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Stationarity and control of a tandem fluid network with fractional Brownian motion input

Part of: Operations research and management science Computer system organization Special processes

Published online by Cambridge University Press:  01 July 2016

Chihoon Lee  and
Ananda Weerasinghe
Chihoon Lee*
Affiliation:
Colorado State University
Ananda Weerasinghe*
Affiliation:
Iowa State University
*
∗ Postal address: Department of Statistics, Colorado State University, Fort Collins, CO 80523, USA. Email address: [email protected]
∗∗ Postal address: Department of Mathematics, Iowa State University, Ames, IA 50011, USA.
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Abstract

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We consider a stochastic control model for a queueing system driven by a two-dimensional fractional Brownian motion with Hurst parameter 0 < H < 1. In particular, when H > ½, this model serves to approximate a controlled two-station tandem queueing model with heavy-tailed ON/OFF sources in heavy traffic. We establish the weak convergence results for the distribution of the state process and construct an explicit stationary state process associated with given controls. Based on suitable coupling arguments, we show that each state process couples with its stationary counterpart and we use it to represent the long-run average cost functional in terms of the stationary process. Finally, we establish the existence result of an optimal control, which turns out to be independent of the initial data.

Keywords

Stochastic controlcontrolled queueing systemheavy traffic theoryfractional Brownian motiontandem queuelong-range dependenceself-similarity

MSC classification

Primary: 60K25: Queueing theory 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service
Secondary: 90B18: Communication networks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 3 , September 2011 , pp. 847 - 874
Copyright
Copyright © Applied Probability Trust 2011 

References

Ata, B. and Kumar, S. (2005). Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies. Ann. Appl. Prob. 15, 331–391.CrossRefGoogle Scholar
Bernard, A. and el Kharroubi, A. (1991). Régulations déterministes et stochastiques dans le premier ‘orthant’ de R n . Stoch. Stoch. Rep. 34, 149–167.Google Scholar
Biagini, F., Hu, Y., Øksendal, B. and Sulem, A. (2002). A stochastic maximum principle for processes driven by fractional Brownian motion. Stoch. Process. Appl. 100, 233–253.Google Scholar
Biagini, F., Hu, Y., Øksendal, B. and Zhang, T. (2008). Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London.CrossRefGoogle Scholar
Cooper, W. L., Schmidt, V. and Serfozo, R. F. (2001). Skorohod-Loynes characterizations of queueing, fluid, and inventory processes. Queueing Systems 37, 233–257.Google Scholar
Delgado, R. (2007). A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic. Stoch. Process. Appl. 117, 188–201.Google Scholar
Duffield, N. G. and O'Connell, N. (1995). Large deviations and overflow probabilities for the general single-server queue, with applications. Math. Proc. Camb. Phil. Soc. 118, 363–374.CrossRefGoogle Scholar
Duncan, T. E. (2007). Some stochastic systems with a fractional Brownian motion and applications to control. In Proc. American Control Conf. (New York, July 2007), pp. 1110–1114.Google Scholar
Duncan, T. E. and Jin, Y. (2008). Maximum queue length of a fluid model with an aggregated fractional Brownian input. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Inst. Math. Statist. Collections 4), Institute of Mathematical Statistics, Beachwood, Ohio, pp. 235–251.Google Scholar
Ghosh, A. P., Roitershtein, A. and Weerasinghe, A. (2010). Optimal control of a stochastic network driven by a fractional brownian motion input. Adv. Appl. Prob. 42, 183–209.Google Scholar
Gong, W.-B., Liu, Y., Misra, V. and Towsley, D. (2005). Self-similarity and long range dependence on the internet: a second look at the evidence, origins and implications. Comput. Networks 48, 377–399.Google Scholar
Hairer, M. (2005). Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Prob. 33, 703–758.Google Scholar
Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley, New York.Google Scholar
Harrison, J. M. and Shepp, L. A. (1984). A tandem storage system and its diffusion limit. Stoch. Process. Appl. 16, 257–274.Google Scholar
Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in ON/OFF processes and associated fluid models. Math. Operat. Res. 23, 145–165.Google Scholar
Hu, Y. and Zhou, X. Y. (2005). Stochastic control for linear systems driven by fractional noises. SIAM J. Control Optimization 43, 2245–2277.Google Scholar
Hu, Y., Nualart, D. and Song, J. (2009). Fractional martingales and characterization of the fractional Brownian motion. Ann. Prob. 37, 2404–2430.CrossRefGoogle Scholar
Jain, R. and Routhier, S. A. (1986). Packet trains: measurements and a new model for computer network traffic. IEEE J. Sel. Areas Commun. 4, 986–995.Google Scholar
Kleptsyna, M. L., Le Breton, A. and Viot, M. (2003). About the linear-quadratic regulator problem under a fractional Brownian perturbation. ESAIM Prob. Statist. 7, 161–170.Google Scholar
Konstantopoulos, T. and Lin, S.-J. (1996). Fractional Brownian approximations of queueing networks. In Stochastic Networks (New York, 1995, Lecture Notes Statist. 117), Springer, New York, pp. 257–273.Google Scholar
Loynes, R. M. (1962). The stability of a queue with non-independent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
Loynes, R. M. (1964). The stability of a system of queues in series. Proc. Camb. Phil. Soc. 60, 569–574.Google Scholar
Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437.Google Scholar
Mandjes, M., Mannersalo, P. and Norros, I. (2007). Gaussian tandem queues with an application to dimensioning of switch fabric interfaces. Comput. Networks 51, 781–797.Google Scholar
Massoulie, L. and Simonian, A. (1999). Large buffer asymptotics for the queue with fractional Brownian input. J. Appl. Prob. 36, 894–906.Google Scholar
Norros, I. (1995). On the use of fractional Brownian motion in the theory of connectionless networks. IEEE J. Sel. Areas Commun. 13, 953–962.Google Scholar
Novikov, A. and Valkeila, E. (1999). On some maximal inequalities for fractional Brownian motions. Statist. Prob. Lett. 44, 47–54.CrossRefGoogle Scholar
Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin.Google Scholar
Taqqu, M. S., Willinger, W. and Sherman, R. (1997). Proof of a fundamental result in self-similar traffic modeling. Comput. Commun. Rev. 27, 5–23.Google Scholar
Whitt, W. (2002). Stochastic-Process Limits. Springer, New York.Google Scholar
Williams, R. J. (1998). An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems 30, 5–25.Google Scholar
Willinger, W., Paxson, V. and Taqqu, M. S. (1998). Self-similarity and heavy tails: structural modeling of network traffic. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Birkhäuser, Boston, pp. 27–53.Google Scholar
Willinger, W., Taqqu, M. S., Sherman, R. and Wilson, D. V. (1997). Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Trans. Networking 5, 71–86.Google Scholar
Yan, L. (2004). Maximal inequalities for the iterated fractional integrals. Statist. Prob. Lett. 69, 69–79.Google Scholar
Zeevi, A. J. and Glynn, P. W. (2000). On the maximum workload of a queue fed by fractional Brownian motion. Ann. Appl. Prob. 10, 1084–1099.Google Scholar