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Capacity of Wireless Networks:Protocol and Physical Models

P. R. Kumar

Dept. of Electrical and Computer Engineering, and Coordinated Science Lab University of Illinois, Urbana-Champaign

Email: prkumar@illinois.eduWeb: http://decision.csl.illinois.edu/~prkumar

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License. Based on a work at decision.csl.illinois.edu See last page and http://creativecommons.org/licenses/by-nc-nd/3.0/

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How much traffic can wireless networks carry?

(Or what is the capacity of wireless networks?)

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And how should information be transferred in wireless networks?

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Outline   Models 5   Best Case Analysis of Protocol Model 15   Sharpest results for Best Case of Protocol Model 44   Best Case Analysis of Physical Model 46   Sharpest results for Best Case of Physical Models 50   Analysis of Random Case 52   Some experimentation: A scaling law 85   Summary 87   References 91

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Models of Wireless Networks

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Multi-hop Wireless networks

  “Multi-hop transport” –  Nodes relay packets until they reach their destinations

  Communication networks formed by nodes with radios –  Spontaneously deployable anywhere –  Automatically adaptive to number of nodes,

traffic requirements, locations

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Two fundamental properties of the wireless medium   It is subject to fading and attenuation

–  Signals get distorted –  Time varying channel –  Unreliable

  It is a shared medium –  Users share the same spectrum –  Users are located next to each other –  Transmissions can interfere with each other –  So users need to cooperate to use the medium

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Spatial reuse of spectrum

  Spatial reuse of frequency in cellular systems

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Shared nature of wireless medium   Packets can “collide” destructively

–  Destructive interference –  Nothing can be decoded from two concurrent transmissions

in same region

or

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One model for successful sharing

r2 r1

(1+Δ) r1

(1+Δ)r2

  Receiver not in vicinity of an interfering transmission

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Spatial reuse of spectrum   Spatial reuse of frequency in cellular systems

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Other models for successful sharing

Rate = B log 1+ Piri−α

N + Pjrj−α

j≠i∑

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

bps

  Signal to Noise Ratio (SNR) Signal to Noise Ratio = SNR := Received Signal Strength

Noise =PiriαN

  Signal to Interference plus Noise Ratio (SINR)

SINR := Received Signal StrengthInterference Strength + Noise =

Piriα

Pjrjα

j≠i∑ +N

  Model 2: Reception successful if SINR exceeds a threshold:

SINR =Piriα

Pjrjα

j≠i∑ +N

≥ β

  Model 3: Transmitter-to-Receiver Communication Rate depends on SINR:

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A framework for studying wireless networks

  Model –  Disk of area A sq.m –  n nodes –  Each can transmit at W bits/sec

  Wireless channel is a shared medium –  Packets are successfully received when

there is no local interference

  How much information can such wireless networks carry? -  Throughput for each node: Measured in Bits/Sec –  Transport capacity of entire network: Measured in Bit-Meters/Sec –  Scaling with the number of nodes n

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  Protocol Model

Receiver R should be

(i) within range r of its own transmitter T �

(ii) outside footprint (1+Δ)r’ of any other transmitter T’ using range r’ �

Model for successful decoding of packet

r R

T r’

(1+Δ) r T’

(1+Δ)r’

R’

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Best Case Analysis of Protocol Model

