We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times—the total latency—is minimized. In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a “selfishly motivated ” assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance. In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total

Modern communication networks are able to respond to randomly uctuating demands and failures by adapting rates, by rerouting traffic and by reallocating resources. They are able to do this so well that, in many respects, large-scale networks appear as coherent, almost intelligent, organisms. The design and control of such networks present challenges of a mathematical, engineering and economic nature. This paper outlines how mathematical models are being used to address current issues concerning the stability and fairness of rate control algorithms for the Internet.

Prompted by the advent of QoS routing in the Internet, we investigate the properties that path weight functions must have so that hop-by-hop routing is possible and optimal paths can be computed with a generalized Dijsktra's algorithm. For this purpose we define an algebra of weights which contains a binary operation, for the composition of link weights into path weights, and an order relation. Isotonicity is the key property of the algebra. It states that the order relation between the weights of any two paths is preserved if both of them are either prefixed or appended by a common, third, path. We show that isotonicity is both necessary and sufficient for a generalized Dijkstra's algorithm to yield optimal paths. Likewise, isotonicity is also both necessary and sufficient for hop-by-hop routing. However, without strict isotonicity, hop-by-hop routing based on optimal paths may produce routing loops. They are prevented if every node computes what we call lexicographic-optimal paths. These paths can be computed with an enhanced Dijkstra's algorithm that has the same complexity as the standard one. Our findings are extended to multipath routing as well. As special cases of the general approach, we conclude that shortestwidest paths can neither be computed with a generalized Dijkstra's algorithm nor can packets be routed hop-by-hop over those paths. In addition, loop-free hop-by-hop routing over widest and widest-shortest paths requires that each node computes lexicographic-optimal paths, in general.

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Modern communication networks are able to respond to randomly uctuating demands and failures by allowing bu ers to ll, by rerouting tra c and by reallocating resources. They are able to do this so well that, in many respects, largescale networks appear as coherent, almost intelligent, organisms. The design and control of such networks present challenges of a mathematical, engineering and economic nature. In this lecture I describe some of the models that have proved useful in the analysis of stability, statistical sharing and pricing, in systems ranging from the telephone networks of today to the information superhighways of tomorrow.

In large-scale communication networks, like the Internet, it is usually impossible to globally manage network traffic. In the absence of global control it is typically assumed in traffic modeling that network users follow the most rational approach, that is, they behave selfishly to optimize their own individual welfare. This motivates the analysis of network traffic using models from Game Theory in which each player is aware of the situation facing all other players and tries to minimize its cost. Under these assumptions, the routing process should arrive into a so-called Nash equilibrium in which no network user has an incentive to change its strategy. It is well known (and easy to see) that Nash equilibria do not always optimize the overall performance of the system. In this survey we investigate the relation between these two notions for traffic routing: network performance in the Nash equilibria and the optimal performance of the system. Our main focus is on the analysis of the coordination ratio, which is the ratio between the worst possible Nash equilibrium and the overall optimum. In other words, this analysis seeks the price of uncoordinated

Most of the end-to-end congestion control schemes are "voluntary" in nature and critically depend on enduser cooperation. We show that in the presence of selfish users, all such schemes will inevitably lead to a congestion collapse. Router and switch mechanisms such as service disciplines and buffer management policies determine the sharing of resources during congestion. We show, using a Game Theoretic approach, that all the currently proposed mechanisms, either encourage the behaviour that leads to congestion ("Evil behaviour") or are oblivious to it. We propose a sample...

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We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportionally fair allocation. This work combines classical queueing networks with more recent work on stochastic flow level models and proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic, queueing level description; a macroscopic, flow level description and a teleological, optimization description. 1. Introduction. In

Various fairness objectives are studied in relation to Pareto optimal sets and Nash equilibria. We examine the already discussed general parameterized fairness objective that covers a variety of fairness criteria and the newly introduced Nash-proportionate-fairness objective. We study them mainly numerically on a simple static load balancing model with two identical servers (computers) each of which has an independent arrival process and its own queue. Through the numerical results, several intuitive results are shown. For example, we observe that the points that achieve the general parameterized fairness objectives may cover a part but not all of the Pareto set, and at times, do not cover the Nash-proportionate-fair Pareto optimal point.