1. The Capacity of Wireless Networks
Piyush Gupta and P. R. Kumar
Presented by Zhoujia Mao

2. Outline Arbitrary networks Two models: protocol and physical
An upper bound on transport capacity
Constructive lower bound on transport capacity
Random networks
Constructive lower bound on throughput capacity
Conclusions

3. Arbitrary Networks n nodes are arbitrary located in a unit area disc
Each node can transmit at W bits/sec over the channel
Destination is arbitrary
Rate is arbitrary
Transmission range is arbitrary
Omni directional antenna
When does a transmission received successfully ?
Allowing for two possible models for successful reception over one hop: The protocol model and the Physical model
We will consider two types of networks: arbitrary networks and random networks , where the nodes and their destinations are randomly chosen
Arbitrary network is defined as …
The reason to look at those networks is to not to be confined with any parameters of the model, and thus to find upper and lower bounds that suitable to any network model.
We will result in an upper bound, meaning that even under optimal circumstances, the throughput will never exceed this bound
We will look at those two models for successful reception of a packet: protocol model and physical model.

4. Protocol Model Let Xi denote the location of a node
A transmission is successfully received by Xj if:
For every other node Xk simultaneously transmitting
D is the guarding zone specified by the protocol

5. Physical Model Let Be a subset of nodes simultaneously transmitting
Let Pk be the power level chosen at node Xk
Transmission from node Xi is successfully received at node Xj if:
Models a situation where minimum SIR is necessary for successful receptions
Signal decays with distance as 1/r^alpha

6. Transport Capacity of Arbitrary Networks
Network transport one bit-meter when one bit transported one meter toward its destination
Main result :
Under the Protocol Model the transport capacity is ( as n )
if nodes are optimally placed, the traffic pattern is optimally chosen and the range of each transmission is optimally chosen
Transport capacity is measured with bit meters.

7. Arbitrary Network – upper bound on transport capacity
Assumptions:
There are n nodes arbitrarily located in a disk of unit area on the plane
The network transport lnT bits over T seconds
The average distance between source and destination of a bit is L
Transport capacity of lambda*n*L is achieved
In the time slot each node transmitting according to its arbitrary rate
The slotting assumption can be eliminated.

8. Theorem 2.1
In the protocol model, the transport capacity lnL is bounded as follows:
In the physical model,
We will prove only the protocol model bound.

9. Remarks
The upper bound in Protocol Model only depends on dispersion in the neighborhood of the receiver
The upper bound in Physical Model improves when α is large, i.e., when the signal power decays more rapidly with distance
When the domain is of A squares meters rather than 1 m^2, then all the upper bounds above are scaled by

10. Arbitrary Network – constructive lower bound
Theorem 3.1: There is a placement of nodes and an assignment of traffic patterns such that the network can achieve under protocol model
Proof – define r :=
Place transmitters at locations:
Place receivers at locations:

11. A constructive lower bound on capacity of arbitrary networkDr
>(1+D)r
(( ))
(( ))
(( )) r
2Dr
(( ))

12. Random Networks
n nodes are randomly located on S2 (the surface of a sphere of area 1sq m) or in a disk of area 1sq m in the plane
Each node has randomly chosen destination to send l(n) bits/sec
All transmissions employ the same nominal range or power
Two models: Protocol and Physical
S^2 introduced here to separate edge effects from other phenomena

13. Protocol Model
Let Xi denote the location of a node and r the common range
A transmission is successfully received by Xj if:
For every other Xk simultaneously transmitting
Very similar to the arbitrary case, but here we have got a constant r

14. Physical Model Let Be a subset of nodes simultaneously transmitting
Let P be the common power level
Transmission from node Xi is successfully received at node Xj if:
Here there is only one power level: P common to everybody

15. Throughput Capacity of Random Networks
Feasible throughput: λ(n) bits per second is feasible if there is a spatial and temporal scheme for scheduling transmissions such that every node can send λ(n) bits per second on average to its chosen destination
Throughput capacity: throughput capacity of the class of random network is of order θ(f(n)) bits per second if there are constants c > 0, c’ < ∞ such that

16. Spatial tessellation Let {a1,a2,….ap} be a set of p points on S2
The Voronoi cell V(ai) is the set of all points which are closer to ai than any of the other aj’s i.e.:
Point ai is called the generator of the Voronoi cell V(ai)

17. A Voronoi tessellation of S2

18. Tessellation properties
For each e>0, There is a Voronoi tessellation such that Each cell contains a disk of radius e and is contained in a disk of radius 2e
We will use a Voronoi tessellation for which :
Every Voronoi cell contains a disk of area 100logn/n . Let r(n) be its radius
Every Voronoi cell is contained in a disk of radius 2r(n)
For a1, choose any point in S^2.
Suppose there are already a1…ap that each two are distant at least 2e. Two cases:
If there is a point x such that D(x,e) not intersecting any of D(ai,e) add x
Otherwise stop
It finished because it removes area of disk with radii e>0
When stop, we have a set that each are at list 2e apart, and all other points on s^2 are within a distance of 2e from one of them.
Note that the area of a disk of radius r on S^2 is less than pi*r^2

19. Adjacency and interference
Adjacent cells are two cells that share a common point.
We will choose the range of transmission r(n) so that:
With this range, every node in a cell is within a distance r(n) from every node in its own cell or adjacent cell
8r(n)
2r(n)

20. Theorem 4.1
For Random Networks on in the Protocol Model, there is a deterministic constant c > 0 such that bits per second is feasible whp
For Physical Model, there are c’, c” such that
is feasible whp

21. Proof:
Lemma: in the Protocol Model there is a schedule for transmitting packets such that in every (1+ ) slots, each cell in the tessellation gets one slot in which to transmit, and such that all transmissions are successfully received within a distance r(n) from their transmitters
From the above lemma, the rate at which each cell gets to transmit is W/(1+ ) per second

22. Lemma: There is a δ’(n)→0 such that Prob ( (Traffic needing to be carried by cell V) ≤ c5λ(n) ) ≥ 1- δ’(n)
From the above lemma, the rate at which each cell needs to transmit is less than c5λ(n) whp. With high probability, this rate can be accommodated by all cells if it is less than the available rate, i.e., if

23. Within a cell, the traffic to be handled by the entire cell can be handled by any one node in the cell, since each node can transmit at rate W bits per second whenever necessary
Lemma: Every cell in has no more than interfering neighbors. depends only on ∆ and grows no faster than linearly in (1+∆)^2
Thus, for Protocol Model,
is feasible whp

24. Lemma: if ∆ is chosen to satisfy
then the above result of Protocol Model also holds for Physical Model
Plug the expression of ∆ into , we get

25. Theorem 5.1
For Random Networks on under the Protocol Model, there is a deterministic c’ < ∞ such that

26. Proof:
Lemma: the number of simultaneous transmission on any particular channel is no more than in the Protocol Model
Let L denote the mean length of the path of packets, then the mean number of hops taken by a packet is at least

27. Since each source generates λ(n) bits per second, there are n sources and each bit needs to be relayed on average by at least nodes, so the total number of bits per second needs to be at least Also, each transmission over a single channel is of W bits per second, so from the above lemma the number of bits can not be more than
bits per second, so

28. Lemma: the asymptotic probability that graph G(n, r(n)) has an isolated node and is disconnected is strictly positive if and
By the definition of feasible throughput, the absence of isolated node is a necessary condition for feasibility of any throughput. Thus, is necessary to guarantee connectivity whp. We obtain the upper bound for Protocol Model

29. Conclusion Implication for design Not considered Number of nodes
Signal decay rate …
Not considered
Delay
Mobility

30. Thanks ~ 

31. A graph of degree no more than c1 can have its vertices colored by using no more than (1+c1) colors
So color the graph such that no two interfering neighbors have the same color, so in each slot all the nodes with the same color transmit

32. If V’ is an interfering neighbor of V, then V’ and similarly every other interfering neighbor, must be contained within a common large disk D of radius 6r(n)+ (2+D)r(n)