Worst-Case Efficiency Analysis of Queueing Disciplines

• Authors: Damon Mosk-Aoyama;Tim Roughgarden
• Venue: ICALP '09 Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming: Part II
• Year: 2009

Consider n users vying for shares of a divisible good. Every user i wants as much of the good as possible but has diminishing returns, meaning that its utility U i (x i ) for x i *** 0 units of the good is a nonnegative, nondecreasing, continuously differentiable concave function of x i . The good can be produced in any amount, but producing $X = \sum_{i=1}^n x_i$ units of it incurs a cost C (X ) for a given nondecreasing and convex function C that satisfies C (0) = 0. Cost might represent monetary cost, but other interesting interpretations are also possible. For example, x i could represent the amount of traffic (measured in packets, say) that user i injects into a queue in a given time window, and C (X ) could denote aggregate delay (X ·c (X ), where c (X ) is the average per-unit delay). An altruistic designer who knows the utility functions of the users and who can dictate the allocation x = (x 1 ,...,x n ) can easily choose the allocation that maximizes the welfare $W(x) = \sum_{i=1}^n U_i(x_i) - C(X)$, where $X = \sum_{i=1}^n x_i$, since this is a simple convex optimization problem.