queueing analysis of a feedback- controlled (tcp/ip) network gaurav rainadamon wischikmark handley...

Queueing analysis of a feedback-controlled (TCP/IP) network

Gaurav Raina Damon Wischik Mark HandleyCambridge UCL UCL

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

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 In this paper we argue that the rule-of-thumb (B = RTT*C) is now outdated and incorrect for backbone routers.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 Appenzeller Isaac Keslassy Nick McKeownStanford University Stanford University Stanford University

http://tiny-tera.stanford.edu/~nickm/papers/index.html

TCPif (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;


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);


_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]

band

wid

th [

0-10

0 kB

/sec

]

How TCP shares capacity

sum of flowbandwidths

time

availablebandwidth

individualflow

bandwidths

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/RTT2 [pkts/sec]

and reduces x by x/2 for every packet loss• average increase in rate = average decrease in rate:

1/RTT2 = (p x) x/2

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/RTT2 [pkts/sec]

and reduces x by x/2 for every packet loss• average increase in rate = average decrease in rate:

1/RTT2 = (p x) x/2

• Consider a link with N identical flowsLet 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

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 MacfadyenStatistical 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

Fixed-point analysis

0.5 1 1.5 2

-4

-3

-2

-1

C*RTT=4 pkts

C*RTT=20 pkts

C*RTT=100 pkts

traffic intensity x/C

log10 ofpkt loss

probability

Queue simulations

queuesize 10

20

10

20

30

10

20

30

40

50

60

10

20

30

40

50

60

70

80

90

20 40 60 80

0.5

1

20 40 60 80

0.5

1

20 40 60 80

0.5

1

20 40 60 80

0.5

1arrivalrate x time

N=50 N=100 N=500 N=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 N1/2B (B=3 pkts)

A Poisson Limit for Buffer Overflow Probabilities SIGCOMM 2002

Jin Cao, Kavita RamananBell 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

Mean-field limit• Consider a link with N flows

and capacity NC and buffer N1/2B• Let xt be the average bandwidth at time t

Let pt be the packet loss probability at time t• As N we believe a mean-field limit holds.

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

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

20 40 60 80 1000.60.8

1.21.4

20 40 60 80 1000.60.8

1.21.4

time

arrivalrate x/C

Instability plot

0.5 1 1.5 2

-4

-3

-2

-1

C*RTT=4 pkts

C*RTT=20 pkts

C*RTT=100 pkts

traffic intensity x/C

log10 ofpkt loss

probability

Illustration: 20 flows

Standard TCP, single bottleneck link, no AQMservice 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)





B=2,400 pkts(rule of thumb)





B=24,000 pkts(rule of thumb)

Alternative buffer-sizing rules

b25 b100 b400

0.5 1 1.5

-4

-3

-2

-1

b10 b20 b50

0.5 1 1.5

-4

-3

-2

-1

0.5 1 1.5 2

-4

-3

-2

-1

b50 b1000

0.5 1 1.5-6

-5

-4

-3

-2

-1

r

p

Stanford rule buffer = bandwidth*delay / sqrt(#flows)

or Rule of thumb, no AQM buffer = bandwidth*delay

Rule of thumb with REDbuffer=bandwidth*delay*{¼,1,4}

Kelly rule, no AQMbuffer={10,20,50} pkts

Kelly rule, no AQM, ScalableTCPbuffer={50,1000} pkts

Scalable TCP: improving performance in highspeed wide area networks SIGCOMM CCR 2003

Tom KellyCERN -- 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/

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.TanStatistical 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

Teleological description

• Consider several TCP flows sharing a single link

• Let xr 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 solvemaximise r U(xr) - P(y,C) over xr0 where y=

r xr

x

U(x

)

y

P(y

,C)

C

Teleological description

• Consider several TCP flows sharing a single link

• Let xr 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 solvemaximise r U(xr) - P’(y,C) over xr0 where y=

r xr

x

U(x

)

y

P’(

y,C

)

C

By reducing buffer size, we increase the penalty for high utilization.

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 sizelead 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.

queueing analysis of a feedback- controlled (tcp/ip) network gaurav rainadamon wischikmark handley...

Documents