TOOLS 2002: Computer Performance Evaluation: Modelling Techniques and Tools pp 1-30 | Cite as
Heavy Tails: The Effect of the Service Discipline
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Abstract
This paper considers the M/G/1 queue with regularly varying service requirement distribution. It studies the effect of the service discipline on the tail behavior of the waiting- or sojourn time distribution, demonstrating that different disciplines may lead to quite different tail behavior. The orientation of the paper is methodological: We outline three different methods of determining tail behavior, illustrating them for service disciplines like FCFS, Processor Sharing and LCFS.
Keywords
Sojourn Time Busy Period Heavy Tail Service Requirement Service Discipline
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References
- 1.Abate, J., Whitt, W. (1997). Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queueing Systems 25, 173–233.MATHCrossRefMathSciNetGoogle Scholar
- 2.Anantharam, V. (1988). How large delays build up in a GI/G/1 queue. Queueing Systems 5, 345–368.CrossRefMathSciNetGoogle Scholar
- 3.Anantharam, V. (1999). Scheduling strategies and long-range dependence. Queueing Systems 33, 73–89.MATHCrossRefMathSciNetGoogle Scholar
- 4.Baccelli, F., Foss, S. (2001). Moments and tails in monotone-separable stochastic networks. Research Report RR 4197, INRIA Rocquencourt.Google Scholar
- 5.Beran, J., Sherman, R., Taqqu, M.S., Willinger, W. (1995). Long-range dependence in variable-bit-rate video traffic. IEEE Trans. Commun. 43, 1566–1579.CrossRefGoogle Scholar
- 6.Bingham, N.H., Doney, R.A. (1974). Asymptotic properties of super-critical branching processes. I: The Galton-Watson process. Adv. Appl. Prob. 6, 711–731.MATHCrossRefMathSciNetGoogle Scholar
- 7.Bingham, N.H., Goldie, C.M., Teugels, J.L. (1987). Regular Variation (Cambridge University Press, Cambridge, UK).MATHGoogle Scholar
- 8.Borst, S.C., Boxma, O.J., Jelenković, P.R. (2000). Coupled processors with regularly varying service times. In: Proc. Infocom 2000 Conference, Tel-Aviv, Israel, 157–164.Google Scholar
- 9.Borst, S.C., Boxma, O.J., Jelenković, P.R. (2000). Reduced-load equivalence and induced burstiness in GPS queues. Queueing Systems, to appear.Google Scholar
- 10.Borst, S.C., Boxma, O.J., Van Uitert, M.G.J. (2001). Two coupled queues with heterogeneous traffic. In: Teletraffic Engineering in the Internet Era, Proc. ITC-17, Salvador da Bahia, Brazil, eds. J. Moreira de Souza, N.L.S. da Fonseca, E.A. de Souza e Silva (North-Holland, Amsterdam), 1003–1014.Google Scholar
- 11.Borst, S.C., Boxma, O.J., Van Uitert, M.G.J. (2002). The asymptotic workload behavior of two coupled queues. CWI Report PNA-R0202. Submitted for publication.Google Scholar
- 12.Borst, S.C., Zwart, A.P. (2001). Fluid queues with heavy-tailed M/G/∞ input. SPOR-Report 2001-02, Department of Mathematics and Computer Science, Eindhoven University of Technology. Submitted for publication.Google Scholar
- 13.Boxma, O.J., Cohen, J.W. (2000). The single server queue: Heavy tails and heavy traffic. In: S elf-similar Network Traffic and Performance Evaluation, eds. K. Park, W. Willinger (Wiley, New York), 143–169.CrossRefGoogle Scholar
- 14.Boxma, O.J., Cohen, J.W., Deng, Q. (1999). Heavy-traffic analysis of the M/G/1 queue with priority classes. In: Teletraffic Engineering in a Competitive World, Proc. ITC-16, Edinburgh, UK, eds. P. Key, D. Smith (North-Holland, Amsterdam), 1157–1167.Google Scholar
- 15.Boxma, O.J., Deng, Q., Resing, J.A.C. (2000). Polling systems with regularly varying service and/or switchover times. Adv. Perf. Anal. 3, 71–107.Google Scholar
- 16.Boxma, O.J. Dumas, V. (1998). The busy period in the fluid queue. Perf. Eval. Review 26, 100–110.CrossRefGoogle Scholar
- 17.Cohen, J.W. (1973). Some results on regular variation for distributions in queueing and fluctuation theory. J. Appl. Prob. 10, 343–353.CrossRefMATHGoogle Scholar
- 18.Cohen, J.W. (1982). The Single Server Queue (North-Holland Publ. Cy., Amsterdam; revised edition).MATHGoogle Scholar
- 19.Cohen, J.W., Boxma, O.J. (1983). Boundary Value Problems in Queueing System Analysis (North-Holland Publ. Cy., Amsterdam).MATHCrossRefGoogle Scholar
- 20.Crovella, M., Bestavros, A. (1996). Self-similarity in World Wide Web Traffic: evidence and possible causes. In: Proc. ACM Sigmetrics’ 96, 160–169.Google Scholar
- 21.Deng, Q. (2001). The two-queue E/1 -L polling model with regularly varying service and/or switchover times. SPOR-Report 2001-09, Department of Mathematics and Computer Science, Eindhoven University of Technology.Google Scholar
- 22.Fayolle, G., Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47, 325–351.MATHCrossRefMathSciNetGoogle Scholar
