×

zbMATH — the first resource for mathematics

On busy period and sojourn time distributions in the M/G/1-EPS queue with catastrophes. (English. Russian original) Zbl 1180.90083
Autom. Remote Control 70, No. 12, 2061-2072 (2009); translation from Avtom. Telemekh. 2009, No. 12, 134-146 (2009).
Summary: We derive the Laplace-Stiltjes transforms of busy period and sojourn time distributions in the M/G/1 queue with egalitarian processor sharing (EPS) and the possibility of (external) catastrophes. Each arrival of the catastrophes immediately removes all the positive jobs (and hence unfinished work) in this non work-conserving queueing system. One of the main results is obtained by means of the so-called method of decomposition into delay elements introduced by the first author.
MSC:
90B22 Queues and service in operations research
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Li, Q.-L., Constructive Computation in Stochastic Models with Applications, Berlin: Springer, 2009.
[2] Zhen, Q. and Knessl, C., Asymptotic Expansions for the Conditional Sojourn Time Distribution in the M/M/1-PS Queue, Queueing Syst., 2007. vol. 57, pp. 157–168. · Zbl 1176.90135 · doi:10.1007/s11134-007-9054-5
[3] Brandt, A. and Brandt, M., Waiting Times for M/M Systems under State-Dependent Processor Sharing, Queueing Syst., 2008, vol. 59, pp. 297–319. · Zbl 1162.90392 · doi:10.1007/s11134-008-9086-5
[4] Brandt, A. and Brandt, M., Insensitivity Bounds for the Moments of the Sojourn Times in M/GI Systems under State-Dependent Processor Sharing, Report 09-02, Zuse Inst. Berlin (ZIB), 2009, pp. 1–25. · Zbl 1197.60085
[5] Kleinrock, L., Analysis of a Time-Shared Processor, Naval Res. Logistics Quart., 1964, vol. 11, pp. 59–73. · Zbl 0129.30902 · doi:10.1002/nav.3800110105
[6] Kleinrock, L., Queueing Systems, vol. 2: Computer Applications, New York: Wiley, 1976. Translated under the title Vychislitel’nye sistemy s ocheredyami, Moscow: Mir, 1979.
[7] Yashkov, S.F., The Non-Stationary Distribution of Numbers of Calls in the M/G/1 Processor-Sharing Queue, Advances in Simulation, P.A. Lukar and B. Schmidt, Eds., Berlin: Springer, 1988, vol. 2, pp. 158–162.
[8] Yashkov, S.F., Analiz ocheredei v EVM (Analysis of Queues in Computers), Moscow: Radio i Svyaz’, 1989.
[9] Yashkov, S.F., Mathematical Problems in the Theory of Shared-Processor Systems, J. Soviet Math., 1992, vol. 58, no. 2, pp. 101–147. · Zbl 0735.68010 · doi:10.1007/BF01097426
[10] Kitayev, M.Yu. and Yashkov, S.F., Distribution of the Conditional Sojourn Time in a System with Division of Time of Servicing, Eng. Cybernetics, 1978, vol. 16, no. 4, pp. 162–167.
[11] Yashkov, S.F., A Derivation of Response Time Distribution for an M/G/1 Processor-Sharing Queue, Problems Control Inform. Theory, 1983, vol., 12, no. 2, pp. 133–148. · Zbl 0519.68052
[12] Yashkov, S.F. and Yashkova, A.S., Some Insight into the Time-Dependent Properties of the Queue-Length Process in the M/G/1-EPS and LCFS-P Queues, Inform. Proc., 2005, vol. 5, no. 2, pp. 102–105 (available at http://www.jip.ru/ ).
[13] Yashkov, S.F. and Yashkova, A.S., Processor Sharing: A Survey of the Mathematical Theory, Autom. Remote Control, 2007, vol. 68, no. 9, pp. 1662–1731. · Zbl 1147.93003 · doi:10.1134/S0005117907090202
[14] Sevast’yanov, B.A., An Ergodic Theorem for Markov Processes and Its Application to Telephone Systems with Refusals, Theor. Prob. Appl., 1957, vol. 2, no. 1, pp. 104–112. · doi:10.1137/1102005
[15] Jaiswal, N.K., Priority Queues, New York: Academic, 1968. Translated under the title Ocheredi s prioritetami, Moscow: Mir, 1973. · Zbl 0179.47904
[16] Yashkov, S.F., Processor-Sharing Queues: Some Progress in Analysis (Invited Paper), Queueing Syst., 1987, vol. 2, no. 1, pp. 1–17. · Zbl 0648.68050 · doi:10.1007/BF01182931
[17] Gelenbe, E., Glynn, P., and Sigman, K., Queues with Negative Arrivals, Appl. Probab., 1991, vol. 28, no. 1, pp. 245–250. · Zbl 0744.60110 · doi:10.1017/S0021900200039589
[18] Bocharov, P.P. and Vishnevskii, V.M., G-Networks: Development of the Theory of Multiplicative Networks, Autom. Remote Control, 2003, vol. 64, no. 5, pp. 714–739. · Zbl 1066.90009 · doi:10.1023/A:1023606704003
[19] Jain, G. and Sigman, K., A Pollaczek-Khinchine Formula for M/G/1 Queues with Disasters, J. Appl. Probab., 1996, vol. 33, pp. 1191–1200. · Zbl 0867.60082
[20] Li, Q.-L. and Lin, C., The M/G/1 Processor-Sharing Queue with Disasters, Comput. Math. Appl., 2006, vol. 51, no. 6–7, pp. 987–998. · Zbl 1180.90072 · doi:10.1016/j.camwa.2005.10.012
[21] Yashkov, S.F., A Note on Application of the Method of Supplementary Variables to the Analysis of a Processor Sharing System, Autom. Remote Control, 2008, vol. 69, no. 9, pp. 1662–1669. · Zbl 1167.68350
[22] Bhat, U.N., An Introduction to Queueing Theory. Modeling and Analysis in Applications., New York: Springer, 2008. · Zbl 1152.60001
[23] Takács, L., Introduction to the Theory of Queues, Oxford: Oxford Univ. Press, 1962. · Zbl 0106.33502
[24] Yashkov, S.F., Properties of Invariance of Probabilistic Models of Adaptive Scheduling in Shared-Use Systems, Autom. Control Comput. Sci., 1980, vol. 14, no. 6, pp. 46–51. · Zbl 0479.68046
[25] Kelly, F.P., Reversibility and Stochastic Networks, New York: Wiley, 1979.
[26] Abramowitz, M. and Stegun, L.A., Handbook of Mathematical Functions, New York: Dover, 1972. · Zbl 0543.33001
[27] Yashkov, S.F., The M/D/1-EPS Queue Revisited, Proc. Dobrushin Int. Conf., Moscow: Inst. Inform. Transm. Problems, 2009, pp. 179–185 (available as full paper on CD-ROM).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.