Capacity of Ad Hoc Networks

1
Capacity of Ad Hoc Networks
2
The Attenuation Model

• Path loss
• Ratio of received power to transmitted power
• Function of medium properties and propagation
  distance
• If PR is received power, PT is the transmitted
  power, and d is distance
• Where ? ranges from 2 to 4

3
Interference Models

• In addition to path loss, bit-error rate of a
  received transmission depends on
• Noise power
• Transmission powers and distances of other
  transmitters in the receivers vicinity
• Two models GK00
• Physical model
• Protocol model

4
The Physical Model

• Let Xi denote set of nodes that are
  simultaneously transmitting
• Let Pi be the transmission power of node Xi
• Transmission of Xi is successfully received by Y
  if
• Where ? is the min signal-interference ratio
  (SIR)

5
The Protocol Model

• Transmission of Xi is successfully received by Y
  if for all k
• where ? is a protocol-specified guard-zone to
  prevent interference

6
Measures for Network Capacity

• Throughput capacity GK00
• Number of successful packets delivered per second
• Dependent on the traffic pattern
• What is the maximum achievable, over all
  protocols, for a random node distribution and a
  random destination for each source?
• Transport capacity GK00
• Network transports one bit-meter when one bit has
  been transported a distance of one meter
• Number of bit-meters transported per second
• What is the maximum achievable, over all node
  locations, and all traffic patterns, and all
  protocols?

7
Transport Capacity Assumptions

• n nodes are arbitrarily located in a unit disk
• We adopt the protocol model
• Each node transmits with same power
• Condition for successful transmission from Xi to
  Y for any k
• Transmissions are in synchronized slots

8
Transport Capacity Lower Bound

• What configuration and traffic pattern will yield
  the highest transport capacity?
• Distribute n/2 senders uniformly in the unit disk
• Place n/2 receivers just close enough to senders
  so as to satisfy threshold

9
Transport Capacity Lower Bound
10
Transport Capacity Lower Bound

• Sender-receiver distance is
• Assuming channel bandwidth W, transport capacity
  is
• Thus, transport capacity per node is

11
Transport Capacity Upper Bound

• For any slot, we will upper bound the total
  bit-meters transported
• For a receiver j, let r_j denote the distance
  from its sender
• If channel capacity is W, then bit-meters
  transported per second is

12
Transport Capacity Upper Bound

• Consider two successful transmissions in a slot

13
Transport Capacity Upper Bound

• Balls of radii around , for all , are
  disjoint
• So bit-meters transported per slot is

14
Throughput Capacity of Random Networks

• The throughput capacity of an -node random
  network is
• There exist constants c and c such that

15
Implications of Analysis

• Transport capacity
• Per node transport capacity decreases as
• Maximized when nodes transmit to neighbors
• Throughput capacity
• For random networks, decreases as
• Near-optimal when nodes transmit to neighbors
• Designers should focus on small networks and/or
  local communication

16
Remarks on Capacity Analysis

• Similar claims hold in the physical model as well
• Results are unchanged even if the channel can be
  broken into sub-channels
• More general analysis
• Power law traffic patterns LBD03
• Hybrid networks KT03, LLT03, Tou04
• Asymmetric scenarios and cluster networks Tou04

17
Asymmetric Traffic Scenarios

• Number of destinations smaller than number of
  sources
• nd destinations for n sources 0 lt d lt 1
• Each source picks a random destination
• If 0 lt d lt 1/2, capacity scales as nd
• If 1/2 lt d lt 1, capacity scales as n1/2
• Tou04

18
Power Law Traffic Pattern

• Probability that a node communicates with a node
  x units away is
• For large negative , destinations clustered
  around sender
• For large positive , destinations clustered at
  periphery
• As goes from lt -2 to gt -1, capacity scaling
  goes from to LBD03

19
Relay Nodes

• Offer improved capacity
• Better spatial reuse
• Relay nodes do not count in
• Expensive addition of nodes as pure relays
  yields less than -fold increase
• Hybrid networks n wireless nodes and nd access
  points connected by a wired network
• 0 lt d lt 1/2 No asymptotic benefit
• 1/2 lt d lt 1 Capacity scaling by a factor of nd

20
Mobility and Capacity

• A set of nodes communicating in random
  source-destination pairs
• Expected number of hops is
• Necessary scaling down of capacity
• Suppose no tight delay constraint
• Strategy packet exchanged when source and
  destination are near each other
• Fraction of time two nodes are near one another
  is
• Refined strategy Pick random relay node (a la
  Valiant) as intermediate destination GT01
• Constant scaling assuming that stationary
  distribution of node location is uniform