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Transmissions consume area

n nodes

A sq.m

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Transmissions consume area

n nodes

A sq.m

r1

r2 (1+Δ)r1

(1+Δ)r2

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Transmissions consume area

r1

r2 (1+Δ)r1

(1+Δ)r2

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Transmissions consume area

r1

(1+Δ)r1

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Transmissions consume area

r1

≥ (1+Δ)r1

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Transmissions consume area

r1

≥ (1+Δ)r1

≥ Δr1

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Transmissions consume area

≥ Δr1

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Transmissions consume area

r1

r2 (1+Δ)r1

(1+Δ)r2

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Transmissions consume area

r2 (1+Δ)r2

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Transmissions consume area

r2 ≥ (1+Δ)r2

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Transmissions consume area

r2 ≥ (1+Δ)r2

≥ Δr2

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Transmissions consume area

≥ Δr2

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Transmissions consume area

≥ Δr1

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Transmissions consume area

≥ Δ (r1+r2)/2 Δr2/2

Δr1/2

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Transmissions consume area

Δr2/2

Δr1/2

r1

r2

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Transmissions consume area

Δr1/2

r1

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Transmissions consume area

Δr1/2

r1

n nodes

A sq.m

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Transmissions consume area

Δr2/2

Δr1/2

r1

r2

n nodes

A sq.m

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Transmissions consume area

Δr2/2

Δr1/2

r1

r2

n nodes

A sq.m

r3

Δr3/2

r4

Δr4/2

r5 Δr5/2

r6 Δr6/2

But Total Area = A

So πΔ2ri

2

16i=1

n /2

∑ ≤ A

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The area constraint πΔ2ri

2

16i=1

n/2

∑ ≤ A

ri2

i=1

n/2

∑ ≤ 16AπΔ2

1n/2 ri

2

i=1

n/2

∑ ≤ 32AπΔ2 ⋅

1n

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Convexity   A convex function: f(r) = r2

1n/2 ri

i=1

n /2

∑⎛⎝⎜

⎞⎠⎟

2

≤ 1n/2 ri

2

i=1

n /2

∑

Square of Average ≤ Average of squares

Eg.

≤ 32AπΔ2 ⋅

1n

So

3+ 52

⎛⎝⎜

⎞⎠⎟

2

≤ 32 + 52

2 (or 16 ≤17)

3 5 4

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Convexity   A convex function: f(r) = r2

Square of Average ≤ Average of squares

Eg.

rii=1

n /2

∑ ≤ 8AπΔ2 ⋅n

Hence

3 5 4

3+ 52

⎛⎝⎜

⎞⎠⎟

2

≤ 32 + 52

2 (or 16 ≤17)

1n/2 ri

i=1

n /2

∑⎛⎝⎜

⎞⎠⎟

2

≤ 1n/2 ri

2

i=1

n /2

∑ ≤ 32AπΔ2 ⋅

1n

So

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Bound on Transport capacity:Bit-meters/second pumped by network

W bps ri meters

Wri bit-meters/second Wrii=1

n/2

∑  Total bit-meters/second pumped

by the network is

rii=1

n/2

∑ ≤ 8AπΔ2 ⋅n  Remember

Wrii=1

n/2

∑ ≤W 8AπΔ2 ⋅n  So

≤W 8πΔ2 ⋅A ⋅n

  Hence

Bit-meters pumped by the network

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  n/2 nearest neighbor connections

Feasibility of bit-meters/sec Θ W An( )

(n / 2)Wr = 18W An

2A / n  Each of distance r =

  Total bit meters of entire network

  Basically the wireless network provides every usera throughput of W bps to its nearest neighbor

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Scaling law for capacity

  How much information can wireless networks transfer with this mode of operation?

–  Implications of square-root law

–  If equitably divided, each node can send bit-meters/sec

–  Law of diminishing returns in this scaling

Θ W An

⎛

⎝⎜⎞

⎠⎟

–  Transport capacity is

–  Aggregate pumping capacity of the network (Gupta & K ʻ00)

Θ W An( ) bit-meters/second

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Best possible scenario   Optimal network

»  Optimally located nodes, destinations, demands for OD-pairs »  Optimal spatial and temporal scheduling, routes, ranges for each transmission

  Protocol Model: Network can transport bit-meters/sec Best case capacity

for Protocol Model

  If equitably divided, each node can send bit-meters/sec

Θ W An( )

Θ W A

n⎛

⎝ ⎜

⎞

⎠ ⎟

W1+ 2Δ

nn + 8π

≤ ≤ 8

πWΔ

n bit-meters/sec

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Another intuitive way of understanding result

  Best Range: Tradeoff between short and long hops

–  If range is r, distance traveled on each hop is r

–  So number of hops to travel one unit of distance ≈ (1/r)

–  Space used by each transmission = r2

–  Total Space used ≈ (1/r)· r2 = r

–  �

–  So best to use smallest range r

–  However smallest range that keeps network connected = 1/√n

r2 r2 r2 r2

Or Distance-Rate product

Transmission range Number of simultaneous transmissions

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  Domain is a disk of unit area

  There are n nodes in the domain

  Each node can transmit at W bits/sec

  Channel can be split into M sub-channels of capacities

W1, W2, …, WM bits/sec with

  Assume slots of length τ -  Wmτ bits can be transmitted in slot s in subchannel m

Same result also holds if channel is split into several sub-channels

�

Wm =Wm=1

M∑

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Sharpest results for Best Case of Protocol Model

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Sharpest results for Best Case:Protocol Model

  Protocol Model (Agarwal & K ʼ03)

1π

W

(1+ Δ) Δ 2 + Δ⋅ A ⋅ n ≤ ≤ 8

πW

(1+ Δ) Δ 2 + Δ⋅ A ⋅ nTransport

capacity

  Upper and lower bounds differ by only a factor of

-  A sharper characterization of“exclusion region”

-  Study of general antenna patterns,directional antennas, etc

8

(Agarwal and K ʼ03)

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Best Case Analysis of Physical Model

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  Physical Model: Signal-to-Interference-Plus Noise Ratio (SIR) Model

The Physical SINR Model

SINR Ratio = Piri−α

N + Pjrj−α

j≠i∑ ≥ β

-  Pi = power of i-th node –  N = Noise power –  rj = Distance of j -th transmitter from given receiver –  r-α : Signal Power Path Loss, α >2 -  β = SIR for successful reception

-  Will show a simple derivation of a O(n(α-1)/α) bound (Gupta & K ʻ04)

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O(nα−1

α ) bound:Physical Model

Idea

SINR constraint also implies a space constraint:

Piri−α

N + Pjrj−α

j≠i∑

≥ βRecall SINR requirement:

Piri−α

N + Pjrj−α

j∑

≥ ββ +1Including signal power in denominator:

riα ≤ (β + 1)Pi

βN + β Pjrj−α

j∑  So

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SIR requirement consumes space

rj ≤

2π

(since area of disk = 1)  However

riα ≤ (β + 1)Pi

βN + β π4

⎛ ⎝ ⎜

⎞ ⎠ ⎟

α2

Pjj∑  Hence

Ri is range of i-th transmission∑ Ri

α ≤ (β +1)

β π4

⎛

⎝ ⎜

⎞

⎠ ⎟

α2

  So

  Now use convexity of Rα rather than that of r2

riα ≤ (β + 1)Pi

βN + β Pjrj−α

j∑  So

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Sharpest results for Best Case ofPhysical Models

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Sharpest results for Best Case:Physical Models

  Physical Model (Agarwal and K ʼ04)

-  Transport capacity is

(Agarwal and K ʻ04)

Θ n( )

  Generalized Physical Model (Agarwal and K ʼ04) -  Adaptive coding is used so that bit-rate depends on SINR

-  Transport capacity is still

Rate = B log 1+ Piriδ

N + Pjrjδ

j≠i∑

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

bps(Shannonʼs formula for capacity of a channel with additive white Gaussian noise)

Θ n( )

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Analysis of The Random Case

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Scaling Law for Random Networks   n nodes randomly located in disk of unit area

–  Each node chooses random destination –  Equal throughput λ bits/sec for all OD pairs –  Each node chooses same range r

limn→∞

Pr(λ(n) = cn logn

is feasible) = 1, and

limn→∞

P(λ(n) = ′ c n logn

is feasible) = 0

Θ 1

n logn

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟   Each node can send bits/sec

even with -  With best choice of spatio-temporal scheduling,

ranges and routes

  Theorem: Throughput that can be supported (Gupta & K ʻ00)

(i)  Random case = Nearly best case

(ii)  It is nearly optimal to use common range for all nodes

Sharp cutoff phenomenon

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Model of randomly formed networks   Domain = Surface of sphere of unit area

–  No edge effects (can generalize to plane)   n identical nodes randomly located (uniform, iid)

  Each node can transmit at W bits/sec

  Range of each nodeʼs transmission is r(n)

  Destination nodes randomly chosen –  Node closest to a randomly chosen (uniform, iid) point

  Each node sends λ(n) bits/sec to its destination

  Theorem The capacity of a randomly formed network is

λ(n)

Θ 1

n logn

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ bits/sec

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Constructive proof of capacity   Voronoi Tessellation

–  Generators a1, a2, … , ap –  Cell V(ai) = region closest to ai

  Choose a tessellation so that –  Every cell contains a disk of radius ρ�

–  Every cell is contained in a disk of radius 2ρ -  Procedure: Add a generator 2ρ away from other generators -  Stop when no more possible

–  Cells neither too fat nor too thin

  Choose ρ(n) so Area of disk ≈ πρ2 (n)( ) = 100 lognn

a1 a2

a3

V(a1)

ρ 2ρ

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  Choose transmission range r(n) = 8ρ(n)

  Neighboring cells can communicate

–  Each cell is contained in disk of radius 2ρ(n) –  Range is 8ρ(n)

Neighboring cells can communicate

2ρ(n)

8ρ(n)

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Interfering neighbors

  Definition Interfering Neighbors

–  Cell c and Cell cʼ are interfering neighbors if

–  Some point in one cell is within a distance (2+Δ) r(n) of some point in other cell

  Transmissions from cell c and cʼ can never collide if they are not interfering neighbors

r(n) r(n) Δr(n)

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Number of interfering neighbors

  Every cell has no more than c1 interfering neighbors

ρ(n)

(2+Δ)16ρ(n)

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A transmission schedule for cells   There is a transmission schedule such that every cell

gets one transmission slot in every (1+c1) slots

–  Proof based on graph coloring –  Draw an edge between two cells if they are interfering

neighbors –  Degree of graph ≤ c1 –  Can vertex color the graph with no more than (1+c1) colors –  All nodes of same color transmit in a slot

–  Proof for SIR model based on separation distances of other receivers

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Xdest(i)

Xi

Yi

OD Pairs, Traffic and “Lines”   Nodes Randomly located {X1, X2, … , Xn}

  Destinations –  Yi randomly located point (uniform iid) –  Xdest(i) = nearest node to Yi

  OD Pair = (Xi , Xdest(i))   Traffic of OD Pair = λ

  Li = line joining Xi and Yi   {Li} are iid

Xi

Yi

Li

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Routes

  Packets hop from one cell intersecting Li to next cell intersecting Li

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When is scheme feasible?

  Scheme is feasible if

–  Every cell contains at least one nodeto act as a relay

–  Each cell can handle all the trafficpassing through it

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How many “lines” can a cell handle?   Each cell gets one slot out of every (1+c1) slots

  When it transmits it can transmit at W bits/sec

  Hence each cell can carry bits/sec

  Each line Li passing through cell consumes λ bits/sec

  Hence each cell can handle at most lines

W

(1+ c1)

Wλ(1+ c1)

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To show feasibility of scheme …. We need to show

  Every cell has at least one node

  Every cell has less than lines passing

through it

Wλ(1+ c1)

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Every cell contains a node   Every cell contains a disk of probability (100 log n)/n

  So

  Every cell contains a node with probability approaching one as n → ∞ (whp)

Prob(Given cell contains no nodes) ≤ (1− 100lognn

)n

Prob(Every cell contains a node) ≥ 1− n100 logn

(1− 100 lognn

)n

Number of cells ≤ n100 logn

limn→∞

Prob(Every cell contains a node) = 1

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Weak Law of large numbers

  {X1, X2, …, Xn} are iid, common distribution P

  When n is large enough

Pr # of points in Gn

− P(G) > ε⎛⎝⎜

⎞⎠⎟< δ

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Shattering

  ℑ = set of subsets

  B = {z1, z2, …, zp} = a finite set of p points

  B is shattered by ℑ if –  For every subset A of B there is a G∈ ℑ with G ∩B=A

  Example ℑ = set of all rectangles B = {z1, z2, z3, z4}

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Vapnik-Chervonenkis dimension   Vapnik-Chervonenkis Dimension VC(ℑ )

–  Size of largest set than can be shattered by ℑ

  Example ℑ = set of all rectangles There is no set with 5 points that can be shattered VC(ℑ) = 4

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Uniform law of large numbers   Suppose VC(ℑ ) = d

  When

 

  Every set in ℑ has nearly the mean number of points

Pr SupG∈ℑ

# of points in Gn

− P(G) > ε⎛⎝⎜

⎞⎠⎟

≤ 4 (2n)2d+1 +1( )e−ε2n

8

Uniformity over ℑ

When n > n(ε,δ ,d), Pr SupG∈ℑ

# of points in Gn

− P(G) > ε⎛⎝⎜

⎞⎠⎟≤ δ

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VC dimension of disks on the plane   Theorem VC(set of disks in the plane) = 3

  Proof

Suppose {x1, x2, x3, x4} can be shattered

Suppose ∠x1 + ∠x3 ≥ 180o

Then ∠ x1 < ∠a and ∠ x3 < ∠c

But ∠a + ∠c = 180o

So ∠x1 + ∠x3 < 180o

b

d

x1 x2

x3 x4

x1 x2

x3 x4

a c x1 x3

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From S2 to R2 and back:The inversion map   Let the inversion map

  Properties:

  Theorem

VC(disks smaller than hemisphere on S2) = 3

Proof If 4 points shattered, rotate to lower hemisphere

f−1(z) = f (z)

f : Punctured sphere → Plane

f (z) = zz 2

f : Disks on S2 → Disks on R2 Plane

(0,0,0)

(0,0,-1)

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Probability of a line passing through a cell

  Each cell is contained in a disk of area (400 log n)/n

  Radius of disk γ (n) ≤ 400 lognn

Pr(Line L intersects cell)

≤ Min c, cx

lognn

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢

⎤ ⎦ ⎥ c x + 400logn

n⎛ ⎝ ⎜

⎞ ⎠ ⎟ dx0

c∫

≤ cx

lognn

⎛ ⎝ ⎜

⎞ ⎠ ⎟ xdx0

c∫ + c 400 logn

ndx0

c∫

≤ c lognn

⎛ ⎝ ⎜

⎞ ⎠ ⎟ dx0

c∫ + c 400 lognn

dx0c∫

≤ c lognn

+ c lognn

≤ c lognn

Angle

= cxlognn x

Length = c lognn

Area ≤ Min c, cx

lognn

⎛ ⎝ ⎜

⎞ ⎠ ⎟

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Expected traffic through cell  

 

  Question: What is the actual traffic passing through cell?

  Question: What is the actual number of lines through each cell?

E # of Lines passing through cell[ ] ≤ c n logn

E Traffic passing through cell[ ] ≤ cλ(n) logn

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VC dimension for lines intersecting disks   Lemma

D = set of all disks of radius ζ

C(D) = All great circles which intersect D∈D

VC dim(C(D): D∈D) = VC dim(F(D): D∈D)

where F(D) = Intersection of two large disks each larger than a hemisphere

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The traffic through each cell   VC dim (Set of sets) = VC dim (Set of complements)   VC dim(G ∩Gʼ: G∈G,Gʼ∈G) ≤ 10 VC dim(G)   So VC-dim(C(D):D∈D) ≤ 30�

  Hence �

Pr(Every cell has less than c n logn lines passing through it) ≥ 1− δ (n) where δ (n)→ 0

Pr(Every cell has traffic less than cλ(n) n logn through it) ≥ 1− δ (n) where δ (n)→ 0

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Feasible level of traffic   Every cell has to handle bits/sec

  Every cell can handle bits/sec

  So λ(n) can be handled if

  Theorem

Pr λ(n) = cn logn

is feasible⎛

⎝⎜⎞

⎠⎟→ 1

cλ(n) n logn

W1+ c1

cλ(n) n logn ≤W1+ c1

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Upper bound on capacity:What is possible?

