Communication in wireless ad hoc networks

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0 Aug. 14 2013 Leonid Chaichenets - Communication in wireless ad hoc networks

Institute for Applied and Numerical Mathematics – RG Numerical Simulation, Optimization and High Performance Computing

Communication in wireless ad hoc networks

a stochastic geometry model

KIT – University of the State of Baden-Wuerttemberg and

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Introduction

Ad hoc networks

No (hard-wired) infrastructure Self-organization

Cooperation among the subscribers (≡ nodes)

Applications

Military and law enforcement communication Rapidly deployed temporary networks Sensor networks

Modeling

Avoid expensive mistakes Test for suitability

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Considered scenario

Inhomogeneous Mobile ad hoc network Nearest Receiver (IMNR)

Node positions:finite, inhomogeneous Poisson point process

Channel access method:slotted ALOHA

Routing:nearest receiver

Transmission success:signal-to-noise ratio condition

Transmit power:constant

Path loss model:omnidirectional power law

Performance metrics

Local:probability of coverage

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Classification & literature survey

Literature survey

Baccelli and Błaszczyszyn (2009):MNR model

Node positions:homogeneousPPP

Performance metrics:probability of coverage & density of successful transmissions

Haenggi and Ganti (2008):analysis of interference

Node positions:Poisson cluster processes & general motion-invariant PP

Performance metric:probability of coverage

Haenggi, Andrews, Baccelli et al. (2009):homogeneous case

Weber, Andrews and Jindal (2010):Transmission Capacity

Classification

inhomogeneous, finitePPP

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Structure

System model Performance metrics Results

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Network geometry & channel access

PP of node positionsΦ

Φfinite, inhomogeneous PPP onR2 Λ=λGintensity measure (IM)

λ>0 expected total number of nodes

Gdistribution of the position of a typical node

gprobability density function (PDF) ofG

PPs of rolesΦ_R, transmittersΦ_Tx and receiversΦ_Rx

ΦRindependentF-marking ofΦ F=pMACδ1+ (1−pMAC)δ0

pMAC∈(0,1)probability of media access

ΦTx=ΦR(· × {1})PP of transmitters

IM:ΛTx=pMACΛ=λTxG

ΦRx= ΦR(· × {0})PP of receivers, IM:ΛRx=λRxG

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Routing Model

Definition: nearest neighbour functionY∗

x∈R2 _{(position of the transmitter)}

µ=∑I_i₌₁δyi boundedly finite counting measure onR 2

(realization ofΦRx ≡positions of receivers)

DefineY∗: R2×N_R2 →R2 by: Y∗(x,µ) =    argmin y∈µ

{kx−yk} if existent and unique

∞ otherwise

PP of transmitter-receiver pairsΦTxRx ΦTx=∑Ii=1δXi ≡Positions of transmitters

Y_i =Y∗(X_i,Φ_Rx)≡targeted receiver of the transmitter atX_i ΦTxRx= ∑Ii=1δ(X_i,Y_i)≡PP of transmitter-receiver pairs

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Success of a transmission

signal-to-interference ratio SIR= R `(kx−yk) `(ky−zk)ν(dz) = kx−yk−α R ky−zk−α ν(dz)

`=`(r)omnidirectional path loss

`(r) =r−α_{power law where 2}_≤_α_≤_{4 loss exponent}

xposition of the transmitter yposition of the receiver

ν=∑I_i₌₁δz_i positions of competing transmitters SIR success functions

Defines: R2×_R2_×_N

R2 → {0,1}by:

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Structure

Done: system model

Network geometry:inhomogeneous PPP

Channel access method:slotted ALOHA

Routing model:nearest receiver routing

Success of a transmission:SIR success function

Next: performance metrics

Local performance metric:probability of coverage

Global performance metric:mean number of successful

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Local performance metric

probability of coveragepc

(

x,y

)

pc(x,y) =

Z

s(x,y,ν(· ×R2))P₍!TxRx_x_,_y₎(dν)

x∈_R2 _≡_{position of the transmitter} y∈_R2_≡_{position of the targeted receiver} ssuccess function

P!TxRx

(x,y) local modified Palm distribution ofΦ

TxRx_at₍_x,_y₎_≡

distribution of the PP of transmissions competing with(x,y)

