Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-23T22:18:49.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

Published online by Cambridge University Press:  14 July 2016

Guy Fayolle  and
Rudolph Iasnogorodski
Guy Fayolle*
Affiliation:
INRIA
Rudolph Iasnogorodski*
Affiliation:
Université d'Orléans
*
∗Postal address: INRIA, Domaine de Voluceau — Rocquencourt, BP 105, 78153Le Chesnay Cedex, France.
∗∗Postal address: Département de Mathématiques, Université d'Orléans, 45046 Orléans la Source, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

Keywords

ERGODICITYSUBMARTINGALES
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 2 , June 1987 , pp. 347 - 354
Copyright
Copyright © Applied Probability Trust 1987 

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] Aldous, D. (1985) Ultimate instability of EBO protocol for acknowledgment-based transmission control of random access communication channels. Int. Report, Dept. of Statistics, U. C. Berkeley.Google Scholar
[2] Capetanakis, J. I. (1979) Tree algorithms for packet broadcast channels. IEEE Trans. Inf. Theory 25, 505–515.CrossRefGoogle Scholar
[3] Kaplan, ?. (1979) A sufficient condition for nonergodicity of a Markov chain. IEEE Trans. Inf. Theory 25, 470–471.CrossRefGoogle Scholar
[4] Kelly, F. B. and Mac Phee, I. M. The number of packets transmitted by collision detect random access schemes. Unpublished.Google Scholar
[5] Lamperti, J. (1960) Criteria for the recurrence or transience of stochastic process, I. J. Math. Anal. Appl. 1, 314–330.CrossRefGoogle Scholar
[6] Malyshev, V. A. and Mensikov, ?. V. (1981) Ergodicity, continuity and analyticity of countable Markov chains. Trans. Moscow Math. Soc. 1.Google Scholar
[7] Rosenkrantz, W. A. (1984) Some theorems on the instability of the EBO protocol. Performance 84, North-Holland, Amsterdam.Google Scholar
[8] Sennott, L., Humblet, P. A. and Tweedie, R. L. (1983) Mean drifts and the non-ergodicity of Markov chains. Operat. Res. 31, 783–789.Google Scholar
[9] Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar