We consider a multidimensional inhomogeneous birth-death process and obtain bounds for the probabilities of the corresponding one-dimensional processes.
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Ju.L. Daleckij and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, AMS, Providence (1974).
G. Fayolle, P. King, and F. Mitrani, “The solution of certain two-dimensional Markov models,” Adv. Appl. Prob., 14, 295–308 (1982).
B. Granovsky and A. Zeifman, “Nonstationary queues: estimation of the rate of convergence,” Queueing Syst., 46, 363–388 (2004).
M. Jonckheere and S. Shneer, “Stability of multi-dimensional birth-and-death processes with statedependent 0-homogeneous jumps,” Adv. Appl. Probab., 46, No. 1, 59–75 (2014).
J. Keilson and L.D. Servi, “The matrix M/M/∞ system: retrial models and Markov modulated sources,” Adv. Appl. Prob., 25, 453–471 (1993)
G. S. Tsitsiashvili, M. A. Osipova, N. V. Koliev, and D. Baum, “A product theorem for Markov chains with application to PF-queueing networks,” Ann. Oper. Res., 113, No. 1–4, 141–154 (2002).
A.I. Zeifman, “On the estimation of probabilities for birth and death processes,” J. Appl. Probab., 32, 623–634 (1995).
A. I. Zeifman, “Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes,” Stoch. Proc. Appl., 59, 157–173 (1995).
A. Zeifman, S. Leorato, E. Orsingher, Ya. Satin, and G. Shilova, “Some universal limits for nonhomogeneous birth and death processes,” Queueing Syst., 52, 139–151 (2006).
A. I. Zeifman, V. E. Bening, and I. A. Sokolov, Markov Chains and Models in Continuous Time, Elex-KM, Moscow (2008).
A. Zeifman and A. Korotysheva, “Perturbation bounds for mt/mt/n queue with catastrophes,” Stoch. Mod., 28, No. 1, 49–62 (2012).
A. I. Zeifman and V. Y. Korolev, “On perturbation bounds for continuous-time Markov chains,” Stat. Probab. Let., 88, 66–72 (2014).
A. Zeifman, Y. Satin, V. Korolev, and S. Shorgin, “On truncations for weakly ergodic inhomogeneous birth and death processes,” Int. J. Appl. Math. Comput. Sci., 24, 503–518 (2014).
A. Zeifman, A. Sipin, V. Korolev, and V. Bening, “Estimates of some characteristics of multidimensional birth-and-death processes,” Dokl. Math., 92, 695–697 (2015).
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* This work was supported by the Russian Foundation for Basic Research, projects No. 13–07–00223, 14–07–00041, 15–01–01698, 15–07–02341; and by Ministry of Education and Science.
Proceedings of the XXXII International Seminar on Stability Problems for Stochastic Models, Trondheim, Norway, June 16–21, 2014.
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Zeifman, A.I., Sipin, A.S., Korotysheva, A.V. et al. Estimation of Probabilities for Multidimensional Birth-Death Processes. J Math Sci 218, 238–244 (2016). https://doi.org/10.1007/s10958-016-3025-5
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DOI: https://doi.org/10.1007/s10958-016-3025-5