1. Capacity of Ad Hoc Networks
Wireless Networks Spring 2005

2. Wireless Networks Spring 2005
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
Wireless Networks Spring 2005

3. Wireless Networks Spring 2005
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 receiver’s vicinity
Two models [GK00]:
Physical model
Protocol model
Wireless Networks Spring 2005

4. Wireless Networks Spring 2005
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)
Wireless Networks Spring 2005

5. Wireless Networks Spring 2005
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
Wireless Networks Spring 2005

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?
Wireless Networks Spring 2005

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
Wireless Networks Spring 2005

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
Wireless Networks Spring 2005

9. Transport Capacity: Lower Bound
sender
receiver
Wireless Networks Spring 2005

10. Transport Capacity: Lower Bound
Sender-receiver distance is
Assuming channel bandwidth W, transport capacity is
Thus, transport capacity per node is
Wireless Networks Spring 2005

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
Wireless Networks Spring 2005

12. Transport Capacity: Upper Bound
Consider two successful transmissions in a slot:
Wireless Networks Spring 2005

13. Transport Capacity: Upper Bound
Balls of radii around , for all , are disjoint
So bit-meters transported per slot is
Wireless Networks Spring 2005

14. Throughput Capacity of Random Networks
The throughput capacity of an -node random network is
There exist constants c and c’ such that
Wireless Networks Spring 2005

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
Wireless Networks Spring 2005

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 [LBD+03]
Hybrid networks [KT03, LLT03, Tou04]
Asymmetric scenarios and cluster networks [Tou04]
Wireless Networks Spring 2005

17. Asymmetric Traffic Scenarios
Number of destinations smaller than number of sources
nd destinations for n sources; 0 < d <= 1
Each source picks a random destination
If 0 < d < 1/2, capacity scales as nd
If 1/2 < d <= 1, capacity scales as n1/2
[Tou04]
Wireless Networks Spring 2005

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 < -2 to > -1, capacity scaling goes from to [LBD+03]
Wireless Networks Spring 2005

19. Wireless Networks Spring 2005
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 < d < 1/2: No asymptotic benefit
1/2 < d <= 1: Capacity scaling by a factor of nd
Wireless Networks Spring 2005

20. Wireless Networks Spring 2005
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
Wireless Networks Spring 2005