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CONDITIONS FOR RECURRENCE AND TRANSIENCE FOR TIME-INHOMOGENEOUS BIRTH-AND-DEATH PROCESSES

Published online by Cambridge University Press:  23 June 2023

VYACHESLAV M. ABRAMOV*
Affiliation:
24 Sagan Drive, Cranbourne North, Victoria 3977, Australia

Abstract

We derive conditions for recurrence and transience for time-inhomogeneous birth-and-death processes considered as random walks with positively biased drifts. We establish a general result, from which the earlier known particular results by Menshikov and Volkov [‘Urn-related random walk with drift $\rho x^\alpha /t^\beta $’, Electron. J. Probab. 13 (2008), 944–960] follow.

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Abramov, V. M., ‘A large closed queueing network with autonomous service and bottleneck’, Queueing Syst. 35(1–2) (2000), 2354.CrossRefGoogle Scholar
Abramov, V. M., ‘A large closed queueing network containing two types of node and multiple customer classes: one bottleneck station’, Queueing Syst. 48(1–2) (2004), 4573.CrossRefGoogle Scholar
Abramov, V. M., ‘Analysis of multiserver retrial queueing system: a martingale approach and an algorithm of solution’, Ann. Oper. Res. 141 (2006), 1950.CrossRefGoogle Scholar
Abramov, V. M., ‘Conservative and semiconservative random walks: recurrence and transience’, J. Theoret. Probab. 31 (2018), 19001922; Correction, J. Theoret. Probab. 32 (2019), 541–543.CrossRefGoogle Scholar
Abramov, V. M., ‘Extension of the Bertrand–De Morgan test and its application’, Amer. Math. Monthly 127(5) (2020), 444448.CrossRefGoogle Scholar
Abramov, V. M., ‘Necessary and sufficient conditions for the convergence of positive series’, J. Class. Anal. 19 (2022), 117125.CrossRefGoogle Scholar
Abramov, V. M., ‘Crossings states and sets of states in random walks’, Methodol. Comput. Appl. Probab. 25(1) (2023), Article no. 28.CrossRefGoogle Scholar
Feller, W., An Introduction to Probability Theory and Its Applications, Vol. 2 (John Wiley, New York, 1971).Google Scholar
Jacod, J. and Shiryaev, A. N., Limit Theorems for Stochastic Processes, 2nd edn (Springer, New York, 2003).CrossRefGoogle Scholar
Karlin, S. and McGregor, J., ‘The classification of the birth-and-death processes’, Trans. Amer. Math. Soc. 86(2) (1957), 366400.CrossRefGoogle Scholar
Kogan, Y. and Liptser, R. S., ‘Limit non-stationary behavior of large closed queueing networks with bottlenecks’, Queueing Syst. 14 (1993), 3355.CrossRefGoogle Scholar
Kogan, Y., Liptser, R. S. and Shenfild, M., ‘State dependent Benes buffer model with fast loading and output rates’, Ann. Appl. Probab. 5 (1995), 97120.CrossRefGoogle Scholar
Liptser, R. S. and Shiryaev, A. N., Theory of Martingales (Kluwer, Dordrecht, 1989).CrossRefGoogle Scholar
Menshikov, M. and Volkov, S., ‘Urn-related random walk with drift $\rho x^\alpha /t^\beta$ ’, Electron. J. Probab. 13 (2008), 944960.CrossRefGoogle Scholar
Protter, P. E., Stochastic Integration and Differential Equations, 2nd edn (Springer, New York, 2005).CrossRefGoogle Scholar