1. Queueing analysis of a feedback- controlled (TCP/IP) network Gaurav RainaDamon WischikMark Handley CambridgeUCLUCL

2. Some Internet History 1974: First draft of TCP/IP “A protocol for packet network interconnection”, Vint Cerf and Robert Kahn 1983: ARPANET switches on TCP/IP 1986: Congestion collapse 1988: Congestion control for TCP “Congestion avoidance and control”, Van Jacobson “A Brief History of the Internet”, the Internet Society

3. Sizing router buffers SIGCOMM 2004 Abstract. All Internet routers contain buffers to hold packets during times of congestion. Today, the size of the buffers is determined by the dynamics of TCP's congestion control algorithm. In particular, the goal is to make sure that when a link is congested, it is busy 100% of the time; which is equivalent to making sure its buffer never goes empty. A widely used rule-of-thumb states that each link needs a buffer of size B = RTT*C, where RTT is the average round-trip time of a flow passing across the link, and C is the data rate of the link. For example, a 10Gb/s router linecard needs approximately 250ms*10Gb/s = 2.5Gbits of buffers; and the amount of buffering grows linearly with the line-rate. Such large buffers are challenging for router manufacturers, who must use large, slow, off-chip DRAMs. And queueing delays can be long, have high variance, and may destabilize the congestion control algorithms. In this paper we argue that the rule-of-thumb (B = RTT*C) is now outdated and incorrect for backbone routers. This is because of the large number of flows (TCP connections) multiplexed together on a single backbone link. Using theory, simulation and experiments on a network of real routers, we show that a link with N flows requires no more than B = (RTT*C)/  N, for long-lived or short-lived TCP flows. The consequences on router design are enormous: A 2.5Gb/s link carrying 10,000 flows could reduce its buffers by 99% with negligible difference in throughput; and a 10Gb/s link carrying 50,000 flows requires only 10Mbits of buffering, which can easily be implemented using fast, on-chip SRAM. Guido AppenzellerIsaac KeslassyNick McKeown Stanford UniversityStanford UniversityStanford University http://tiny-tera.stanford.edu/~nickm/papers/index.html

4. TCP if (seqno > _last_acked) { if (!_in_fast_recovery) { _last_acked = seqno; _dupacks = 0; inflate_window(); send_packets(now); _last_sent_time = now; return; } if (seqno < _recover) { uint32_t new_data = seqno - _last_acked; _last_acked = seqno; if (new_data < _cwnd) _cwnd -= new_data; else _cwnd=0; _cwnd += _mss; retransmit_packet(now); send_packets(now); return; } uint32_t flightsize = _highest_sent - seqno; _cwnd = min(_ssthresh, flightsize + _mss); _last_acked = seqno; _dupacks = 0; _in_fast_recovery = false; send_packets(now); return; } if (_in_fast_recovery) { _cwnd += _mss; send_packets(now); return; } _dupacks++; if (_dupacks!=3) { send_packets(now); return; } _ssthresh = max(_cwnd/2, (uint32_t)(2 * _mss)); retransmit_packet(now); _cwnd = _ssthresh + 3 * _mss; _in_fast_recovery = true; _recover = _highest_sent; } time [0-8 sec] bandwidth [0-100 kB/sec]

5. How TCP shares capacity sum of flow bandwidths time available bandwidth individual flow bandwidths

6. Macroscopic description of TCP Let x be the mean bandwidth of a flow [pkts/sec] Let RTT be the flow’s round-trip time [sec] Let p be the packet loss probability The TCP algorithm increases x at rate 1/RTT 2 [pkts/sec] and reduces x by x/2 for every packet loss average increase in rate = average decrease in rate: 1/RTT 2 = (p x) x/2

7. Macroscopic description Let x be the mean bandwidth of a flow [pkts/sec] Let RTT be the flow’s round-trip time [sec] Let p be the packet loss probability The TCP algorithm increases x at rate 1/RTT 2 [pkts/sec] and reduces x by x/2 for every packet loss average increase in rate = average decrease in rate: 1/RTT 2 = (p x) x/2 Consider a link with N identical flows Let NC be the capacity of the link [pkts/sec] packet loss ratio = fraction of work that exceeds service rate: p = (Nx-NC) + /Nx = (x-C) + /x

8. Fixed-Point Models for the End-to-End Performance Analysis of IP Networks ITC 2000 RJ Gibbens, SK Sargood, C Van Eijl, FP Kelly, H Azmoodeh, RN Macfadyen, NW Macfadyen Statistical Laboratory, Cambridge; and BT, Adastral Park Abstract. This paper presents a new approach to modeling end-to-end performance for IP networks. Unlike earlier models, in which end stations generate traffic at a constant rate, the work discussed here takes the adaptive behaviour of TCP/IP into account. The approach is based on a fixed-point method which determines packet loss, link utilization and TCP throughput across the network. Results are presented for an IP backbone network, which highlight how this new model finds the natural operating point for TCP, which depends on route lengths (via round-trip times and number of resources), end-to-end packet loss and the number of user sessions. http://www.statslab.cam.ac.uk/~frank/PAPERS/fpmee.html

9. Fixed-point analysis C*RTT=4 pkts C*RTT=20 pkts C*RTT=100 pkts traffic intensity x/C log 10 of pkt loss probability

10. Queue simulations queue size arrival rate x time N=50N=100N=500N=1000 What’s the queueing theory behind p = (x-C) + /x ? Where does buffer size come in? Simulate a queue fed by N Poisson flows, each of rate x pkts/sec (x=0.95 then 1.05 pkts/sec) served at rate NC (C=1 pkt/sec) with buffer size N 1/2 B (B=3 pkts)

11. A Poisson Limit for Buffer Overflow Probabilities SIGCOMM 2002 Jin Cao, Kavita Ramanan Bell Labs Abstract. A key criterion in the design of high-speed networks is the probability that the buffer content exceeds a given threshold. We consider n independent traffic sources modelled as point processes, which are fed into a link with speed proportional to n. Under fairly general assumptions on the input processes we show that the steady state probability of the buffer content exceeding a threshold b>0 tends to the corresponding probability assuming Poisson input processes. We verify the assumptions for a large class of long-range dependent sources commonly used to model data traffic. Our results show that with superposition, significant multiplexing gains can be achieved for even smaller buffers than suggested by previous results, which consider O(n) buffer size. Moreover, simulations show that for realistic values of the exceedance probability and moderate utilisations, convergence to the Poisson limit takes place at reasonable values of the number of sources superposed. This is particularly relevant for high-speed networks in which the cost of high-speed memory is significant. http://www.ieee-infocom.org/2002/papers/655.pdf

12. Mean-field limit Consider a link with N flows and capacity NC and buffer N 1/2 B Let x t be the average bandwidth at time t Let p t be the packet loss probability at time t As N  we believe a mean-field limit holds.

13. Mean-field limit Fluid-based Analysis of a Network of AQM Routers Supporting TCP Flows with an Application to RED SIGCOMM 2000 Vishal Misra, Wei-Bo Gong, Don Towsley Rate-based versus queue-based models of congestion control ACM Sigmetrics 2004 Supratim Deb, R. Srikant Mean field convergence of a rate model of multiple TCP connections through a buffer implementing RED To appear in Annals of Applied Probability David McDonald, Julien Reynier

14. Stability/instability of fluid model For some values of C*RTT, the differential equation is stable For others it is unstable and there are oscillations (i.e. the flows are partially synchronized) When it is unstable, we can calculate the amplitude of the oscillations time arrival rate x/C

