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Birth-death processes with an instantaneous reflection barrier

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Anyue Chen  and
Kai Liu
Anyue Chen*
Affiliation:
University of Greenwich
Kai Liu*
Affiliation:
University of Liverpool
*
∗ Postal address: School of Computing and Mathematical Science, University of Greenwich, Maritime Greenwich Campus, Old Royal Naval College, Park Row, Greenwich, London SE10 9LS, UK. Email address: [email protected]
∗∗ Postal address: Division of Statistics and OR, Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool L69 7ZL, UK.
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Abstract

A new structure with the special property that an instantaneous reflection barrier is imposed on the ordinary birth—death processes is considered. An easy-checking criterion for the existence of such Markov processes is first obtained. The uniqueness criterion is then established. In the nonunique case, all the honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. It is proved that honest processes are always ergodic without necessarily imposing any extra conditions. Equilibrium distributions for all these ergodic processes are established. Several examples are provided to illustrate our results.

Keywords

Birth—death processunstable continuous-time Markov chainexistenceuniquenessergodicityequilibrium distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 163 - 179
Copyright
Copyright © Applied Probability Trust 2003 

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References

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, Berlin.CrossRefGoogle Scholar
Chen, A. Y., and Liu, K. (2003). Piecewise birth–death processes. To appear in Markov Process. Relat. Fields.Google Scholar
Chen, A. Y., and Renshaw, E. (1990). Markov branching processes with instantaneous immigration. Prob. Theory Relat. Fields 87, 209–240.CrossRefGoogle Scholar
Chen, A. Y., and Renshaw, E. (1993). Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Prob. Theory Relat. Fields 94, 427–456.CrossRefGoogle Scholar
Chen, A. Y., and Renshaw, E. (1995). Markov branching processes regulated by emigration and large immigration. Stoch. Process. Appl. 57, 339–359.CrossRefGoogle Scholar
Chen, A. Y., and Renshaw, E. (1997). The M/M/1 queue with mass exodus and mass arrivals when empty. J. Appl. Prob. 34, 192–207.CrossRefGoogle Scholar
Chen, A. Y., and Renshaw, E. (2000). Existence, recurrence and equilibrium properties of Markov branching processes with instantaneous immigration. Stoch. Process. Appl. 88, 177–193.CrossRefGoogle Scholar
Chen, M. F. (1992). From Markov Chains to Nonequilibrium Particle Systems. World Scientific, Singapore.CrossRefGoogle Scholar
Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities. Springer, New York.Google Scholar
Feller, W. (1959). The birth and death processes as diffusion processes. J. Math. Pures Appl. 38, 301–345.Google Scholar
Freedman, D. (1983). Approximating Countable Markov Chains. Springer, New York.CrossRefGoogle Scholar
Hou, Z. T., and Guo, Q. F. (1988). Homogeneous Denumerable Markov Processes. Springer, Berlin.Google Scholar
Kendall, D. G., and Reuter, G. E. H. (1954). Some pathological Markov processes with a denumerable infinity of states and the associated semigroups of operators on l . In Proc. Internat. Cong. Math. (2–9 September 1954, Amsterdam), Vol. 3, North-Holland, Amsterdam, pp. 377–415.Google Scholar
Kingman, J. F. C. (1972). Regenerative Phenomena. Wiley, New York.Google Scholar
Kolmogorov, A. N. (1951). On the differentiability of the transition probabilities in stationary Markov processes with a denumerable number of states. Moskov Gos. Univ. Uĉenye Zapiski Matematika 148, 53–59 (in Russian).Google Scholar
Parthasarathy, P. R., and Krishna, K. B. (1991). Density-dependent birth and death process with state-dependent immigration. Math. Comput. Modelling 15, 11–16.CrossRefGoogle Scholar
Reuter, G. E. H. (1959). Denumerable Markov processes. II. J. London Math. Soc. 34, 81–91.CrossRefGoogle Scholar
Reuter, G. E. H. (1962). Denumerable Markov processes. III. J. London Math. Soc. 37, 63–73.CrossRefGoogle Scholar
Reuter, G. E. H. (1969). Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitsth. 13, 315–320.CrossRefGoogle Scholar
Rogers, L. C. G., and Williams, D. (1986). Construction and approximation of transition matrix functions. Adv. Appl. Prob. Spec. Suppl. December 1986, 133–160.Google Scholar
Rogers, L. C. G., and Williams, D. (1987). Diffusions, Markov Processes and Martingales, Vol. 2. John Wiley, Chichester.Google Scholar
Rogers, L. C. G., and Williams, D. (1994). Diffusions, Markov Processes and Martingales, Vol. 1, 2nd edn. John Wiley, Chichester.Google Scholar
Van Doorn, E. (1981). Stochastic Monotonicity and Queueing Applications of Birth–Death Processes (Lecture Notes Statist. 4). Springer, Berlin.Google Scholar
Wang, Z. K., and Yang, X. Q. (1992). Birth and Death Processes and Markov Chains. Springer, Berlin.Google Scholar
Williams, D. (1967). A note on the Q-matrices of Markov chains. Z. Wahrscheinlichkeitsth. 7, 116–121.CrossRefGoogle Scholar
Williams, D. (1976). The Q-matrix problem. In Séminaire de Probabilités X (Lecture Notes Math. 511), ed. Meyer, P. A., Springer, Berlin, pp. 216–234.Google Scholar
Yamazato, M. (1975). Some results on continuous time branching processes with state-dependent immigration. J. Math. Soc. Japan 27, 479–496.CrossRefGoogle Scholar
Yang, X. Q. (1990). The Construction Theory of Denumerable Markov Processes. John Wiley, New York.Google Scholar