a linear immigration-death process. Various numerical computations are performed to illustrate the obtained results.

Suggested Citation

  • Di Crescenzo, Antonio & Giorno, Virginia & Nobile, Amelia G., 2016. "Constructing transient birth–death processes by means of suitable transformations," Applied Mathematics and Computation, Elsevier, vol. 281(C), pages 152-171.
    • Handle: RePEc:eee:apmaco:v:281:y:2016:i:c:p:152-171
    DOI: 10.1016/j.amc.2016.01.058
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    1. Callaert, Herman & Keilson, Julian, 1973. "On exponential ergodicity and spectral structure for birth-death processes I," Stochastic Processes and their Applications, Elsevier, vol. 1(2), pages 187-216, April.
    2. Antonio Crescenzo & Virginia Giorno & Balasubramanian Krishna Kumar & Amelia G. Nobile, 2012. "A Double-ended Queue with Catastrophes and Repairs, and a Jump-diffusion Approximation," Methodology and Computing in Applied Probability, Springer, vol. 14(4), pages 937-954, December.
    3. Spiros Dimou & Antonis Economou, 2013. "The Single Server Queue with Catastrophes and Geometric Reneging," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 595-621, September.
    4. Economou, Antonis & Fakinos, Demetrios, 2003. "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, Elsevier, vol. 149(3), pages 625-640, September.
    5. Di Crescenzo, A. & Giorno, V. & Nobile, A.G. & Ricciardi, L.M., 2008. "A note on birth-death processes with catastrophes," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2248-2257, October.
    6. Callaert, Herman & Keilson, Julian, 1973. "On exponential ergodicity and spectral structure for birth-death processes, II," Stochastic Processes and their Applications, Elsevier, vol. 1(3), pages 217-235, July.
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    Cited by:

    1. Zeifman, A. & Satin, Y. & Kiseleva, K. & Korolev, V. & Panfilova, T., 2019. "On limiting characteristics for a non-stationary two-processor heterogeneous system," Applied Mathematics and Computation, Elsevier, vol. 351(C), pages 48-65.
    2. Virginia Giorno & Amelia G. Nobile, 2020. "Bell Polynomial Approach for Time-Inhomogeneous Linear Birth–Death Process with Immigration," Mathematics, MDPI, vol. 8(7), pages 1-29, July.
    3. Zeifman, A.I. & Satin, Y.A. & Kiseleva, K.M., 2020. "On obtaining sharp bounds of the rate of convergence for a class of continuous-time Markov chains," Statistics & Probability Letters, Elsevier, vol. 161(C).
    4. Giorno, Virginia & Nobile, Amelia G., 2022. "On some integral equations for the evaluation of first-passage-time densities of time-inhomogeneous birth-death processes," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    5. Giorno, Virginia & Nobile, Amelia G., 2020. "On a class of birth-death processes with time-varying intensity functions," Applied Mathematics and Computation, Elsevier, vol. 379(C).
    6. Zeifman, A.I. & Korolev, V.Yu. & Satin, Ya.A. & Kiseleva, K.M., 2018. "Lower bounds for the rate of convergence for continuous-time inhomogeneous Markov chains with a finite state space," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 84-90.
    7. Antonio Di Crescenzo & Virginia Giorno & Balasubramanian Krishna Kumar & Amelia G. Nobile, 2018. "A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation," Mathematics, MDPI, vol. 6(5), pages 1-23, May.

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