Proportionally fair pricing: dynamics, stability and pathology

A potential key to the future control on the Internet of high quality-of-service traffic is dynamic pricing of bandwidth allocations, with `proportional fairness´ as one attractive possibility. For the first time to the authors´ knowledge, `small-changes´ and complex frequency-domain techniques (as used in classical control theory) are used to study dynamics of the relationships between bid-prices, proportionally fair bandwidth allocations and resource prices; instability, stability and convergence rates; integration from one (approximate) solution to another (accurate) solution allowing assessment of stability of major changes of traffic on the network; how coalitions of customers might distort the market by changing their bids but not getting any change in their allocation of bandwidth. Both standard proportional fairness and price-limited proportional fairness are discussed: the latter having floor prices for the resources. These prevent large unwanted allocations of bandwidth that may be needed at a later time, provide a smooth transition between low and high demands and greatly speed up dynamic allocations at low utilisation