Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T17:35:10.227Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

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

Published online by Cambridge University Press:  01 July 2016

Kurt Majewski
Kurt Majewski*
Affiliation:
Siemens AG
*
∗ Postal address: Siemens AG, CT SE 6, 81730 München, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider a constant rate traffic which shares a buffer with a random cross traffic. A first come first served or priority service discipline is applied at the buffer. After service at the first buffer the constant rate traffic moves to a play-out buffer. Both buffers provide service at constant rate and infinite waiting room. We investigate logarithmic large and moderate deviation asymptotics for the tail probabilities of the steady-state queue length distribution at the play-out buffer for long-range dependent cross traffic in critical loading. We characterize the asymptotic behavior of the cross traffic which leads to a large queue length at the play-out buffer and compare it to the one for renewal cross traffic.

Keywords

Moderate deviationsfirst come first servedpriority service disciplineheavy trafficqueueing network

MSC classification

Primary: 60F10: Large deviations
Secondary: 60K25: Queueing theory 90B15: Network models, stochastic 68M20: Performance evaluation; queueing; scheduling
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 407 - 428
Copyright
Copyright © Applied Probability Trust 2007 

References

Addie, R., Mannersalo, P. and Norros, I. (2002). Most probable paths and performance formulae for buffers with Gaussian input traffic. Europ. Trans. Telecommun. 13, 183–196.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. (New York) 51), 2nd edn. Springer, New York.Google Scholar
Avram, F., Dai, J. and Hasenbein, J. J. (2001). Explicit solutions for variational problems in the quadrant. Queueing Systems 37, 261–291.CrossRefGoogle Scholar
Beran, J. (1994). Statistics for Long-Memory Processes. Chapman and Hall, New York.Google Scholar
Berger, A. W. and Whitt, W. (1998). Effective bandwidth with priorities. IEEE/ACM Trans. Networking 6, 447–460.CrossRefGoogle Scholar
Cox, D. (1984). Long-range dependence: a review. In Statistics: An Appraisal (Proc. 50th Anniv. Conf. Iowa State Statist. Lab.), ed. David, H., The Iowa State University Press, Ames, pp. 55–74.Google Scholar
Dembo, A. and Zajic, T. (1997). Uniform large and moderate deviations for functional empirical processes. Stoch. Process. Appl. 71, 195–211.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, London.Google Scholar
Dupuis, P. and Ishii, H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stoch. Stoch. Reports 35, 31–62.CrossRefGoogle Scholar
Dupuis, P. and Ramanan, K. (2002). A time-reversed representation for the tail probabilities of stationary reflected Brownian motion. Stoch. Process. Appl. 98, 253–288.CrossRefGoogle Scholar
Ganesh, A. and O'Connell, N. (2002). A large deviation principle with queueing applications. Stoch. Stoch. Reports 73, 25–35.CrossRefGoogle Scholar
Ganesh, A., O'Connell, N. and Wischik, D. (2004). Big Queues (Lecture Notes Math. 1838). Springer, Berlin.CrossRefGoogle Scholar
Harrison, J. M. and Williams, R. (1992). Brownian models of feedforward queueing networks: quasi- reversibility and product form solutions. Ann. Appl. Prob. 2, 263–293.CrossRefGoogle Scholar
Harrison, J. M. and Reiman, M. I. (1981). Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.CrossRefGoogle Scholar
Hogg, R. V. and Klugman, S. A. (1984). Loss Distributions. John Wiley, New York.CrossRefGoogle Scholar
Konstantopoulos, T. and Lin, S.-J. (1996). Fractional Brownian motion approximations of queueing networks. In Stochastic Networks: Stability and Rare Events (Lecture Notes Statist. 117), eds Glasserman, P., Sigman, K. and Yao, D. D., Springer, New York, pp. 257–274.CrossRefGoogle Scholar
