Congestion Control Games Michael Schapira¤ Aviv Zohary Abstract Handling congestion in the Internet is vital to its functioning. End-to-end congestion-control mechanisms such as those embedded in the TCP protocol are in charge of addressing this crucial challenge. However, it may not always be in the best interest of end-points to comply with the protocol. I.e., it may be the case that end-points can better their throughput by not adhering to TCP. We address this problem by presenting an economic, or mechanism design, approach. We present a TCP-inspired game-theoretic model of congestion-control. This model captures the asynchronous nature of the network, the partial information of the end-points, and important aspects of the myopic TCP dynamics. In particular, we show that following a compliant strategy in our model is incentive-compatible, and optimal with respect to max-min fairness. In fact, we show that this is true even for deviations by coalitions of any size. \The Internet is an equilibrium, we just have to identify the game." [17] \While TCP may be the answer, we have yet to de¯ne the question." [9] 1 Introduction 1.1 Our Results Congestion control is a crucial task in today's Internet. The predominant form of congestion control is embodied in the Transmission Control Protocol (TCP) [8], which is one of the most widely deployed distributed algorithms in existence. However, TCP is still little understood from a theory of computer science, and economic theory perspectives. Researchers have put forth various theoretical frameworks aimed to enable the exploration of crucial aspects of TCP. Most notable, are the works of Kelly (see [11, 10] and references therein) and of Karp et al. [9]. In this paper we present a simpli¯ed model of TCP. Using this model, we address various computational/networking and game-theoretic TCP-inspired problems. In particular, our model captures the asynchrony of the network, the partial information that end-points have about the network and each other, and the greedy/myopic nature of TCP dynamics. At the heart of our model is a network graph G = (V; E) where V represents routers and E represents data links. Tra±c between every two end-points is modeled by a connection, repre- i 1 sented by a ¯xed path P in the network that leads from a source vertex si to a target vertex ti . Connections can transmit tra±c at di®erent rates along their ¯xed paths. Edges have capacities, and when more incoming tra±c enters an edge than can traverse that edge, routers drop some ¤The School of Computer Science and Engineering, The Hebrew University of Jerusalem, Israel. [email protected]. Supported by a grant from the Israel Science Foundation. yThe School of Computer Science and Engineering, The Hebrew University of Jerusalem, Israel. [email protected]. Supported by a grant from the Israel Science Foundation. 1Paths are ¯xed because in the Internet congestion control is independent of route establishment. 1 of the incoming packets. Routers can have very di®erent packet-dropping policies (e.g, they can handle congestion by dropping packets randomly, by prioritizing tra±c that originates in certain connections over tra±c of others, etc.). The choice of these packet-dropping policies is known have a signi¯cant e®ect on the properties and performance of the network. Our exploration of this basic model and its various extensions, as reflected in the organization of this paper, consists of three steps, each building upon the results of the previous one: 1) Flow Dynamics. (Section 2) We ¯rst consider congestion control when the end-points (con- nections) send tra±c at a constant rate. We show that even this restricted case is non-trivial as, in general, it is possible that the flow rates never \converge". I.e., it is possible that the network will never stabilize, and the capacity shares of edges allocated to di®erent connections continue to fluctuate inde¯nitely. In contrast, we prove the following result for the well known packet-dropping policy called Fair Queueing [2], that has been proposed (and implemented to some extent) for managing TCP tra±c on the Internet. Theorem: (Informal) If all routers in the network use Fair Queueing then the flow dynamics are guaranteed to converge to a unique \stable" solution that maximizes the max-min fairness. 