Mathematical Analysis of Throughput Bounds in Random Access with ZigZag Decoding

Jeongyeup Paek, Michael J. Neely

University of Southern California

CSI Seminar Series - June 2, 2009(To Appear in WiOpt 2009)

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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?

3

802.11 MAC and Collision

Alice Bob

AP

CSMA is not perfect…

Collision

Repeatedly collide… with some random jitter

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ZigZag Decoding

AP

Alice Bob

Pa1 3 Pa1 3

Pb2 4Pb42∆1 ∆2

∆1- ∆2

1st collision 2nd collision

0

Can reconstruct both packets Pa and Pb!!

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System ModelsThree Idealized Multi-Access Models(Bertsekas and Gallager, Data Networks)

[1] Slotted random access[2] Slotted Aloha (stabilized)[3] Slotted CSMA (with mini-slot )

Common assumptionsSlotted time (t{0,1,2,…})Fixed size packets TX time 1 slotCollided packets must be retransmittedIf 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 slotDecodable 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 AccessN-users with infinite backlog of data to sendTransmit with probability ‘q’

….

….

N

8

Slotted Random Access

6688.0lim

5.0

5.1

1 prob. with ,}E{

}E{

2}E{

1)1(2}E{

)1(2

,)1(1

*

*

1

2211

N

ZigZag

ZigZagZigZagZigZag

NZigZag

N

Nq

frame size

a framepackets in# success

PP a framepackets in# success

PPPframe size

qqN

PqqN

P

81.8% improvement compared to the bound

without Zigzag (e-1 = 0.3678)

Using Renewal Theory,

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Slotted Random Access

Numerical solution matches the derived bound for N

qNMax Throughput

10

Using “Drift Theorem” for system stability…

[2] Slotted AlohaNew users arrive at Poisson rate , and immediately transmit in the next slotBacklogged users transmit with probability q(i)

5123.0lim

69.0

31.1

)1(2

)1()1()1(

2

*

*

1212

n

nnnZigZag

nq

qqnn

eqnqeqeP

But not as good as hoped!

… better than the bound w/o Zigzag (e-1 = 0.3678)

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Slotted Aloha - ModifiedNew 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

6688.0lim

6442.0

3558.1

*

*

N

nq 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 slotNew users arriving during transmission slot are backloggedBacklogged users transmit with probability q(i)

zigzag

zigzag

PP

PP

0

1*

1

2

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

N * qN Max Throughput

0045.05966.01

zigzag

zigzag

PP

PP

0

1*

1

2

ZigZag

w/o ZigZag

Better throughputTransmit more aggressively!

Curve fitted

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CSMA - Simulation ResultSimulation results match the numerically solved bound

ZigZag decoding improves maximum throughput by ~20%

~20%

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Experimental Results from the original ZigZag paper [Gollakota and Katabi]

ImplementationGNU Radio, 14-node 802.11b testbed10% of sender-receiver pairs are hidden terminal,10% sense each other partially.Only receiver (AP) modifications.

ResultsAvg. loss rate (over 20% pairs): 72.6% 0.7%Avg. throughput (over all pairs): improved by 25.2%

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Conclusion

ZigZag decoding improves maximum throughput significantly.

Model w/o ZigZag with ZigZag % gain

Random Access 0.3678 0.6688 81.8

Aloha 0.3678 0.6688 81.8

CSMA ( = 0.05) 0.7298 0.8759 20.0

Thank you.