1. CAPACITY OF POWER CONSTRAINED AD-HOC NETWORKS
Prof. Rohit Negi
Arjunan Rajeswaran
Presented by Jin Heo
* The slides are from the authors and modified by Jin Heo

2. Background – what has been done, how, results ?
Outline
Introduction – what, why is it a hard problem ?
Background – what has been done, how, results ?
Recent Research – why, novelty, results ?

3. Introduction ad hoc wireless networks
Infrastructure
Last hop is wireless
High Cost, maintenance
Data rates known
Reliable
Ad hoc
Multi-hop wireless
Low cost, maintenance
Supported data rate ?
Reliability ?
Cellular : Key point: Wireless is simply a last-hop link in a primarily wired (telephone/internet) network. Basically for un-tethering.
High cost in deployment, signaling billing etc.
Ad-hoc : key point : All wireless links – possibly many links till reach infrastructure or not at all. : NO infrastructure.
Many variations : mobility , capabilities, traffic patterns, node placement.

4. Introduction ad hoc wireless networks
Consider model with low spectral efficiency
Arbitrary large bandwidth
Power constrained
 Interference becomes relatively negligible!
Two applications
UWB
Sensor network
The result is that throughput increases with node enter the network

5. Capacity of UWB ad-hoc wireless networks
Summary
Capacity of UWB ad-hoc wireless networks
Previous Result : bits/s
Our result : bits/s
under alternate Communication model (UWB)
n is the node density
distance loss exponent

6. Design problem overview
source 1
destination 1
This shared wireless channel
is scheduled into links by
the MAC (link layer)
Routes : set of links from
each source to destination
Created by the routing layer
destination 2
source 2
Aim : maximize the uniform throughput per node r (bits/s) ( fairness, uniform capabilities )
Uniform throughput – the throughput achieved by all nodes in the network
Two Control Parameters to maximize throughput – Routing and MAC
Routing (choosing sets of links) – MAC (scheduling access of shared channel, generates the links)  Strong interaction
Definition of uniform throughput – the throughput achieved by all nodes in the network : fairness
Now the aim is to maximize this uniform throughput
We have the MAC and Routing to control. Variables
MAC : scheduling access of the shared channel to the competing nodes.
MAC creates the links over which routing may optimize.
Routing – scheme to relay data packets form the source to destination in many hops – all wireless ?
Strong interaction because the routes decide the link capacities so let us look at a simpler problem – only MAC

7. Design problem MAC
Original ad-hoc network 1 2 3
Flow contention graph
Link
Contention
MAC (in simplest case) ~ MAC scheduling problem to a Flow contention graph: Graph Coloring NP complete
Complicated due to Routing - MAC interaction
Formally : optimization problem – intractable
Exact solution : very hard
Need some kind of order bound as a function of number of nodes!
MAC problem – the scheduling problem may be made explicit by a conversion to a Flow contention graph where each link is represented by a node and the contention by an edge between them.
So now we need a coloring of the flow contention graph which would be the schedule solving the MAC problem, where the number of colors for a node proportional to the required capacity of the link.
So, how do we analyze this problem ??
The idea is rather than find an exact value we need some kind of order bound ?? So we can at least say it is increasing or decreasing as a function of some variable ( number of nodes n )??

8. Background1 – what has been done, how, results ?
Outline
Introduction – what, why is it a hard problem ?
Background1 – what has been done, how, results ?
Recent Research – why, novelty, results ?
1. P.Gupta, P.R. Kumar, “Capacity of Wireless Networks,”
IEEE Trans. Information Theory, Vol.46, March 2000.

9. Background overview Assumptions
Nodes(n) location Xi : identical, independent uniform
Unit area : sphere’s surface
Homogenous nodes : rate r(n) , power P
Destination : random independent
Control : MAC, routing
Here n nodes are assumed to be iid on the surface of a sphere. Thus it is a random network !!
Surface of a sphere is assume to avoid the edge effects you would see on a plane – later it is shown that the same results hold for a plane too
Identical nodes are assumed so it is typical of sensor networks ….
So if the network is random what is under our control in the design – MAC and routing
Network is a particular sample form the underlying distribution of nodes on the unit area.
The network is random so is the capacity and hence capacity is a random variable – note the definition
With this definition we shall now on call throughput capacity as capacity ( fro simplicity) note IT IS NOT SHANNON CAPACITY.
Aim : provide order bounds on the capacity as a function of n .
c>0 c’<infinity
It is the time average bits/s per source destination pair.

10. Background overview Assumptions Metric
Nodes(n) location Xi : identical, independent uniform
Homogenous nodes : rate r(n), power P
Unit area : sphere’s surface
Destination : random independent
Control : MAC, routing
Metric
Finding order bounds on the UNIFORM THROUGHPUT CAPACITY r(n)
It is the time average bits/s per source destination pair.
Here n nodes are assumed to be iid on the surface of a sphere. Thus it is a random network !!
Surface of a sphere is assume to avoid the edge effects you would see on a plane – later it is shown that the same results hold for a plane too
Identical nodes are assumed so it is typical of sensor networks ….
So if the network is random what is under our control in the design – MAC and routing
Network is a particular sample form the underlying distribution of node son the unit area.
The network is random so is the capacity and hence capacity is a random variable – note the definition
With this definition we shall now on call throughput capacity as capacity ( fro simplicity) note IT IS NOT SHANNON CAPACITY.
Aim : provide order bounds on the capacity as a function of n .
c>0 c’<infinity
It is the time average bits/s per source destination pair.

