Worst-Case Efficiency Analysis of Queueing Disciplines

Abstract

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.

The results in Section [3] of this work also appear in Chapter 5 of the first author’s PhD thesis [4]