Hierarchical Cooperation Achieves Linear Scaling in Ad Hoc Wireless

NetworksDavid Tse

Wireless FoundationsU.C. Berkeley

AISP Workshop May 2, 2007

Joint work with Ayfer Ozgur and Olivier Leveque at EPFL

Scaling of Ad Hoc Wireless Networks

• 2n nodes randomly located in a fixed area.

• n randomly assigned source-destination pairs.

• Each S-D pair demands the same data rate.

• How does the total throughput T(n) of the network scale with n?

How much can cooperation help?

??T(n) = £(1) T(n) = £(p n)Gupta-Kumar 00

?

Main Result

Linear capacity scaling is achievable with intelligent cooperation.

More precisely:

For every > 0, we construct a cooperative scheme that can achieve a total throughput T(n) = n1-.

Channel Model

• Baseband channel gain between node k and l:

where rkl is the distance apart and kl is the random phase (iid across nodes).

• is the path loss exponent (in power)

®¸ 2

hkl = Gr¡ ®2

kl ej µk l

Gupta-Kumar Capacity is Interference-Limited

• Long-range transmission causes too much interference.

• Nearest-neighbor transmission means each packet is transmitted times (multi-hop).

• To get linear scaling, must be able to do many simultaneous long-range transmissions.

• How to deal with interference?• A natural idea: distributed MIMO (Aeron &Saligrama 06).

p n

MIMO:Multiple Transmit Multiple Receive Antennas

• The random MxM channel matrix allows transmission of M parallel streams of data.

• Originally conceived for antennas co-located at the same device.

Distributed MIMO

• MIMO effect can be simulated if nodes within each cluster can cooperate.

• But cooperation overhead limits performance.• What kind of architecture minimizes overhead?

A 3-Phase Scheme

• Divide the network into clusters of size M nodes.

• Focus first on a specific S-D pair. source s wants to send M bits to destination d.

Phase 1 :Setting up Tx cooperation:1 bit to each node in Tx cluster

Phase 2:Long-rangeMIMO between s and d clusters.

Phase 3:Each node in Rx clusterquantizes signal into k bitsand sends to destination d.

Parallelization across S-D PairsPhase 1:Clusters work in parallel.Sources in each cluster taketurn distributing their bits.

Total time = M2

Phase 2:1 MIMO trans.at a time.

Total time = n

Phase 3:Clusters work in parallel.Destinations in each clustertake turn collecting their bits.

Total time = kM2

Back-of-the-Envelope Throughput Calculation

total number of bits transferred = nM

total time in all three phases = M2 +n + kM2

throughput: bits/second

Optimal cluster size

Best throughput:

nMM 2 + n + kM 2

M ¤ = p np n

Further Parallelization

• In phase 1 and 3, M2 bits have to be exchanged within each cluster, 1 bit per node pair.

• Previous scheme exchanges these bits one at a time (TDMA), takes time M2.

• Can we increase the spatial reuse ?• Can break the problem into M sessions, each

session involving M S-D pairs communicating 1 bit with each other:

cooperation = communication

• Any better scheme for the small network can build a better scheme for the original network.

Recursion

Lemma: A scheme with thruput Mb for the smaller network yields for the original network a thruput:

with optimal cluster size:

n 12¡ b

M ¤ = n 12¡ b

f (b) = 12¡ b

MIMO + Hierarchical Cooperation-> Linear Scaling

.

Setting up Tx cooperation

Long-range MIMO

Cooperateto decode

At the highest level hierarchy, cluster size is of the order n1- => near network-wide MIMO cooperation.

Upper Bound

• A simple upper bound: each source node has the benefit of all other nodes in the network cooperating to receive without interference from other nodes.

• Each source gets a rate of at most order log n.• Yields an upper bound on network throughput

• The hierarchical scheme is nearly information theoretically optimal.

Cn · O(n logn)

From Dense to Extended Networks

• So far we have looked at dense networks, where the total area is fixed.

• Another natural scaling is to keep the density of nodes fixed and the network covers an increasing area.

• Here power limitation comes into play in addition to interference limitation.

Main Results for Extended Networks

• For path loss exponent between 2 and 3:

a modification of the hierarchical scheme can achieve:

T(n) = n2-/2

We also showed this is the best that one can do.

• For > 3:

Multihop is optimal: T(n) = p n

Conclusion

• Hierachical cooperation allows network-wide MIMO without significant cooperation overhead.

• Network wide MIMO achieves a linear number of degrees of freedom.

• This yields a linear scaling law for dense networks.

• Also yields optimal scaling law for extended networks.