Queueing Systems

, Volume 59, Issue 1, pp 63–86 | Cite as

Continuity theorems for the M/M/1/n queueing system

Article

Abstract

In this paper continuity theorems are established for the number of losses during a busy period of the M/M/1/n queue. We consider an M/GI/1/n queueing system where the service time probability distribution, slightly different in a certain sense from the exponential distribution, is approximated by that exponential distribution. Continuity theorems are obtained in the form of one or two-sided stochastic inequalities. The paper shows how the bounds of these inequalities are changed if further assumptions, associated with specific properties of the service time distribution (precisely described in the paper), are made. Specifically, some parametric families of service time distributions are discussed, and the paper establishes uniform estimates (given for all possible values of the parameter) and local estimates (where the parameter is fixed and takes only the given value). The analysis of the paper is based on the level crossing approach and some characterization properties of the exponential distribution.

Keywords

Continuity theorems Loss systems M/GI/1/n and M/M/1/n queues Busy period Branching process Number of level crossings Kolmogorov (uniform) metric Stochastic ordering Stochastic inequalities 

Mathematics Subject Classification (2000)

60K25 60B05 60E15 62E17 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramov, V.M.: Investigation of a Queueing System with Service Depending on Queue Length. Donish, Dushanbe (1991a) (in Russian) Google Scholar
  2. 2.
    Abramov, V.M.: Asymptotic properties of lost customers for one queueing system with refusals. Kibernetika (Ukr. Acad. Sci.) 2, 123–124 (1991b) (in Russian) Google Scholar
  3. 3.
    Abramov, V.M.: On the asymptotic distribution of the maximum number of infectives in epidemic models with immigration. J. Appl. Probab. 31, 606–613 (1994) CrossRefGoogle Scholar
  4. 4.
    Abramov, V.M.: On a property of a refusals stream. J. Appl. Probab. 34, 800–805 (1997) CrossRefGoogle Scholar
  5. 5.
    Abramov, V.M.: Inequalities for the GI/M/1/n loss system. J. Appl. Probab. 38, 232–234 (2001a) CrossRefGoogle Scholar
  6. 6.
    Abramov, V.M.: Some results for large closed queueing networks with and without bottleneck: Up- and down-crossings approach. Queueing Syst. 38, 149–184 (2001b) CrossRefGoogle Scholar
  7. 7.
    Abramov, V.M.: On losses in M X/GI/1/n queues. J. Appl. Probab. 38, 1079–1080 (2001c) CrossRefGoogle Scholar
  8. 8.
    Abramov, V.M.: Asymptotic analysis of the GI/M/1/n loss system as n increases to infinity. Ann. Oper. Res. 112, 35–41 (2002) CrossRefGoogle Scholar
  9. 9.
    Abramov, V.M.: Asymptotic behavior of the number of lost messages. SIAM J. Appl. Math. 64, 746–761 (2004) CrossRefGoogle Scholar
  10. 10.
    Abramov, V.M.: Stochastic inequalities for single-server loss queueing systems. Stoch. Anal. Appl. 24, 1205–1221 (2006) CrossRefGoogle Scholar
  11. 11.
    Abramov, V.M.: Optimal control of a large dam. J. Appl. Probab. 44, 249–258 (2007a) CrossRefGoogle Scholar
  12. 12.
    Abramov, V.M.: Optimal control of a large dam, taking into account the water costs. arXiv:math/0701458 (2007b)
  13. 13.
    Azlarov, T.A., Volodin, N.A.: Characterization Problems Associated with Exponential Distribution. Springer, Berlin (1986) Google Scholar
  14. 14.
    Choi, B.D., Kim, B., Wee, I.-S.: Asymptotic behavior of loss probability in GI/M/1/K queue as K tends to infinity. Queueing Syst. 36, 437–442 (2000) CrossRefGoogle Scholar
  15. 15.
    Ciesielski, K.C.: Set Theory for the Working Mathematician. Cambridge University Press, London (1997) Google Scholar
  16. 16.
    Cooper, R.B., Tilt, B.: On the relationship between the distribution of maximal queue-length in the M/G/1 queue and the mean busy period in the M/G/1/n queue. J. Appl. Probab. 13, 195–199 (1976) CrossRefGoogle Scholar
  17. 17.
    Cooper, R.B., Niu, S.-C., Srinivasan, M.M.: Some reflections of the renewal theory. Paradox in queueing theory. J. Appl. Math. Stoch. Anal. 11, 355–368 (1998) CrossRefGoogle Scholar
  18. 18.
    Daley, D.J.: Queueing output processes. Adv. Appl. Probab. 8, 395–415 (1976) CrossRefGoogle Scholar
  19. 19.
    Dudley, R.: (1976). Probability and Metrics. Lecture Notes, vol. 45, Aarhus University Google Scholar
  20. 20.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971) Google Scholar
  21. 21.
    Gakis, K.G., Sivazlian, B.D.: A generalization of the inspection paradox in an ordinary renewal process. Stoch. Anal. Appl. 11, 43–48 (1993) CrossRefGoogle Scholar
  22. 22.
    Gordienko, E.I., Ruiz de Chávez, J.: New estimates for continuity in M/GI/1/∞ queues. Queueing Syst. 29, 175–188 (1998) CrossRefGoogle Scholar
  23. 23.
    Gordienko, E.I., Ruiz de Chávez, J.: A note on continuity of M/G/1 queues. Int. J. Pure Appl. Math. 18, 535–539 (2005) Google Scholar
  24. 24.
    Herff, W., Jochems, B., Kamps, U.: The inspection paradox with random time. Stat. Pap. 38, 103–110 (1997) CrossRefGoogle Scholar
  25. 25.
    Kalashnikov, V.V.: The analysis of continuity of queueing systems. In: Itô, K., Prokhorov, Yu.V. (eds.) Probability Theory and Mathematical Statistics. Lecture Notes in Mathematics, vol. 1021, pp. 268–278. Springer, New York (1983) CrossRefGoogle Scholar
  26. 26.
    Kalashnikov, V.V., Rachev, S.T.: Mathematical Methods for Constructing of Queueing Models. Wadsworth and Brooks, Cole (1990) Google Scholar
  27. 27.
    Kennedy, D.P.: The continuity of single-server queues. J. Appl. Probab. 9, 370–381 (1972) CrossRefGoogle Scholar
  28. 28.
    Kremers, W.: An extension and implications of the inspection paradox. Stat. Probab. Lett. 6, 269–273 (1988) CrossRefGoogle Scholar
  29. 29.
    Peköz, E.A., Righter, R., Xia, C.H.: Characterizing losses in finite buffer systems. J. Appl. Probab. 40, 242–249 (2003) CrossRefGoogle Scholar
  30. 30.
    Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester (1991) Google Scholar
  31. 31.
    Resnick, S.I.: Adventures in Stochastic Processes. Birkhäuser, Boston (1992) Google Scholar
  32. 32.
    Righter, R.: A note on losses in the M/GI/1/n queue. J. Appl. Probab. 36, 1240–1244 (1999) CrossRefGoogle Scholar
  33. 33.
    Ross, S.M.: The inspection paradox. Probab. Eng. Inf. Sci. 17, 47–51 (2003) Google Scholar
  34. 34.
    Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983) Google Scholar
  35. 35.
    Tomko, J.: One limit theorem in queueing problem as input rate increases infinitely. Stud. Sci. Math. Hung. 2, 447–454 (1967) (in Russian) Google Scholar
  36. 36.
    Whitt, W.: The continuity of queues. Adv. Appl. Probab. 6, 175–183 (1974) CrossRefGoogle Scholar
  37. 37.
    Wolff, R.W.: Losses per cycle in a single server queue. J. Appl. Probab. 39, 905–909 (2002) CrossRefGoogle Scholar
  38. 38.
    Zolotarev, V.M.: On stochastic continuity of queueing systems of type G/G/1. Theor. Probab. Appl. 21, 250–269 (1976) CrossRefGoogle Scholar
  39. 39.
    Zolotarev, V.M.: Quantitative estimates of continuity of queueing systems of type G/G/∞. Theor. Probab. Appl. 22, 679–691 (1977) CrossRefGoogle Scholar
  40. 40.
    Zolotarev, V.M.: Probability metrics. Theor. Probab. Appl. 28, 278–302 (1983) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia

Personalised recommendations