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LOWER BOUNDS FOR LRD/GI/1 QUEUES WITH SUBEXPONENTIAL SERVICE TIMES

Published online by Cambridge University Press:  22 January 2004

Cathy H. Xia ,
Zhen Liu ,
Mark S. Squillante  and
Li Zhang
Cathy H. Xia
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
Zhen Liu
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
Mark S. Squillante
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
Li Zhang
Affiliation:
IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, E-mail: [email protected]
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Abstract

We investigate the tail distribution of the virtual waiting times in a LRD/GI/1 queue where the arrival process is long-range dependent (LRD) and the service times are independent and identically distributed (i.i.d.) random variables. We present two lower bounds on the stationary waiting time tail asymptotics, which illustrate the different dominating components that influence server performance under various conditions. In particular, we show that the tail distribution of the stationary waiting time is bounded below by that of the associated LRD/D/1 queues resulting from replacing all random service times by the mean. This shows the performance impact purely due to the long-range dependency of the arrival process. On the other hand, when the service times are subexponential, we show that the tail distribution of the stationary waiting time is bounded below by that of the corresponding D/GI/1 queue by replacing the dependent arrival process with its associated independent version. This shows the minimum performance impact due to the tail distribution of the service times. The above two lower bounds indicate that the performance of LRD/GI/1 queues will be dominated by the heavier tail of the corresponding LRD/D/1 and D/GI/1 queues. These features are further illustrated and quantified through examples and via numerous simulation experiments.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 1 , January 2004 , pp. 87 - 101
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Arlitt, M.F. & Williamson, C.L. (1997). Internet Web servers: Workload characterization and performance implications. IEEE/ACM Transactions on Networking 5(5): 631–645.Google Scholar
Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: Extremeal behaviour, stationary distributions and first passage probabilities. Annals of Applied Probability 8: 354–374.Google Scholar
Asmussen, S., Schmidli, H., & Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential jumps. Advances in Applied Probability 31: 422–447.Google Scholar
Chistakov, V.P. (1964). A theorem on sums of independent positive random variables and its application to branching random process. Theory of Probability and Its Applications 9: 640–648.Google Scholar
Corvella, M.E. & Bestavros, A. (1996). Self-similarity in World Wide Web traffic: Evidence and possible causes. Performance Evaluation Review 24: 160–169.Google Scholar
Downey, A. (2001). The structural cause of file size distributions. Proceedings of the International Symposium on Modeling Analysis and Simulation of Computer and Telecommunication Systems.
Duffield, N.G. & O'Connell, N. (1995). Large deviation and overflow probabilities for the general single-server queue, with applications. Mathematical Proceedings of the Cambridge Philosophical Society 118: 363–375.Google Scholar
Iyengar, A.K., Squillante, M.S., & Zhang, L. (1999). Analysis and characterization of large-scale web server access patterns and performance. World Wide Web 2: 85–100.Google Scholar
Jelenkovic, P. & Lazar, A.A. (1998). Subexponential asymptotics of a Markov-modulated random walk with queueing applications. Journal of Applied Probility 35(2): 325–347.Google Scholar
Leland, W.E., Taqqu, M.S., Willinger, W., & Wilson, D.V. (1994). On the self-similarity nature of Ethernet traffic (extended version). IEEE/ACM Transactions on Networking 2(1): 1–15.Google Scholar
Liu, Z., Niclausse, N., & Jalpa-Villanueva, C. (2001). Traffic model and performance evaluation of Web servers. Performance Evaluation 46(2–3): 77–100.Google Scholar
Loynes, R.M. (1968). The stability of a queue with non-independent inter-arrival and service times. Proceedings of Cambridge Philosophical Society 58: 497–520.Google Scholar
Norros, I. (1994). A storage model with self-similar input. Queueing Systems 16: 387–396.Google Scholar
Pakes, A. (1975). On the tails of waiting time distributions. Journal of Applied Probability 12: 555–564.Google Scholar
Park, K. & Willinger, W. (eds.). (2002). Self-similar network traffic and performance evaluation. New York: Wiley.
Paxson, V. (1995). Fast approximation of self-similar network traffic. Technical Report LBL-36750/UC-405, Lawrence Berkeley Laboratory.
Sigman, K. (1999). Appendix: A primer on heavy-tailed distributions. Queueing Systems 33: 261–275.Google Scholar
Vanichpun, S. & Makowski, A.M. (2002). Positive correlations and buffer occupancy: Lower bounds via supermodular ordering. Proceedings of IEEE INFOCOM 2002, pp. 1298–1306.
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 Transactions on Networking 5(1): 71–86.Google Scholar
Wolff, R.W. (1988). Stochastic modeling and the theory of queues. Englewood Cliffs, NJ: Prentice-Hall.
Xia, C.H., Liu, Z., Squillante, M.S., Zhang, L., & Malouch, N. (2003). Analysis of performance impact of drill-down techniques for Web traffic models. Proceedings of the 18th International Teletraffic Congress (ITC-18), pp. 1–10.