Abstract
First we define a regenerative flow and describe its properties. Then a single-server queueing system with regenerative input flow and an unreliable server are considered. By applying coupling we establish the ergodicity condition and prove the limit theorem in the heavy traffic situation (traffic coefficient \(\rho <1, \rho \uparrow 1\)). The asymptotic analysis of the super-heavy traffic situation (\(\rho \ge 1\)) is also realized.
Similar content being viewed by others
References
Afanasyeva, L.G.: Queueing systems with cyclic control processes. Cybern. Syst. Anal. 41(1), 43–55 (2005)
Afanasyeva, L.G., Bashtova, E.E.: Limit theorems for queuing systems with doubly stochastic poisson arrivals (heavy traffic conditions). Probl. Inf. Transm. 44(4), 352–369 (2008)
Afanasyeva, L.G., Bashtova, E.E.: Limit theorems for queues in heavy traffic situation. Mod. Probl. Math. Mech. IV, 1 140–54 (in Russian) (2009)
Afanasyeva, L., Bashtova, E., Bulinskaya, E.: Limit theorems for semi-Markov queues and their applications. Commun. Stat. Simul. Comput. 41(6), 688–709 (2012)
Afanasyeva, L.G., Bulinskaya, E.V.: Stochasic models of transport flows. Commun. Stat. Theory Methods 40(16), 2830–2846 (2011)
Afanasyeva, L.G., Rudenko, I.V.: Queueing systems \(GI|G|\infty \) and there applications to transport models analysis. Theory Probab. Appl. 57(3), 427–452 (2012)
Asmussen, S.: Applied Probability and Queues. Wiley, Chichester (1987)
Bikjalis, A.: Estimates of the remainder term in the central limit theorem. (Russian). Litovsk Mat. Sb. 6, 323–346 (1966)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
Borovkov, A.A.: Some limit theorems in the theory of mass service, II. Theory Probab. Appl. 10, 375–400 (1965)
Borovkov, A.A.: Stochastic Processes in Queuing Theory. Springer, Berlin (1976)
Borovkov, A.A.: Asymptotic Methods in Queuing Theory. Wiley, Chichester (1984)
Cox, D.R.: Renewal Theory, London: Methuen and Co LTD. Wiley, New York (1962)
Djellab, N.V.: On the \(M|G|1\) retrial queue subjected to breakdowns RAIRO. Oper. Res. 36, 299–310 (2002)
Gaver, D.P.: A waiting line with interrupted service, including priorities. J. R. Soc. B24, 73–90 (1962)
Foss, S., Kovalevskii, A.: A stability criterion via fluid limits and its application to a polling model. Queueing Syst. 32, 131–168 (1999)
Foss, S.G., Kalashnikov, V.V.: Regeneration and renovation in queues. Queueing Syst. 8(3), 211–223 (1991)
Foster, F.G.: On the stochastic matrices associated with queueing processes. Ann. Math. Stat. 24(3), 355–360 (1953)
Gideon, R., Pyke, R.: Markov Renewal Modeling of Poisson Traffic at Intersection having Separate Turn Lanes. Semi-Markov Models and Applications. Kluwek Academic Publishers, Dordrecht (1999)
Grandell, J.: Double Stochastic Poisson Processes. Lect. Notes Math., vol. 529. Springer, Berlin (1976)
Haight, F.A.: Mathematical Theories of Traffic Flow, Mathematical in Science and Eng., vol. 7. Academic Press, New York (1963)
Iglehart, D.L., Witt, L.W.: Multipple cannel queues in heavy traffic, I and II. Adv. Appl. Prob. 2(150–177), 355–369 (1970)
Lindvall, T.: The probabilistic proof of Blackwell’s renewal theorem. Ann. Probab. 5(3), 482–485 (1977)
Lindvall, T.: Lectures on the Coupling Method. J. Wiley, New York (1992)
Loynes, R.: The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58(3), 497–520 (1962)
Malyshev, V.A., Men’shikov, M.V.: Ergodicity continuity and analyticity of countable Markov chains. Trans. Moscow Math. 1, 1–48 (1982)
Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, New York (1994)
Morozov, E.: The stability of non-homogeneous queueing system with regenerative input. J. Math. Sci. 89, 407–421 (1997)
Mousstafa, M.D.: Input-output Markov processes. Proc. Koninkijke Nederlands Akad. Wetenschappen 60, 112–118 (1957)
Serfozo, R.: Applications of the key renewal theorem: crudely regenerative process. J. Appl. Probab. 29, 384–395 (1992)
Sherman, N., Kharoufen, J., Abramson, M.: An \(M|G|1\) retrial queue with unreliable server for streaming multimedia applications. Prob. Eng. Inf. Sci. 23, 281–304 (2009)
Smith, W.L.: Renewal theory and its ramifications. J. R. Stat. Soc. B,20, N2. (1958)
Tanner, J.C.: The delay to pedestrians crossing a road. Biometrika 38, 383–392 (1951)
Thorisson, H.: Coupling, Stationary and Regeneration. Springer, New York (2000)
Whitt, W.: Heavy Traffic Limit Theorems for Queues: A Survey. Lecture Notes in Economics and Math. Systems, vol. 98. Springer, Berlin (1974)
Acknowledgments
This work is partially supported by RFBR-grant 13-01-00653.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Afanasyeva, L.G., Bashtova, E.E. Coupling method for asymptotic analysis of queues with regenerative input and unreliable server. Queueing Syst 76, 125–147 (2014). https://doi.org/10.1007/s11134-013-9370-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-013-9370-x