Transcript WiOpt'09 talk - 'Mathematical Analysis ofThroughput Bounds
Mathematical Analysis of Throughput Bounds in Random Access with ZigZag Decoding
Jeongyeup Paek, Michael J. Neely University of Southern California WiOpt 2009
ZigZag
“
ZigZag Decoding: Combating Hidden Terminals in Wireless Networks
”
, Shyamnath Gollakota and Dina Katabi, SIGCOMM 2008.
802.11 receiver design that allows successful reception of packets despite collision Ha! Then can we get better max. throughput?
By how much?
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802.11 MAC and Collision
Collision AP Repeatedly collide … with some random jitter Alice Bob
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ZigZag Decoding
0 1 3 P a ∆1 2 4 Alice 1 st collision ∆1- ∆2 AP P b 2 nd collision 1 3 P a ∆2 2 4 P b Bob Can reconstruct both packets P a and P b !!
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System Models
Three Idealized Multi-Access Models (Bertsekas and Gallager, Data Networks) [1] Slotted random access [2] Slotted Aloha (stabilized) [3] Slotted CSMA (with mini-slot
) Common assumptions Slotted time (t
{0,1,2,
…
}) ….
Fixed size packets TX time
1 slot ….
Collided packets must be retransmitted If only one node sends a packet in a slot, the packet is always received correctly
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Definitions and Assumptions
‘ Collision ’ : when 3 or more users transmit in a slot ‘ ZigZag ’ : if exactly 2 users transmit in a slot Decodable using ZigZag decoding ‘ 0 ’, ‘ 1 ’, ‘ Zigzag ’ , or ‘ C ’ immediate feedback If ‘ ZigZag ’ occurs in a slot, That slot is automatically extended into 2 slots Two colliding users retransmit in the next slot, and others never retransmit in the next slot Exactly 2 packets are perfectly received at the receiver during 2 slots
throughput during this period = 1pkt/slot Ignore decoding failure and 3 packet decoding
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[1] Slotted Random Access
N-users with infinite backlog of data to send Transmit with probability ‘q’ N ….
….
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Slotted Random Access
P
1
N
1
q
( 1
q
)
N
1 ,
P ZigZag
N
2
q
2 ( 1
q
)
N
2 E{
frame size
} 2
P ZigZag
( 1
P ZigZag
) 1
P ZigZag
E{
# success packets in a frame
}
P
1 2
P ZigZag
Using Renewal Theory,
E{
# success packets in a frame
} , with E{
frame size
} prob.
1
q
*
N
1 .
5 0 .
5
N
lim * 0 .
6688
Do some math… 81.8% improvement compared to the bound without Zigzag (e -1 = 0.3678)
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Slotted Random Access
Max Throughput Numerical solution the derived bound matches for N
q N
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[2] Slotted Aloha
New users arrive at Poisson rate
, and immediately transmit in the next slot Backlogged users transmit with probability q(i)
P ZigZag
2 2
e
( 1
q
)
n
e
nq
( 1
q
)
n
1
e
n
(
n
1 )
q
2 ( 1
q
)
n
1 2
q
*
n
1 .
31 0 .
69
than the bound w/o Zigzag (e -1 = 0.3678)
n
lim * 0 .
5123
But not as good as hoped!
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Slotted Aloha - Modified
New users arriving during the ZigZag frame does not transmit in the second slot of ZigZag frame Listen for feedback and become backlogged if in Zigzag
q
*
n
1 .
3558 0 .
6442
N
lim * 0 .
6688
81.8% improvement compared to the bound without Zigzag (e -1 = 0.3678) Simulation result ( 0.6675
) matches the bound
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[3] Slotted CSMA with mini-slots
New users arriving during mini-slot transmit in the next slot New users arriving during transmission slot are backlogged Backlogged users transmit with probability q(i)
*
P
1 1 2
P zigzag P
0
P zigzag
Exact µ * given in terms of q(i) A bit too complicated to find closed form formula for optimal q(i) and optimal throughput ….
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CSMA - Numerical
Better throughput
*
P
1 1 2
P zigzag P
0
P zigzag
Transmit more aggressively!
Curve fitted ZigZag w/o ZigZag Max Throughput
1 0 .
5966 0 .
0045
N * q N
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CSMA - Simulation Result
Simulation results match the numerically solved bound ~25% ZigZag decoding improves maximum throughput by ~25%
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Experimental Results from the original ZigZag paper [Gollakota and Katabi] Implementation GNU Radio, 14-node 802.11b testbed 10% of sender-receiver pairs are hidden terminal, 10% sense each other partially.
Only receiver (AP) modifications.
Results Avg. loss rate (over 20% pairs): 72.6%
0.7% Avg. throughput (over all pairs): improved by 25.2%
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Conclusion
Model Random Access Aloha CSMA (
= 0.1 ) CSMA (
= 0.05) w/o ZigZag 0.3678
0.3678
0.6417
0.7298
with ZigZag 0.6688
0.6688
0.8122
0.8759
ZigZag decoding improves maximum throughput significantly.
% gain 81.8
81.8
26.5
20.0
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