1. Wireless Networks Spring 2005 Capacity of Ad Hoc Networks

2. Wireless Networks Spring 2005 The Attenuation Model  Path loss: oRatio of received power to transmitted power oFunction of medium properties and propagation distance  If P R is received power, P T is the transmitted power, and d is distance  Where  ranges from 2 to 4

3. Wireless Networks Spring 2005 Interference Models  In addition to path loss, bit-error rate of a received transmission depends on: oNoise power oTransmission powers and distances of other transmitters in the receiver’s vicinity  Two models [GK00]: oPhysical model oProtocol model

4. Wireless Networks Spring 2005 The Physical Model  Let {X i } denote set of nodes that are simultaneously transmitting  Let P i be the transmission power of node X i  Transmission of X i is successfully received by Y if:  Where  is the min signal-interference ratio (SIR)

5. Wireless Networks Spring 2005 The Protocol Model  Transmission of X i is successfully received by Y if for all k  where  is a protocol-specified guard-zone to prevent interference

6. Wireless Networks Spring 2005 Measures for Network Capacity  Throughput capacity [GK00]: oNumber of successful packets delivered per second oDependent on the traffic pattern oWhat is the maximum achievable, over all protocols, for a random node distribution and a random destination for each source?  Transport capacity [GK00]: oNetwork transports one bit-meter when one bit has been transported a distance of one meter oNumber of bit-meters transported per second oWhat is the maximum achievable, over all node locations, and all traffic patterns, and all protocols?

7. Wireless Networks Spring 2005 Transport Capacity: Assumptions  n nodes are arbitrarily located in a unit disk  We adopt the protocol model oEach node transmits with same power oCondition for successful transmission from X i to Y: for any k  Transmissions are in synchronized slots

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

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

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

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

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

15. Wireless Networks Spring 2005 Implications of Analysis  Transport capacity: oPer node transport capacity decreases as oMaximized when nodes transmit to neighbors  Throughput capacity: oFor random networks, decreases as oNear-optimal when nodes transmit to neighbors  Designers should focus on small networks and/or local communication

16. Wireless Networks Spring 2005 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: oPower law traffic patterns [LBD + 03] oHybrid networks [KT03, LLT03, Tou04] oAsymmetric scenarios and cluster networks [Tou04]

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

18. Wireless Networks Spring 2005 Power Law Traffic Pattern  Probability that a node communicates with a node x units away is oFor large negative, destinations clustered around sender oFor large positive, destinations clustered at periphery  As goes from -1, capacity scaling goes from to [LBD + 03]

19. Wireless Networks Spring 2005 Relay Nodes  Offer improved capacity: oBetter spatial reuse oRelay nodes do not count in oExpensive: addition of nodes as pure relays yields less than -fold increase  Hybrid networks: n wireless nodes and n d access points connected by a wired network o0 < d < 1/2: No asymptotic benefit o1/2 < d <= 1: Capacity scaling by a factor of n d

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 oFraction 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