  Recall spaceconstraint

  Space usedper transmission≤ π Δ2r2(ν)/16

  Total area available = 1

  Number of simultaneous transmissions

  Total transmitted by all nodes

≤16

πΔ2r2 (n)

≤16W

πΔ2r2 (n) bits/sec

D(Xj,Δr/2) and D(Xm, Δr’/2)�are disjoint

r (1+Δ)r

r’ (1+Δ)r’

r Xi

Xj

Xk

Xm

Δr/2

Δr’/2

Xj does not lie in D(Xk,(1+Δ)r’) Xm does not lie in D(Xi,(1+Δ)r)

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How much do we need?   E[Length of line Li ] = L   Mean number of hops per OD pair ≥ L/r(n)   There are n OD pairs   Total number of hops ≥ Ln/r(n)   Each OD pair requires λ(n) bits/sec   Total transmission required over all nodes ≥ Lnλ (n)/r(n)

  Feasibility condition

  Upper bound

Lnλ(n)r(n)

≤ 16WπΔ2r2 (n)

λ(n) ≤ 16WπΔ2Lnr(n)

bits/sec Make r(n) small

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Connections based on distance [Penrose ʼ97, Gupta & K ʼ98]

  Model: n i.i.d. points in an unit square or disk of unit area

  On average, each node should connect to more than 1·log(n) neighbors

  Connection rule: Connect each node to every node that is within distance r(n)

  Theorem limn→∞

Prob(Network is connected) = 1 if and only

πr2 (n) = log(n)+ f (n)n

with f (n)→∞

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Network disconnectivity at small r(n)   n iid, uniformly located nodes

  Join two nodes by an edge if distance less than r(n)

  G(n, r(n)) = resulting random graph

  Let P(n, r(n)) = Probability of an isolated node

  Theorem

Consider .

Then lim P(n, r(n)) > 0 if and only if limsup k(n) < +∞.

Also condition for connectedness of random geometric graph.

πr2 (n)= logn + k(n) n

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Asymptotic probability of isolated node   Idea of proof

P(n,r(n)) ≥ Pr(i is the only isolated node)i=1

n

∑

≥ Pr(i is isolated)i=1

n

∑ − Pr(ij≠i∑

i=1

n

∑ and j are isolated)

≥ n(1-A(r))n-1 − n(n −1)((A(2r)− A(r))(1− 32A(r))n−2

+ (1− A(2r))(1− 2A(r))n−2 )

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Upper bound on capacity

  We thus need for feasibility

  Thus

r(n) ≥ logn πn

λ(n) ≤ 16WΔ2L πn logn

bits/sec

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Capacity of randomly formed wireless networks   Theorem

Capacity

  Note: Worse by factor than optimally designed networks

  Results also hold on plane: Use inversion map!

λ(n) = Θ 1n logn

⎛

⎝⎜⎞

⎠⎟ bits/sec

1logn

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Physical Model: Random Network   n nodes randomly located

–  Each node chooses random destination –  Equal throughput λ bits/sec for all OD pairs –  Each node chooses same power level P

  Theorem

λ(n) bits/sec

-  With best choice of routes, hops, spatio-temporal scheduling

  Franceschetti, Dousse, Tse, Thiran ʼ07: –  In the Generalized Physical Model:

≤ Θ 1

n

⎛

⎝ ⎜

⎞

⎠ ⎟

Θ 1

n logn

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ≤

(Gupta and K ʼ00) Θ

1n

⎛⎝⎜

⎞⎠⎟

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Some Experimentation: A Scaling Law

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Some experimentation: A scaling law   Throughput = 2.6/n1.68 Mbps per node

-  No mobility -  No routing protocol overhead

- Routing tables hardwired –  No TCP overhead

– UDP –  IEEE 802.11

  Why 1/n1.68?

