The Annals of Applied Probability

Large deviation properties of data streams that share a buffer

Paul Dupuis and Kavita Ramanan

Full-text: Open access


Using large deviation techniques, we analyze the tail behavior of the stationary distribution of the buffer content process for a two-station communication network. We also show how the associated rate function can be expressed as the solution to a finite-dimensional variational problem. Along the way, we develop a number of results and techniques that are of independent interest, including continuity results for the input-output mapping for certain multiclass fluid models and a new technique for obtaining large deviation principles for invariant distributions from sample path large deviation results.

Article information

Ann. Appl. Probab., Volume 8, Number 4 (1998), 1070-1129.

First available in Project Euclid: 9 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 90B12 60K25: Queueing theory [See also 68M20, 90B22]

Large deviations data networks effective bandwidths fluid models Skorokhod problem rate function


Ramanan, Kavita; Dupuis, Paul. Large deviation properties of data streams that share a buffer. Ann. Appl. Probab. 8 (1998), no. 4, 1070--1129. doi:10.1214/aoap/1028903374.

Export citation


  • [1] Anick, D., Mitra, D. and Sondhi, M. M. (1982). Stochastic theory of a data handling sy stem with multiple sources. Bell Sy st. Tech. J. 61 1871-1894.
  • [2] Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA.
  • [3] Chang, C.-S. (1993). Approximations of ATM networks: effective bandwidths and traffic descriptors. Technical report IBM RC 18954, T.J. Watson Research Center.
  • [4] Dembo, A. and Zeitouni, O. (1993). Large Deviations Techniques and Applications. Jones and Bartlett, Boston.
  • [5] de Veciana, G., Courcoubetis, C. and Walrand, J. (1993). Decoupling bandwidths for networks: a decomposition approach to resource management. Memorandum UCB/ERL M93/50, Univ. California, Berkeley, CA.
  • [6] Dupuis, P. and Ellis, R. S. (1996). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
  • [7] Dupuis, P. and Ishii, H. (1991). On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics Stochastics Rep. 35 31-62.
  • [8] Dupuis, P. and Nagurney, A. (1993). Dy namical sy stems and variational inequalities. Ann. Oper. Res. 44 9-42.
  • [9] Dupuis, P. and Williams, R. (1994). Ly apunov functions for semimartingale reflecting Brownian motions. Ann. Probab. 22 680-702.
  • [10] Elwalid, A. I. and Mitra, D. (1993). Effective bandwidth of general Markovian traffic sources and admission control of high speed networks. IEEE/ACM Trans. Networking 1 329-343.
  • [11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [12] Freidlin, M. I. and Wentzell, A. D. (1984). Random Perturbations of Dy namical Sy stems. Springer, New York.
  • [13] Hsu, I. and Walrand, J. (1995). Admission control for ATM networks. In Stochastic Networks (F. P. Kelly and R. J. Williams, eds.) 411-427. Springer, New York.
  • [14] Kelly, F. P. (1991). Effective bandwidths at multi-class queues. Queueing Sy stems 9 5-16.
  • [15] Kesidis, G., Walrand, J. and Chang, C. S. (1993). Effective bandwidths for multiclass Markov fluids and other ATM sources. IEEE/ACM Trans. Networking 1 424-428.
  • [16] Loy nes, R. M. (1962). The stability of a queue with non-indepedent inter-arrivals and service times. Math. Proc. Cambridge Philos. Soc. 58 497-520.
  • [17] Majewski, K. (1996). Large deviations of feedforward queueing networks. Ph.D. thesis, Ludwig-Maximilian-Univ., M ¨unchen.
  • [18] O'Brien, G. L. and Vervaat, W. (1991). Capacities, large deviations and loglog laws. In Stable Processes and Related Topics (S. Cambanis, G. Samorodnitsky and M. Taqqu, eds.) 43-84. Birkh¨auser, Boston.
  • [19] O'Connell, N. (1995). Large deviations in queueing networks. Preprint.
  • [20] Puhalskii, A. A. (1991). On functional principles of large deviations. In New Trends in Probability and Statistics (V. Sazonov and T. Shervashidze, eds.) 198-218. VSP-Mokslas, Utrecht.
  • [21] Ramanan, K. and Dupuis, P. (1995). Large deviation properties of data streams that share a buffer. LCDS Technical Report 95-8, Brown Univ.
  • [22] Shwartz, A. and Weiss, A. (1995). Large Deviations for Performance Analy sis: Queues, Communication, and Computing. Chapman and Hall, New York.
  • [23] Stroock, D. W. (1984). An Introduction to the Theory of Large Deviations. Springer, New York.
  • [24] Weiss, A. (1986). A new technique for analyzing large traffic sy stems. Adv. in Appl. Probab. 21 506-532.