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April  2016, 12(2): 637-652. doi: 10.3934/jimo.2016.12.637

Tail asymptotics of fluid queues in a distributed server system fed by a heavy-tailed ON-OFF flow

Byeongchan Lee 1, , Jonghun Yoon 1, , Yang Woo Shin 2, and  Ganguk Hwang 1,

1. 

Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 305-701, South Korea, South Korea, South Korea
2. 

Department of Statistics, Changwon National University, Changwon 641-773, South Korea

Received  September 2014 Revised  March 2015 Published  June 2015

The appearance of heavy-tailedness in users' traffic significantly degrades the performance of communication systems, and a distributed server system is considered as a good solution to this problem because of its distributed service characteristic by multiple servers. So we tackle the question in this paper that a distributed server system can alleviate heavy-tailedness, so that users experience good QoS as if there were no heavy-tailedness. To this end, we first mathematically model a distributed server system and obtain a heavy-tailed random sum with the help of the theory of perturbed random walk. We then analyze the tail asymptotic of the heavy-tailed random sum to find a condition with which the distributed server system can alleviate heavy-tailedness.
Keywords: heavy-tailedness, distributed server systems., random sums, Internet traffic, tail asymptotics.
Mathematics Subject Classification: Primary: 68M20, 68M14; Secondary: 60G5.
Citation: Byeongchan Lee, Jonghun Yoon, Yang Woo Shin, Ganguk Hwang. Tail asymptotics of fluid queues in a distributed server system fed by a heavy-tailed ON-OFF flow. Journal of Industrial & Management Optimization, 2016, 12 (2) : 637-652. doi: 10.3934/jimo.2016.12.637
References:
[1]

A. Aleškevičien.e, R. Leipus and J. Šiaulys, Tail behavior of random sums under consistent variation with applications to the compound renewal risk model,, Extremes, 11 (2008), 261.  doi: 10.1007/s10687-008-0057-3.  Google Scholar

[2]

S. Asmussen, H. Schmidli and V. Schmidt, Tail probabilities for non-standard risk and queueing processes with subexponential jumps,, Advances in Applied Probability, (1999), 442.  doi: 10.1239/aap/1029955142.  Google Scholar

[3]

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[5]

S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions,, $2^{nd}$ edition, (2013).  doi: 10.1007/978-1-4614-7101-1.  Google Scholar

[6]

W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson, On the self-similar nature of Ethernet traffic,, ACM SIGCOMM Computer Communication Review, 23 (1993), 183.   Google Scholar

[7]

K. W. Ng, Q. H. Tang and H. Yang, Maxima of sums of heavy-tailed random variables,, Astin Bulletin, 32 (2002), 43.  doi: 10.2143/AST.32.1.1013.  Google Scholar

[8]

V. Paxson and S. Floyd, Wide area traffic: the failure of Poisson modeling,, IEEE/ACM Transactions on Networking (ToN), 3 (1995), 226.  doi: 10.1145/190314.190338.  Google Scholar

[9]

Z. Palmowski and B. Zwart, Tail asymptotics of the supremum of a regenerative process,, Journal of applied probability, (2007), 349.  doi: 10.1239/jap/1183667406.  Google Scholar

[10]

C. Y. Robert and J. Segers, Tails of random sums of a heavy-tailed number of light-tailed terms,, Insurance: Mathematics and Economics, 43 (2008), 85.  doi: 10.1016/j.insmatheco.2007.10.001.  Google Scholar

[11]

F. Semchedine, L. Bouallouche-Medjkoune, and D. Aïssani, Task assignment policies in distributed server systems: A survey,, Journal of Network and Computer Applications, 34 (2011), 1123.  doi: 10.1016/j.jnca.2011.01.011.  Google Scholar

[12]

K. Sigman, Appendix: A primer on heavy-tailed distributions,, Queueing Systems, 33 (1999), 261.  doi: 10.1023/A:1019180230133.  Google Scholar

[13]

G. E. Willmot and H. Yang, Martingales and ruin probability,, Actuarial Research Clearing House, 1 (1996), 521.   Google Scholar

show all references

References:
[1]

A. Aleškevičien.e, R. Leipus and J. Šiaulys, Tail behavior of random sums under consistent variation with applications to the compound renewal risk model,, Extremes, 11 (2008), 261.  doi: 10.1007/s10687-008-0057-3.  Google Scholar

[2]

S. Asmussen, H. Schmidli and V. Schmidt, Tail probabilities for non-standard risk and queueing processes with subexponential jumps,, Advances in Applied Probability, (1999), 442.  doi: 10.1239/aap/1029955142.  Google Scholar

[3]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events: For Insurance and Finance,, Springer-Verlag, (1997).  doi: 10.1007/978-3-642-33483-2.  Google Scholar

[4]

G. Faÿ, B. González-Arévalo, T. Mikosch and G. Samorodnitsky, Modeling teletraffic arrivals by a Poisson cluster process,, Queueing Systems, 54 (2006), 121.  doi: 10.1007/s11134-006-9348-z.  Google Scholar

[5]

S. Foss, D. Korshunov, and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions,, $2^{nd}$ edition, (2013).  doi: 10.1007/978-1-4614-7101-1.  Google Scholar

[6]

W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson, On the self-similar nature of Ethernet traffic,, ACM SIGCOMM Computer Communication Review, 23 (1993), 183.   Google Scholar

[7]

K. W. Ng, Q. H. Tang and H. Yang, Maxima of sums of heavy-tailed random variables,, Astin Bulletin, 32 (2002), 43.  doi: 10.2143/AST.32.1.1013.  Google Scholar

[8]

V. Paxson and S. Floyd, Wide area traffic: the failure of Poisson modeling,, IEEE/ACM Transactions on Networking (ToN), 3 (1995), 226.  doi: 10.1145/190314.190338.  Google Scholar

[9]

Z. Palmowski and B. Zwart, Tail asymptotics of the supremum of a regenerative process,, Journal of applied probability, (2007), 349.  doi: 10.1239/jap/1183667406.  Google Scholar

[10]

C. Y. Robert and J. Segers, Tails of random sums of a heavy-tailed number of light-tailed terms,, Insurance: Mathematics and Economics, 43 (2008), 85.  doi: 10.1016/j.insmatheco.2007.10.001.  Google Scholar

[11]

F. Semchedine, L. Bouallouche-Medjkoune, and D. Aïssani, Task assignment policies in distributed server systems: A survey,, Journal of Network and Computer Applications, 34 (2011), 1123.  doi: 10.1016/j.jnca.2011.01.011.  Google Scholar

[12]

K. Sigman, Appendix: A primer on heavy-tailed distributions,, Queueing Systems, 33 (1999), 261.  doi: 10.1023/A:1019180230133.  Google Scholar

[13]

G. E. Willmot and H. Yang, Martingales and ruin probability,, Actuarial Research Clearing House, 1 (1996), 521.   Google Scholar

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