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

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.
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]

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

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

[1]

D. Alderson, H. Chang, M. Roughan, S. Uhlig, W. Willinger. The many facets of internet topology and traffic. Networks & Heterogeneous Media, 2006, 1 (4) : 569-600. doi: 10.3934/nhm.2006.1.569

[2]

Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial & Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593

[3]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165

[4]

Guang-hui Cai. Strong laws for weighted sums of i.i.d. random variables. Electronic Research Announcements, 2006, 12: 29-36.

[5]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[6]

Fabio Della Rossa, Carlo D’Angelo, Alfio Quarteroni. A distributed model of traffic flows on extended regions. Networks & Heterogeneous Media, 2010, 5 (3) : 525-544. doi: 10.3934/nhm.2010.5.525

[7]

Yang Yang, Kam C. Yuen, Jun-Feng Liu. Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims. Journal of Industrial & Management Optimization, 2018, 14 (1) : 231-247. doi: 10.3934/jimo.2017044

[8]

Antonia Katzouraki, Tania Stathaki. Intelligent traffic control on internet-like topologies - integration of graph principles to the classic Runge--Kutta method. Conference Publications, 2009, 2009 (Special) : 404-415. doi: 10.3934/proc.2009.2009.404

[9]

Yuebao Wang, Qingwu Gao, Kaiyong Wang, Xijun Liu. Random time ruin probability for the renewal risk model with heavy-tailed claims. Journal of Industrial & Management Optimization, 2009, 5 (4) : 719-736. doi: 10.3934/jimo.2009.5.719

[10]

Sharif E. Guseynov, Shirmail G. Bagirov. Distributed mathematical models of undetermined "without preference" motion of traffic flow. Conference Publications, 2011, 2011 (Special) : 589-600. doi: 10.3934/proc.2011.2011.589

[11]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[12]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[13]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[14]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[15]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355

[16]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

[17]

Veronika Schleper. A hybrid model for traffic flow and crowd dynamics with random individual properties. Mathematical Biosciences & Engineering, 2015, 12 (2) : 393-413. doi: 10.3934/mbe.2015.12.393

[18]

Brooke L. Hollingsworth, R.E. Showalter. Semilinear degenerate parabolic systems and distributed capacitance models. Discrete & Continuous Dynamical Systems - A, 1995, 1 (1) : 59-76. doi: 10.3934/dcds.1995.1.59

[19]

Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187

[20]

Drew Fudenberg, David K. Levine. Tail probabilities for triangular arrays. Journal of Dynamics & Games, 2014, 1 (1) : 45-56. doi: 10.3934/jdg.2014.1.45

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (0)

[Back to Top]