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One aspect of my PhD research that is not commercially sensitive is that of deciding what it means to allocate bandwith fairly. This was the topic of my Electronics Letters paper.

It is an issue that has also been dealt with in considerable detail by Frank Kelly of the Statistics Laboratory at Cambridge University. He has published several papers on this topic. A good introduction can be found in Charging and rate control for elastic traffic which was published in European Transactions on Telecommunications volume 8(1997) pp 33-37. It is also available in postscript form on his web pages: here.

In Frank's paper, the concept of proportional fairness is introduced. A set of flows (xs) is proportionally fair if the equation Sum over s in S of [(xs* - xs)/xs] &lt= 0 is satisfied for all alternative feasible sets of flows (x*s). (A set of flows is feasible if none of the network's resources are overloaded.)

In the published version of this paper, there was a minor error. The inequality in the equation above was shown as being a strict inequality. However, it is fairly straightforward to see that the inequality should not be strict. I pointed this out to Frank in December 1997, and I am grateful to Frank for acknowledging my input in the corrected version of the paper, which is available here, on his website.

If you want to see why the inequality should not be strict, read my query - available as postscript or PDF - the question I raised over the definition of proportional fairness.

In fact, I think the issue is more complicated. Some interesting questions are raised in relation to proportional fairness. As it stands, the definition can't be used as a measure of how proportionally fair something might be. It can only tell you if something is the proportioanally fair solution to a problem. You get some odd results.

For example in the case of the network considered in my query it is not difficult to determine the proportionally fair flows. However, moving to any other point at which the constraints are satisfied - in other words, at which both network resources are fully utilised - results in no aggregate proportional change: the summation is zero.

But if, instead of considering a change from the proportionally fair solution to an alternative, you consider a change from the alternative to the proportionally fair solution, the aggregate proportional change is non-zero. In fact, it is positive - indicating that the alternative is definitely not the proportionally fair solution.

In other words, all feasible solutions are demonstrably "less" proportionally fair than the proportionally fair solution, even though there is a whole class of solutions for which the proportionally fair solution is not demonstrably "more" proportionally fair.

This strikes me as odd. It must be related to the assymetry in the definition of proportional fairness. A possible alternative definition might be: Sum over s in S of [(xs* - xs)/sqrt(xs.xs*)] < 0 This definition certainly holds true for the simple cases I have considered, giving an unambiguous "measure" of proportional fairness.

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