The Capacity of The Capacity of Wireless Wireless NetworksNetworks

Danss Course, Sunday, Danss Course, Sunday, 23/11/0323/11/03

Wireless ad hoc networkWireless ad hoc network

• No wired backboneNo wired backbone• No centralized controlNo centralized control• Nodes may cooperate in routing each other’s Nodes may cooperate in routing each other’s

data packetsdata packets• At the Network Layer – problems are in At the Network Layer – problems are in

routing, mobility of nodes and power routing, mobility of nodes and power constraintsconstraints

• At the MAC layer – problems with protocols At the MAC layer – problems with protocols such as TDMA, FDMA,CDMA such as TDMA, FDMA,CDMA

• At the Physical layer – problems in power At the Physical layer – problems in power controlcontrol

Lecture MinutesLecture Minutes

Arbitrary networksArbitrary networks

1.1. Two models: protocol and physicalTwo models: protocol and physical

2.2. An upper bound on transport capacityAn upper bound on transport capacity

3.3. Constructive lower bound on transport Constructive lower bound on transport capacitycapacity

Random networksRandom networks

1.1. Two models: protocol and physicalTwo models: protocol and physical

2.2. Constructive lower bound on throughput Constructive lower bound on throughput capacitycapacity

ConclusionsConclusions

Arbitrary NetworksArbitrary Networks• nn nodes are arbitrary located in a unit area disc nodes are arbitrary located in a unit area disc• Each node is can transmit at W Each node is can transmit at W bits/secbits/sec over the over the

channelchannel• Destination is arbitraryDestination is arbitrary• Rate is arbitraryRate is arbitrary• Transmission range is arbitraryTransmission range is arbitrary• Will later add some assumptions on the networkWill later add some assumptions on the network• When does a transmission received When does a transmission received

successfully ?successfully ?Allowing for two possible models for successful Allowing for two possible models for successful

reception over one hop: The protocol model and reception over one hop: The protocol model and the Physical modelthe Physical model

Protocol ModelProtocol Model

• Let Let XXii denote the location of a nodedenote the location of a node

• A transmission is successfully received by A transmission is successfully received by XXjj if:if:

r 1

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

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

XX XX jijk 1

r

jx

ixkx

r 1

lx

Physical ModelPhysical Model

• Let Let kX k ;

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:

2

XX

PN

XX

P

Tkik

jk

k

ji

i

Transport Capacity of Transport Capacity of Arbitrary NetworksArbitrary Networks

• Network transport one Network transport one bit-meterbit-meter when one bit when one bit transported one meter toward its destinationtransported one meter toward its destination

• Main result 1:Main result 1: Under the Protocol Model the transport capacity isUnder the Protocol Model the transport capacity is

meters/sec-bit nW )(

meters/sec-bit ncW

meters/sec-bit Wnc 1

'

• Main result 2:Main result 2: Under the Physical Model,Under the Physical Model,

WhileWhile is notis not

is feasibleis feasible

Arbitrary Network – upper Arbitrary Network – upper bound on transport bound on transport

capacitycapacityAssumptions:Assumptions:• There are There are nn nodes arbitrarily located in a nodes arbitrarily located in a

disk of unit area on the planedisk of unit area on the plane• The network transport The network transport nT bits over T nT bits over T

seconds, i.e. each node generate bits at seconds, i.e. each node generate bits at rate rate

• The average distance between source and The average distance between source and destination of a bit is Ldestination of a bit is L

• Transmissions are slotted into Transmissions are slotted into synchronized slots of length synchronized slots of length sec sec

TheoremTheorem

• In the protocol model, the transport capacity In the protocol model, the transport capacity nL nL is bounded as follows:is bounded as follows:

meters/sec-bit nW8

nL

1

• In the physical model, In the physical model,

meters/sec-bit nW1

22

nL

11

Arbitrary Network – Arbitrary Network – constructive lower boundconstructive lower bound

• There is a placement of nodes and an assignment There is a placement of nodes and an assignment of traffic patterns such that the network can of traffic patterns such that the network can achieve under protocol modelachieve under protocol model

meters/sec-bit n

nW

821

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

24

1

21

1

Place transmitters at locations:Place transmitters at 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 locations:Place receivers at locations:

A constructive lower bound A constructive lower bound on capacity of arbitrary on capacity of arbitrary

networknetwork

rr

rr

)) (()) ((rrrr

(>(>11++))rr

)) (()) ((

)) (()) ((

)) (()) ((

Random NetworksRandom Networks

• nn nodes are randomly located on S nodes are randomly located on S22 (the surface (the surface of a sphere of area 1sq m) or in a disk of area of a sphere of area 1sq m) or in a disk of area 1sq m in the plane1sq m in the plane

• Each node has randomly chosen destination to Each node has randomly chosen destination to send send (n) bits/sec(n) bits/sec

• All transmissions employ the same nominal range All transmissions employ the same nominal range or poweror power

• Two models: Protocol and PhysicalTwo models: Protocol and Physical

Protocol ModelProtocol Model• Let Let XXii denote the location of a node and denote the location of a node and rr the common range the common range

