The Capacity of Wireless Networks

Piyush Gupta and P. R. Kumar

Presented by Zhoujia Mao

Outline

Arbitrary networks 1. Two models: protocol and physical2. An upper bound on transport capacity3. Constructive lower bound on transport capacityRandom networks1. Two models: protocol and physical2. Constructive lower bound on throughput capacityConclusions

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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

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Protocol Model

• Let Xi denote the location of a node

• A transmission is successfully received by Xj if:

r 1

XX XX jijk 1

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For every other node For every other node XXkk simultaneously simultaneously transmitting transmitting

• is the guarding zone specified by the protocolis the guarding zone specified by the protocol

r

jx

ixkx

r 1

lx

Physical Model• Let kX k ;

Pi

Tkik

jk

k

ji

XX

PN

XX

5

Be a subset of nodes simultaneously transmitting Be a subset of nodes simultaneously transmitting

• Let PLet Pkk be the power level chosen at node Xbe the power level chosen at node Xkk

• Transmission from node XTransmission from node Xii is successfully is successfully received at node Xreceived at node Xjj if: if:

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 )

meters/sec-bit nW )(

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if nodes are optimally placed, the if nodes are optimally placed, the traffic pattern is optimally chosen and traffic pattern is optimally chosen and the range of each transmission is the range of each transmission is optimally chosen optimally chosen

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 nT bits over T seconds• The average distance between source and

destination of a bit is L

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Theorem 2.1 • In the protocol model, the transport capacity nL is bounded

as follows:

meters/sec-bit nW8

nL

1

8

• In the physical model, In the physical model,

meters/sec-bit nW1

22

nL

11

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

9

A

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

meters/sec-bit n

nW

821

n

24

1

21

1

10

• Proof Proof –– define define r :=r :=

Place transmitters at Place transmitters at locations:locations: even is kj wherer))r2(1 k)r,2(j(1 and )r)2(1 kr,)r2(j(1

odd is kj wherer))r2(1 k)r,2(j(1 and )r)2(1 kr,)r2(j(1 Place receivers at Place receivers at

locations:locations:

A constructive lower bound on capacity of arbitrary network

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rr

rr

)) (()) ((rrrr

(>(>11++))rr

)) (()) ((

)) (()) ((

)) (()) ((

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 (n)

bits/sec• All transmissions employ the same nominal range or power• Two models: Protocol and Physical

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Protocol Model

• Let Xi denote the location of a node and r the common range

• A transmission is successfully received by Xj if:

r XX

r XX

jk

ji

1.2

.1

13

For every other For every other XXk k simultaneously transmittingsimultaneously transmitting

Physical Model• Let kX k ;

Tkik

jk

ji

XX

PN

XX

P

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Be a subset of nodes simultaneously transmitting Be a subset of nodes simultaneously transmitting

• Let PLet P be the common power level be the common power level

• Transmission from node XTransmission from node Xii is successfully is successfully received at node Xreceived at node Xjj if: if:

Throughput Capacity of Random Networks

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• 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 1)feasible is )(')((Prinflim

nfcnob

n

1)feasible is )()((Prlim

ncfnobn

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.:

jpji2

i axMinaxSx aV 1::)(

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Point Point aaii is called the generator of the Voronoi is called the generator of the Voronoi cell V(cell V(aaii) )

A Voronoi tessellation of S2

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• For each >0, There is a Voronoi tessellation such that Each cell contains a disk of radius and is contained in a disk of radius 2

We will use a Voronoi tessellation for which :1. Every Voronoi cell contains a disk of area 100logn/n . Let (n) be its

radius 2. Every Voronoi cell is contained in a disk of radius 2(n)

Tessellation properties

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Adjacency and interference• Adjacent cells are two cells that share a common point.• We will choose the range of transmission r(n) so that:

(n)8 nr )(

With this range, every node in a cell is within a With this range, every node in a cell is within a distance r(n) from every node in its own cell or distance r(n) from every node in its own cell or adjacent celladjacent cell

2(n)

8(n)

• 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

Theorem 4.12S

nn

cWn

log)1()(

2

nnc

Wcn

log)1))2

21

13("(2(

')(

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• 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

1c

n

1c

– 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

nV sup

nn log

nn log

15 1

log)(c

Wnnnc

– 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

n 1c

1c

)log()1()log)(1()(

215 nn

cW

nncc

Wn

– 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

21

2 )1))2

2

1

13((2(1

c）（

nn

cWn

log)1()(

2

nnc

Wcn

log)1))2

21

13("(2(

')(

21

• For Random Networks on under the Protocol Model, there is a deterministic c’ < ∞ such that

Theorem 5.12S

0)feasible is log

')((Pr lim

2

nn

Wcnob

n

• 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

)(

422

11 nrc

)(

L

nr

– 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

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)(

L

nr

)(

(n)Ln

nr

)(

4W22

11 nrc

)(

c)(

212

nnr

Wn

– 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

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n

knnr n

log)(2 nn ksuplim

n

nnr

log

)(

0)feasible is log

')((Pr lim

2

nn

Wcnob

n

Conclusion

• Implication for design– Number of nodes– Signal decay rate– …

• Not considered– Delay– Mobility– …

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Thanks ~

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• 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

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• 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 6(n)+ (2+)r(n)

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