The Capacity of Wireless Networks

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The Capacity of Wireless Networks

• Paper by Piyush Gupta P.R. Kumar
• Slides by Danss Course
• Modified by Sanquan Song
• Presenter Sanquan Song

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Wireless ad hoc network

• No wired backbone
• No centralized control
• Nodes may cooperate in routing each others data
  packets
• At the Network Layer problems are in routing,
  mobility of nodes and power constraints
• At the MAC layer problems with protocols such
  as TDMA, FDMA,CDMA
• At the Physical layer problems in power control

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Lecture Minutes

• 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 1
• Under the Protocol Model the transport capacity
  is (? as n ?)

• Main result 2
• Under the Physical Model,

is feasible
While
is not (? as a ?)
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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,
  i.e. each node generate bits at rate l
• The average distance between source and
  destination of a bit is L
• Transmissions are slotted into synchronized slots
  of length t sec

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Theorem

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

• In the physical model,

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Arbitrary Network constructive lower bound

• 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

• Main result 1
• Under the Protocol Model the order of the
  throughput capacity

• Main result 2
• Under the Physical Model,

is feasible
While
is not
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Random Networks A constructive lower bound on
capacity

• We will show a scheme such that each
  source-destination pair can be guaranteed a
  channel of capacity

With probability approaching 1 as

• Steps
• Define the Voronoi tessellation
• Bound the number of interfering neighbors of a
  Voronoi cell
• Bound the length of an all-cell transmission
  schedule
• Define the routes of a packet on the Voronoi
  tessellation
• Prove that each cell contains at least one node
• Calculate the expected routes that pass through a
  cell and infer the expected traffic of each node

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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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A bound on the number of Interfering cells

• Two cells are interfering neighbors if there is a
  point in one cell which is within a distance of
  (2D)r(n) of some point in the other cell
• Lemma Every cell in Vn has no more than c1
  interfering cells. c1 grows no faster than
  linearly in (1D)2
• Proof 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)

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A bound on the length of an all-cell inclusive
transmission schedule

• Lemma - In the protocol model, there is a
  schedule for transmitting packets such that in
  every (1c1) slots, each cell in Vn gets one slot
  in which to transmit
• Proof 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
• There is a schedule also for the physical model.

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The routes of packets

• Source destination pairs let Yi be a randomly
  chosen location such that Xi and Yi are
  independent. The destination Xdest(i) is chosen
  as the node Xj which is closest to Yi
• Corollary The random sequence Li straight
  line connecting Xi and Yi is i.i.d.
• Routes of packets will be choose to approximate
  these straight line segments
• Final destination will be one hop away from Yi ,
  with high probability

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Each cell contains at least one node

• Definition 1
• Let F be a set of subset. A finite set of points
  A is said to be shattered by F if for every
  subset B of A there is a set F in F such that
• Definition 2
• The VC-dimension of F , denoted by VC- dim(F ) ,
  is defined as the supremum of the sizes of all
  finite sets that can be shattered by F

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Vapnic-Chervonenkis Theorem

• If F is a set of finite VC dimension d and Xiis
  a sequence of i.i.d. random variables with common
  probability distribution P, then for every e,d gt
  0,

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VCdim of the set of disks in R2
x2
x1
x3
x4
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A cell contains at least one node

• Let F denote the class of disks of area
  100logn/n.
• So VCdim(F) is 3. Let V be a cell contained in a
  disk D. Hence

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Mean number of routes served by each cell

• First calculate the probability that a line Li or
  great circle intersect a cell V
• Lemma for every line Li and cell V

• So the expected number of lines Li that intersect
  a cell is bounded as

• The same as for great circles !

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Actual traffic served by each cell

• We bounded the mean number of routes passing
  through each cell. However, we need to bound the
  actual random number of routes served by each
  cell !!
• Remember the sequence Xi ,Yi is i.i.d.
• Therefore , we can appeal to uniform convergence
• We will show that each great circle that
  intersect a disc D, can be mapped to a point on
  the band F(D) that is equidistant from the center
  of D
• Then we can bound the VCdim of the band and so of
  the great circles

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Lower bound on throughput capacity

• Because of uniform convergence, we obtain

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Lower bound on throughput capacity

• We have shown that there is a schedule for
  transmitting packets such that in every (1c1)
  slots, each cell can transmit.
• Thus the rate at which each cell transmit is
  W/(1c1) bits/sec
• On the other hand, the rate a cell needs to
  transmit is less than

• So with high probability, and because c1 is grow
  linearly with (1D)2 we have

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Conclusions

• Designers may want to consider designing networks
  with small number of nodes
• Communication with nearby nodes at constant bit
  rates can be provided in a dense clusters of
  nodes, since the source destination distance
  shrink as

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Questions ?