Kozachenko, Y., Vasylyk, O. and Sottinen, T. (2002). Path space large deviations of a large buffer with Gaussian input traffic. Queueing Systems 42, 113–129.CrossRefGoogle Scholar
Loynes, R. (1962). The stability of a queue with nonindependent inter-arrivals and service times. Math. Proc. Camb. Philos. Soc. 58, 497–520.CrossRefGoogle Scholar
Majewski, K. (1998). Large deviations of the steady-state distribution of reflected processes with applications to queueing systems. Queueing Systems 29, 351–381.CrossRefGoogle Scholar
Majewski, K. (2003). Large deviations for multi-dimensional reflected fractional Brownian motion. Stoch. Stoch. Reports 75, 233–257. (Correction: 76 (2004), 479.)CrossRefGoogle Scholar
Majewski, K. (2005). Fractional Brownian heavy traffic approximations of multiclass feedforward queueing networks. Queueing Systems 50, 199–230.CrossRefGoogle Scholar
Majewski, K. (2005). Sample path moderate deviations for a family of long-range dependent traffic and associated queue length processes. Stochastics 77, 81–107.CrossRefGoogle Scholar
Majewski, K. (2006). Sample path large deviations for a family of long-range dependent traffic and associated queue length processes. Queueing Systems 52, 105–118.CrossRefGoogle Scholar
Majewski, K. (2006). Sample path large deviations for multiclass feedforward queueing networks in critical loading. Ann. Appl. Prob. 16, 1893–1924.CrossRefGoogle Scholar
Majewski, K. (2007). Minimizing large deviation paths for a family of long-range dependent processes and their fractional Brownian approximations. Stoch. Models 23, 49–77.CrossRefGoogle Scholar
Mandjes, M. and van Uitert, M. (2005). Sample-path large deviations for tandem and priority queues with Gaussian inputs. Ann. Appl. Prob. 15, 1193–1226.CrossRefGoogle Scholar
Mandjes, M., Mannersalo, P., Norros, I. and van Uitert, M. (2006). Large deviations of infinite intersections of events in Gaussian processes. Stoch. Process. Appl. 116, 1269–1293.CrossRefGoogle Scholar
Mannersalo, P. and Norros, I. (2002). A most probable path approach to queueing systems with general Gaussian input. Comput. Networks 40, 399–411.CrossRefGoogle Scholar
Mikosch, T., Resnick, S., Rootzén, H. and Stegeman, A. (2002). Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Ann. Appl. Prob. 12, 23–68.CrossRefGoogle Scholar
Mogulskii, A. (1976). Large deviations for trajectories of multi-dimensional random walks. Theory Prob. Appl. 21, 300–315.CrossRefGoogle Scholar
Norros, I. (1995). On the use of fractional Brownian motions in the theory of connectionless networks. IEEE J. Selected Areas Commun. 13, 953–962.CrossRefGoogle Scholar
Norros, I. (1999). Busy periods of fractional Brownian storage: a large deviations approach. Adv. Performance Anal. 2, 1–19.Google Scholar
Norros, I. (2002). Large deviations of queues with long-range dependent input. In Theory and Applications of Long-Range Dependence, Birkhäuser, Boston, MA, pp. 409–415.Google Scholar
Puhalskii, A. A. (1994). Large deviations of semimartingales via convergence of the predictable characteristics. Stoch. Stoch. Reports 49, 27–85.CrossRefGoogle Scholar
Puhalskii, A. A. (1995). Large deviation analysis of the single server queue. Queueing Systems 21, 5–66.CrossRefGoogle Scholar
Puhalskii, A. A. (1999). Moderate deviations for queues in critical loading. Queueing Systems 31, 359–392.CrossRefGoogle Scholar
Ramanan, K. and Dupuis, P. (1998). Large deviation properties of data streams that share a buffer. Ann. Appl. Prob. 8, 1070–1129.Google Scholar
Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.Google Scholar
Sigman, K. (1999). Appendix: a primer on heavy-tailed distributions. Queueing Systems 33, 261–275.CrossRefGoogle Scholar