Congestion control

Congestion control adjusts user transmission rates in response to load signals from shared resources. A resource can compute its price from local aggregate load, while each user sums prices along its route. Primal congestion potential and dual congestion potential formulations explain equilibrium and stability for suitable monotone response functions.

A resource congestion price signals scarcity at a capacity-constrained link. Under multiplicative resource-price dynamics, its increase indicates excess demand and its decrease indicates spare capacity. A route pays the sum of these prices, its route congestion price. A zero resource congestion price at an optimum satisfies complementary slackness with possible spare capacity.

The total price seen by a route is the sum of resource congestion prices on its links. With weighted logarithmic utility $w_r\log x_r$, maximizing utility minus linear expenditure gives the demand $x_r=w_r/p_r$. This relation links user decisions to multiplicative resource-price dynamics.

For route prices $p=A^T\mu$, logarithmic users demand $x_r=w_r/p_r$, while resource prices adjust asThis model raises prices under excess demand and lowers them under spare capacity. Positive capacities and nonempty routes are standard well-definedness assumptions. Positive initial prices approach the dual optimum; zero prices are invariant and can create boundary equilibria of multiplicative price dynamics that overload a resource.

For a complementary-slackness optimum ${\mu }^{\ast }$, defineinterpreting a zero-reference summand as ${\mu }_j/{\kappa }_j$. This is a nonnegative generalized reverse-relative-entropy expression for positive resource prices. Along multiplicative resource-price dynamics, with $p=A^T\mu$ and $g=Ax-C$,Its sublevel bounds keep all route prices away from zero and permit the LaSalle invariance principle, including optima with some zero individual prices.

For multiplicative resource-price dynamics with positive capacities, nonempty routes and strictly positive initial prices, full row rank of the link-route incidence matrix makesstrictly concave. There is a unique optimizing price vector, possibly on the boundary, and the relative-entropy Lyapunov function for resource prices proves convergence to it. Row-rank deficiency still gives unique optimal route prices and rates, but resource prices may depend on the initial state. Full column rank alone and unrestricted zero initial prices do not yield the same uniqueness theorem.

In ${\dot{\mu }}_j={\kappa }_j{\mu }_j(y_j-q_j({\mu }_j))$, a zero price remains zero even if resource demand exceeds supply. Such a boundary equilibrium need not minimize the dual congestion potential; uniqueness of its positive minimizer does not imply uniqueness on the nonnegative orthant.

For strictly increasing continuous supply functions $q_j$ with $q_j(0)=0$, this potential is strictly convex on prices with positive route-price sums. Its minimum is interior for every used resource and gives the unique positive-price equilibrium. Multiplicative price dynamics can also have boundary equilibria of multiplicative price dynamics if zero prices are permitted.

This potential subtracts integrated resource prices from weighted logarithmic utility. Continuous nonnegative nondecreasing prices make it strictly concave on positive rates. If every route meets a resource with a nonzero price function, its superlevel sets are compact and its maximizer is unique. The rate adjustment ${\dot{x}}_r={\kappa }_r{x}_r{\partial }_{r}U$ increases it strictly away from equilibrium.