15. Instability plot C*RTT=4 pkts C*RTT=20 pkts C*RTT=100 pkts traffic intensity x/C log 10 of pkt loss probability

16. Illustration: 20 flows Standard TCP, single bottleneck link, no AQM service C=60 pkts/sec/flow, RTT=200 ms, #flows N=20 B=20 pkts (Kelly rule) B=54 pkts (Stanford rule) B=240 pkts (rule of thumb)

17. Illustration: 200 flows Standard TCP, single bottleneck link, no AQM service C=60 pkts/sec/flow, RTT=200 ms, #flows N=200 B=20 pkts (Kelly rule) B=170 pkts (Stanford rule) B=2,400 pkts (rule of thumb)

18. Illustration: 2000 flows Standard TCP, single bottleneck link, no AQM service C=60 pkts/sec/flow, RTT=200 ms, #flows N=2000 B=20 pkts (Kelly rule) B=537 pkts (Stanford rule) B=24,000 pkts (rule of thumb)

19. Alternative buffer-sizing rules Stanford rule buffer = bandwidth*delay / sqrt(#flows) or Rule of thumb, no AQM buffer = bandwidth*delay Rule of thumb with RED buffer=bandwidth*delay*{¼,1,4} Kelly rule, no AQM buffer={10,20,50} pkts Kelly rule, no AQM, ScalableTCP buffer={50,1000} pkts

20. Scalable TCP: improving performance in highspeed wide area networks SIGCOMM CCR 2003 Tom Kelly CERN -- IT division Abstract. TCP congestion control can perform badly in highspeed wide area networks because of its slow response with large congestion windows. The challenge for any alternative protocol is to better utilize networks with high bandwidth-delay products in a simple and robust manner without interacting badly with existing traffic. Scalable TCP is a simple sender-side alteration to the TCP congestion window update algorithm. It offers a robust mechanism to improve performance in highspeed wide area networks using traditional TCP receivers. Scalable TCP is designed to be incrementally deployable and behaves identically to traditional TCP stacks when small windows are sufficient. The performance of the scheme is evaluated through experimental results gathered using a Scalable TCP implementation for the Linux operating system and a gigabit transatlantic network. The preliminary results gathered suggest that the deployment of Scalable TCP would have negligible impact on existing network traffic at the same time as improving bulk transfer performance in highspeed wide area networks. http://www-lce.eng.cam.ac.uk/~ctk21/scalable/

21. Rate control in communication networks: shadow prices, proportional fairness and stability Journal of the Operational Research Society, 1998 F.P.Kelly, A.K.Maulloo, D.K.H.Tan Statistical Laboratory, Cambridge Abstract. This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalizations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterized by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimization problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimization problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalized to include routing control, and provide natural implementations of proportionally fair pricing. http://www.statslab.cam.ac.uk/~frank/rate.html

22. Teleological description Consider several TCP flows sharing a single link Let x r be the mean bandwidth of flow r [pkts/sec] Let y be the total bandwidth of all flows [pkts/sec] Let C be the total available capacity [pkts/sec] TCP and the network act so as to solve maximise  r U(x r ) - P(y,C) over x r  0 where y=  r x r x U(x)U(x) y P(y,C) C

23. Teleological description Consider several TCP flows sharing a single link Let x r be the mean bandwidth of flow r [pkts/sec] Let y be the total bandwidth of all flows [pkts/sec] Let C be the total available capacity [pkts/sec] TCP and the network act so as to solve maximise  r U(x r ) - P’(y,C) over x r  0 where y=  r x r x U(x)U(x) y P’(y,C) C By reducing buffer size, we increase the penalty for high utilization.

24. Conclusion Analysis: –Use fixed-point model to find the equilibrium point; –Find a mean-field limit, and calculate how stable it is. Three rules for choosing buffer size lead to three different mean-field limits. –Rule of thumb e.g. 10 Gbytes –Stanford rule e.g. 100 Mbytes –Kelly rule e.g. 20 kbytes The network acts to solve an optimization problem. –It may or may not attain the solution. –We can choose which optimization problem, by choosing the right buffer size & changing TCP’s code.