The Dynamics of a Model of A Computer Protocol

The Internet has become a central means of communication on a global scale. Its emergence over the last two decades has been facilitated by the introduction of congestion control methodologies such as TCP, that allow millions of users to share network resources without causing congestion collapse. Beginning with the work of Frank Kelly [2], researchers have discovered that many currently used protocols designed to minimize congestion and to provide some notion of "fairness" can be viewed as a distributed, iterative algorithm for solving a global optimization problem. With this it is now possible to use mathematical analysis to help study, characterize and design protocols. The model we will study is based on the work of Wang, Li, Low and Doyle [5] who developed distributed algorithms for maximizing network utility through joint control of congestion and routing. They considered two routing allocation strategies. The rst, multipath routing occurs when a number of paths connecting a source and destination may be available and the goal is to assign tra c and bandwidth rates along the routes so that network utility is maximized. The second approach, single path routing, is on the other end of the spectrum of path speci cation. Again there are multiple paths linking a source and destination, but only a single path is selected, for example a path with minimum cost. Wang et al proved that there is no duality gap for the multipath optimization problem. However in general there is a duality gap for the single path problem. They established the equivalence between the existence of an equilibrium for the algorithm and the absence of a duality gap for the single path problem. Yet even though there is an equilibrium it can be unstable. Wang et al proposed the addition of a static component to the link costs in order to stabilize it. The following equations model a congestion control protocol with a random route allocation scheme that relaxes the requirement for a single path of minimum cost, but also reduces the number of eligible paths that would have to be considered in the multipath case. This is done by constraining the entropy of the route distribution. By varying the entropy, the route allocation scheme dials between single path and multipath routing. We want to determine the dynamics of this model as the entropy varies. We represent a computer network as a planar graph with nodes representing locations in the network. In the literature, the total amount of bandwidth to send information from one part of the network to another is controlled by an agent called a source. There are several routes a source can take to its destination and we assume each route is de ned by a set of uni-directional links indexed by l = 1, · · ·L. These links are represented in the graph as weighted edges, where an edge weight is the link capacity. Sources are indexed by s, and tra c is assigned to route r ∈ R(s) with probability βrs where R(s) is the set of all routes used by source s. If pl(k) is the link cost at time k and cl is the capacity of the lth link then the equations of the model are