Variance Estimation for Networking Congestion Avoidance Algorithm. - - PowerPoint PPT Presentation

Variance Estimation for Networking Congestion Avoidance Algorithm.

Olga I. Bogoiavlenskaia PetrSU, Department of Computer Science

• lbgvl@cs.karelia.ru

TCP Congestion Control

• The paradigm of Distributed Control in Packet Switching Network
• Transmission Control Program, 1974.
• Congestion collapse
• Variance not important yet

1

• S. Low tree

2

Example of Tahoe Trajectory

3

TCP Congestion Control Development

• Jitter sensitive applications
• TCP vs UDP
• High BDP links utilization vs Congestion Control
• Best effort vs QoS guarantees

4

Variety of Experimental Versions

• TCP CUBIC - cubical growth period. RTT independent
• High Speed TCP (HSTCP), S. Floyd 2003. Congestion Avoidance
• coeff. of linear growth and multiplicative decrease are convex functions
• f current window size
• Scalable TCP (STCP) T. Kelly, 2003. Decreases time of data recovery
• H-TCP, Hamilton Institute, Ireland, 2004. Intended for links with

high BDP value. Uses RTT size to react on losses

• TCP Hybla 2003-04. Developed for satellite links. Scales throughput

to mimic NewReno and utilize link at the same time.

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Variety of Experimental Versions

• TCP Westwood, 2001. Tries to identify the reason behind losses.

Developed for wireless links.

• TCP-Illinois uses dynamic function for defining Congestion Avoidance

parameters

• TCP-LP (Low Priority)
• TCP-YeAH

6

TCP Variance

• Why important?
• A lot of models of average window size — Reno, NewReno, CUBIC
• Asymptotic studies etc.
• D. Towsley group for p > 0.025

T = MSS RTT

• 2p

3 + RTOmin(1, 3

• 3p

8 )p(1 + 32p2)

(1)

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TCP NewReno ’Saw’

wi

i+1

τ

w

α

t w(t)

w td td

i i+1 i i+1

τ

td i−1

’triple−duplicate’ periods

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Variance evaluation

• Altman E., Avrachenkov K., Barakat C. A Stochastic model of TCP/IP

with Stationary Random Losses, Proceedings of ACM SIGCOMM’00. Stockholm, 2000. pp. 231-242.

• Some ‘popular’ assumptions provide difficulties in estimating TCP

variance, e.g.

• Variance is V ar[X] = E[X2] − (E[X]2)
• Root square lows are derived thorough Goelders’s inequality i.e.

E[X] ≤

• E[X2]

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Variance estimation

• Using geometrical considerations one get from TCP ‘saw’

X2

n+1 = αX2 n + 2bSn,

• Expanding, one gets

X2

k+n = 2b n

• i=0

α2iSk+n−i

• r
• Xk+n =
• 2b

n

• i=0

α2iSk+n−i

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Variance estimation E[Xk+n] = E  

• 2b

n

• i=0

α2iSk+n−i   ≤ √ 2b

n

• i=0

E

• α2iSk+n−i
• Now when n → ∞ one gets the following

E[X] = lim

n→∞ E[Xn] ≤ lim

√ 2b

n

• i=0

E

• α2iSi
• =

√ 2b 1 − αE[

• Sn]

For b = 1 and α = 1

2 and hence

E[X] ≤ 2 √ 2E[

• Sn]

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Variance evaluation Now we have two estimations of sliding window size expectation

• E[X] ≤ A =
• 8

3E[S2 n]

• E[X] ≤ B = 2

√ 2E[√Sn] Reminder. If B < A then it could be used for estimation variance V ar[X] ≤ A2 − B2. This holds if E[

• Sn] <
• E[Sn]

3 .

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Variance evaluation. Examples Lets p.d.f. of Sn is F(x) = 1 − eλx then E[Sn] = 1 λ and E[

• Sn] =

√π 2

• 1

λ. Condition does not hold.

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Variance evaluation. Examples. Lets p.d.f of Sn is Pierson root square distribution then its moments can be calculated through Γ-function and E[Sn] = 2

1 2Γ(n+1

2 )

Γ(n

2)

and E[

• Sn] = 2

1 4Γ(2n+1

4 )

Γ(n

2) .

The result depends on parameter n. Condition holds for e.g. n = 10.

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Variance evaluation Notice that lim

n→∞ E[X2 n] = lim n→∞ E[X2 n+1]

and lim

n→∞ E[X2 n] = 2bE[Sn]

1 − α2 . Hence there might take place lim

n→∞ E[Xn] =

• 2b

1 − α2E[

• Sn].

GetTCP kernel level monitor for OS Linux

15

Conclusion

• The Development of Congestion Control schemes is considered
• The importance of TCP variance evaluation is demonstrated
• Possible approaches to the problem are analyzed
• Variance estimation is proposed

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