- 23.Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. II (Wiley, New York).MATHGoogle Scholar
- 24.Jelenković, P.R., Momcilović, P. (2002). Resource sharing with subexponential distributions. In: Proc. IEEE Infocom 2002, New York NY, USA, to appear.Google Scholar
- 25.Karamata, J. (1930). Sur un mode de croissance régulière des fonctions. Mathematica (Cluj) 4, 38–53.MATHGoogle Scholar
- 26.Kleinrock, L. (1976). Queueing Systems, Vol. II: Computer Applications (Wiley, New York).MATHGoogle Scholar
- 27.Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.MATHCrossRefGoogle Scholar
- 28.Konheim, A.G., Meilijson, I., Melkman, A. (1981). Processor sharing of two parallel lines. J. Appl. Prob. 18, 952–956.MATHCrossRefMathSciNetGoogle Scholar
- 29.Leland, W.E., Taqqu, M.S., Willinger, W., Wilson, D.V. (1994). On the self-similar nature of Ethernet traffic (extended version). IEEE/ACM Trans. Netw. 2, 1–15.CrossRefGoogle Scholar
- 30.De Meyer, A., Teugels, J.L. (1980). On the asymptotic behaviour of the distribution and the service time in M/G/1. J. Appl. Prob. 17, 802–813.MATHCrossRefGoogle Scholar
- 31.Mikosch, T. (1999). Regular variation, subexponentiality and their applications in probability theory. EURANDOM Report 99-013.Google Scholar
- 32.Núñez-Queija, R.(2000). Processor-Sharing Models for Integrated-Services Networks. Ph.D. thesis, Eindhoven University of Technology, ISBN 90-646-4667-8 (also available from the author upon request).Google Scholar
- 33.Núñez-Queija, R. (2000). Sojourn times in a processor-sharing queue with service interruptions. Queueing Systems 34, 351–386.MATHCrossRefGoogle Scholar
- 34.Núñez-Queija, R. (2002). Queues with equally heavy sojourn time and service requirement distributions. CWI Report PNA-R0201. Submitted for publication.Google Scholar
- 35.Ott, T.J. (1984). The sojourn-time distribution in the M/G/1 queue with processor sharing. J. Appl. Prob. 21, 360–378.MATHCrossRefMathSciNetGoogle Scholar
- 36.Pakes, A.G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555–564.CrossRefMathSciNetMATHGoogle Scholar
- 37.Parekh, A.K., Gallager, R.G. (1993). A generalized processor sharing approach to flow control in integrated services networks: the single-node case. IEEE/ACM Trans. Netw. 1, 344–357.CrossRefGoogle Scholar
- 38.Paxson, A., Floyd, S. (1995). Wide area traffic: the failure of Poisson modeling. IEEE/ACM Trans. Netw. 3, 226–244.CrossRefGoogle Scholar
- 39.Resnick, S., Samorodnitsky, G. (1999). Activity periods of an infinite server queue and performance of certain heavy-tailed fluid queues. Queueing Systems 33, 43–71.MATHCrossRefMathSciNetGoogle Scholar
- 40.Sakata, M., Noguchi, S., Oizumi, J. (1971). An analysis of the M/G/1 queue under round-robin scheduling. Oper. Res. 19, 371–385.MATHCrossRefGoogle Scholar
- 41.Schrage, L.E., Miller, L.W. (1966). The queue M/G/1 with the shortest remaining processing time discipline. Oper. Res. 14, 670–684.MATHMathSciNetGoogle Scholar
- 42.Schassberger, R. (1984). A new approach to the M/G/1 processor sharing queue. Adv. Appl. Prob. 16, 802–813.CrossRefMathSciNetGoogle Scholar
- 43.Willinger, W., Taqqu, M.S., Sherman, R., Wilson, D.V. (1997). Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Trans. Netw. 5, 71–86.CrossRefGoogle Scholar
- 44.Wolff, R.W. (1989). Stochastic Modeling and the Theory of Queues (Prentice Hall, Englewood Cliffs).Google Scholar
- 45.Yashkov, S.F. (1983). A derivation of response time distribution for a M/G/1 processor-sharing queue. Prob. Control Inf. Theory 12, 133–148.MATHMathSciNetGoogle Scholar
- 46.Yashkov, S.F. (1987). Processor-sharing queues: Some progress in analysis. Queueing Systems 2, 1–17.MATHCrossRefMathSciNetGoogle Scholar
- 47.Zwart, A.P. (1999). Sojourn times in a multiclass processor sharing queue. In: Teletraffic Engineering in a Competitive World, Proc. ITC-16, Edinburgh, UK, eds. P. Key, D. Smith (North-Holland, Amsterdam), 335–344.Google Scholar
- 48.Zwart, A.P. (2001). Queueing Systems with Heavy Tails. Ph.D. thesis, Eindhoven University of Technology.Google Scholar
- 49.Zwart, A.P. (2001). Tail asymptotics for the busy period in the GI/G/1 queue. Math. Oper. Res. 26, 485–493.MATHCrossRefMathSciNetGoogle Scholar
- 50.Zwart, A.P., Borst, S.C., Mandjes, M. (2001). Exact asymptotics for fluid queues fed by multiple heavy-tailed On-Off flows. Ann. Appl. Prob., to appear. Shortened version in: Proc. Infocom 2001, Anchorage AK, USA, 279–288.Google Scholar
- 51.Zwart, A.P., Boxma, O.J. (2000). Sojourn time asymptotics in the M/G/1 processor sharing queue. Queueing Systems 35, 141–166.MATHCrossRefMathSciNetGoogle Scholar
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