Theory of Probability and Mathematical Statistics

Author(s): D. B. Andreev; M. A. Elesin; E. A. Krylov; A. V. Kuznetsov; A. I. Zeifman
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 68 (2003).
Journal: Theor. Probability and Math. Statist. No. 68 (2004), 1-10.
MSC (2000): Primary 60J27, 60J80
Posted: May 11, 2004
Retrieve article in: PDF DVI PostScript

Abstract: We study stability and ergodicity of a special class of nonhomogeneous birth-death processes and consider applications of estimates for queue-length process for

and

queues.

1.: D. B. Andreev, M. A. Elesin, E. A. Krylov, A. V. Kuznetsov, and A. I. Zeifman, On ergodicity and stability estimates for some nonhomogeneous Markov chains, J. Math. Sci. 112 (2002), 4111-4118. MR 2003m:60204
2.: J. R. Artalejo and M. J. Lopez-Herrero, Analysis of the busy period for the queue: an algorithmic approach, J. Appl. Prob. 38 (2001), 209-222. MR 2001k:60130
3.: J.-D. Deuschel and C. Mazza, convergence of time nonhomogeneous Markov processes: I. Spectral estimates, Ann. Appl. Prob. 4 (1994), no. 4, 1012-1056. MR 96b:60188
4.: A. Di Crescenzo and A. G. Nobile, Diffusion approximation to a queueing system with time dependent arrival and service rates, QUESTA 19 (1995), 41-62. MR 96b:60238
5.: Yu. L. Daletski ${\u{\i}}\kern.15em$ and M. G. Krein, Stability of Solutions of Differential Equations in Banach Spaces, ``Nauka'', Moscow, 1970; English transl., Amer. Math. Soc., Providence, RI, 1974. MR 50:5125; MR 50:5126
6.: van E. Doorn, Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process, Adv. Appl. Prob. 17 (1985), 504-530. MR 86m:60183
7.: C. Fricker, P. Robert, and D. Tibi, On the rate of convergence of Erlang's model, J. Appl. Prob. 36 (1999), 1167-1184. MR 2000k:60186
8.: B. V. Gnedenko and I. P. Makarov, Properties of solutions of a problem with losses in the case of periodic intensities, Differentsial'nye Uravneniya 7 (1971), 1696-1698. (Russian) MR 45:2254
9.: B. V. Gnedenko and A. D. Solov'ev, On conditions of the existence of final probabilities for a Markov process, Math. Operationsforsch. Statist. 4 (1973), 379-390. (Russian) MR 52:15680
10.: B. V. Gnedenko, On a generalization of Erlang's formulae, Zastos. Mat. 12 (1971), 239-242. (Russian) MR 47:4351
11.: V. Giorno and A. Nobile, On some time-nonhomogeneous diffusion approximations to queueing systems, Adv. Appl. Prob. 19 (1987), 974-994. MR 89a:60218
12.: B. L. Granovsky and A. I. Zeifman, Nonstationary Markovian queues, J. Math. Sci. 99 (2000), no. 4, 1415-1438. MR 2001g:60206
13.: L. Green, P. Kolesar, and A. Svornos, Some effects of nonstationarity on multiserver Markovian queueing systems, Oper. Res. 39 (1991), 502-511.
14.: D. P. Heyman and W. Whitt, The asymptotic behaviour of queues with time-varying arrival rates, J. Appl. Prob. 21 (1984), 143-156. MR 85c:60158
15.: V. V. Kalashnikov, Qualitative analysis of complex systems behavior by the test functions method, ``Nauka'', Moscow, 1978. (Russian) MR 82b:90041
16.: N. V. Kartashov, Strong stable Markov chains, Problems of the Stability for Stochastic Models, VNIISI, Moscow, 1981, pp. 54-59; English transl., J. Soviet Math. 34 (1986), 1493-1498. MR 84b:60089
17.: J. B. Keller, Time-dependent queues, SIAM Rev. 24 (1982), 401-412. MR 85c:60160
18.: M. Kijima, On the largest negative eigenvalue of the infinitesimal generator associated with queues, Oper. Res. Let. 9 (1990), 59-64. MR 91f:60171
19.: A. Mandelbaum and W. Massey, Strong approximations for time-dependent queues, Math. Oper. Res. 20 (1995), 33-64. MR 96b:60240
20.: W. A. Massey and W. Whitt, On analysis of the modified offered-load approximation for the nonstationary Erlang loss model, Ann. Appl. Prob. 4 (1994), 1145-1160. MR 95m:60147
21.: M. H. Rothkopf and S. S. Oren, A closure approximation for the nonstationary queue, Management Sci. 25 (1979), 522-534. MR 81c:60101
22.: W. Stadie and P. R. Parthasarathy, On the convergence to stationarity of the many-server Poisson queue, J. Appl. Prob. 36 (1999), 546-557. MR 2000i:60109
23.: W. Stadie and P. R. Parthasarathy, Generating function analysis of some joint distributions for Poisson loss systems, QUESTA 34 (2000), 183-197. MR 2001g:60238
24.: M. Voit, A note of the rate of convergence to equilibrium for Erlang's model in the subcritical case, J. Appl. Prob. 37 (2000), 918-923. MR 2001e:60183
25.: A. I. Zeifman, Stability for continuous-time nonhomogeneous Markov chains, Lect. Notes Math. 1155 (1985), 401-414. MR 87h:60136
26.: A. I. Zeifman, Qualitative properties of nonhomogeneous birth-death processes, Problems of the Stability for Stochastic Models, VNIISI, Moscow, 1988, pp. 32-40; English transl., J. Soviet Math. 57 (1991), 3217-3224. MR 92b:60084
27.: A. I. Zeifman, Properties of a loss system in the case of variable rate, Avtomat. i Telemekh. 1 (1989), 107-113; English transl., Automat. Remote Control 50 (1989), 82-87. MR 90g:60088
28.: A. I. Zeifman, Some estimates of the rate of convergence for birth and death processes, J. Appl. Prob. 28 (1991), 268-277. MR 92f:60147
29.: A. I. Zeifman, Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes, Stoch. Proc. Appl. 59 (1995), 157-173. MR 96g:60109
30.: A. I. Zeifman and D. Isaacson, On strong ergodicity for nonhomogeneous continuous-time Markov chains, Stoch. Proc. Appl. 50 (1994), 263-273. MR 95c:60068

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60J27, 60J80

D. B. Andreev
Affiliation: Vologda State Pedagogical University, Vologda, Russia

M. A. Elesin
Affiliation: Vologda State Pedagogical University, Vologda, Russia

E. A. Krylov
Affiliation: Vologda State Pedagogical University, Vologda, Russia

A. V. Kuznetsov
Affiliation: Vologda State Pedagogical University, Vologda, Russia

A. I. Zeifman
Affiliation: Vologda State Pedagogical University, Vologda, Russia
Address at time of publication: Vologda Scientific Coordinate Centre of Central Economics and Mathematics Institute, Russian Academy of Sciences, Vologda, Russia
Email: zai@uni-vologda.ac.ru

DOI: 10.1090/S0094-9000-04-00594-0
PII: S 0094-9000(04)00594-0
Received by editor(s): 4/APR/2002
Posted: May 11, 2004
Copyright of article: Copyright 2004, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN: 1547-7363(e) 0094-9000(p)

Previous number \| Table of contents \| Next number Recently posted articles \| Most recent number \| All numbers Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS