On-Line End-to-End Congestion Control

Congestion control in the current Internet is accomplished mainly by TCP/IP. To understand the macroscopic network behavior that results from TCP/IP and similar endto-end protocols, one main analytic technique is to show that the the protocol maximizes some global objective function of the network traffic. Here we analyze a particular end-to-end, MIMD (multiplicative-increase, multiplicative-decrease) protocol. We show that if all users of the network use the protocol, and all connections last for at least logarithmically many rounds, then the total weighted throughput (value of all packets received) is near the maximum possible. Our analysis includes round-trip-times, and (in contrast to most previous analyses) gives explicit convergence rates, allows connections to start and stop, and allows capacities to change. 1. Congestion control and optimization Congestion control in the current Internet is accomplished mainly by TCP/IP — 90% of Internet traffic is TCP-based [41]. Meanwhile the design and analysis of TCP and other end-to-end congestion-control protocols are only partially understood and are becoming the subject of increasing attention [25, 28]. One main analytic technique is to interpret the protocol as solving some underlying combinatorial optimization problem on the network — to show that the protocol causes the traffic distribution, over time, to optimize some global objective function [41, 29, 26, 2, 40, 17, 30, 52, 9, 46, 44, 54]. For example, a continuous analogue of TCP-Reno (under various assumptions about the network) maximizes ∑ i τ −1 i arctan(τixi), where τi is the (constant) round-trip time of packets sent by the ith user and the variable xi is that user’s transmission rate [41]. (Each term in the sum is a smoothed threshold function.) Similarly, a continuous analogue of TCP-Vegas maximizes ∑ i αidi logxi where di is the round-trip propagation delay of packets sent by the ith user and αi is a protocol parameter. Typically these results concern a continuous analogue of the protocol. They analyze a system of differential equations where time is continuous and each rate xi is a continuous function of time. They show (e.g. using a Lyapunov function [22, 50]) that, as time tends to infinity, the vector x tends to an equilibrium point that maximizes the objective function in question. This approach is very general. It has been used to design and analyze protocols other than TCP, including protocols that require the network routers to explicitly transmit congestion information to the users by means other than packet loss and latency. (Typically the congestion signals are dual variables — Lagrange multipliers, or “shadow prices” — with interesting economic interpretations.) For a survey of results of this kind, see [41, 29]. A few specific technical papers include [26, 2, 40, 17, 30, 52, 9]. Our interest in this paper is in protocols that are both online and end-to-end: the number of packets sent on a path p at time t is determined solely by the number of packets sent and received on p in previous rounds. (The protocol has no a-priori knowledge of the network or how the paths relate to it, and learns about the network only through packet loss.) Such protocols are implementable in the current Internet, without modifications to routers. The protocol we analyze in this paper, which we call the Linear MIMD Protocol, is an example (see Fig. 1). It can be implemented by modifying only the TCP server. Generally, existing works (that formally analyze end-toend protocols implementable in the current Internet) assume that all connections start at time 0 and continue indefinitely in a static network. They show that the objective function is optimized in the limit as time tends to infinity. (See Low’s survey and Kelly’s survey [29].) The only exceptions that we are aware of are for the special case of a singlebottleneck network [18, 9]. Thus, we do not yet have a complete theoretical understanding of speed of convergence (noted as important by Low in his survey [41]) or of the effects of dynamic connections and changing network conditions. (These issues have been studied empirically, e.g. Proceedings of the 43 rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’02) 0272-5428/02 $17.00 © 2002 IEEE Inputs for connection p: starting rate f0(p) > 0, active time interval Tp = {sp, sp + 1, . . . , ep}. Parameters: αp > 0, βp ∈ (0, 1) At times t = sp, sp + 1, . . . , sp + τp take sent(p, t) := f0(p). At times t = sp + 1 + τp, . . . , ep, take sent(p, t) := sent(p, t − 1 − τp) × [ 1 + αp − βplsr(p, t− 1) ]