Ergodicity and stability of nonstationary queueing systems
Authors:
D. B. Andreev, M. A. Elesin, E. A. Krylov, A. V. Kuznetsov and A. I. Zeifman
Translated by:
The authors
Journal:
Theor. Probability and Math. Statist. 68 (2004), 1-10
MSC (2000):
Primary 60J27, 60J80
DOI:
https://doi.org/10.1090/S0094-9000-04-00594-0
Published electronically:
May 11, 2004
MathSciNet review:
2000389
Full-text PDF Free Access
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Additional Information
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 $M_t/M_t/S$ and $M_t/M_t/S/S$ queues.
References
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- J.-D. Deuschel and C. Mazza, $L^2$ convergence of time nonhomogeneous Markov processes: I. Spectral estimates, Ann. Appl. Prob. 4 (1994), no. 4, 1012–1056. MR 96b:60188
- 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
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- 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
- 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
- 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
- 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
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- B. L. Granovsky and A. I. Zeifman, Nonstationary Markovian queues, J. Math. Sci. 99 (2000), no. 4, 1415–1438. MR 2001g:60206
- L. Green, P. Kolesar, and A. Svornos, Some effects of nonstationarity on multiserver Markovian queueing systems, Oper. Res. 39 (1991), 502–511.
- 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
- V. V. Kalashnikov, Qualitative analysis of complex systems behavior by the test functions method, “Nauka”, Moscow, 1978. (Russian) MR 82b:90041
- 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
- J. B. Keller, Time-dependent queues, SIAM Rev. 24 (1982), 401–412. MR 85c:60160
- M. Kijima, On the largest negative eigenvalue of the infinitesimal generator associated with $M/M/n/n$ queues, Oper. Res. Let. 9 (1990), 59–64. MR 91f:60171
- A. Mandelbaum and W. Massey, Strong approximations for time-dependent queues, Math. Oper. Res. 20 (1995), 33–64. MR 96b:60240
- 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
- M. H. Rothkopf and S. S. Oren, A closure approximation for the nonstationary $M/M/s$ queue, Management Sci. 25 (1979), 522–534. MR 81c:60101
- 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
- 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
- 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
- A. I. Zeifman, Stability for continuous-time nonhomogeneous Markov chains, Lect. Notes Math. 1155 (1985), 401–414. MR 87h:60136
- 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
- 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
- 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
- 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
- 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
References
- 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
- J. R. Artalejo and M. J. Lopez-Herrero, Analysis of the busy period for the $M/M/c$ queue: an algorithmic approach, J. Appl. Prob. 38 (2001), 209–222. MR 2001k:60130
- J.-D. Deuschel and C. Mazza, $L^2$ convergence of time nonhomogeneous Markov processes: I. Spectral estimates, Ann. Appl. Prob. 4 (1994), no. 4, 1012–1056. MR 96b:60188
- 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
- Yu. L. Daletskiĭ 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
- 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
- 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
- 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
- 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
- B. V. Gnedenko, On a generalization of Erlang’s formulae, Zastos. Mat. 12 (1971), 239–242. (Russian) MR 47:4351
- V. Giorno and A. Nobile, On some time-nonhomogeneous diffusion approximations to queueing systems, Adv. Appl. Prob. 19 (1987), 974–994. MR 89a:60218
- B. L. Granovsky and A. I. Zeifman, Nonstationary Markovian queues, J. Math. Sci. 99 (2000), no. 4, 1415–1438. MR 2001g:60206
- L. Green, P. Kolesar, and A. Svornos, Some effects of nonstationarity on multiserver Markovian queueing systems, Oper. Res. 39 (1991), 502–511.
- 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
- V. V. Kalashnikov, Qualitative analysis of complex systems behavior by the test functions method, “Nauka”, Moscow, 1978. (Russian) MR 82b:90041
- 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
- J. B. Keller, Time-dependent queues, SIAM Rev. 24 (1982), 401–412. MR 85c:60160
- M. Kijima, On the largest negative eigenvalue of the infinitesimal generator associated with $M/M/n/n$ queues, Oper. Res. Let. 9 (1990), 59–64. MR 91f:60171
- A. Mandelbaum and W. Massey, Strong approximations for time-dependent queues, Math. Oper. Res. 20 (1995), 33–64. MR 96b:60240
- 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
- M. H. Rothkopf and S. S. Oren, A closure approximation for the nonstationary $M/M/s$ queue, Management Sci. 25 (1979), 522–534. MR 81c:60101
- 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
- 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
- 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
- A. I. Zeifman, Stability for continuous-time nonhomogeneous Markov chains, Lect. Notes Math. 1155 (1985), 401–414. MR 87h:60136
- 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
- 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
- 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
- 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
- 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
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Additional Information
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
Received by editor(s):
April 4, 2002
Published electronically:
May 11, 2004
Article copyright:
© Copyright 2004
American Mathematical Society