Abstract
We show, under regularity conditions, that a counting process satisfies a large deviations principle in ℝ or the Gärtner-Ellis condition (convergence of the normalized logarithmic moment generating functions) if and only if its inverse process does. We show, again under regularity conditions, that embedded regenerative structure is sufficient for the counting process or its inverse process to have exponential asymptotics, and thus satisfy the Gärtner-Ellis condition. These results help characterize the small-tail asymptotic behavior of steady-state distributions in queueing models, e.g., the waiting time, workload and queue length.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen,Applied Probability and Queues (Wiley, 1987).
J.A. Bucklew,Large Deviation Techniques in Decision, Simulation and Estimation (Wiley, 1990).
C.-S. Chang, Stability, queue length and delay of deterministic and stochastic queueing networks, IEEE Trans. Auto. Contr., to appear.
C.-S. Chang, P. Heidelberger, S. Juneja and P. Shahabuddin, Effective bandwidth and fast simulation of ATM intree networks, IBM T.J. Watson Research Center, Yorktown Heights, NY (1992).
A. Dembo and O. Zeitouni,Large Deviation Techniques and Applications (Jones and Bartlett, Boston, MA, 1992).
R. Ellis, Large deviations for a general class of random vectors, Ann. Prob. 12 (1984) 1–12.
J. Gärtner, On large deviations from the invariant measure, Theory Prob. Appl. 22 (1977) 24–39.
P.W. Glynn, A.A. Puhalskii and W. Whitt, A functional large deviation principle for first passage time processes, in preparation.
P.W. Glynn and W. Whitt, Ordinary CLT and WLLN versions ofL=λW, Math. Oper. Res. 13 (1988) 674–692.
P.W. Glynn and W. Whitt, Limit theorems for cumulative processes, Stoch. Proc. Appl. 47 (1993) 299–314.
P.W. Glynn and W. Whitt, Logarithmic asymptotics for steady-state tail probabilities in a single-server queue, in:Studies in Applied Probability, Festschrift in honor of Lajos Takács in honor of his 70th birthday, eds. J. Galambos and J. Gani, to appear.
T. Kuczek and K.N. Crank, A large-deviation result for regenerative processes, J. Theory Prob. 4 (1991) 551–561.
W.A. Massey and W. Whitt, Unstable asymptotics for nonstationary queues, Math. Oper. Res. 19 (1994) 267–291.
A. Shwartz and A. Weiss,Large Deviations for Performance Analysis: Queues, Communication and Computing, in preparation.
W. Whitt, Some useful functions for functional limit theorems, Math. Oper. Res. 5 (1980) 67–85.
W. Whitt, Tail probabilities with statistical multiplexing and effective bandwidths in multiclass queues, Telecommun. Syst. 2 (1993) 71–107.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Glynn, P.W., Whitt, W. Large deviations behavior of counting processes and their inverses. Queueing Syst 17, 107–128 (1994). https://doi.org/10.1007/BF01158691
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01158691