Optimal Partitioning of Multicast Receivers Min Sik Kim - - PowerPoint PPT Presentation
Optimal Partitioning of Multicast Receivers Min Sik Kim minskim@cs.utexas.edu Co-authors: Simon Lam and Yang Yang Outline of Presentation Motivation Optimal Receiver Partitioning Max-min Fair Rate Optimal Partitioning
minskim@cs.utexas.edu
Co-authors: Simon Lam and Yang Yang
Motivation Optimal Receiver Partitioning Max-min Fair Rate Optimal Partitioning Experimental Evaluations Conclusion
Heterogeneous receiver capacities
– Receiver host restrictions – Network path: Modem, ISDN, Cable Modem, LAN
Sender R1 R2 R3
Single-rate
– The lowest rate – To maximize inter-receiver fairness
Multi-rate
– Q1: How many groups? – Q2: How to determine the rates?
Sender Sender
L
s
R1 R2 R3 R1 R2 R3 Sender R1 R2 R3
As many as receivers A fixed number of groups, e.g., 4 groups The more groups, the higher
– the sender overhead to encode – the network overhead to keep the states – the receiver overhead to decode
Our result: 4-5 groups for majority of
benefits
Static
– Independent of receiver capacities – Determined by encoding scheme
Dynamic
an optimal solution
Terminology
– Isolated rate ri – Receiver utility function u(r, g) – Group utility
=
G i i g
r u g G U ) , ( ) , (
Sender R1 R2 R3 2 1 3 0.5 1.0 0.3 ri u(ri, g) U(G, g) = 1.0 + 0.5 + 0.3 = 1.8 g = 1
u(r, g) Properties
– Non-decreasing when r and g approach to each other. – Maximum when r = g.
Examples
g r u u(r, g) = min(r, g) / max(r, g) g r u u(r, g) = min(r, g)
Session utility Optimal partition
– Maximizes the session utility.
∑
=
=
K k k k K K
g G U g G g G V
1 1 1
) , ( )}) , ( , ), , ({(
R1 R2 R3 2 1 3 1.0 ri u(ri, g1) U(G1, g1) = 1.0 g1 = 1 g2 = 2 1.0 0.7 u(ri, g2) U(G2, g2) = 1.0 + 0.7 = 1.7 V = 1.0 + 1.7 = 2.7
Ordered partition
– If and , then .
There exists an ordered partition that is
r1 r2 r3 r4 r5
Ordered partition Unordered partition
r1 r2 r3 r4 r5
k
G i ∈
1 +
∈
k
G j
j i
r r ≤
r1 r2 r3 r4 r5 r1 r2 r3 r4 r5 r1 r2 r3 r4 r5 r1 r2 r3 r4 r5
Dynamic programming algorithm
– Finds the ordered optimal partition
( )
}) , , 1 ({ ) 1 , ( max ) , (
* * 1 *
i j U m j V m i V
i j
+ − =
< ≤
r1 r2 … rj … ri m-1 groups m-th group
}) , , 1 ({
*
i j U
) 1 , (
*
− m j V
4 groups for 80% of the maximum.
20 40 60 80 100 1 2 3 4 5 6 7 8 9 Utility (%) The number of groups Uniform Normal Bi-modal 20 40 60 80 100 1 2 3 4 5 6 7 8 9 Utility (%) The number of groups Uniform Normal Bi-modal
u(r, g) = min(r, g) / max(r, g) u(r, g) = min(r, g)
Max-min fair rates as isolated rates Aggregation
1 2 5 4 7 [4, 5.5): 1 [5.5, 7]: 1 [1, 3): 2 [3, 5]: 1 [1, 4): 2 [4, 7]: 3
Aggregation error < 3% with 4 intervals.
152 154 156 158 160 162 1 2 3 4 5 6 7 8 9 10 V The number of intervals 780 785 790 795 800 805 810 815 820 825 1 2 3 4 5 6 7 8 9 10 V The number of intervals 156 158 160 162 164 166 168 1 2 3 4 5 6 7 8 9 10 V The number of intervals 990 1000 1010 1020 1030 1040 1050 1 2 3 4 5 6 7 8 9 10 V The number of intervals
u(r, g) = min(r, g) / max(r, g) u(r, g) = min(r, g) Normal Bi-modal
Heterogeneous receiver capacities. Determine the optimal receiver partition
in multi-rate multicasts.
Achieve 80% of the maximal utility. Future work
– Extension to other network fairness.