Wireless Networks Spring 2005

Capacity of Ad Hoc Networks

Wireless Networks Spring 2005

The Attenuation Model

Path loss: o Ratio of received power to transmitted powero 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

)( dPOP T

R =

Wireless Networks Spring 2005

Interference Models

In addition to path loss, bit-error rate of a received transmission depends on:o Noise powero Transmission powers and distances of other

transmitters in the receiver’s vicinity

Two models [GK00]:o Physical modelo Protocol model

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)

≥+ ∑

≠ikk

k

i

i

YXdP

N

YXdP

),(

),(

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

),()1(

),( YXdP

YXdP

k

k

i

i Δ+≥

Wireless Networks Spring 2005

Measures for Network Capacity

Throughput capacity [GK00]:o Number of successful packets delivered per secondo Dependent on the traffic patterno What is the maximum achievable, over all protocols, for a

random node distribution and a random destination for each source?

Transport capacity [GK00]: o Network transports one bit-meter when one bit has been

transported a distance of one metero Number of bit-meters transported per secondo What is the maximum achievable, over all node locations,

and all traffic patterns, and all protocols?

Wireless Networks Spring 2005

Transport Capacity: Assumptions

n nodes are arbitrarily located in a unit disk We adopt the protocol model

o Each node transmits with same powero Condition for successful transmission from Xi to Y: for

any k

Transmissions are in synchronized slots

),()1(),( YXdYXd ki δ+≥

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

Wireless Networks Spring 2005

Transport Capacity: Lower Bound

sender

receiver

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

)( nWΩ

)/1( nΩ

)(nWΩ

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

∑≤jjrW

receiver )(

Wireless Networks Spring 2005

Transport Capacity: Upper Bound

Consider two successful transmissions in a slot:

j

k

l

i

€

i→ j and k → l

€

d( j, l ) ≥ (1+δ)d(i, j)− d(l ,k)

€

d(l , j) ≥ (1+δ)d(k, l )− d(i, j)

€

d(l , j) ≥δ2

(d(i, j)+ d(k, l ))

Wireless Networks Spring 2005

Transport Capacity: Upper Bound

Balls of radii around , for all , are disjoint

So bit-meters transported per slot is

)1(2 Orj

j =∑

)( jrΘ j j

)( nWO

)()()( 2 nOhOrj

j ==∑

)( nOrj

j =∑

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

0]log

'Pr[lim

1]log

Pr[lim

=

=

∞→

∞→

feasible is

feasible is

nnW

c

nnW

c

n

n

)log

(nn

WΘ

n

Wireless Networks Spring 2005

Implications of Analysis

Transport capacity:o Per node transport capacity decreases as o Maximized when nodes transmit to neighbors

Throughput capacity:o For random networks, decreases aso Near-optimal when nodes transmit to neighbors

Designers should focus on small networks and/or local communication

n1

nn log1

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:

o Power law traffic patterns [LBD+03]o Hybrid networks [KT03, LLT03, Tou04]o Asymmetric scenarios and cluster networks [Tou04]

Wireless Networks Spring 2005

Asymmetric Traffic Scenarios

Number of destinations smaller than number of sourceso nd destinations for n sources; 0 < d <= 1o 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

Power Law Traffic Pattern

Probability that a node communicates with a node x units away is

o For large negative , destinations clustered around sender

o For large positive , destinations clustered at periphery As goes from < -2 to > -1, capacity scaling goes

from to [LBD+03]

∫=

1)(

εα

α

dttx

xp

€

)1(O )/1( nO

Wireless Networks Spring 2005

Relay Nodes

Offer improved capacity:o Better spatial reuse o Relay nodes do not count in o 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 o 0 < d < 1/2: No asymptotic benefito 1/2 < d <= 1: Capacity scaling by a factor of nd

nkn

1+k

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 othero 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

€

n

€

n

€

n

€

1/n