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Theoretical Capacity of Multi-hop Wireless Ad Hoc Networks

Yue Fang

A.Bruce McDonald

R-WIN Lab

ECE Department

Northeastern University

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Outline

Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

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Outline

Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

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Wireless Ad Hoc Networks

Easy to setup, no wiring required Provides support of mobile (and ubiquitious)

computing Limited resources Lower capacity Dynamic characteristics

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Capacity Analysis of Wireless Ad Hoc Networks

The capacity of wireless network (Gupta & Kumar)

Theoretical maximum throughput of 802.11 Channel capacity of multi-hop wireless ad

hoc network

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Network Capacity

Network Capacity The ability of data exchange the whole network can bear

at any time. No universal semantic is available. Two interpretations of network capacity

Maximum instantaneous capacity (MIC) • Ideal routing and scheduling

Network saturation capacity (NSC)• Uniformly distributed nodes and traffic independent of

routing and scheduling.

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Topology Generation

Network topology is generated by repeating specific patterns to avoid unnecessary randomness.

nnavgavg=3=3 nnavgavg=6=6

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Outline

Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

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Previous Work

Novel concepts --- “deferral set” and “equivalent competitor” are proposed to facilitate multi-hop capacity analysis. Deferral set: the set of all nodes and links that will affect

the ongoing communication Equivalent Competitor: The amount of competition faced

by the ongoing communication in terms of single node.

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Previous Work (2) Node being two hop neighbor

depends on whether it has a neighbor which is direct neighbor of ongoing communication.

Only the communication from two hop neighbor to one hop neighbor will affect the ongoing communication.

Channel capacity (Schan) is derived based on node behavior model.

F

I

G

H

A

C E

D

B

Only communication between C and F will affect the communication between A and B, thus the equivalent competitor is 1/3.

X

X

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Network Saturation Capacity (NSC)

It is necessary to study the relation between the capacity and node location.

navg N NSC(Mb/s) navg N NSC(Mb/s)

Nl x Schan

Sim-ulation

Nl x Schan

Sim-ulation

3 49 1.75 1.779 4 64 1.56 1.636

6 81 1.06 1.52 8 81 0.957 1.17

11 75 0.417 0.86 12 81 0.405 0.82

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Boundary Condition Nodes that close to the boundary of the network

have fewer neighbors. Hence less channel contention, consequently, greater available capacity.

Boundary zone is defined as the doughnut shaped region occupied by all nodes that are more closer to the boundary (Xi < 2r, where Xi is the distance from node i to the network boundary).

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Phantom Node Nodes in the boundary zone

(A, B) tend to have higher capacity than the nodes in the center of the network.

Nodes in the shaded area are called “phantom” nodes

In order to have an accurate estimation of network saturation capacity, the percentage of “phantom node” to the number of nodes in the network should below a threshold.

AA

BB2r-d2r-d

dd

αα

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NSC: How big is big?

Boundary condition effect can be regarded as negligible when then network radius is at least 10 times of transmission range.

Additional parameters may affect network capacity: such as spatial and temporal variation of distribution of nodes, traffic, channel quality, mobility, etc.

Number of phantom nodes vs. Number of phantom nodes vs. R/rR/r

Percentage of phantom nodes Percentage of phantom nodes to Numto Num

Num

ber

of P

hant

om N

odes

# of

pha

ntom

nod

es/ t

otal

nod

es

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Outline

Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

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Maximum Instantaneous Capacity (MIC)

MIC reflects the bottleneck throughput between any set of sources and destinations.

MIC can only be achieved under ideal scheduling --- every link either transmitting/receiving, or in deferral state.

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Maximum Instantaneous Capacity (MIC)

The objective is to find a sequence of simultaneously active links --- aggregate link set that cover the connected work. MIC is the bottleneck of the aggregate link set.

MIC can be approximated in two steps: Find the maximum aggregate link set --- NP

problem Find the bottleneck

c1 simultaneous c1 simultaneous linkslinks

c2 simultaneous c2 simultaneous linkslinks

c3 simultaneous c3 simultaneous linkslinksc1>= c2>=c3c1>= c2>=c3

Bottleneck Bottleneck is the MIC: is the MIC:

c3c3

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44

11

22

33 55

NP completeness By appropriate means, the problem of finding maximum aggregate

link set in the network can be transformed to classic maximum independent set problem [6].

(1,2(1,2))

(4.5(4.5))

(3,4(3,4))

(2,5(2,5))

(2,3(2,3))

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MIC: The greedy Algorithm

1. List all the feasible deferral sets in ascending order by the number of links in each set.

2. Pick the first deferral set in the list , transmission along corresponding link can be granted.

3. If more than one deferral set have same size, the tie is broken by activating the link with minimum LOS (line-of-sight) distance.

4. Update the candidate deferral list.

5. Update the size of remaining feasible deferral sets.

6. Repeat 1-5 until the candidate deferral set list is empty.

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Random Link Selection

Randomly select the link to be activated. Faster than greedy heuristic Results are of the same order.

N Concurrent Links

Greedy Random

432 3 118 99

530 4 119 87

737 6 139 108

952 8 163 115

nnavgavg

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Bottleneck Aggregate Link Set

Every link has to be activate at least once. The MIC is the bottleneck the maximum aggregate link

set. The optimal solution is NP-complete. Greedy algorithm has polynomial bounded number of

iterations. Random selection.

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Experimental Result

N Greedy Algorithm Random Selection

Round Bottleneck Round Bottleneck

432 3 116 111 45 94

530 4 241 110 95 82

737 6 591 128 178 101

952 8 1899 152 348 107

nnavgavg

Number of requited iterations and corresponding lower bound

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Outline

Introduction Network Saturation Capacity Maximum Instantaneous Capacity Discussion

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Discussion The semantic of network capacity itself is analyzed to

provide a clearer understanding and basis for comparison. The analysis is central of a broad cross-layer framework Extensible in terms of access protocols, generalization and

application to real control problems. The results, while sub-optimal and on worst-case analysis

improve on the most often cited results from [2]

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Discussion (2) From [2], using protocol model, the capacity of a random

network is Number of concurrent active link in a saturated network

can be approximated by the number of non-overlapping level-1 interference sets. Which can be obtained by:

Network capacity then is:

(lo g

)W

n n

R

r

N

nr

r

N

navg

avg

2

2

2

20 9 8

1

0 9 8 0 9 8 1( . ) ( . ) ( . ) ( )

1 '

/ / 2 ( 1)avg

avg avg

NC W

n C W

N n N N n

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Conclusion

The extension of the channel capacity analysis [?] The semantic of network capacity is discussed with two

interpretation --- NSC and MIC. Agreement of the results reported in [2] mutually

validated the two models Provides insight regarding how to more effectively

leverage available network capacity.

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Reference

[1] “Theoretical channel capacity in multi-hop ad hoc networks” by Yue Fang and A. Bruce McDonald

[2] “ The capacity of wireless networks’’ by P. Gupta and P.R. Kumar

[3] “Finding a maximum independent set” by R. E. Tarjan and A. E. Trojanowski.

[4] “Theoretical Maximum Throughput of IEEE 802.11 and its Applications” by Jangeun Jun, Pushkin Peddabachagari and Miail Sichitiu