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

We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a model of selfish routing in which the latency experienced by network traffic on an edge of the network is a function of the edge congestion, and network users are assumed to selfishly route traffic on minimumlatency paths. The quality of a routing of traffic is measured by the sum of travel times, also called the total latency. The outcome of selfish routing—a Nash equilibrium—does not in general minimize the total latency; hence, selfish behavior carries the cost of decreased network performance. We quantify this degradation in network performance via the price of anarchy, the worst-possible ratio between the total latency of a Nash equilibrium and of an optimal routing of the traffic. We show the price of anarchy is determined only by the simplest of networks. Specifically, we prove that under weak hypotheses on the class of allowable edge latency functions, the worst-case ratio between the total latency of a Nash equilibrium and of a minimum-latency routing for any multicommodity flow network is achieved by a singlecommodity

In noncooperative networks users make control decisions that optimize their own performance measure. Focusing on routing, we devise two methodologies for architecting noncooperative networks, that improve the overall network performance. These methodologies are motivated by problem settings arising in the provisioning and the run time phases of the network. For either phase, Nash equilibria characterize the operating point of the network. The goal of the provisioning phase is to allocate link capacities that lead to systemwide efficient Nash equilibria. In general, the solution of such design problems is counterintuitive, since adding link capacity might lead to a degradation of user performance. We show that, for systems of parallel links, such paradoxes cannot occur and the optimal solution coincides with the solution in the single-user case. We derive some extensions to general network topologies. During the run time phase, a manager controls the routing of part of the network flow. The manager is aware of the noncooperative behavior of the users and makes its routing decisions based on this information while aiming at improving the overall system performance. We obtain necessary and sufficient conditions for enforcing an equilibrium that coincides with the global systemwide optimum, and indicate that these conditions are met in many cases of interest.

AbstractWe study the problem of optimizing the performance of a system shared by selfish, noncooperative users. We consider the concrete setting of scheduling jobs on a set of shared machines with load-dependent latency functions specifying the length of time necessary to complete a job; we measure system performance by the total latency of the system. Assigning jobs according to the selfish interests of individual users (who wish to minimize only the latency that their own jobs experience) typically results in suboptimal system performance. However, in many systems of this type there is a mixture of "selfishly controlled " and "centrally controlled " jobs; as the assignment of centrally controlled jobs will influence the subsequent actions by selfish users, we aspire to contain the degradation in system performance due to selfish behavior by scheduling the centrally controlled jobs in the best possible way. We formulate this goal as an optimization problem via Stackelberg games, games in which one player acts a leader (here, the centralized authority interested in optimizing system performance) and the rest as followers (the selfish users). The problem is then to compute a strategy for the leader (a Stackelberg strategy) that induces the followers to react in a way that (at least approximately) minimizes the total latency in the system. In this paper, we prove that it is NP-hard to compute the optimal Stackelberg strategy and present simple strategies with provable performance guarantees. More precisely, we give a simple algorithm that computes a strategy inducing a job assignment with total latency no more than a constant times that of the optimal assignment of all of the jobs; in the absence of centrally controlled jobs and a Stackelberg strategy, no result of this type is possible. We also prove stronger performance guarantees in the special case where every machine latency function is linear in the machine load.

In noncooperative networks users make control decisions that optimize their individual performance objectives. Nash equilibria characterize the operating points of such networks. Nash equilibria are generically inefficient and exhibit suboptimal network performance. Focusing on routing, a methodology is devised for overcoming this deficiency, through the intervention of the network manager. The manager controls part of the network flow, is aware of the noncooperative behavior of the users and performs its routing aiming at improving the overall system performance. The existence of maximally efficient strategies for the manager, i.e., strategies that drive the system into the global network optimum, is investigated. A maximally efficient strategy of the manager not only optimizes the overall performance of the network, but also induces an operating point that is efficient with respect to the performance of the individual users (Pareto efficiency). Necessary and sufficient conditions for...

The recent move towards market deregulation and open competition has sparked a wave of serious introspection in the telecommunications service industry. Telecom providers and operators are now required to open up their primary revenue channels to competing industries. In this paper, we examine the service structure of two major global communication networks - the Telephone Network and the Internet and explore their weaknesses and strengths. Building upon the experience we gained during the development of the initial xbind prototypes, we discuss the realization of an open programable networking environment based on a new service architecture for advanced telecommunication services. Our approach to the problem stems from two angles - one conceptual, the other implementational. In the first, we attempt to develop a service model that is open and reflects the economic market structure of the future telecommunications service industry. We believe that investigating such a model will help cl...

We study the price of selfish routing in non-cooperative networks like the Internet. In particular, we investigate the price...

Abstract We consider a directed network in which every edge possesses a latency function specifying the time needed to traverse the edge given its congestion. Selfish, noncooperative agents constitute the network traffic and wish to travel from a source s to a sink t as quickly as possible. Since the route chosen by one network user affects the congestion (and hence the latency) experienced by others, we model the problem as a noncooperative game. Assuming each agent controls only a negligible portion of the overall traffic, Nash equilibria in this noncooperative game correspond to s-t flows in which all flow paths have equal latency. A natural measure for the performance of a network used by selfish agents is the common latency experienced by each user in a Nash equilibrium. It is a counterintuitive but well-known fact that removing edges from a network may improve its performance; the most famous example of this phenomenon is the so-called Braess's Paradox. This fact motivates the following network design problem: given such a network, which edges should be removed to obtain the best possible flow at Nash equilibrium? Equivalently, given a large network of candidate edges to be built, which subnetwork will exhibit the best performance when used selfishly? We give optimal inapproximability results and approximation algorithms for several network design problems of this type. For example, we prove that for networks with n vertices and continuous, nondecreasing latency functions, there is no approximation algorithm for this problem with approximation ratio less than n/2 (unless P = N P). We also prove this hardness result to be best possible by exhibiting an n/2-approximation algorithm. For networks in which the latency of each edge is a linear function of the congestion, we prove that there is no ( 43- ffl)-approximation algorithm for the problem (for any ffl> 0, unless P = N P); the existence of a 43-approximation algorithm follows easily from existing work, proving this hardness result sharp. Moreover, we prove that an optimal approximation algorithm for these problems is what we call the trivial algorithm: given a network of candidate edges, build the entire network. A consequence of this result is that Braess's Paradox (even in its worst-possible manifestation) is impossible to detect efficiently.

The exponential growth of computer networking demands massive upgrades of capacity in existing networks. Traditional capacity design methodologies, developed with the single-class networking paradigm in mind, overlook the noncooperative structure of modern networks. Consequently, such design approaches entail the danger of degraded performance when resources are added to a network, a phenomenon known as the Braess paradox. The present paper proposes methods for efficiently adding resources to a noncooperative network of general topology. It is shown that the paradox is avoided when resources are added across the network, rather than on a local scale, and when upgrades are focused on direct connections between sources and destinations. The relevance of these findings to modern networks is demonstrated.

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