Clustering of Source/Channel Rate Allocations for Receiver-driven Multicast

under a Limited Number of Streams

Philip A. Chou, Microsoft Research

Kannan Ramchandran, UC Berkeley

ClusteringReceivers

• Questions to be answered:– How large should M be to serve “most” receivers well?– How can we design the collection of M streams?– How can a receiver decide which of the M streams to use?

• We will assume streams are all at the same bitrate.• Redundancy is provided by FEC.

Space of associatedchannels

Distribution over channels

1

sourceparity

2 … M

Existing FEC Systems

• Commercial systems (e.g., Windows Media) use systematic Reed-Solomon code to produce N-K parity packets for every K source packets

• The parameters (N,K) are chosen to match the packet loss characteristics for the channel

sour

ce

sour

ce

sour

ce

sour

ce

pari

ty

pari

ty

pari

ty

100Kbps 100Kbps 100Kbps 100Kbps

KN

for more reliable channels for less reliable channels

Existing FEC Systems

• Priority Encoding Transmission (PET, Albanese et al., 1996) is similar, but it allows K to change across source layers with different importance.

100Kbps 100Kbps 100Kbps 100Kbps

K2

K3

K1 pari

ty

pari

ty

pari

ty

sour

ce

sour

ce

sour

ce

N

PET packetization

• Property: recover layer i iffreceive ≥ Ki packets (out of N)

• Albanese et al. (1996) use 3 layers (I,P,B), don’t optimize

• Davis & Danskin (1996) optimize Ki s for any number of layers for minimum distortion

• Mohr, Riskin, & Ladner (1999) assume fine grain scalability (e.g., SPIHT) and adjust breakpoints using greedy search

• Puri & Ramchandran (1999) optimize breakpoints using O(N) algorithm

1 ... K 1 N

1 ... K 1 N

1 ... K i N

1 ... K i N

1 ... K N N

...

...

...

...

...

1 ... K N N...

K 1 b yte s

K 1 b yte s

K i b yte s

K i b yte s

K N b yte s

K N b yte s

......

...

layer 1layer i

layer N...

...unused

Packet 1

Packet N

R N -1

R N

R i

R i-1

R 1

R 0

N b yte sG O F G O F G O F

...... tim e

Optimal Stream for a Known Channel• Wolog assume N layers, layer i 1,…,N coded with Ki=i.• Let R=(R0,R1,…,RN) be breakpoint vector, where R0 ≡ 0 and R1,…,RN index the last

byte in layers 1,…,N respectively.• Let D(R0), D(R1), …, D(RN) be the corresponding vector of distortions if R0, R1,…,

RN source bytes are recovered.

• Let q=(q0,q1,…,qN) be probability mass vector, whereqk=Pr{1st k of N layers recovered}=Pr{k of N packets received}.

• The effect of any stationary packet erasure channel on the receiver’s expected distortion is through q=(q0,q1,…,qN).

R0 R1 RN

R

D(R0)

D(R1)D(RN)

OperationalD(R)

function

Expected Distortion and Rate

• Expected Distortion is

• Transmission rate (bytes per GOF) is

where k=N/(k(k+1)) for k=1,…,N-1 and N=1. • Finding R=(R0,R1,…,RN) that minimizes D(R) s.t. R(R) ≤ R*

can be found by minimizing D(R)+R(R) for some using the O(N) algorithm of Puri & Ramchandran.

N

kNNkk RDRDqqRDqD

000 ))(,),((),,()()( R

N

kkk

N

kkk RkRRNR

111 α/)()(R

Optimal Stream for a Collection of Channels

• Let {q} be a collectionof channels indexed by Є over which thereis a distribution

• Expected distortion of PET packetization R for channel q is

• Overall expected distortion (w.r.t. ) is

• Hence to min D(R) s.t. R(R) ≤ R*, find q=∫q and use P&R.

N

kkkθθ RDqD

0, )()(R

N

kkk

N

kk

q

kθθ RDqRDθdνqθdνDD

k

00, )()()()()()(

RR

Multiple Optimal Streamsfor a Collection of Channels

• Start with M streams with PETpacketizations R1,…,RM.

• Let m() be stream numberto which receiver withchannel q should subscribe.

• Optimal m() (minimizing overall expected distortion) is

m() = argminm D(Rm) = argminm ∑ q,kD(Rm,k),

which induces partition cells m={:m()=m}.• Optimal PET packetization Rm for cell m is

which can be solved by the Puri-Ramchandran algorithm.• Repeat.

1 2… M

)()(minarg θdνDm

θm RR R

Simulation Setup

• We simulate collection ofiid packet erasure channels

with N=40, ~ Beta(1,b),mean =1/(1+b)=.10,.15,.20.

• We assume D(R)=2-2cR,R = #bytes per GOF, R*=7/c.

• Find clusters with M=1,2,4,8,16,32.

]1,0[,)1(, θθθq kNkNkkθ

Beta(1,1/-1) distribution

Simulation Results

Conclusion

• We have presented a clustering algorithm that finds the set of M streams having source/channel rate allocations (PET packetizations) that optimally covers the space of packet erasure channels under an arbitrary distribution– “nearest neighbor” performed by N-dim dot product– “centroid” is performed by O(N) algorithm

• For typical (?) distribution of channels, 4 streams can gain 4 out of a possible 5 dB (i.e., loses only 1 dB compared to an infinite number of streams).