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 (x_{s}) is proportionally fair if the equation
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

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: This definition certainly holds true for the simple cases I have considered, giving an unambiguous "measure" of proportional fairness.

If you have got

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