Hostname: page-component-89b8bd64d-46n74 Total loading time: 0 Render date: 2026-05-07T14:54:21.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Functional Large Deviation Principles for Waiting and Departure Processes

Published online by Cambridge University Press:  27 July 2009

Anatolii A. Puhalskii  and
Ward Whitt
Anatolii A. Puhalskii
Affiliation:
Department of Mathematics, University of Colorado at Denver, Denver, Colorado 80217-3364, puhalski@math.cudenver.edu, Institute for Problems in Information Transmission, 19 Bolshoi Karetnii, Moscow 101447, Russia
Ward Whitt
Affiliation:
AT&T Labs, Florham Park, New Jersey 07932-0971, wow@research.att.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish functional large deviation principles (FLDPs) for waiting and departure processes in single-server queues with unlimited waiting space and the first-in first-out service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the nonuniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition, and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit expressions for the rate function of the departure process, but not more generally. In general, the rate function for the departure process evidently must be calculated numerically. We also obtain an FLDP for the departure process of completed work, which has important application to the concept of effective bandwidths for admission control and capacity planning in packet communication networks.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 12 , Issue 4 , October 1998 , pp. 479 - 507
Copyright
Copyright © Cambridge University Press 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1.Anantharam, V. (1989). How large delays build up in a GI/G/I queue. Queueing Systems 5: 345–367.Google Scholar
2.Aubin, J.-P. & Ekeland, I. (1984). Applied nonlinear analysis. New York: Wiley.Google Scholar
3.Berger, A.W. & Whitt, W. (1997). A general framework for effective bandwidths with priority and loss criteria. Florham Park, NJ: AT&T Labs.Google Scholar
4.Berger, A.W. & Whitt, W. (1998). Effective bandwidths with priorities. IEEE/ACM Transactions on Networking, to appear.CrossRefGoogle Scholar
5.Bertsimas, D., Paschalidis, I., & Tsitsiklis, J. (1995). On the large deviations behavior of acyclic networks of G/G/1 queue. Cambridge, MA: MIT.Google Scholar
6.Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.Google Scholar
7.Chang, C.S. (1995). Sample path large deviations and intree networks. Queueing Systems 20: 7–36.CrossRefGoogle Scholar
8.Chang, C.S., Heidelberger, P., Juneja, S., & Shahabuddin, P. (1994). Effective bandwidth and fast simulation of ATM intree networks. Performance Evaluation 20: 45–66.CrossRefGoogle Scholar
9.Chang, C.S. & Thomas, J.A. (1995). Effective bandwidths in high-speed digital networks. IEEE Journal of Selected Areas in Communications 13: 1091–1100.CrossRefGoogle Scholar
10.Chang, C.S. & Zajic, T. (1995). Effective bandwidths of departure processes from queues with time-varying capacities. Proceedings of IEEE INFOCOM 95: 1001–1009.Google Scholar
11.Chen, H. (1996). Rate of convergence of the fluid approximation for generalized Jackson networks. Journal of Applied Probability 33: 804–814.CrossRefGoogle Scholar
12.Dembo, A. & Zajic, T. (1995). Large deviations: From empirical mean and measure to partial sums processes. Stochastic Processes and Their Applications 57: 91–124.Google Scholar
13.Dembo, A. & Zeitouni, O. (1993). Large deviations, techniques and applications. Boston: Jones and Bartlett.Google Scholar
14.Dobrushin, R.L. & Pechersky, E.A. (1995). Large deviations for tandem queueing systems. Journal of Applied Mathematics and Stochastic Analysis 7: 301–330.Google Scholar
15.Glynn, P.W. & Whitt, W. (1994). Large deviations behavior of counting processes and their inverses. Queueing Systems 17: 107–128.CrossRefGoogle Scholar
16.Glynn, P.W. & Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. Journal of Applied Probability 31A: 131–156.Google Scholar
17.Harrison, J.M. (1985). Brownian motion and stochastic flow systems. New York: Wiley.Google Scholar
18.Iglehart, O.L. & Whin, W. (1970). Multiple channel queues in heavy traffic, I and II. Advances in Applied Probability 2: 150–177, 355–369.Google Scholar
19.Kelly, F.P. (1996). Notes on effective bandwidths. In Kelly, F.P., Zachary, S., & Ziedins, I. (eds.). Stochastic networks. Oxford: Clarendon Press, pp. 141–168.Google Scholar
20.Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space D[0,∞). Journal of Applied Probability 10: 109–121.CrossRefGoogle Scholar
21.Lynch, J. & Sethuraman, J. (1987). Large deviations for processes with independent increments. Annals of Probability 15: 610–627.Google Scholar
22.Mogulskii, A. (1993). Large deviations for processes with independent increments. Annals of Probability 21: 202–215.CrossRefGoogle Scholar
23.O'Connell, N. (1997). Large deviations for departures from a shared buffer. Journal of Applied Probability 34: 753–766.Google Scholar
24.Pomarede, J.L. (1976). A unified approach via graphs to Skorohod's topologies on the function space D. Ph.D. dissertation, Yale University, New Haven, CT.Google Scholar
25.Puhalskii, A. (1991). On functional principle of large deviations. In Sazonov, V. & Shervashidze, T. (eds.), New trends in probability and statistics, Vol. 1, VSP/Mokslas, pp. 198–218.Google Scholar
26.Puhalskii, A. (1994). Large deviations of semimartingales via convergence of the predictable characteristics. Stochastics 49: 27–85.Google Scholar
27.Puhalskii, A. (1994). The method of stochastic exponentials for large deviations. Stochastic Processes and Their Applications 54: 45–70.CrossRefGoogle Scholar
28.Puhalskii, A. (1995). Large deviation analysis of the single server queue. Queueing Systems 21: 5–66.CrossRefGoogle Scholar
29.Puhalskii, A. (1997). Large deviations of semimartingales: A maxingale problem approach. I. Limits as solutions to a maxingale problem. Stochastics 61: 141–243.Google Scholar
30.Puhalskii, A. & Whitt, W. (1997). Functional large deviation principles for first-passage-time processes. Annals of Applied Probability 7: 362–381.Google Scholar
31.Ross, K.W. (1995). Multiservice loss models for broadband telecommunications. London: Springer.CrossRefGoogle Scholar
32.Shwartz, A. & Weiss, A. (1995). Large deviations for performance analysis. London: Chapman and Hall.Google Scholar
33.Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theory of Probability and Its Applications 1: 261–290.CrossRefGoogle Scholar
34.Tsoucas, P. (1992). Rare events in series of queues. Journal of Applied Probability 29: 168–175.Google Scholar
35.Varadhan, S.R.S. (1966). Asymptotic probabilities and differential equations. Communications in Pure Applied Mathematics 19: 261–286.CrossRefGoogle Scholar
36.Varadhan, S.R.S. (1984). Large deviations and applications. Philadelphia: SIAM.Google Scholar
37.de Veciana, G., Courcoubetis, C., & Walrand, J. (1994). Decoupling bandwidths for networks: A decomposition approach to resource management for networks. Proceedings of IEEE INFOCOM '94 2: 466–474.CrossRefGoogle Scholar
38.de Veciana, G., Kesidis, G., & Walrand, J. (1995). Resource management in wide-area ATM networks using effective bandwidths. IEEE Journal of Selected Areas in Communications 13: 1081–1090.CrossRefGoogle Scholar
39.Whitt, W. (1980). Some useful functions for functional limit theorems. Mathematics of Operations Research 1: 67–85.CrossRefGoogle Scholar
40.Whitt, W. (1993). Tail probabilities with statistical multiplexing and effective bandwidths in multiclass queues. Telecommunications Systems 2: 71–107.Google Scholar