Advertisement

Asymptotic Analysis of Queueing Models Based on Synchronization Method

  • L. G. AfanasyevaEmail author
Article
  • 10 Downloads

Abstract

This paper is focused on the stability conditions for a multiserver queueing system with heterogeneous servers and a regenerative input flow X. The main idea is constructing an auxiliary service process Y which is also a regenerative flow and definition of the common points of regeneration for both processes X and Y. Then the traffic rate is defined in terms of the mean of the increments of these processes on a common regeneration period. It allows to use well-known results from the renewal theory to find the instability and stability conditions. The possibilities of the proposed approach are demonstrated by examples. We also present the applications to transport system capacity analysis.

Keywords

Regenerative flow Synchronization Stability condition Service discipline 

Mathematics Subject Classification (2010)

60K25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

Work is partially supported by RFBR grant 17-01-00468.

References

  1. Afanasyeva L (2005) Queueing systems with cyclic control sequences. Institute of Cybernetic. Ukraina, 41(1):54–69. [in Russian]Google Scholar
  2. Afanasyeva L, Bashtova E (2014) Coupling method for asymptotic analysis of queues with regenerative input and unreliable server. Queueing Systems 76(2):125–147CrossRefzbMATHGoogle Scholar
  3. Afanasyeva L, Bashtova E, Bulinskaya E (2012) Limit theorems for Semi-Markov queues and their applications. Communications in Statistics Part B: Simulation and Computation 41(6):688–709CrossRefzbMATHGoogle Scholar
  4. Afanasyeva L, Bulinskaya EV (2013) Asymptotic analysis of traffic lights performance under heavy-traffic assumption. Methodol Comput Appl Probab 15 (4):935–950CrossRefzbMATHGoogle Scholar
  5. Afanasyeva L, Rudenko IV (2012) G I|G| queueing systems and their applications to the analysis of traffic models. Theory of Probability Applications 57(3):427–452Google Scholar
  6. Afanasyeva L, Bulinskaya EV (2009) Some problems for the flow of interacting particles. Modern Problems of Mathematics and Mechanics 2:55–68Google Scholar
  7. Afanasyeva L, Bulinskaya EV (2010) Mathematical models of transport systems based on queueing systems methods. Proceedings of Moscow Institute of Physics and Technology 2(4):6–21Google Scholar
  8. Afanasyeva L, Bulinskaya EV (2011) Stochastic models of transport flows. Commun Stat Theory Methods 40(16):2830–2846CrossRefzbMATHGoogle Scholar
  9. Afanasyeva L, Mihaylova IV (2015) Two models of the highway intersected by a crosswalk. Survey of applied and industrial mathematics 22(5):520–532Google Scholar
  10. Afanasyeva L, Tkachenko A (2014) Multichannel queueing systems with regenerative input flow. Theory of Probability and Its Applications 58(2):174–192CrossRefzbMATHGoogle Scholar
  11. Afanasyeva L, Tkachenko A (2016) Stability analysis of multi-server discrete-time queueing systems wth interruptions and regenerative input flow. In: Manca R, McClean S, Skiadas C (eds) New trends in stochastic modeling and data analysis, pp 13–26Google Scholar
  12. Asmussen S (2003) Applied probability and queues. Springer, BerlinzbMATHGoogle Scholar
  13. Avi-Itzhak B, Naor P (1963) Some queueing problems with the service station subject to breakdown. Oper Res 11(3):303–320CrossRefzbMATHGoogle Scholar
  14. Avrachenkov K, Morozov E, Steyaert B (2016) Sufficient stability conditions for multiclass constant retrial rate systems. Queueing Systems 82:149–171CrossRefzbMATHGoogle Scholar
  15. Baycal-Gursoy M, Xiao W (2004) Stochastic decomposition in M|M| queues with Markov-modulated service rates. Queueing Systems 48:75–88CrossRefzbMATHGoogle Scholar
  16. Baycal-Gursoy M, Xiao W, Ozbay K (2009) Modeling traffic flow interrupted by incidents. Eur J Oper Res 195:127–138CrossRefzbMATHGoogle Scholar
  17. Belorusov T (2012) Ergodicity of a multichannel queueing system with balking. Theory of Probability and Its Applications 56(1):120–126CrossRefzbMATHGoogle Scholar
  18. Blank M (2003) Ergodic properties of a simple deterministic traffic flow model. J Stat Phys 111:903–930CrossRefzbMATHGoogle Scholar
  19. Borovkov AA (1976) Stochastic processes in queueing theory. Springer, BerlinCrossRefzbMATHGoogle Scholar
  20. Caceres FC, Ferrari PA, Pechersky E (2007) A slow to start traffic model related to a M|M|1 queue. J. Stat. Mech. PO7008, arXiv:cond-mat/0703709 v2 [cond-mat.stat-mech] 31 May 2007
  21. Chen H (1995) Fluid approximation and stability of multiclass queueing networks: work-conserving disciplines. Ann Appl Probab 5:637–665CrossRefzbMATHGoogle Scholar
  22. Chen H, Yao D (2001) Fundamentals of queueing networks. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. Chowdhury D (1999) Vehicular traffic: a system of interacting particles driven far from equilibrium. arXiv:cond-mat/9910173 v1 [cond-mat.stat-mech] 12 Oct 1999
