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TRANSIENT ANALYSIS OF LINEAR BIRTH–DEATH PROCESSES WITH IMMIGRATION AND EMIGRATION

  • Yuxi Zheng* (a1), Xiuli Chao[dagger] (a2) and Xiaomei Ji (a1)

Abstract

Linear birth–death processes with immigration and emigration are major models in the study of population processes of biological and ecological systems, and their transient analysis is important in the understanding of the structural behavior of such systems. The spectral method has been widely used for solving these processes; see, for example, Karlin and McGregor [11]. In this article, we provide an alternative approach: the method of characteristics. This method yields a Volterra-type integral equation for the chance of extinction and an explicit formula for the z-transform of the transient distribution. These results allow us to obtain closed-form solutions for the transient behavior of several cases that have not been previously explicitly presented in the literature.

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REFERENCES

Bailey, N.T.J. (1964). Elements of stochastic processes. New York: Wiley.
Bartlett, M.S. (1978). An introduction to stochastic processes, 3rd ed. Cambridge: Cambridge University Press.
Chao, X. & Zheng, Y. (2003). Transient analysis of immigration birth–death processes with total catastrophes. Probability in the Engineering and Informational Sciences 17: 83106.
Cox, D.R. & Miller, H.D. (1965). The theory of stochastic processes. London: Chapman & Hall.
DiBenedetto, E. (1995). Partial differential equations. Boston: Birkhäuser.
Halfin, S. & Whitt, W. (1981). Heavy-traffic limits for queues with many exponential servers. Operations Research 29: 567588.
Iosifescu, M. & Tautu, P. (1973). Stochastic processes and applications in biology and medicine, Vol. II. Berlin: Springer-Verlag.
Ismail, M.E.H., Letessier, J., & Valent, G. (1988). Linear birth and death models and associated Laguerre and Meixner polynomials. Journal of Approximation Theory 55: 337348.
John, F. (1982). Partial differential equations. New York: Springer-Verlag.
Karlin, S. & McGregor, J.L. (1957). Many server queueing processes with Poisson input and exponential service times. Pacific Journal of Mathematics 7: 87118.
Karlin, S. & McGregor, J.L. (1958). Linear growth, birth and death processes. Journal of Mathematics and Mechanics 7: 643662.
Kelton, W.D. & Law, A.M. (1985). The transient behavior of the M/M/s queue, with implication for steady-state simulation. Operations Research 33: 378396.
Kendall, D.G. (1949). Stochastic processes and population growth. Journal of the Royal Statistical Society B 11: 230264.
Keyfitz, N. (1977). Introduction to the mathematics of population with revision. Reading, MA: Addison-Wesley.
Kleinrock, L. (1975). Queueing systems, Vol. I, Theory. New York: Wiley.
Kumar, B.K. & Arivudainambi, D. (2000). Transient solution of an M/M/1 queue with catastrophes. Computers and Mathematics with Applications 40: 12331240.
Kyriakidis, E.G. (1994). Stationary probabilities for a simple immigration-birth–death process under the influence of total catastrophes. Statistics and Probability Letters 20: 239240.
Kyriakidis, E.G. (2001). The transient probabilities of the simple immigration–catastrophe process. The Mathematical Scientist 26: 5658.
Massey, W.A. & Whitt, W. (1993). Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems: Theory and Applications 13: 183250.
Morse, P.M. (1955). Stochastic properties of waiting lines. Journal of Operations Research Society of America 3: 255261.
Pegden, C.D. & Rosenshine, M. (1982). Some new results for the M/M/1 queue. Management Science 28: 821828.
Pipkin, A. C. (1991). A course on integral equations. Text in Applied Mathematics Volume 9. New York: Springer-Verlag.
Pogorzelski, W. (1966). Integral equations and their applications, Vol. 1. New York: Pergamon Press.
Ross, S. (2003). Introduction to probability models, 8th ed. San Diego, CA: Academic Press.
Rothkopf, M.H. & Oren, S.S. (1979). A closure approximation for the nonstationary M/M/s queue. Management Science 25: 522534.
Saaty, T.L. (1960). Time-dependent solution of the server Poisson queue. Operations Research 8: 773781.
van Doorn, E. (1981). Stochastic monotonicity and queueing applications in birth–death processes. Lecture Notes in Statistics Volume 4. New York: Springer-Verlag.
Weast, R.C. (1975). Handbook of tables for mathematics, rev. 4th ed. Cleveland, OH: CRC Press.
Whitt, W. (1981). Comparing counting processes and queues. Advances in Applied Probability 13: 207220.

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