• A transmission is successfully received by A transmission is successfully received by XXjj if:if:

r XX

r XX

jk

ji

1.2

.1

For every other For every other XXk k simultaneously transmittingsimultaneously transmitting

Physical ModelPhysical Model

• Let Let kX k ;

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:

Tkik

jk

ji

XX

PN

XX

P

Throughput Capacity of Throughput Capacity of Random NetworksRandom Networks

• Main result 1:Main result 1:

Under the Protocol Model the order of the Under the Protocol Model the order of the throughput capacitythroughput capacity

bit/sec nn

Wn )

log()(

bit/sec n

Wcn

')(

• Main result 2:Main result 2: Under the Physical Model,Under the Physical Model,

WhileWhile is notis not

is feasibleis feasiblebit/sec nn

cWn

log)(

Random Networks: A Random Networks: A constructive lower bound constructive lower bound

on capacityon capacityWe will show a scheme such that each source-We will show a scheme such that each source-

destination pair can be guaranteed a channel of destination pair can be guaranteed a channel of capacity capacity

bit/sec nn

cW

log)1( 2With probability approaching 1 asWith probability approaching 1 as n

StepsSteps• Define the Voronoi tessellationDefine the Voronoi tessellation• Bound the number of interfering neighbors of a Voronoi cellBound the number of interfering neighbors of a Voronoi cell• Bound the length of an all-cell transmission scheduleBound the length of an all-cell transmission schedule• Define the routes of a packet on the Voronoi tessellationDefine the routes of a packet on the Voronoi tessellation• Prove that each cell contains at least one nodeProve that each cell contains at least one node• Calculate the expected routes that pass through a cell and Calculate the expected routes that pass through a cell and

infer the expected traffic of each nodeinfer the expected traffic of each node

Spatial tessellationSpatial tessellation

• Let {Let {aa11,,aa22,….,….aapp} be a set of } be a set of pp points on S points on S2 2

• The Voronoi cell V(The Voronoi cell V(aaii) is the set of all points ) is the set of all points which are closer to which are closer to aaii than of the other than of the other aajj’s i.e.:’s i.e.:

jpji2

i axMinaxSx aV 1::)(

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

A Voronoi tessellation of A Voronoi tessellation of SS22

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

We will use a Voronoi tessellation for which :We will use a Voronoi tessellation for which :1.1. Every Voronoi cell contains a disk of area 100logn/n . Every Voronoi cell contains a disk of area 100logn/n .

Let Let (n) be its radius (n) be its radius

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

Tessellation propertiesTessellation properties

Adjacency and Adjacency and interferenceinterference

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

8(n)

2(n)

A bound on the number of A bound on the number of Interfering cellsInterfering cells

• Two cells are interfering neighbors if there is a point in Two cells are interfering neighbors if there is a point in one cell which is within a distance of (2+one cell which is within a distance of (2+))rr(n) of some (n) of some point in the other cellpoint in the other cell

• Lemma – Every cell in VLemma – Every cell in Vnn has no more than has no more than cc11 interfering interfering cells. cells. cc1 1 grows no faster than linearly in (1+grows no faster than linearly in (1+))22

Proof – Proof – if V’ is an interfering neighbor of V, then V’ and if V’ is an interfering neighbor of V, then V’ and similarly every other interfering neighbor, must be similarly every other interfering neighbor, must be contained within a common large disk D of radius 6contained within a common large disk D of radius 6(n)+ (n)+ (2+(2+))rr(n) (n)

A bound on the length of an A bound on the length of an all-cell inclusive all-cell inclusive

transmission scheduletransmission schedule• LemmaLemma - In the protocol model, there is a schedule - In the protocol model, there is a schedule

for transmitting packets such that in every (1+cfor transmitting packets such that in every (1+c11) ) slots, each cell in Vslots, each cell in Vnn gets one slot in which to transmit gets one slot in which to transmit

Proof Proof – A graph of degree no more than c– A graph of degree no more than c11 can have its can have its vertices colored by using no more than (1+cvertices colored by using no more than (1+c11) colors.) colors.

So color the graph such that no two interfering So color the graph such that no two interfering neighbors have the same color, so in each slot all the neighbors have the same color, so in each slot all the nodes with the same color transmitnodes with the same color transmit

• There is a schedule also for the physical model….There is a schedule also for the physical model….

The routes of packetsThe routes of packets

• Source – destination pairs – Source – destination pairs – let Ylet Yii be a be a randomly chosen location such that Xrandomly chosen location such that Xii and Y and Yii are independent. The destination Xare independent. The destination Xdest(i)dest(i) is is chosen as the node Xchosen as the node Xjj which is closest to Y which is closest to Yii

• Corollary: The random sequence {LCorollary: The random sequence {Lii} = } = {straight line connecting X{straight line connecting Xii and Y and Yii } is i.i.d. } is i.i.d.