2) End-to-End Congestion Control. (Section 3) Now, we allow connections (the source vertices) to adjust their transmission rates dynamically. We show that for some packet-dropping policies these dynamic-rate adjustments might be crucial for ensuring high network performance. We present a TCP-inspired protocol called Probing-Increase-Educated-Decrease (PIED). This protocol instructs each connection to gradually increase its transmission rate until it learns that the network is congested, at which case it must decrease its sending rate quickly and drastically. We show that: Theorem: (Informal) If all routers use Fair Queueing then PIED converges to a \good" solution even in a fully asynchronous network. (For other packet-dropping policies the transmission rates of the connections may oscillate inde¯nitely). Formally, the asynchrony of the network is modeled via an adversarial entity called the scheduler [13] that decides when connections can change their rates and when update messages reach their destinations. 3) Incentives and Congestion Control. (Section 4) Finally, we think of the connections as controlled by strategic agents with partial information. Each connection is assumed to be sel¯sh and wishes to maximize its average throughput over time. We ask the following question: Is it worth while to execute PIED? That is, can a connection better its throughput by not executing PIED? We show that in general this may not be the case. However, once again Fair Queueing helps. We prove our main result: Theorem: (Informal) If every router uses Fair Queueing then PIED is incentive-compatible in ex-post Nash. That is, if all connections but one are executing PIED, then the remaining connection cannot gain by not doing so. In fact, we show that PIED is even resilient to deviations by coalitions of connections (no collaboration results in a strict gain of all members in the coalition). We also show that PIED is Pareto optimal (a well-known notion of economic e±ciency). We stress that (unlike most works in mechanism design) all these results do not involve monetary transfers { a luxury we cannot permit in realistic \online" congestion-control protocols. Other works, like Akella et al. [1]), have also examined TCP from a game-theory perspective. However, while the question there was that of examining the ine±ciency of Nash equilibria (\price of anarchy" [12, 16]) we take a mechanism design approach to congestion control. This continues the line of research presented in the seminal papers of Nisan and Ronen [14], and of Feigenbaum 2 et al. [4] (and bears a resemblance to the work of Gibbens and Kelly [7]). Similar mechanism design questions have been considered in the context of another central Internet protocol, the Border Gateway Protocol (BGP) that handles interdomain routing (see Feigenbaum et al. [3] and subsequent works). 1.2 The Practical Implications of Our Results We do not presume to realistically model TCP. The major shortcoming of our model (and the ones that precede it) has to do with the fact that TCP is sensitive to the loss of packets (while in our model connections only care about throughput). This sensitivity implies that if packets have been dropped at some point along the path that belong to a connection, then they must be retransmitted (perhaps along with other packets that precede them). The resulting pattern of retransmission and loss of packets further aggravates the congestion along the path and may trigger a so-called \congestion collapse" in the network [6]. Exhibiting models in which connections are penalized in a more realistic way is a major open question. However, our model does enable the study of some of TCP's important aspects (asynchrony, partial information, greedy heuristic, etc.). In particular, there are reasons for congestion collapse that are captured by our model. For instance, Floyd and Fall [6] mention that congestion may occur due to packets that are dropped close to their targets after having consumed resources along their path (perhaps having been transmitted that far at the expense of other packets). Additionally, our model is useful in the analysis of new congestion control paradigms, like the one suggested by Raghavan and Snoeren in [15]. In [15], it is suggested to use coding techniques to handle packet losses. This results in a setting similar to our model (connections only care about throughput). 2 Flow Dynamics In this section, we consider congestion-control when the end-points send tra±c at a constant rate. We show that even analyzing this restricted model (de¯ned in Subsection 2.1) is non-trivial as it is possible that the flow rates never \converge"(see Subsection 2.2). We prove that when all routers in the network drop packets according to Fair Queueing [2] then these flow dynamics are guaranteed to converge to a unique solution that maximizes the max-min fairness (Subsection 2.3). 2.1 The Model We shall now formally present our model. Let G = (V; E) be the directed network graph. We assume that each directed edge e 2 E in the graph has a capacity denoted by ce 2 R+, and a 2 latency denoted by ¢e. Each¡ vertex v 2 V¢ represents a router in the network , and each edge represents a data link. Let P 1;P 2;:::;P n be a set of simple directed paths in G.