11. Background communication model
Wireless link : T bits/s
Interference Model :
Protocol : interference radius, guard zone of
Transmissions is successful if every other receiver is out of interference range of the receiver
Physical : propagation loss , If SNR > threshold, the receiver can receive packets
Transmitter
Simultaneous Transmitter
The wireless link has w bits/s and so is a communication theoretic perspective
Interference models tell us when a node Xk interferes with a transmission form node Xi to Xj
The guard zone of delta works for the high SNR threshold case since then the interference region
is greater than the transmission region and so 1+delta is valid.
Also works when you have some threshold of SNR for operation and assuming a single interferer.
Receiver

12. Background intuition r(n) is MAC vs Routing
tr(n) - transmission range/power
L – mean source-destination distance
MAC vs Routing
Reduces to Generates bits/s
Available capacity Generated Traffic
Hence
Mean number of hops to destination
Capacity loss due to interferences
Here provide the intuition about how MAC and the Routing need to be traded off
Only then can we justify the voronoi tessellation and the chosen sizes for uniform convergence.
Let L be the mean source destination distance can be figured out right – a constant
So now number of hops = l/ Tr trafiic generated is l* r/tr
Each node has T ? No relay has T/tr^2 and so the tradeoff
Upper bound is a limit on tr – above a threshold – else implies loss of connectivity

13. traffic to be carried available capacity
Background capacity
Lower bound (achievable) : use the a cellular like scheme
traffic to be carried available capacity
Upper bound : using transmission radius
transmission radius lower bound for connectivity
Order Bound :
r(n) is
Describe the bound briefly –

14. Background – what has been done, how, results ?
Outline
Introduction – what, why is it a hard problem ?
Background – what has been done, how, results ?
Recent Research – why, novelty, results ?

15. Recent Research overview
Communication model : UWB (ultra wideband systems)
limited power
large bandwidth
MAC : CDMA is optimal
At least as good as any other algorithm in this communication model
Routing : power constrained routing
R. Negi and Arjunan – The Capacity of power constrained ad-hoc networks
Note the difference an increasing function of n

16. Recent Research communication model
power spectral density
Power : constrained to P0
Medium loss
Bandwidth: arbitrarily large W
large w.r.t data rate – low spectral effeciency
Examples : UWB ( a), sensors
Link operates at Shannon Capacity
Power and data rate adaptation according to the link condition
Bandwidth scaling is one and the large is another
As a function of n you will require bandwidth to make interference negligible
This tells you growth rate as a function of n
But to make it hold it needs to be large – how large – approximation of shannon capacity should hold
This gives absolute values.

17. Recent Research PHY - interference
SINR = Signal / (Noise + Interference)
Noise = noise density * bandwidth
In bandwidth-constrained scenario, SINR is dominated by interference
In low spectral efficiency, SINR is mainly affected by ambient noise
And also that Xi is transmitter and Xj is receiver and the rest are interferers

18. Recent Research PHY - interference
Noise
interference
Bandwidth scaling : Makes W arbitrarily large  interference becomes negligible w.r.t ambient noise
Interference negligible  no scheduling !
CDMA MAC is optimal, compared to FDMA/TDMA
And also that Xi is transmitter and Xj is receiver and the rest are interferers

19. Recent Research PHY –link capacity
What is the link capacity –
The Shannon capacity for a link with Gaussian noise and interference sources: W log(1+SINR)
Capacity is LINEAR in allotted power
Rate/Capacity : adapts to link
The link capacity is bounded due to the power constraint
And also that Xi is transmitter and Xj is receiver and the rest are interferers

20. Recent Research routing
Relaxed power constraint (upper bound) :
routing can be decoupled from power constraint of each node
Power on a route :
Optimal Route :
r(n) can be very large : each hop arbitrarily high capacity !
Number of hops is limited : hop length is bounded
Note mention that the relaxed power constraint is assumed to derive an upper bound
Also that this implies a power adaptation / control and since power and rate are connected a rate adaptation : results in route decoupling
So basically each source destination attempts to use minimum power : resulting in minimum overall power for the network since no constraint on any node. Thus this solution maybe scaled for the appropriate maximum uniform throughput capacity.

21. Recent Research capacity
Upper bound : relaxed power constraint
Power is bounded by the average
Number of hops (K) : bounded by tessellation
Distance term : bounded by average source destination distance
Lower bound : extend the previous scheme:
traffic to be carried < available capacity
Order bound : (soft order neglecting log terms)
r(n) is
Here in the bounding of the distance the problem is that
Pessimistically – the number of nodes could be just one and so the distance is order 1^alpha - very bad for capacity.
Optimistically – it could be all nodes and then the distance term is (1/n)^alpha * n too good for capacity
So what do MOST or rather almost all routes actually see – calculate the number of nodes and bound it by the average distance between source and destination.
IN these cases mostly the routes are all close to the average ……………
We are designing for the worst case …to ensure that ever node will satisfy its minimum criterion that is why it is hard.

22. Capacity of UWB ad-hoc wireless networks
Conclusion
Capacity of UWB ad-hoc wireless networks
Previous Result : bits/s
Our result : bits/s
under alternate Communication model (UWB)
n is the node density
distance loss exponent
GAIN
The gain comes two fold : 1. The infinite bandwidth washes away interference ( in the previous case basically washes away noise)
in our case the uwb washes away interference.
The second gain is that the large bandwidth gives us an explicit relationship between the capacity and power ie we exploit the fact that a node close by could be transmitting at high rate and so we need not be designing it for the farthest node like in the previous communication model
Gives us the gain of \sqrt(n) ^alpha.

23. Information order bounds
Intuitively f grows no faster than g
Make a note as a function of naturals etc
Basically growth rates
Intuitively f,g grow at the same rate