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Journal of Applied Mathematics and Stochastic Analysis
Volume 2008 (2008), Article ID 518214, 25 pages
Research Article

A Fluid Model for a Relay Node in an Ad Hoc Network: Evaluation of Resource Sharing Policies

1Korteweg-de Vries Institute, University of Amsterdam, Plantage Muidergracht 24, 1018 Amsterdam, The Netherlands
2CWI, P.O. Box 94079, 1090 Amsterdam, The Netherlands
3EURANDOM, Eindhoven, The Netherlands
4University of Twente, P.O. Box 217, 7500 , Enschede, The Netherlands

Received 3 August 2007; Accepted 6 May 2008

Academic Editor: Hans Daduna

Copyright © 2008 Michel Mandjes and Werner Scheinhardt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fluid queues offer a natural framework for analyzing waiting times in a relay node of an ad hoc network. Because of the resource sharing policy applied, the input and output of these queues are coupled. More specifically, when there are 𝑛 users who wish to transmit data through a specific node, each of them obtains a share 1 / ( 𝑛 + 𝑊 ) of the service capacity to feed traffic into the queue of the node, whereas the remaining fraction 𝑊 / ( 𝑛 + 𝑊 ) is used to serve the queue; here 𝑊 > 0 is a free design parameter. Assume now that jobs arrive at the relay node according to a Poisson process, and that they bring along exponentially distributed amounts of data. The case 𝑊 = 1 has been addressed before; the present paper focuses on the intrinsically harder case 𝑊 > 1 , that is, policies that give more weight to serving the queue. Four performance metrics are considered: (i) the stationary workload of the queue, (ii) the queueing delay, that is, the delay of a “packet” (a fluid particle) that arrives at an arbitrary point in time, (iii) the flow transfer delay, (iv) the sojourn time, that is, the flow transfer time increased by the time it takes before the last fluid particle of the flow is served. We explicitly compute the Laplace transforms of these random variables.