Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T19:37:31.450Z Has data issue: false hasContentIssue false
  • English
  • Français

On the asymptotic relationship between the overflow probability and the loss ratio

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Han S. Kim  and
Ness B. Shroff
Han S. Kim*
Affiliation:
Purdue University
Ness B. Shroff*
Affiliation:
Purdue University
*
∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA.
∗ Postal address: School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907-1285, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log P{Q > x} and logPL(x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them.

Keywords

Overflow probabilityloss ratioasymptotic relationship

MSC classification

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 836 - 863
Copyright
Copyright © Applied Probability Trust 2001 

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.)

References

[1] Abate, J., Choudhury, G. L. and Whitt, W. (1994). Asymptotics for steady-state tail probabilities in structured Markov queueing models. Commun. Statist. Stoch. Models 10, 99–143.CrossRefGoogle Scholar
[2] Addie, R. G. and Zukerman, M. (1994). An approximation for performance evaluation of stationary single server queues. IEEE Trans. Commun. 42, 3150–3160.Google Scholar
[3] Anick, D., Mitra, D. and Sondhi, M. M. (1982). Stochastic theory of a data handling system with multiple sources. Bell System Tech. J. 61, 1871–1894.Google Scholar
[4] Brandt, A., Franken, P. and Lisek, B. (1990). Stationary Stochastic Models. John Wiley, New York.Google Scholar
[5] Chang, C.-S. (1994). Stability, queue length, and delay of deterministic and stochastic queueing networks. IEEE Trans. Automatic Control 39, 913–931.Google Scholar
[6] Choe, J. and Shroff, N. B. (1999). On the supremum distribution of integrated stationary Gaussian processes with negative linear drift. Adv. Appl. Prob. 31, 135–157.Google Scholar
[7] Cohen, J. W. (1969). The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[8] Daigle, J. N. and Langford, J. D. (1986). Models for analysis of packet voice communication systems. IEEE J. Selected Areas Commun. 4, 847–855.Google Scholar
[9] Duffield, N. G. and O'Connell, N. (1995). Large deviations and overflow probabilities for the general single server queue, with application. Proc. Camb. Phil. Soc. 118, 363–374.Google Scholar
[10] Elwalid, A. I. (1991). Markov modulated rate processes for modeling, analysis and control of communication networks. , Columbia University.Google Scholar
[11] Glynn, P. W. and Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. In Studies in Applied Probability (J. Appl. Prob. Spec. Vol. 31A), eds Galambos, J. and Gani, J., Applied Probability Trust, Sheffield, pp. 131–155.Google Scholar
[12] Heyman, D. and Whitt, W. (1989). Limits for queues as the waiting room grows. Queueing Systems 5, 381–392.Google Scholar
[13] Jelenkovic, P. R. (1999). Subexponential loss rate in a GI/GI/1 queue with applications. Queueing Systems 33, 91–123.CrossRefGoogle Scholar
[14] Kim, H. S. and Shroff, N. B. (1999). An approximation of the loss probability. Tech. Rep., Purdue University.Google Scholar
[15] Kim, H. S. and Shroff, N. B. (1999). Loss probability calculations in a finite buffer queueing system. Tech. Rep., Purdue University.Google Scholar
[16] Kim, H. S. and Shroff, N. B. (2001). Loss probability calculations and asymptotic analysis for finite buffer multiplexers. To appear in IEEE/ACM Trans. Networking.Google Scholar
[17] Likhanov, N. and Mazumdar, R. R. (1998). Cell-loss asymptotics in buffers fed with a large number of independent stationary sources. In Proc. IEEE INFOCOM (San Francisco, CA, 1998), pp. 339–346. Available at http://www.ieee-infocom.org/1998/.Google Scholar
[18] Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
[19] Shroff, N. B. and Schwartz, M. (1998). Improved loss calculations at an ATM multiplexer. IEEE/ACM Trans. Networking 6, 411–422.CrossRefGoogle Scholar
[20] Sriram, K. and Whitt, W. (1986). Characterizing superposition arrival processes in packet multiplexer for voice and data. IEEE J. Selected Areas Commun. 4, 833–846.Google Scholar
[21] Zwart, A. P. (2000). A fluid queue with a finite buffer and subexponential input. Adv. Appl. Prob. 32, 221–243.Google Scholar