Link-Traffic Loop-Free Property in Optimal Routing for Multi-Class Networks Hisao Kameda, Jie Li, and Eitan Altman Abstract— Communications networks where nodes are in- terconnected by a generally configured manner and wherein there are multiple classes of users, each of which has its distinct generalized link costs (communication delays), are considered. Individually and overall optimal routing problems for multi- class networks are formulated along with the discussions on mutual equivalence between both problems, on the existence and uniqueness of solutions, and on the relation between the formulations with path and link flow patterns. We show that a link-traffic loop-free property holds within each class for both individually and overall optimal routing in a wide range of networks, and obtain the condition that characterizes the category of networks for which the link-traffic loop-free property holds. I. I Optimal routing has been important in the field of com- puter networks. We have two typical approaches for optimal routing. (1) One arises in a context of minimizing the overall cost (overall mean delay) of all users (or packets) from the arrival (origin) node of each user to its destination node through a number of communication links over the entire network. The optimal routing policy with this framework is called the overall optimal routing policy. (2) A second approach is a distributed one in which one seeks for a set of routing strategies for all users such that no user can decrease its cost (expected delay) by deviating from its strategy unilaterally. This could be viewed as a result of leaving each user the decision on which path to route through. This approach is called the individually optimal routing or selfish routing [15], [16]. The situation where each user has unilaterally minimized its cost is called a Wardrop equilibrium [13], [17] or a Nash equilibrium where no user has any incentive to make a unilateral decision to change its route. Traditionally, most work has focused on overall optimal routing in computer networks (e.g., [2], [7], [8]). For computer networks, however, minimizing the cost of each user from its arrival (origin) node to its destination node is a major concern of the user. Especially in the Internet, a user may wish to have its minimum cost for a given traffic pattern that results from the routing decisions of all other users. Thus, individually optimal routing has attracted in- creasing attention of researchers and practitioners in modern Hisao Kameda and Jie Li are with the Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba Science City, Ibaraki 305-8573, Japan. Email: {kameda,lijie}@cs.tsukuba.ac.jp. Eitan Altman is with INRIA, BP93, 06902 Sophia Antipolis Cedex, France. Email: Eitan.Altman@sophia.inria.fr. computer networks, and some research results have been obtained on individually optimal routing (or selfish routing) [1], [3], [12], [15], [16]. In most studies on optimal routing problems for com- munication networks in the literature (e.g., [1]–[3], [7], [8], [12]), the link cost is modeled as a simple function dependent only on the link flow itself. For convenience, we call it the traditional link-cost model. However, the link cost may not depend only the flow of the link itself but also on the flow through other links. For example, we consider a modern computer network implemented by a wireless network. In a wireless network, when a link connecting two nodes has more flow and, thus, uses more power, links neighboring the link may have less capacity. This is because there should be some mutual interference among the neighboring links, which may occur even in the case where electro-magnetic wave channels are best assigned to links, or more likely, in the case of ad hoc networks where nodes may move arround. Therefore, unlike the traditional link-cost model, in this paper, the cost on a link of a network is modeled by a function of the flows of all links of the entire network. We call it a general link-cost model. For convenience, we may call a network with multi-class users a multi-class network. We note, however, that, in these optimization problems, the cost to be optimized (the objective function) depends only on the link flow pattern while the instrument (the set of decision variables) is the path flow pattern. In this paper, we discuss individually and overall optimal routing problems in general link-cost models of multi-class networks, on which Dafermos has obtained some basic results [4]–[6]. Our treatment is, however, more general than hers in the following sense: Our model allows the origin- destination scheme that would allow each user of a class to enter any origin and leave any destination both available to the class with/without fixing the arrival rate at each origin and the departure rate at each destination; the link loop-free property is discussed; the relation between the case where the instrument is the path flow pattern and the case where it is the link flow pattern. We confirm that, for multiclass networks under our assumptions, there is an associate overall optimal routing problem that corresponds to the individually optimal routing problem, and that both have the same solution. We discuss the existence and uniqueness of the solution to the overall optimal routing and, then, on the basis of the associate relation, of the solution to the individually optimal routing, 43rd IEEE Conference on Decision and Control December 14-17, 2004 Atlantis, Paradise Island, Bahamas 0-7803-8682-5/04/$20.00 ©2004 IEEE FrA03.5 4217