Mobile Ad hoc Networks COE 549
Capacity Regions
Tarek Sheltami
KFUPM
CCSE
COE
http://faculty.kfupm.edu.sa/coe/tarek/coe549.htm
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Outline
 Capacity of Wireless Ad Hoc Networks
 Why do we want to know the capacity of the
network?
 Basic Rate Matrices
 Definition of Convex Hull
 Convex Combinations of Basic Rate Matrices
 Successive Interference Cancellation
 Comparison Between Different Schemes
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Capacity of Wireless Ad Hoc Networks
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Assume perfect coordination among nodes:
 Perfect medium access control
 Perfect routing
 Perfect queuing
Every node knows exactly what to do
How much traffic can the network support?
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Why do we want to know the
capacity of the network?
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If we know the theoretical limits, we can compare
them with the performance of protocols we design
and know how much we could improve
If we know how the theoretical limits are achieved,
we can improve our protocols
Unfortunately, we still do not have any good answers
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A Special Case
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Assumptions:
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All nodes can transmit with rate W bps
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Any two nodes can communicate directly
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Any two transmissions will not collide
What is the network capabilities?
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Transmission Schemes and Rate Matrices
Transmission Schemes  Basic Rate
Matrices
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Rows represent original data source.
Columns represent receiver or transmitter of information.
Negative entries represent the rate of bits sent.
Positive entries represent the rate of bits received.
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Transmission Schemes  Basic Rate
Matrices..
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K1= 0.5 R1 + 0.5 R2
K1= 0.75 R1 + 0.25 R2
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Transmission Schemes..
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How many transmission schemes are
there?
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There are finitely many transmission schemes:
 Each node is either transmitting or staying quiet
 Each transmitter is transmitting data to one out of n nodes
The precise number of transmission schemes depends on capabilities of
nodes:
 Can nodes forward other nodes’ packets?
 Is spatial reuse allowed?
 Is power control allowed?
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Properties of Rate Matrices
 Sum of elements along same row is 0.
 We have n2 − n = n(n − 1) degrees of freedom.
 Each of the n(n − 1) non-diagonal element corresponds to
one of the n(n − 1) source-destination pairs.
 Rate matrices describe the information exchange in a
convenient format.
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Convex Combinations of Basic Rate Matrices
(time division routing)
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Definition of Convex Hull
 With words: Convex hull of a collection of vectors (or matrices!) is
all their convex combinations (weighted sums where all the
coefficients are positive and sum up to 1).
 With math:
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Does it make sense?
 A node must receive a packet before it transmits it (unless the node is also
the source)
 Therefore rate matrices with negative off-diagonal components must be
excluded
 All other rate matrices make sense
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Capacity Region Definition
 With words: All the convex combinations of the basic rate
matrices {Ri}, provided all off-diagonal components are nonnegative.
 With math:
Pn is the subset of all n × n matrices with all their
off-diagonal components non-negative
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Feasibility Problem
Feasibility Problem: Given a set of end-to-end communication rates
Rij, we want to know if these rates are supportable by the network.
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Specifying the Basic Rate Matrices
 So far: capacity region is convex hull of basic rate matrices.
 Still need to specify them!
 Each network has a repertoire of basic rate matrices depending
on:
 Multiple hops allowed?
 Multiple transmissions allowed?
 Power Control?
 Successive Interference Cancellation?
 More capabilities  more basic rate matrices  larger capacity
region.
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Five Sets of Rules
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Successive Interference Cancellation (SIC)
 Without SIC, each node treats signals intended for other users
as noise.
 With SIC, nodes may decode interference signals and subtract
them out.
 Advantage: Subtracting interference increases the data rate
that the receiver can handle.
 Disadvantage: SIC imposes a constraint on the rate of the
interfering link
 Disclaimer: The decoded interfering signal is not forwarded
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Network Model
 Single channel (no frequency division), half-duplex nodes.
 All nodes transmit with maximum power P (no power control).
 No multicasting/broadcasting (each created packet has a single
destination).
 Channel described by gain matrix G = [Gij].
 Receivers are hampered by thermal noise with power
 Given the SINR, the received rate will have to be less than:
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Network Model..
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Example
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Example..
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Capacity Region
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References
[1] Stavros Toumpis and Andrea Goldsmith, “Capacity Regions for Wireless Ad Hoc Networks”
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