ν(· ×R2)≡positions of interferers (do not care about positions of the

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Global performance metric

Mean number of successful transmissionsT

T =E

Z

s(x,Y∗(x,ΦRx),ΦTx\x)ΦTx(dx)

ssuccess function

x≡position of the transmitter

y=Y∗(x,ΦRx)position of the receiver point ofΦRxnearest tox

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Structure

Done: performance metrics

Local performance metric:probability of coverage

Global performance metric:mean number of successful

transmissions

Next: Results

Uniqueness of the nearest receiver Distribution of the nearest receiver Estimate of the probability of coverage Formula for the global performance metric

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Uniqueness of the nearest receiver

Y∗(x,ΦRx) =    argmin y∈ΦRx

{kx−yk} if existent and unique

∞ otherwise

x≡position of the transmitter

Y∗(x,ΦRx)≡targeted receiver of the transmitter atx ΦRx _≡_{positions of the receivers}

Theorem It is: P Y∗(x,ΦRx) =∞ =P ΦRx =∅ =e−λRx P Y∗(x,ΦRx) =∞|_ΦRx₆₌_∅ =0

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Distribution of the nearest receiver

Theorem: distribution of the distance to the nearest receiver

Distributions of

x−Y∗(x,ΦRx)

form a stochastic kernelHwith source R2 _{and target}_R_{. For all}_x_∈_R2_:

H(x,·) =e−λRx_δ_∞₍_·_{) +}_H_ac₍_x,_·₎

The PDF ofHac(x,·)is given by: h(x,r) =λRx

R

∂Br(x)g dσ·e

−λRxG(Br(x))

Theorem: distribution of nearest receiver

Distributions ofY∗(x,Φ_Rx)form a stochastic kernelG∗with sourceR2 and targetR2_{. For all}_x_∈_R2_:

G∗(x,·) =e−λRx_δ_∞₍·_{) +}_G∗

ac(x,·)

The PDF ofG_ac∗ (x,·)is given by:

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Estimate of the probability of coverage

pc(x,y) = Z s(x,y,ν(· ×R2))P₍!TxRx_x_,_y₎(dν) = Z s(x,y,ν)PTx(dν) Complicated dependence onν: s(x,y,ν) =0⇔ kx−yk −α R ky−zk−α ν(dz) <β

Estimate ofs(x,y,ν): dominant interferers (Weber et al.)

Consider only the case, that one interferer is enough to disturb the transmission: ⇒ ky−zk< β 1 αkx−yk pc(x,y)≤P ΦTx₍_B β 1 αkx−yk(y)) =0 =e −λTxG(B β 1 αkx−yk (y))

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Formula for the global performance metric

T = E Z s(x,Y∗(x,ΦRx),ΦTx\x)ΦTx(dx) ΦTx,ΦRx = indep. Z Z Z s(x,Y∗(x,µ),ν\x)ν(dx)PRx(dµ)PTx(dν) Tonelli = Z Z Z s(x,Y∗(x,µ),ν\x)PRx(dµ)ν(dx)PTx(dν) Distribution = of NR Z Z Z s(x,y,ν\x)G∗(x,dy)ν(dx)PTx(dν) = E Z Z s(x,y,ΦTx\x)G∗(x,dy)ΦTx(dx) refined = Campbell Z Z Z s(x,y,ν)G∗(x,dy)P!xTx(dν)ΛTx(dx) Slivnyak = Tonelli Z Z Z s(x,y,ν)PTx(dν)G∗(x,dy)ΛTx(dx) = Z Z pc(x,y)G∗(x,dy)ΛTx(dx)

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Structure

Done: results

Uniqueness of the nearest receiver Distribution of the nearest receiver Estimate of the probability of coverage Formula for the global performance metric

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Résumé

Comparison with the homogeneous case (MNR Model)

Finite network⇒global performance metric Price to pay is higher complexity:

pc(r)⇒pc(x,y)

G∗(dr)⇒G∗(x,dy)

Suitability of the IMNR model for describing sensor networks

Complete independence of node positions⇒PPP Simple channel access method⇒slotted ALOHA Undirected transmissions⇒omnidirectional path loss Energy Efficiency⇒nearest receiver routing (extreme case)

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Outlook

Next steps

Numerical evaluation of the integral formula Optimization of the system parameters Further refinement of the model:

Ambient & thermal noise Fading

Other path loss models (fix unphysical singularity at 0) Estimate the global performance metric from below Power control

Big picture

IMNR: single time slot

⇒correlation between time slots IMNR: only physical- and data link layer ⇒network layer (routing)

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Basic definitions

(X,d_X)nonempty complete separable metric space

B_XBorel-σ-algebra,Bb_Xfamily of bounded measureable sets µboundedly finite counting measure on(X,BX), if

µ(A)∈N0

for eachA∈ Bb X

N_Xfamily of boundedly finite counting measures N_X=σ(µ7→µ(A); A∈ B)itsσ-Algebra

Φis a point process (PP), if it is a(N_X,N_X)-valued random variable

Conclusion:point processes are random boundedly finite counting

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Representation of point processes

Example: binomial process

Let(Xi)i∈Nbe i.i.d. onXandn∈Nbe fixed. Then Φ= n

∑

i=1 δXi is a point process. Representation theorem

Each point processΦhas a representation Φ=

I

∑

i=1 δX_i

whereI∈N0∪ {∞},(Xi)i∈N ∈Xare random variables.

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Moment measures and Campbell‘s theorem

Definition: first and second moment measures

First moment measure (Intensity Measure, IM)Λis defined by Λ(A) =EΦ(A)

for eachA∈ B_X(mean number of points inA). Second moment measureΛ2 is defined by

Λ2₍_A_×_B_{) =}_EΦ₍_A₎_Φ₍_B₎

for eachA×B ∈X2 ₍_⇒_{covariance of}_Φ₍_A₎_and_Φ₍_B₎_). Campbell‘s theorem

Letf : X→Rbe measureable and nonnegative orΛ-integrable. Then: E Z f(x)Φ(dx) =E I

∑

i=1 f(Xi) = Z f(x)Λ(dx)

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Poisson point processes

Definition

LetΛbe the intensity measure ofΦ. If

Φ(A₁),. . .,Φ(An)independent for mutually disjointA1,. . .,An ∈ Bb_X

(independent increments)

Φ(A)has Poisson distribution with parameterΛ(A): P(Φ(A) =k) =e−Λ(A)Λ(A)

k

k! for eachk∈_N₀(Poisson statistics)

thenΦis called a Poisson point process (PPP).

Theorem

IfΛis boundedly finite andΦsimple (i.e.Φ({x})≤1 for allx ∈_Xalmost surely w.r.t.P), then:

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Marked point processes

Definition

A marked point processΦ˜ onXwith marks inYis a point process on X×Y, such thatΦ˜(· ×Y)(ground process) is boundedly finite.

Example: independentF-marking (ALOHA)

LetΦ=∑I_i₌₁δX_i be a PPP onX=R2 with IMΛ,Y={0,1}and

F =p_MACδ1+ (1−pMAC)δ0 (0<pMAC<1).

Furthermore letY_i ∈Ybe i.i.d. asF,(Y_i)_i∈Nindependent ofΦ. Then

ΦR =

I

∑

i=1

δ(Xi,Yi) ΦRx =ΦR(· × {0}) ΦTx =ΦR(· × {1})

are PPP with IMsΛ_R =Λ⊗F,Λ_Rx= (1−pMAC)ΛandΛTx=pMACΛ

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Palm theory

Definition: modified Campbell measure and Palm kernels

LetΦbe a simple PP with IMΛ. Its modified Campbell measureC!is defined by

C!(A×B) =E

Z

A1B

(Φ\x)Φ(dx)

for allA×B ∈ B_X⊗ N_X. Furthermore,C!can be factorized: C! =Λ⊗P!_x

The stochastic kernelP!

x with sourceXand targetNXis called modified

Palm kernel.

Refined Campbell‘s theorem

Letf : X×N_X→Rbe mb. and nonnegative orC!integrable. Then: E Z f(x,Φ\x)Φ(dx) = Z f(x,ν)C!(d(x,ν)) = Z Z f(x,ν)P!x(dν)Λ(dx)

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Two theorems on PPPs

Theorem of Slivnyak and Mecke

LetΦbe a simple PP with finite IMΛ. Then: Φis PPP⇔P!

x =PΦalmost everywhere w.r.t.Λ

Theorem: second moment measure of a PPP

LetΦbe a PPP with boundedly finite IMΛ. Its second moment measure Λ2 _is: