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

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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.

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References

  1. Abramov, V.M.: Investigation of a Queueing System with Service Depending on Queue Length. Donish, Dushanbe (1991a) (in Russian)

    Google Scholar 

  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. 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)

    Article  Google Scholar 

  4. Abramov, V.M.: On a property of a refusals stream. J. Appl. Probab. 34, 800–805 (1997)

    Article  Google Scholar 

  5. Abramov, V.M.: Inequalities for the GI/M/1/n loss system. J. Appl. Probab. 38, 232–234 (2001a)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  7. Abramov, V.M.: On losses in M X/GI/1/n queues. J. Appl. Probab. 38, 1079–1080 (2001c)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  9. Abramov, V.M.: Asymptotic behavior of the number of lost messages. SIAM J. Appl. Math. 64, 746–761 (2004)

    Article  Google Scholar 

  10. Abramov, V.M.: Stochastic inequalities for single-server loss queueing systems. Stoch. Anal. Appl. 24, 1205–1221 (2006)

    Article  Google Scholar 

  11. Abramov, V.M.: Optimal control of a large dam. J. Appl. Probab. 44, 249–258 (2007a)

    Article  Google Scholar 

  12. Abramov, V.M.: Optimal control of a large dam, taking into account the water costs. arXiv:math/0701458 (2007b)

  13. Azlarov, T.A., Volodin, N.A.: Characterization Problems Associated with Exponential Distribution. Springer, Berlin (1986)

    Google Scholar 

  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)

    Article  Google Scholar 

  15. Ciesielski, K.C.: Set Theory for the Working Mathematician. Cambridge University Press, London (1997)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  18. Daley, D.J.: Queueing output processes. Adv. Appl. Probab. 8, 395–415 (1976)

    Article  Google Scholar 

  19. Dudley, R.: (1976). Probability and Metrics. Lecture Notes, vol. 45, Aarhus University

  20. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)

    Google Scholar 

  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)

    Article  Google Scholar 

  22. Gordienko, E.I., Ruiz de Chávez, J.: New estimates for continuity in M/GI/1/∞ queues. Queueing Syst. 29, 175–188 (1998)

    Article  Google Scholar 

  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. Herff, W., Jochems, B., Kamps, U.: The inspection paradox with random time. Stat. Pap. 38, 103–110 (1997)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  26. Kalashnikov, V.V., Rachev, S.T.: Mathematical Methods for Constructing of Queueing Models. Wadsworth and Brooks, Cole (1990)

    Google Scholar 

  27. Kennedy, D.P.: The continuity of single-server queues. J. Appl. Probab. 9, 370–381 (1972)

    Article  Google Scholar 

  28. Kremers, W.: An extension and implications of the inspection paradox. Stat. Probab. Lett. 6, 269–273 (1988)

    Article  Google Scholar 

  29. Peköz, E.A., Righter, R., Xia, C.H.: Characterizing losses in finite buffer systems. J. Appl. Probab. 40, 242–249 (2003)

    Article  Google Scholar 

  30. Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester (1991)

    Google Scholar 

  31. Resnick, S.I.: Adventures in Stochastic Processes. Birkhäuser, Boston (1992)

    Google Scholar 

  32. Righter, R.: A note on losses in the M/GI/1/n queue. J. Appl. Probab. 36, 1240–1244 (1999)

    Article  Google Scholar 

  33. Ross, S.M.: The inspection paradox. Probab. Eng. Inf. Sci. 17, 47–51 (2003)

    Google Scholar 

  34. Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983)

    Google Scholar 

  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. Whitt, W.: The continuity of queues. Adv. Appl. Probab. 6, 175–183 (1974)

    Article  Google Scholar 

  37. Wolff, R.W.: Losses per cycle in a single server queue. J. Appl. Probab. 39, 905–909 (2002)

    Article  Google Scholar 

  38. Zolotarev, V.M.: On stochastic continuity of queueing systems of type G/G/1. Theor. Probab. Appl. 21, 250–269 (1976)

    Article  Google Scholar 

  39. Zolotarev, V.M.: Quantitative estimates of continuity of queueing systems of type G/G/∞. Theor. Probab. Appl. 22, 679–691 (1977)

    Article  Google Scholar 

  40. Zolotarev, V.M.: Probability metrics. Theor. Probab. Appl. 28, 278–302 (1983)

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Building 28M, Clayton Campus, Clayton, VIC, 3800, Australia

    Vyacheslav M. Abramov

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  1. Vyacheslav M. Abramov
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Correspondence to Vyacheslav M. Abramov.

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Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.

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Abramov, V.M. Continuity theorems for the M/M/1/n queueing system. Queueing Syst 59, 63–86 (2008). https://doi.org/10.1007/s11134-008-9076-7

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  • DOI: https://doi.org/10.1007/s11134-008-9076-7

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