1 Congestion control with adaptive multipath routing based on optimization Fernando Paganini ORT University, Montevideo, Uruguay Abstract— The paper considers a TCP/IP-style network with flow control at end-systems based on congestion feedback, and routing decisions at network nodes on a per-destination basis. The only generalization with respect to standard IP is that routers split their traffic, for each destination, among their outgoing links. We pose two optimization problems, that generalize and combine those used in the congestion control and traffic engineering literature. In contrast to other work in multipath congestion control, we use variables that are available at each node (source or router). We prove that decentralized algorithms built by combining primal or dual congestion control with adaptation of router splits, converge globally to optimal points. Some comments on practical implications of these results are given. I. I NTRODUCTION The use of optimization and economic ideas in net- work resource allocation has a long history, going back to the study of transportation networks [21]. A classical problem is to minimize a cost of the form  l φ l (y l ) (1) where {y l } represent flows in links of a network, subject to external traffic demands and flow conservation con- straints at network nodes, and φ l (·) is a cost function, often taken to represent delay. A question that has been extensively studied (see [15] and references therein) are the equilibria that result from selfish routing decisions by traffic agents, and their inefficiencies in regard to the above social cost. In IP networks, where routing decisions are made by routers, this type of optimization has been applied to problems of traffic engineering [6], [17], where a network operator seeks routes to serve a “traffic matrix” of demand. This is often done offline and subsequently implemented by some means (creation of MPLS tunnels or selection of weights in IP routing). In [5], an adaptive method based on optimization is proposed, running at the access points to an MPLS network, for a real-time optimization of (1). Most relevant to this paper is the work of Gallager [7], continued in [2], where network nodes perform an adaptive minimization of delay by controlling the split of traffic through outgoing links. Email: paganini@ort.edu.uy. The author is on leave from the Uni- versity of California, Los Angeles. Research supported in part by the David and Lucille Packard Foundation A complementary line of research on congestion con- trol has used optimization tools to control demand into the network. The basic such problem, proposed by Kelly [9], is the maximization of  k U k (x k ) (2) where x k is the input rate of a traffic source, and U k (·) an associated utility function. For single-path routing, there has been substantial progress in finding decentralized algorithms at sources and links to solve this problem, and relating these algorithms to current TCP congestion control (see [9], [12], [18], [13]). Compared to this extensive research on either the supply or the demand sides of the problem, their com- bination (adapting both routing and source traffic) has been less studied. If single-path routing is imposed as in standard IP, it complicates the optimization of (2) through non-convexity [20]. However if we allow multiple routes, the problem is well-behaved, as was already noted in [9]. Here, and in [8], [19], [11], the proposal is to use as adaptation variables the components of rate for each path from source to destination. This gives convergent decentralized algorithms, but appears to be an impractical proposition in a large network. Achieving optimality would require sources to separately control rate on an exponential number of end-to-end paths, based on all combinations of routing choices along the way. This is not scalable, and constraining the set of paths will reduce utility. Also, propagating this control inside the network requires source routing, a significant departure from current Internet practice. In this paper we propose to use only adaptation variables with local meaning: source rates, link conges- tion prices, and the traffic split at each router among its outgoing links for each destination. This combines congestion control with the adaptive routing of [7]. We will present natural optimization problems, and study how they can be solved in a decentralized way by primal and dual congestion control algorithms combined with a suitably chosen adaptation of traffic splits. Other related work we recently became aware of is the upcoming paper [4], that formulates a very similar optimization problem for wireless scheduling. Given the differences in their dynamic solution and implementation context, we do not know at present how both compare.