Worst-Case Efficiency Analysis of Queueing Disciplines

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 Ui(xi) for xi ≥ 0 units of the good is a nonnegative, nondecreasing, continuously differentiable concave function of xi. The good can be produced in any amount, but producing X = ∑n i=1 xi 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, xi 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 = (x1, . . . , xn) can easily choose the allocation that maximizes the welfare W (x) = ∑n i=1 Ui(xi)−C(X), where X = ∑n i=1 xi, since this is a simple convex optimization problem. But what if users are autonomous and choose the quantities xi to maximize their own objectives? The most natural way to proceed is equilibrium analysis, where we model each user as maximizing its own payoff function and consider equilibrium allocations — those from which no user can unilaterally change its quantity to increase its payoff. We can then study the welfare achieved by autonomous and self-optimizing users via the price of anarchy (POA) — the worst (i.e., smallest) ratio between the welfare of an equilibrium (the outcome of selfish behavior) and the maximum-possible welfare (the ideal for an altruistic designer). The POA is a standard measure of inefficiency in game-theoretic systems, with a value near 1 indicating that selfish behavior is essentially benign. Defining user payoffs requires a fundamental modeling decision: how does the joint cost C(X) of producing an allocation translate to negative incentives for