-  Much worse than optimal capacity = c/n1/2 -  Worse even than 1/n timesharing -  Perhaps overhead of MAC layer?

Log(Thpt)

Log( Number of Nodes) (Gupta, Gray & K ʻ01)

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Summarizing ...

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Why multi-hop?

  Multi-hop increases traffic carrying capacity   It may also increase delay

1n 0

cn

Range

Bit-Meters Per SecondPer Node

Broadcast

No connectivity

Multi-hop Networks

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Optimal operation under “collision” model

  Order-optimal spatial reuse architecture

–  Group nodes into cells of size log n�

–  Choose a common power level for all nodes »  Nearly optimal

–  Power should be just enough to guarantee network connectivity

»  Sufficient to reach all points in neighboring cell

–  Route packets along nearly straight line path from cell to cell

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Implications for designers   Design networks with few nodes, or scaled down bandwidth,

or support mainly nearest neighbor communications

  Splitting into several sub-channels (TDMA, FDMA, CDMA)does not help in increasing capacity

  Power consumption: Busy fraction of modems is

  Range of transmissions: Scaled length of hops is

  Architecture for providing optimal capacityGroup nodes into cells of size O(log n) - one node in each cell serving as relay

  kn randomly placed relay nodes increase capacity by factor

  Directed transmissions will help - space is a valuable resource - but only by a constant factor

k

Θ logn

n⎛

⎝ ⎜

⎞

⎠ ⎟

Θ 1

logn⎛ ⎝ ⎜

⎞ ⎠ ⎟

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References-1   M.D. Penrose, “The longest edge of the random minimal spanning

tree,” The Annals of Probability, vol. 7, no. 2, pp. 340-361, 1997.   Piyush Gupta and P. R. Kumar, “Critical Power for Asymptotic

Connectivity in Wireless Networks,” in Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming. Edited by W. M. McEneany, G. Yin, and Q. Zhang, Birkhauser, Boston, MA, pp. 547–566, 1998. ISBN 0-8176-4078-9

  Piyush Gupta and P. R. Kumar, “The Capacity of Wireless Networks,” IEEE Transactions on Information Theory, vol. IT-46, no. 2, pp. 388-404, March 2000

  P. R. Kumar, “A correction to the proof of a lemma in “The capacity of wireless networks”,” IEEE Transactions on Information Theory, vol. 49, no. 11, p. 3117, November 2003.

  Piyush Gupta and P. R. Kumar, “Internets in the Sky: The Capacity of Three Dimensional Wireless Networks,” Communications in Information and Systems, vol. 1, issue 1, pp. 33–49, January 2001.

  P. Gupta, R. Gray and P. R. Kumar, “An Experimental Scaling Law for Ad Hoc Networks,” May 16, 2001.

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  Ashish Agarwal and P. R. Kumar, “Improved Capacity Bounds for Wireless Networks,” Wireless Communications and Mobile Computing, vol. 4, pp. 251–261, 2003.

  Ashish Agarwal and P. R. Kumar, “Capacity Bounds for Ad-Hoc and Hybrid Wireless Networks,” ACM SIGCOMM Computer Communications Review, Special Issue on Science of Networking Design, vol. 34, no. 3, pp. 71–81, July 2004.

  Feng Xue and P. R. Kumar, Scaling Laws for Ad Hoc Wireless Networks: An Information Theoretic Approach. NOW Publishers, Delft, The Netherlands, 2006.

  Massimo Franceschetti, Olivier Dousse, David N. C. Tse, and Patrick Thiran,, Closing the Gap in the Capacity of Wireless Networks Via Percolation Theory, IEEE Transactions on Information Theory, Vol. 53, No. 3, March 2007.

  S. Aeron, V. Saligrama, Wireless Ad-hoc networks: Strategies and scaling laws in Fixed SNR regime, IEEE Trans. on Info Theory (to appear)

References-2

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http://decision.csl.illinois.edu/~prkumar/html_files/talks.html