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Hitting-time and occupation-time bounds implied by drift analysis with applications

Published online by Cambridge University Press:  01 July 2016

Bruce Hajek
Bruce Hajek*
Affiliation:
University of Illinois
*
∗Postal address: Coordinated Science Laboratory, University of Illinois, 1101 West Springfield Ave., Urbana, IL 61801, U.S.A.
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Abstract

Bounds of exponential type are derived for the first-hitting time and occupation times of a real-valued random sequence which has a uniform negative drift whenever the sequence is above a fixed level. The only other assumption on the random sequence is that the increments satisfy a uniform exponential decay condition. The bounds provide a flexible technique for proving stability of processes frequently encountered in the control of queues.

Two applications are given. First, exponential-type bounds are derived for a GI/G/1 queue when the service distribution is exponential type. Secondly, geometric ergodicity is established for a certain Markov chain in which arises in the decentralized control of a multi-access, packet-switched broadcast channel.

Keywords

GEOMETRIC ERGODICITYSTOCHASTIC INEQUALITIESMARTINGALE INEQUALITIESGI/G/1 QUEUESRANDOM ACCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 3 , September 1982 , pp. 502 - 525
Copyright
Copyright © Applied Probability Trust 1982 

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Footnotes

This work was supported by U.S. Navy contract N00014–80-C-0802.

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