Abstract
Using the contraction principle, in this paper we derive a set of closure properties for sample path large deviations. These properties include sum, reduction, composition and reflection mapping. Using these properties, we show that the exponential decay rates of the steady state queue length distributions in an intree network with routing can be derived by a set of recursive equations. The solution of this set of equations is related to the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. We also prove a conditional limit theorem that illustrates how a queue builds up in an intree network.
Similar content being viewed by others
References
V. Anantharam, How large delays build up in a GI/G/1 queue, Queueing Systems 5 (1988) 345–368.
V. Anantharam, P. Heidelberger and P. Tsoucas, Analysis of rare events in continuous time Markov chains via time reversal and fluid approximation, IBM RC 16280 (1990).
M. Avriel,Nonlinear Programming: Analysis and Methods (Prentice-Hall, 1976).
F. Baccelli and P. Bremaud,Elements of Queueing Theory (Springer, New York, 1990).
P. Billingsley,Convergence of Probability Measures (Wiley, New York, 1968).
C.S. Chang, Stability, queue length and delay of deterministic and stochastic queueing networks, IEEE Trans. Aut. Contr. 39 (1994) 913–931.
C.S. Chang, Approximations of ATM networks: effective bandwidths and traffic descriptors, IBM RC 18954 (1993)
C.S. Chang, P. Heidelberger, S. Juneja and P. Shahabuddin, Effective bandwidth and fast simulation of ATM intree networks, Perform. Eval. 20 (1994) 45–66.
A. Dembo and O. Zeitouni,Large Deviations Techniques and Applications (Jones and Barlett, Boston, 1992).
A. Dembo and Tim Zajic, Large deviations from empirical mean and measure to partial sums processes, to appear in Stoch. Proc. Appl.
R. Ellis, Large deviations for a general class of random vectors, Ann. Prob. 12 (1984) 1–12.
A.I. Elwalid and D. Mitra, Effective bandwith of general Markovian traffic sources and admission control of high speed networks, IEEE/ACM Trans. Networking 1 (1993) 329–343.
M.R. Frater, T.M. Lennon and B.D.O. Anderson, Optimally efficient estimation of the statistics of rare events in queueing networks, IEEE Trans. Aut. Contr. 36 (1991) 1395–1405.
J. Gärtner, On large deviations from invariant measure, Th. Prob. Appl. 22 (1977) 24–39.
R.J. Gibbens and P.J. Hunt, Effective bandwidths for the multi-type UAS channel, Queueing Systems 9 (1991) 17–28.
P.W. Glynn and W. Whitt, Logarithmic asymptotics for steady-state tail probabilities in a single-server queue, J. Appl. Prob. 31A (1994) 131–156.
P.W. Glynn and W. Whitt, Large deviations behavior of counting processes and their inverses, Queueing Systems 17 (1994) 107–128.
R. Guérin, H. Ahmadi and M. Naghshineh, Equivalent capacity and its application to bandwidth allocation in high-speed networks, IEEE J. Select. Areas Commun. 9 (1991) 968–981.
I. Iscoe, P. Ney and E. Nummelin, Large deviations of uniformly recurrent Markov Additive Processes, Adv. Appl. Math. 6 (1985) 373–412.
P. Heidelberger, Fast simulation of rare events in queueing and reliability models, IBM RC 19028 (1993).
F.P. Kelly,Reversibility and Stochastic Networks (Wiley, New York, 1979).
F.P. Kelly, Effective bandwidths at multi-class queues, Queueing Systems 9 (1991) 5–16.
G. Kesidis and J. Walrand, Traffic policing and enforcement of effective bandwidth constraints in ATM networks, preprint.
G. Kesidis, J. Walrand and C.S. Chang, Effective bandwidths for multiclass Markov fluids and other ATM sources, IEEE/ACM Trans. Networking 1 (1993) 424–428.
R.M. Loynes, The stability of a queue with non-independent inter-arrival and service times, Proc. Camb. Phil. Soc. 58 (1962) 497–520.
H.D. Miller, A convexity property in the theory of random variables defined on a finite Markov chain, Ann. Math. Stat. 32 (1961) 1260–1270.
A.A. Mogulskii, Large deviations for trajectories of multidimensional random walk, Th. Prob. Appl. 21 (1976) 300–315.
R. Nagarajan, J. Kurose and D. Towsley, Local allocation of end-to-end quality-of-service in high speed networks, preprint.
S. Parekh and J. Walrand, A quick simulation method for excessive backlogs in networks of queues, IEEE Trans. Aut. Contr. 34 (1989) 54–66.
R.T. Rochafellar,Convex Analysis (Princeton University Press, Princeton, 1970).
D. Stoyan,Comparison Methods for Queues and Other Stochastic Models (Wiley, Berlin, 1983).
J. Walrand,An Introduction to Queueing Networks (Prentice-Hall, New Jersey, 1988).
W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5 (1980) 67–85.
W. Whitt, Comparing counting processes and queues, Adv. Appl. Prob. 13 (1981) 207–220.
W. Whitt, Tail probability with statistical multiplexing and effective bandwidths in multiclass queues, Telecom. Syst. 2 (1993) 71–107.
G. De. Veciana and J. Walrand, Effective Bandwidths: call admission, traffic policing & filtering for ATM networks, submitted to IEEE/ACM Trans. Networking (1993).
G. De. Veciana, C. Courcoubetis and J. Walrand, Decoupling bandwidths for networks: a decomposition approach to resource management, submitted to IEEE/ACM Trans. Networking (1993).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chang, C.S. Sample path large deviations and intree networks. Queueing Syst 20, 7–36 (1995). https://doi.org/10.1007/BF01158430
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01158430