The Annals of Applied Probability

A stochastic analysis of resource sharing with logarithmic weights

Philippe Robert and Amandine Véber

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Abstract

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has $x$ requests to transmit, then it receives a fraction of the capacity proportional to $\log(1+x)$, the logarithm of its current load. A detailed fluid scaling analysis of such a network with two nodes is presented. It is shown that the interaction of several time scales plays an important role in the evolution of such a system, in particular its coordinates may live on very different time and space scales. As a consequence, the associated stochastic processes turn out to have unusual scaling behaviors. A heavy traffic limit theorem for the invariant distribution is also proved. Finally, we present a generalization to the resource sharing algorithm for which the $\log$ function is replaced by an increasing function. Possible generalizations of these results with $J>2$ nodes or with the function $\log$ replaced by another slowly increasing function are discussed.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 5 (2015), 2626-2670.

Dates
Received: October 2013
Revised: July 2014
First available in Project Euclid: 30 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1438261050

Digital Object Identifier
doi:10.1214/14-AAP1057

Mathematical Reviews number (MathSciNet)
MR3375885

Zentralblatt MATH identifier
1326.60133

Subjects
Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60F05: Central limit and other weak theorems
Secondary: 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Stochastic networks fluid limits time scales

Citation

Robert, Philippe; Véber, Amandine. A stochastic analysis of resource sharing with logarithmic weights. Ann. Appl. Probab. 25 (2015), no. 5, 2626--2670. doi:10.1214/14-AAP1057. https://projecteuclid.org/euclid.aoap/1438261050


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