The Capacity of Wireless Networks

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The Capacity of Wireless Networks

• Piyush Gupta and P. R. Kumar
• Presented by Zhoujia Mao

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Outline

• Arbitrary networks
• Two models protocol and physical
• An upper bound on transport capacity
• Constructive lower bound on transport capacity
• Random networks
• Two models protocol and physical
• Constructive lower bound on throughput capacity
• Conclusions

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Arbitrary Networks

• n nodes are arbitrary located in a unit area disc
• Each node can transmit at W bits/sec over the
  channel
• Destination is arbitrary
• Rate is arbitrary
• Transmission range is arbitrary
• Omni directional antenna
• When does a transmission received successfully ?
• Allowing for two possible models for
  successful reception over one hop The protocol
  model and the Physical model

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Protocol Model

• Let Xi denote the location of a node
• A transmission is successfully received by Xj if

• For every other node Xk simultaneously
  transmitting
• D is the guarding zone specified by the protocol

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Physical Model

• Let

Be a subset of nodes simultaneously transmitting

• Let Pk be the power level chosen at node Xk
• Transmission from node Xi is successfully
  received at node Xj if

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Transport Capacity of Arbitrary Networks

• Network transport one bit-meter when one bit
  transported one meter toward its destination
• Main result
• Under the Protocol Model the transport capacity
  is (? as n ?)

if nodes are optimally placed, the traffic
pattern is optimally chosen and the range of each
transmission is optimally chosen
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Arbitrary Network upper bound on transport
capacity

• Assumptions
• There are n nodes arbitrarily located in a disk
  of unit area on the plane
• The network transport lnT bits over T seconds
• The average distance between source and
  destination of a bit is L

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Theorem 2.1

• In the protocol model, the transport capacity lnL
  is bounded as follows

• In the physical model,

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Remarks

• The upper bound in Protocol Model only depends on
  dispersion in the neighborhood of the receiver
• The upper bound in Physical Model improves when a
  is large, i.e., when the signal power decays more
  rapidly with distance
• When the domain is of A squares meters rather
  than 1 m2, then all the upper bounds above are
  scaled by

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Arbitrary Network constructive lower bound

• Theorem 3.1 There is a placement of nodes and an
  assignment of traffic patterns such that the
  network can achieve under protocol model

• Proof define r

Place transmitters at locations
Place receivers at locations
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A constructive lower bound on capacity of
arbitrary network
r
Dr
gt(1D)r
(( ))
(( ))
(( ))
r
2Dr
(( ))
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Random Networks

• n nodes are randomly located on S2 (the surface
  of a sphere of area 1sq m) or in a disk of area
  1sq m in the plane
• Each node has randomly chosen destination to send
  l(n) bits/sec
• All transmissions employ the same nominal range
  or power
• Two models Protocol and Physical

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Protocol Model

• Let Xi denote the location of a node and r the
  common range
• A transmission is successfully received by Xj if

For every other Xk simultaneously transmitting
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Physical Model

• Let

Be a subset of nodes simultaneously transmitting

• Let P be the common power level
• Transmission from node Xi is successfully
  received at node Xj if

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Throughput Capacity of Random Networks

• Feasible throughput ?(n) bits per second is
  feasible if there is a spatial and temporal
  scheme for scheduling transmissions such that
  every node can send ?(n) bits per second on
  average to its chosen destination
• Throughput capacity throughput capacity of the
  class of random network is of order ?(f(n)) bits
  per second if there are constants c gt 0, c lt 8
  such that

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Spatial tessellation

• Let a1,a2,.ap be a set of p points on S2
• The Voronoi cell V(ai) is the set of all points
  which are closer to ai than any of the other ajs
  i.e.

Point ai is called the generator of the
Voronoi cell V(ai)
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A Voronoi tessellation of S2
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Tessellation properties

• For each egt0, There is a Voronoi tessellation
  such that Each cell contains a disk of radius e
  and is contained in a disk of radius 2e
• We will use a Voronoi tessellation for which
• Every Voronoi cell contains a disk of area
  100logn/n . Let r(n) be its radius
• Every Voronoi cell is contained in a disk of
  radius 2r(n)

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Adjacency and interference

• Adjacent cells are two cells that share a common
  point.
• We will choose the range of transmission r(n) so
  that

With this range, every node in a cell is within a
distance r(n) from every node in its own cell or
adjacent cell
8r(n)
2r(n)
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Theorem 4.1

• For Random Networks on in the Protocol Model,
  there is a deterministic constant c gt 0 such that
  bits per second
  is feasible whp
• For Physical Model, there are c, c such that
• is feasible whp

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• Proof
• Lemma in the Protocol Model there is a schedule
  for transmitting packets such that in every (1
  ) slots, each cell in the tessellation gets
  one slot in which to transmit, and such that all
  transmissions are successfully received within a
  distance r(n) from their transmitters
• From the above lemma, the rate at which each cell
  gets to transmit is W/(1 ) per second

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• Lemma There is a d(n)?0 such that Prob (
  (Traffic needing to be carried by cell V)
  c5?(n) ) 1- d(n)
• From the above lemma, the rate at which each cell
  needs to transmit is less than c5?(n)
  whp. With high probability, this rate can be
  accommodated by all cells if it is less than the
  available rate, i.e., if

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• Within a cell, the traffic to be handled by the
  entire cell can be handled by any one node in the
  cell, since each node can transmit at rate W bits
  per second whenever necessary
• Lemma Every cell in has no more than
  interfering neighbors. depends only on ? and
  grows no faster than linearly in (1?)2
• Thus, for Protocol Model,
• 
  is feasible whp

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• Lemma if ? is chosen to satisfy
• then the above result of Protocol Model also
  holds for Physical Model
• Plug the expression of ? into
  , we get

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Theorem 5.1

• For Random Networks on under the Protocol
  Model, there is a deterministic c lt 8 such that
  

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• Proof
• Lemma the number of simultaneous transmission on
  any particular channel is no more than
  in the Protocol Model
• Let L denote the mean length of the path of
  packets, then the mean number of hops taken by a
  packet is at least

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• Since each source generates ?(n) bits per second,
  there are n sources and each bit needs to be
  relayed on average by at least nodes, so
  the total number of bits per second needs to be
  at least . Also, each transmission
  over a single channel is of W bits per second, so
  from the above lemma the number of bits can not
  be more than
• bits per second, so
  

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• Lemma the asymptotic probability that graph G(n,
  r(n)) has an isolated node and is disconnected is
  strictly positive if and
  .
• By the definition of feasible throughput, the
  absence of isolated node is a necessary condition
  for feasibility of any throughput. Thus,
  is necessary to guarantee connectivity
  whp. We obtain the upper bound for Protocol Model

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Conclusion

• Implication for design
• Number of nodes
• Signal decay rate
• Not considered
• Delay
• Mobility

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Thanks ?
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• A graph of degree no more than c1 can have its
  vertices colored by using no more than (1c1)
  colors
• So color the graph such that no two interfering
  neighbors have the same color, so in each slot
  all the nodes with the same color transmit

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• If V is an interfering neighbor of V, then V
  and similarly every other interfering neighbor,
  must be contained within a common large disk D of
  radius 6r(n) (2D)r(n)