  24. Dai J (1995) On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models. Ann Appl Probab 5:49–77CrossRefzbMATHGoogle Scholar
  25. Feller W (1966) An introduction to probability theory and its applications, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  26. Fiems D, Bruneel H (2013) Discrete-time queueing systems with Markovian preemptive vacations. Math Comput Model 57(3-4):782–792.  https://doi.org/10.1016/j.mcm.2012.09.003 CrossRefzbMATHGoogle Scholar
  27. Foss S, Konstantopoulos T (2004) An overview on some stochastic stability methods. J Oper Res Soc Jpn 47(4):275–303CrossRefzbMATHGoogle Scholar
  28. Fuks H, Boccara N (2001) Convergence to equilibrium in a class of interacting particle system evolving in descrete time. Phys Rev E 64:016117CrossRefGoogle Scholar
  29. Gaver D Jr (1962) A waiting line with interrupted service, including priorities. J R Stat Soc Ser B (Methodol) 24:73–90zbMATHGoogle Scholar
  30. Georgiadis L, Szpankowski W (1992) Stability of token passing rings. Queueing Systems 11:7–33CrossRefzbMATHGoogle Scholar
  31. Gideon R, Pyke R (1999) Markov renewal modeling of poisson traffic at intersections having separate turn lanes. In: Janssen J, Limneos N (eds) Semi-Markov models and applications. Springer, New York, pp 285–310Google Scholar
  32. Gillent F, Latouche G (1983) Semi-explicit solution for m|P H|1 - like queueing systems. Eur J Oper Res 13(2):151–160CrossRefzbMATHGoogle Scholar
  33. Grandell J (1976) Double stochastic poisson process. Lect Notes Math, vol 529. Springer, BerlinCrossRefGoogle Scholar
  34. Grinbeerg H (1959) An analysis of traffic flows. Oper Res 7:79–85CrossRefGoogle Scholar
  35. Greenshields BD (1935) A study if highway capacity. Proc Highway Res 14:448–477Google Scholar
  36. Helbing D (2001) Traffic and related self-driven many-particle systems. Rev Mod Phys 73:1067–1141CrossRefGoogle Scholar
  37. Inose H, Hamada T (1975) Road traffic control. University of Tokyo Press, TokyozbMATHGoogle Scholar
  38. Krishnamoorthy A, Pramod P, Chakravarthy S (2012) Queues with interruptions: a survey. TOP: 1–31.  https://doi.org/10.1007/s11750-012-0256-6
  39. Kiefer J, Wolfowitz J (1955) On the theory of queues with many servers. Trans Amer Math Soc 78:1–18CrossRefzbMATHGoogle Scholar
  40. Loynes RM (1962) The stability of a queue with non-independent inter-arrival and service times. Proc Cambr Phil Soc 58(3):497–520CrossRefzbMATHGoogle Scholar
  41. Maerivoet S, De Moor B (2005) Cellular automata models of road traffic. Phys Rep 419:1–64CrossRefGoogle Scholar
  42. Malyshev VA, Menshikov MV (1982) Ergodicity continuity and analyticity of countable Markov chains. Trans Moscow Math 1:1–48Google Scholar
  43. Meyn SP, Tweedie RL (2009) Markov chains and stochastic stability. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  44. Morozov E (1997) The stability of a non-homogeneous queueing system with regenerative input. J Math Sci 83(3):407–421CrossRefzbMATHGoogle Scholar
  45. Morozov E (2007) A multiserver retrial queue: regenerative stability analysis. Queueing Systems 56(3-4):157–168CrossRefzbMATHGoogle Scholar
  46. Morozov E, Dimitriou I (2017) Stability analysis of a multiclass retrial system with coupled orbit queues. In: Reinecke P, Di Marco A (eds) Computer performance engineering. EPEW 2017. Lecture notes in Computer Science. Springer, ChamGoogle Scholar
  47. Morozov E, Rumyantsev A (2016) Stability analysis of a M A P|m|s cluster model by matrix-analytic method. European Workshop on Computer Performance Engeneering: 63–76Google Scholar
  48. Morozov E, Fiems D, Bruneel H (2011) Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions. Perform Eval 68(12):1261–1275.  https://doi.org/10.2016/j.peva.2011.07.002 CrossRefGoogle Scholar
  49. Pechinkin A, Socolov I, Chaplygin V (2009) Multichannel queueing system with refusals of servers groups. Informatics and its Applications 3(3):4–15Google Scholar
  50. Rumyantsev A, Morozov E (2017) Stability criterion of a multi-server model with simultaneous service. Ann Oper Res 252(1):29–39CrossRefzbMATHGoogle Scholar
  51. Saaty TL (1961) Elements of queueing theory with applications. McGraw-Hill Book Company, Inc, New YorkzbMATHGoogle Scholar
  52. Sadowsky JS (1995) The probability of large queue lengths and waiting times in a heterogeneous multiserver queue: positive recurrence and logarithmic limits. Adv Appl Prob 27:567–583CrossRefzbMATHGoogle Scholar
  53. Schadschneider A (2000) Statistical physics of traffic flow. arXiv:cond-mat/0007418 v1 [cond-mat.stat-mech] 26 Jul 2000
  54. Smith W (1955) Regenerative stochastic processes. Proc R Soc Lond A Math Phys Sci 232(1188):6–31zbMATHGoogle Scholar
  55. Szpankowski W (1994) Stability conditions for some distributed systems. Buffered random access systems. Adv Appl Prob 26:498–515CrossRefzbMATHGoogle Scholar
  56. Thorisson H (2000) Coupling stationary and regeneration. Springer, New YorkCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Probability TheoryLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations