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Low Delay Random Linear Coding Over a Stream Mohammad Karzand, Douglas J. Leith 1/8

Low Delay Random Linear Coding Over a Stream

Mohammad Karzand, Douglas J. Leith

Trinity College Dublin

January 2015

Low Delay Random Linear Coding Over a Stream Mohammad Karzand, Douglas J. Leith 2/8

Motivation

Observation: TCP performs poorly

In the presence of interference. Large round trip time. High packet loss rates over the networks.

Solution: A transport layer solution referred to as Network Coded TCP . Benefits: Backward compatibility with legacy equipment and easy implementation. Issue: In-order delivery delay, very important for multimedia communications.

Low Delay Random Linear Coding Over a Stream Mohammad Karzand, Douglas J. Leith 3/8

The Idea

Put one coded packet after l − 1 information packets Coded Packet: Random linear combination of all preceding information packets. ci :=

(l−1).i

• j=1

wij · uj wij are drawn from an alphabet with cardinality of Q. Q Large: One coded packet decodes one erasure in the past.

Low Delay Random Linear Coding Over a Stream Mohammad Karzand, Douglas J. Leith 4/8

Delay and Busy Period

Example:

ti Ti ti

~

⎧ ⎨ ⎩

busy period

⎧ ⎨ ⎩

idle period

Erasure: Decoding procedure starts Definition S = k if the k-th coding packet puts an end to the decoding. In this case S = 2 Si, the busy/idle periods, are i.i.d and form a renewal process.

Low Delay Random Linear Coding Over a Stream Mohammad Karzand, Douglas J. Leith 5/8

Main Theorem

Theorem (Busy Time) We have: I. P(S = 0) = (1 − ǫ)l−1 (1) P(S = 1) = (l − 1)ǫ(1 − ǫ)l−1 (2) P(S = k) = l − 1 k ǫk(1 − ǫ)k(l−1) (k − 1)l k − 1

• , ∀k > 1 (3)

II. E(S) = (l − 1)ǫ(1 − ǫ)l−1 1 − lǫ (4) E(S2) = E(S) + l(l − 1)ǫ2(1 − ǫ)l (1 − lǫ)3 (5)

Low Delay Random Linear Coding Over a Stream Mohammad Karzand, Douglas J. Leith 6/8

Simulation Results

Measured in-order delivery delay and upper bound vs coding rate (l − 1)/l and packet erasure rate ǫ.:

0.5 0.6 0.7 0.8 0.9 1 10

−2

10

−1

10 10

1

10

2

10

3

10

4

10

5

coding rate (l−1)/l mean in−order delivery delay (slots) measured delay (ε=0.1) bound (ε=0.1) measured delay (ε=0.05) bound (ε=0.05) measured delay (ε=0.01) bound (ε=0.01)

Low Delay Random Linear Coding Over a Stream Mohammad Karzand, Douglas J. Leith 7/8

Results II

Mean in-order packet delivery delay vs packets transmitted for random linear block code, and for low-delay coding

• scheme. Link rate 25Mbps, RTT 60ms, loss rate 10%, cwnd

fixed at BDP .

1.1 1.12 1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.28 1.3 10 20 30 40 50 60 70 80 90 100 Packets transmitted/Number of information packets Mean in−order delivery delay (ms) low delay code block size 8 block size 16 block size 32 block size 64 block size 128 block size 256