• Routes of packets will be choose to Routes of packets will be choose to approximate these straight line segmentsapproximate these straight line segments

• Final destination will be one hop away from YFinal destination will be one hop away from Yii , with high probability, with high probability

Each cell contains at least Each cell contains at least one nodeone node

Definition 1:Definition 1:

Let Let FF be a set of subset. A finite set of points A is be a set of subset. A finite set of points A is said to be shattered by said to be shattered by FF if for every subset B of A if for every subset B of A there is a set F in there is a set F in FF such that such that

Definition 2:Definition 2:

The VC-dimension of The VC-dimension of FF , denoted by VC- dim( , denoted by VC- dim(FF ) , is ) , is defined as the supremum of the sizes of all finite defined as the supremum of the sizes of all finite sets that can be shattered by sets that can be shattered by FF

B FA

Vapnic-Chervonenkis Vapnic-Chervonenkis TheoremTheorem

If If FF is a set of finite VC dimension is a set of finite VC dimension dd and { and {XXii}is a }is a sequence of i.i.d. random variables with common sequence of i.i.d. random variables with common probability distribution probability distribution PP, then for every , then for every > 0, > 0,

2log,

16log

1))()(1

Pr(sup1

4e8dMaxN

Whenever

FPFXIN

N

ii

VCdim of the set of disks VCdim of the set of disks in Rin R22

x1

x2

x3

x4

4~x

1~x

2~x

3~x

180 xx 31

A cell contains at least A cell contains at least one nodeone node

Let Let FF denote the class of disks of area 100logn/n. denote the class of disks of area 100logn/n.

So VCdim(So VCdim(FF) is 3. Let V be a cell contained in a disk D. Hence:) is 3. Let V be a cell contained in a disk D. Hence:

(n)150logn Vin nodes ofNumber Pr

log50 (n)(n)

whensatisfied is This

)(

2log

)(

4,

)(

16log

)(

24Maxn

)(1)(log100

n

Din nodes of supPr

n

n

nnn

e

n

Whenever

nnn

nNumber

Mean number of routes Mean number of routes served by each cellserved by each cell

• First calculate the probability that a line LFirst calculate the probability that a line Lii or or great circle intersect a cell Vgreat circle intersect a cell V

Lemma: Lemma: for every line Lfor every line Lii and cell V and cell V

n

ncV intersect L line i

logPr

• So the expected number of lines LSo the expected number of lines Lii that intersect that intersect a cell is bounded as: a cell is bounded as:

nncV intersect }{L lines of NumberE i log

• The same as for great circles ! The same as for great circles !

Actual traffic served by Actual traffic served by each celleach cell

• We bounded the We bounded the meanmean number of routes passing number of routes passing through each cell. However, we need to bound the through each cell. However, we need to bound the actualactual random number of routes served by each random number of routes served by each cell !!cell !!

• Remember the sequence {XRemember the sequence {Xi i ,Y,Yii} is i.i.d.} is i.i.d.• Therefore , we can appeal to uniform convergenceTherefore , we can appeal to uniform convergence• We will show that each great circle that intersect a We will show that each great circle that intersect a

disc D, can be mapped to a point on the band F(D) disc D, can be mapped to a point on the band F(D) that is equidistant from the center of Dthat is equidistant from the center of D

• Then we can bound the VCdim of the band and so Then we can bound the VCdim of the band and so of the great circlesof the great circles

Transforming great circles Transforming great circles intersecting disks into intersecting disks into

points lying in equatorial points lying in equatorial bandsbands

Z

C F(D)

Lower bound on throughput Lower bound on throughput capacitycapacity

• Because of uniform convergence, we obtain: Because of uniform convergence, we obtain:

)(1Pr

)(1Pr

0)(

nnlogn(n)c Vby carried be to needed Trafficsup

(n) rate of traffic carriesL line each since

nnlognc V intersect L lines of Numbersup

that suchn sequencea is There

i

i

Lower bound on throughput Lower bound on throughput capacitycapacity

• We have shown that there is a schedule for transmitting We have shown that there is a schedule for transmitting packets such that in every (1+cpackets such that in every (1+c11) slots, each cell can transmit.) slots, each cell can transmit.

• Thus the rate at which each cell transmit is W/(1+cThus the rate at which each cell transmit is W/(1+c11) bits/sec) bits/sec• On the other hand, the rate a cell On the other hand, the rate a cell needsneeds to transmit is less to transmit is less

than:than:

nlogn(n)c

• So with high probability, and because cSo with high probability, and because c11 is grow is grow linearly with (1+linearly with (1+))22 we have: we have:

nlogn

cWn

C1

W nlogn(n)c

1

21)(

ConclusionsConclusions

• Designers may want to consider designing Designers may want to consider designing networks with small number of nodesnetworks with small number of nodes

• Communication with nearby nodes at constant Communication with nearby nodes at constant bit rates can be provided in a dense clusters of bit rates can be provided in a dense clusters of nodes, since the source – destination distance nodes, since the source – destination distance shrink as shrink as

)n

1O(

QuestionsQuestions? ?