1. Queueing theory for TCP Damon Wischik, UCL Gaurav Raina, Cambridge

2. DJW 2007 2/24 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; } TCP transmission rate [0–100 kB/sec] time [0–8 sec]

3. DJW 2007 3/24 Models for TCP Single TCP sawtooth One can explicitly calculate the mean transmission rate, stationary distribution etc.

4. DJW 2007 4/24 Models for TCP * ***** * Single TCP sawtooth Billiards model Baccelli+Hong 2002 Each flow increases its transmission rate linearly... until the total transmission rate hits capacity, when some flows experience dropped packets... and those flows back off. Each flow increases its transmission rate linearly... until the total transmission rate hits capacity, when some flows experience dropped packets... and those flows back off.

5. DJW 2007 5/24 Models for TCP * ***** * Single TCP sawtooth Billiards model Baccelli+Hong 2002 Fluid model Kelly+Maulloo+Tan 1998 Misra+Gong+Towsley 2000 Baccelli+McDonald+Reynier 2002 Approximate the aggregate TCP traffic by means of a differential equation... The differential equation has increase & decrease terms, which reflect the TCP sawtooth. Approximate the aggregate TCP traffic by means of a differential equation... The differential equation has increase & decrease terms, which reflect the TCP sawtooth.

6. DJW 2007 6/24 Models for TCP * ***** * Single TCP sawtooth Billiards model Baccelli+Hong 2002 Fluid model Kelly+Maulloo+Tan 1998 Misra+Gong+Towsley 2000 Baccelli+McDonald+Reynier 2002 QUESTIONS When are these models good? What is the formula for packet drop probability, to plug into the models? How does it depend on buffer size etc.? QUESTIONS When are these models good? What is the formula for packet drop probability, to plug into the models? How does it depend on buffer size etc.?

7. DJW 2007 7/24 Simulation results queue size [0–100%] drop probability [0–30%] traffic intensity [0–100%] time [0–5s] small bufferlarge bufferint. buffer

8. DJW 2007 8/24 Simulation results queue size [0–100%] drop probability [0–30%] traffic intensity [0–100%] time [0–5s] small bufferlarge bufferint. buffer When we vary the buffer size, the system behaviour changes dramatically. WHY? HOW? ANSWER It depends on the timescale and spacescale of fluctuations in the arrival process... ANSWER It depends on the timescale and spacescale of fluctuations in the arrival process...

9. DJW 2007 9/24 Formulate a maths question What is the limiting queue-length process, as N , in the three regimes –large buffers B (N) =BN –intermediate buffers B (N) =B√N –small buffers B (N) =B Why are these the interesting limiting regimes? What are the alternatives? [Bain, 2003] N TCP flows Service rate NC Buffer size B (N) Round trip time RTT

10. DJW 2007 10/24 Intermediate buffers B (N) = B√N Consider a standalone queue –fed by N Poisson flows, each of rate x pkts/sec where x varies slowly, served at constant rate NC, with buffer size B√N –i.e. an M Nx/ /D NC / / 1 /B√N queue –x t =0.95 then 1.05 pkts/sec, C=1 pkt/sec, B=3 pkt Observe –queue size fluctuates very rapidly; busy periods have duration O(1/N) so packet drop probability p t is a function of instantaneous arrival rate x t –queue size flips quickly from near-empty to near-full, in time O(1/√N) –packet drop probability is p t =(x t -C) + /x t N=50N=100N=500N=1000 queue size traffic intensity time

11. DJW 2007 11/24 Intermediate buffers B (N) = B√N Consider a standalone queue –fed by N Poisson flows, each of rate x pkts/sec where x varies slowly, served at constant rate NC, with buffer size B√N –i.e. an M Nx/ /D NC / / 1 /B√N queue –x t =0.95 then 1.05 pkts/sec, C=1 pkt/sec, B=3 pkt Observe –queue size fluctuates very rapidly; busy periods have duration O(1/N) so packet drop probability p t is a function of instantaneous arrival rate x t –queue size flips quickly from near-empty to near-full, in time O(1/√N) –packet drop probability is p t =(x t -C) + /x t N=50N=100N=500N=1000 queue size traffic intensity time This formula was proposed for TCP by Srikant 2004

12. DJW 2007 12/24 Closing the loop In the open-loop queue –we assumed Poisson inputs with slowly varying arrival rate –queue size flips quickly from near- empty to near-full –queue size fluctuates very rapidly, with busy periods of duration O(1/N), so drop probability p t is a function of instantaneous arrival rate x t In the closed-loop TCP system –the round trip time means that we can treat inputs over time [t,t+RTT] as an exogenous arrival process to the queue –the average arrival rate should vary slowly, according to a differential equation –the aggregate of many point processes converges to a Poisson process, and this carries through to queue size [Cao+Ramanan 2002] There is thus a separation of timescales –queue size fluctuates over short timescales, and we obtain p t from the open-loop M/D/1/B (N) model –TCP adapts its rate over long timescales, according to the fluid model differential equation

13. DJW 2007 13/24 Intermediate buffers B (N) = B√N queue size [0–619pkt] drop probability [0–30%] traffic intensity [0–100%] queue size [594–619pkt] time [0–5s] time [50ms] 2Gb/s link, C=50 pkt/sec/flow 5000 TCP flows RTT 150–200ms buffer 619pkt simulator htsim by Mark Handley, UCL

14. DJW 2007 14/24 Intermediate buffers B (N) = B√N queue size [0–619pkt] drop probability [0–30%] traffic intensity [0–100%] queue size [594–619pkt] time [0–5s] time [50ms] 2Gb/s link, C=50 pkt/sec/flow 5000 TCP flows RTT 150–200ms buffer 619pkt simulator htsim by Mark Handley, UCL Queue size fluctuations are O(1), invisible at the O(√N) scale When the traffic intensity changes between capacity, the buffer can fill or drain in time O(1/√N)

15. DJW 2007 15/24 Instability and synchronization When the queue is not full, there is no packet loss, so transmission rates increase additively When the queue becomes full, there are short periods of concentrated packet loss This makes the flows become synchronized queue size [0–619pkt] TCP window sizes [0-25pkt] time [0–5s] This is the billiards model for TCP

16. DJW 2007 16/24 Large buffers B (N) = BN queue size [0–BN pkt] drop probability [0–30%] traffic intensity [0–100%] queue size [(B-25)–B pkt] time [0–5s] time [50ms] Queue size fluctuations are O(1), invisible on the O(N) scale... The total arrival rate is Nx t and the service rate is NC, so it takes time O(1) for the buffer to fill or drain and we can calculate the packet drop probability when the queue is full

17. DJW 2007 17/24 Small buffers B (N) = B queue size [0–BN pkt] drop probability [0–30%] traffic intensity [0–100%] queue size [(B-25)–B pkt] time [0–5s] time [50ms] Queue size fluctuations are of size O(1), over timescale O(1/N), enough to fill or drain the buffer... The queue size changes too quickly to be controlled, but the queue size distribution is controlled by the average arrival rate x t

18. DJW 2007 18/24 Conclusion * ***** * Single TCP sawtooth Billiards model Fluid model Fine for a single flow Good for a small number of flows... This is also an extreme case of the fluid model, when flows are synchronized due to intermediate-sized buffers Good for a small number of flows... This is also an extreme case of the fluid model, when flows are synchronized due to intermediate-sized buffers Arises as a law- of-large- numbers limit, when there are many TCP flows

19. DJW 2007 19/24 Conclusion * ***** * Single TCP sawtooth Billiards model Fluid model Proved by McDonald+Reynier

20. DJW 2007 20/24 Instability & buffer size On the basis of these models, and of a control-theoretic analysis of the limit cycles that ensue, we claimed –a buffer of 30–50 packets should be sufficient –intermediate buffers will make the system become synchronized, and utilization will suffer –large buffers get high utilization, but cause synchronization and delay queue size distribution [0–100%] B=10pkt B=20pkt B=30pkt B=40pkt B=50pkt B=100pkt traffic intensity distribution [85–110%] Plot by Darrell Newman, Cambridge ½=1

21. DJW 2007 21/24 Instability & buffer size On the basis of these models, and of a control-theoretic analysis of the limit cycles that ensue, we claimed –a buffer of 30–50 packets should be sufficient –intermediate buffers will make the system become synchronized, and utilization will suffer –large buffers get high utilization, but cause synchronization and delay queue size distribution [0–100%] B=10pkt B=20pkt B=30pkt B=40pkt B=50pkt B=100pkt traffic intensity distribution [85–110%] Plot by Darrell Newman, Cambridge ½=1 But when other people ran simulations they found... Small buffers lead to terribly low utilization Intermediate buffers work much better But when other people ran simulations they found... Small buffers lead to terribly low utilization Intermediate buffers work much better

22. DJW 2007 22/24 Burstiness The problem is that the Poisson limit breaks down –A key step in the C+R proof is showing that each flow contributes at most one packet in a busy period –But TCP, on a fast access link, will produce bursty traffic: sources can send an entire window-full of packets back-to- back For very fast access speeds, i.e. very bursty TCP traffic, a batch-M/D/1/B model is appropriate

23. DJW 2007 23/24 Burstiness & access speeds Simulations of a system with a single bottleneck link with capacity NC, where each flow is throttled by a slow access link access speed = 5C 40C TCP traffic is bursty over short timescales, so the queue's stochastic fluctuations are large... TCP has continual feedback, and can nearly maintain a steady state TCP traffic is bursty over short timescales, so the queue's stochastic fluctuations are large... TCP has continual feedback, and can nearly maintain a steady state TCP traffic is Poisson-like over short timescales, so the queue's stochastic fluctuations are small... TCP has bang-bang feedback and it is unable to maintain a steady state TCP traffic is Poisson-like over short timescales, so the queue's stochastic fluctuations are small... TCP has bang-bang feedback and it is unable to maintain a steady state

24. DJW 2007 24/24 Burstiness & access speeds Simulations of a system with a single bottleneck link with capacity NC, where each flow is throttled by a slow access link access speed = 5C 40C You have to be careful about short-timescale traffic statistics when you model TCP Queueing theory still has a role! You have to be careful about short-timescale traffic statistics when you model TCP Queueing theory still has a role!