Continuity theorems for the M/ M/1/ n queueing system Authors
First Online: 22 July 2008 Received: 17 January 2008 Revised: 28 June 2008 DOI:
Cite this article as: Abramov, V.M. Queueing Syst (2008) 59: 63. doi:10.1007/s11134-008-9076-7 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
Dedicated to Vladimir Mikhailovich Zolotarev, Victor Makarovich Kruglov, and to the memory of Vladimir Vyacheslavovich Kalashnikov.
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Abramov, V.M.: Investigation of a Queueing System with Service Depending on Queue Length. Donish, Dushanbe (1991a) (in Russian)
Abramov, V.M.: Asymptotic properties of lost customers for one queueing system with refusals. Kibernetika (Ukr. Acad. Sci.)
2, 123–124 (1991b) (in Russian)
Abramov, V.M.: On the asymptotic distribution of the maximum number of infectives in epidemic models with immigration. J. Appl. Probab.
, 606–613 (1994)
Abramov, V.M.: On a property of a refusals stream. J. Appl. Probab.
, 800–805 (1997)
Abramov, V.M.: Inequalities for the
loss system. J. Appl. Probab.
, 232–234 (2001a)
Abramov, V.M.: Some results for large closed queueing networks with and without bottleneck: Up- and down-crossings approach. Queueing Syst.
, 149–184 (2001b)
Abramov, V.M.: On losses in
queues. J. Appl. Probab.
, 1079–1080 (2001c)
Abramov, V.M.: Asymptotic analysis of the
loss system as
increases to infinity. Ann. Oper. Res.
, 35–41 (2002)
Abramov, V.M.: Asymptotic behavior of the number of lost messages. SIAM J. Appl. Math.
, 746–761 (2004)
Abramov, V.M.: Stochastic inequalities for single-server loss queueing systems. Stoch. Anal. Appl.
, 1205–1221 (2006)
Abramov, V.M.: Optimal control of a large dam. J. Appl. Probab.
, 249–258 (2007a)
Abramov, V.M.: Optimal control of a large dam, taking into account the water costs.
Azlarov, T.A., Volodin, N.A.: Characterization Problems Associated with Exponential Distribution. Springer, Berlin (1986)
Choi, B.D., Kim, B., Wee, I.-S.: Asymptotic behavior of loss probability in
tends to infinity. Queueing Syst.
, 437–442 (2000)
Ciesielski, K.C.: Set Theory for the Working Mathematician. Cambridge University Press, London (1997)
Cooper, R.B., Tilt, B.: On the relationship between the distribution of maximal queue-length in the
/1 queue and the mean busy period in the
queue. J. Appl. Probab.
, 195–199 (1976)
Cooper, R.B., Niu, S.-C., Srinivasan, M.M.: Some reflections of the renewal theory. Paradox in queueing theory. J. Appl. Math. Stoch. Anal.
, 355–368 (1998)
Daley, D.J.: Queueing output processes. Adv. Appl. Probab.
, 395–415 (1976)
Dudley, R.: (1976). Probability and Metrics. Lecture Notes, vol. 45, Aarhus University
Feller, W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, New York (1971)
Gakis, K.G., Sivazlian, B.D.: A generalization of the inspection paradox in an ordinary renewal process. Stoch. Anal. Appl.
, 43–48 (1993)
Gordienko, E.I., Ruiz de Chávez, J.: New estimates for continuity in
/1/∞ queues. Queueing Syst.
, 175–188 (1998)
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)
Herff, W., Jochems, B., Kamps, U.: The inspection paradox with random time. Stat. Pap.
, 103–110 (1997)
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)
Kalashnikov, V.V., Rachev, S.T.: Mathematical Methods for Constructing of Queueing Models. Wadsworth and Brooks, Cole (1990)
Kennedy, D.P.: The continuity of single-server queues. J. Appl. Probab.
, 370–381 (1972)
Kremers, W.: An extension and implications of the inspection paradox. Stat. Probab. Lett.
, 269–273 (1988)
Peköz, E.A., Righter, R., Xia, C.H.: Characterizing losses in finite buffer systems. J. Appl. Probab.
, 242–249 (2003)
Rachev, S.T.: Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester (1991)
Resnick, S.I.: Adventures in Stochastic Processes. Birkhäuser, Boston (1992)
Righter, R.: A note on losses in the
queue. J. Appl. Probab.
, 1240–1244 (1999)
Ross, S.M.: The inspection paradox. Probab. Eng. Inf. Sci.
17, 47–51 (2003)
Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983)
Tomko, J.: One limit theorem in queueing problem as input rate increases infinitely. Stud. Sci. Math. Hung.
2, 447–454 (1967) (in Russian)
Whitt, W.: The continuity of queues. Adv. Appl. Probab.
, 175–183 (1974)
Wolff, R.W.: Losses per cycle in a single server queue. J. Appl. Probab.
, 905–909 (2002)
Zolotarev, V.M.: On stochastic continuity of queueing systems of type
/1. Theor. Probab. Appl.
, 250–269 (1976)
Zolotarev, V.M.: Quantitative estimates of continuity of queueing systems of type
/∞. Theor. Probab. Appl.
, 679–691 (1977)
Zolotarev, V.M.: Probability metrics. Theor. Probab. Appl.
, 278–302 (1983)
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