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ZIPF AND LERCH LIMIT OF BIRTH AND DEATH PROCESSES

Published online by Cambridge University Press:  21 December 2009

B. Klar ,
P. R. Parthasarathy  and
N. Henze
B. Klar
Affiliation:
Institut für Stochastik, Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany E-mail: bernhard.klar@kit.edu
P. R. Parthasarathy
Affiliation:
Institut für Stochastik, Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany E-mail: bernhard.klar@kit.edu
N. Henze
Affiliation:
Institut für Stochastik, Karlsruhe Institute of Technology, 76133 Karlsruhe, Germany E-mail: bernhard.klar@kit.edu
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Abstract

Birth and death processes are useful in a wide range of disciplines from computer networks and telecommunications to chemical kinetics and epidemiology. Data from many different areas such as linguistics, music, or warfare fit Zipf's law surprisingly well. The Lerch distribution generalizes Zipf's law and is applicable in survival and dispersal processes. In this article we construct a birth and death process that converges to the Lerch distribution in the limit as time becomes large, and we investigate the speed of convergence. This is achieved by employing continued fractions. Numerical illustrations are presented through tables and graphs.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 1 , January 2010 , pp. 129 - 144
Copyright
Copyright © Cambridge University Press 2009

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References

1.Aksenov, S.V. & Savageau, M.A. (2008). Some properties of the Lerch family of discrete distributions. http://arXiv.org/abs/math/0504485v1.Google Scholar
2.Anderson, W.J. (1991). Continuous-time Markov chains: An applications-oriented approach. New York: Springer.CrossRefGoogle Scholar
3.Bates, D. & Maechler, M. (2008). Matrix: Sparse and dense matrix classes and methods. R package version 0.999375-16. http://www.R-project.orgGoogle Scholar
4.Bowman, K.O. & Shenton, L.R. Continued fractions in statistical applications. Statistics: Textbooks and Monographs, vol. 103. New York: Marcel Dekker.Google Scholar
5.Coolen-Schrijner, P. & Van Doorn, E.A. (2001). On the convergence to stationarity of birth-death processes. Journal of Applied Probability 38: 696–706.Google Scholar
6.Gupta, P.L., Gupta, R.C., Ong, S., & Srivatsava, H.M. (2008). A class of Hurwitz–Lerch zeta distributions and their applications in reliability. Applied Mathematics and Computation 196: 521–531.CrossRefGoogle Scholar
7.Hill, B.M. (1974). The rank-frequency form of Zipf's law. Journal of The American Statistical Association 69: 1017–1026.CrossRefGoogle Scholar
8.Kulasekera, K.B. & Tonkyn, D.W. (1992). A new discrete distribution, with applications to survival, dispersal and dispersion. Communications in Statistics: Simulation Computation 21(2): 499–518.Google Scholar
9.Lorentzen, L. & Waadeland, H. (1992). Continued fractions with applications. Studies in Computational Mathematics, Vol. 3. Amsterdam: North-Holland.Google Scholar
10.Mandelbaum, M., Hylnka, M., & Brill, H.P. (2007). Nonhomogeneous geometric distributions with relations to birth and death processes. TOP 15: 281–296.Google Scholar
11.Moler, C. & Van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45: 3–49.Google Scholar
12.Murphy, J.A. & O'Donohoe, M.R. (1975). Some properties of continued fractions with applications in Markov processes. Journal of Institute of Mathematics and Its Application 16: 57–71.CrossRefGoogle Scholar
13.Naldi, M. (2003). Concentration indices and Zipf's law. Economic Letters 78: 329–334.CrossRefGoogle Scholar
14.Parthasarathy, P.R. & Lenin, R.B. (2004). 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 51. Syracuse, NY: American Science Press.Google Scholar
15.R Development Core Team (2007). R: A language and environment for statistical computing. R Foundation for Statistical Computing. Available from http://www.R-project.org.Google Scholar
16.Stadje, W. & Parthasarathy, P.R. (1999). On the convergence to stationarity of the many-server Poisson queue. Journal of Applied Probability 36: 546–557.Google Scholar
17.Zipf, G.K. (1949). Human behaviour and the principle of the least effort. Reading, MA: Addison Wesley.Google Scholar
18.Zörnig, P. & Altmann, G. (1995). Unified representations of Zipf's distributions. Computational Statistics & Data Analysis 19: 461–473.Google Scholar
19.Jones, W.B. & Thron, W.J. (1980). Continued fractions. Analytic theory and applications. Encyclopedia of Mathematics and its Applications, 11. Reading MA: Addison-Wesley Publishing Co.Google Scholar
20.Rolski, T., Schmidli, H., Schmidt, V. & Teugels, J. (1999). Stochastic processes for insurance and finance. Wiley Series in Probability and Statistics. New York, NY: John Wiley & Sons, Ltd.CrossRefGoogle Scholar