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Stability conditions for some distributed systems: buffered random access systems

Part of: Stability

Published online by Cambridge University Press:  01 July 2016

Wojciech Szpankowski
Wojciech Szpankowski*
Affiliation:
Purdue University
*
* Postal address: Department of Computer Science, Purdue University, W. Lafayette, IN 47907, USA.
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Abstract

We consider the standard slotted ALOHA system with a finite number of buffered users. Stability analysis of such a system was initiated by Tsybakov and Mikhailov (1979). Since then several bounds on the stability region have been established; however, the exact stability region is known only for the symmetric system and two-user ALOHA. This paper proves necessary and sufficient conditions for stability of the ALOHA system. We accomplish this by means of a novel technique based on three simple observations: applying mathematical induction to a smaller copy of the system, isolating a single queue for which Loynes' stability criteria is adopted, and finally using stochastic dominance to verify the required stationarity assumptions in the Loynes criterion. We also point out that our technique can be used to assess stability regions for other multidimensional systems. We illustrate it by deriving stability regions for buffered systems with conflict resolution algorithms (see also Georgiadis and Szpankowski (1992) for similar approach applied to stability of token-passing rings).

Keywords

ALOHAMULTIDIMENSIONAL MARKOV CHAINSSUBSTABILITYLOYNES SCHEMEMATHEMATICAL INDUCTION

MSC classification

Primary: 93D20: Asymptotic stability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 26 , Issue 2 , June 1994 , pp. 498 - 515
Copyright
Copyright © Applied Probability Trust 1994 

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Footnotes

This research was supported in part by NSF grants NCR-9206315 and CCR-9201078, by AFOSR grant 90–0107, and in part by NATO Collaborative Grant 0057/89. This paper was revised while the author was visiting INRIA, Rocquencourt, France, and he wishes to thank INRIA (projects ALGO, MEVAL and REFLECS) for generous support.

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