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

Published online by Cambridge University Press:  16 April 2004

Yuxi Zheng* ,
Xiuli Chao[dagger]  and
Xiaomei Ji
Yuxi Zheng*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Xiuli Chao[dagger]
Affiliation:
Department of Industrial Engineering, North Carolina State University, Raleigh, North Carolina 27695-7906, E-mail: [email protected]
Xiaomei Ji
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
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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.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 2 , April 2004 , pp. 141 - 159
Copyright
© 2004 Cambridge University Press

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References

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: 83–106.Google Scholar
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: 567–588.Google Scholar
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: 337–348.Google Scholar
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: 87–118.Google Scholar
Karlin, S. & McGregor, J.L. (1958). Linear growth, birth and death processes. Journal of Mathematics and Mechanics 7: 643–662.Google Scholar
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: 378–396.Google Scholar
Kendall, D.G. (1949). Stochastic processes and population growth. Journal of the Royal Statistical Society B 11: 230–264.Google Scholar
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: 1233–1240.Google Scholar
Kyriakidis, E.G. (1994). Stationary probabilities for a simple immigration-birth–death process under the influence of total catastrophes. Statistics and Probability Letters 20: 239–240.Google Scholar
Kyriakidis, E.G. (2001). The transient probabilities of the simple immigration–catastrophe process. The Mathematical Scientist 26: 56–58.Google Scholar
Massey, W.A. & Whitt, W. (1993). Networks of infinite-server queues with nonstationary Poisson input. Queueing Systems: Theory and Applications 13: 183–250.Google Scholar
Morse, P.M. (1955). Stochastic properties of waiting lines. Journal of Operations Research Society of America 3: 255–261.Google Scholar
Pegden, C.D. & Rosenshine, M. (1982). Some new results for the M/M/1 queue. Management Science 28: 821–828.Google Scholar
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: 522–534.Google Scholar
Saaty, T.L. (1960). Time-dependent solution of the server Poisson queue. Operations Research 8: 773–781.Google Scholar
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: 207–220.Google Scholar