Estimating Congestion in TCP Traffic Stephan Bohacek and Boris Rozovskii University of Southern California Objective: Develop stochastic model of TCP Necessary ingredients: Models of the network. Specifically packet drop probability and roundtrip time. The model parameters are indicators of congestion.
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Estimating Congestion in TCP Traffic Stephan Bohacek and Boris Rozovskii University of Southern California Objective: Develop stochastic model of TCP Necessary.
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Estimating Congestion in TCP Traffic
Stephan Bohacek and Boris Rozovskii University of Southern California
Objective: Develop stochastic model of TCPNecessary ingredients: Models of the network. Specifically packet drop probability and roundtrip time. The model parameters are indicators of congestion.
Outline
• A very brief introduction to TCP
• Modeling packet drop probability– Modeling roundtrip time– Dynamics of drop probability parameters
• A diffusion model of roundtrip time– Estimating the model parameters
• Stochastic models of TCP
• Results and insights
An Introduction to TCP
• TCP is acknowledgment based. The sender sends a packet of data. When the receiver receives the packet, it responds by sending an acknowledgment packet to the sender.
• If the sender fails to receive an acknowledgment, it assumes that the packet has been dropped and decreases the sending rate.
• If the sender receives an acknowledgment, it assumes that the network is not congested and increases the sending rate.
• The congestion window, X, defines the maximum number unacknowledged packets.
• When the sender receives an acknowledgment it increases the congestion window by 1/[Xt].
• When a packet drop is detected, the sender divides the congestion window in half.
Some TCP Details
Note: the congestion window is not the sending rate. sending rate = congestion window / roundtrip time.
Number of unacknowledgedpackets equals Cwnd. Send no more packets.
Packet arrives at receiver.Receiver sends an acknowledgment.
roundtriptime
Time Series of the Congestion Window(simulation)
50 55 60 65 70 75 80 850
5
10
15
20
25
30
35
40
45
50
Seconds
Con
gest
ion
Win
dow
linear increasewhen no drops occur
divide by two when a drop is detected
Time Series of the Congestion Window (simulation)
50 100 150 200 250 300 350 400 450 5000
5
10
15
20
25
30
35
40
45
50
Seconds
Con
gest
ion
Win
dow
Stochastic model of TCP•Stochastic model of packet drops.•Stochastic model of the roundtrip time.
Drop Models
• Ott (1997) considered deterministic drops.• Padhye (1998) assumed drops to be highly correlated over short
time scales, but independent over longer time scales.• Altman (2000) assumes drops are bursty.• Altman (2000) drop events are modeled as renewal processes with
particular examples, deterministic, Poisson, i.i.d., and Markovian.• Savari (1999) drop events are modeled as Poisson where the
intensity depends on the window size of the TCP protocol.
Models for Packet Drop Probability
0 ,,,,, dRSKRSg tttt
ttt
tt
tt
ttt
ttt RScRbSaSaaRSg 112
210,,
Let St be the sending rate at time t.Let Rt be the roundtrip time experienced by a packet sent at time t.Let t be the congestion level at time t.Let g be the probability that a packet is dropped
general model
memoryless
2210, t
tt
tttt RaRaaRg
depends on roundtrip time only
Preliminary work indicates that, for reasonable sending rates, the drop probability mostly depends on roundtrip time.
Since drops are rare, it is difficult to collect data for slow sending rates.
* *
Determining the Conditional Drop Probability
1
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ktktktk
ktktktktkktktk
RRpRdP
RRpRRdPRRdp
ktktkt
ikt
n
ii
ktktktikt
n
ii
ktktktkt
ktktktktkktktktkktk
dRRRpRa
dRRRpRa
dRRRpRg
dRRRpRdPdRRRdpRdp
10
10
1
111
|
|
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system of linear equations
time.roundtrip for the iesprobabilitn transitio the- |
timeroundtrip a experiencepacket past given the
dropped ispacket next y that theprobabilit the- |
1
1
1
ktkt
kt
ktk
RRp
R
Rdp
observable
assume the congestion level, , is constant
We assume that given Rtk, dk is independent of Rtk-1
End-to-end model with many queues
q1t q2
t qn-1t qn
tsource destination
queues
queuing delay at time t = D1
t
queuing delay at time t = Dn-1
t
kTk DTD 1
2th is queue second in thepacket k by the dexpereincedelay The
kTD1th is queuefirst in thepacket k by the dexpereincedelay The
prop
kTn D kT
n
kTD kT
D kTD kT kT
D kT kTkT D D D D RTT
...
1 ... 12 1
31
2 1...
The kth is sent at time Tk
propagation delay
Modified Diffusion Approximation for a Single Queue
oo dtdDPdtdF ,|,,
otherwise 0
for ,,2
2
,tdd
t
mtdde
t
mtdddtdF o
o
md
o
om
t
mtdd
xo
o
dxet
mtdd
2
2
2
1
Histogram of Observed RTT Increments (real data)
Gaussian (RTT0=28)Queue empties slowly
Queue empties quickly
Agrees with queuing theory(Diffusion Approximation)
Observed and Smoothed Conditional Drop Probabilities
recall that controls the rate at which the transition probability converges to the stationary distribution
0 5 10 15 20 25 30
0
0.02
0.04
0.06
0.08
0.1
0.12Conditional Probability Density p(X | RTT)
Sending Rate
p
compatiblesending rate
Application to Variable Bit Rate Video Transmission
TCP Friendly – Send data at a rate similar to the rate that TCP would.
When the video image changes quickly, the bit rate increases and the sending rate must also increase.The “compatibility” with TCP’s sending rate can the judged by examining the probability density function of the congestion window.
Future Work
• Better models: dynamic models that depend on the past sending rate.– For example:
• Suppose that the sending rate is initially high. Then other TCP flows should decrease their sending rate.
• If the sending rate suddenly decreases, then there is temporarily extra capacity and there should be few drops (maybe).
• Time-out and slow start.• Doubly stochastic processes: Allow the parameters to vary
with time.• More accurate roundtrip time models• Experimental verification• More data, do general models for drop probability exist?
Conclusions
• System theoretical (input/output) view point to the Internet is valid.– Drop probability models– Roundtrip time models (queuing theory works!)
• Stochastic models – seem to accurately predict the TCP in complex
networks.– give useful insight into the performance of TCP
(e.g. the dependence on )– might be for other types of congestion control