Breaking the Poisson inﬁnite delay stranglehold Vaze.pdf · Rahul Vaze and Srikanth Iyer Breaking the Poisson inﬁnite delay stranglehold IISc, Bangalore

Rahul Vaze and Srikanth Iyer

Breaking the Poisson infinite delay stranglehold

IISc, Bangalore

Wireless Network

PPP distributed nodes

Wireless Network


sij = hij`(dij)

sij

Wireless Network


sij = hij`(dij)

sijI =

X

k2�\{i}

hkj`(dkj)

I

Wireless Network


sij = hij`(dij)

sijI =

X

k2�\{i}

hkj`(dkj)

I

SINRij =sij

I +N

Wireless Network


sij = hij`(dij)

sijI =

X

k2�\{i}

hkj`(dkj)

SINRij > �

I

SINRij =sij

I +N

Wireless Network


sij = hij`(dij)

sijI =

X

k2�\{i}

hkj`(dkj)

SINRij > �

I

SINRij =sij

I +N

SINR is a random variable: multiple attempts required

PPP networkNegative result

- Expected delay to any node is infinite

[Baccelli, Blacszczyszyn, Mirsadeghi, Adv. App. Prob. 2010]




T3

T2

T1




T3

T2

T1

min delay has infinite expectation

Negative ResultEven in the absence of interference (AWGN is sufficient)

sijI


sijI


sij


sij

Unbounded sized holes in PPP

More Bad News.

Let be the successful distance traveled by a taggedparticle from origin in time

More Bad News.

d(t)t


More Bad News.

V = limt!1

d(t)

t

d(t)t

Then speed or information velocity is


More Bad News.

V = limt!1

d(t)

t

d(t)t

Then speed or information velocity is

In a PPP network V = 0 [Baccelli, et al, 2010]

Remedy

• Add a regular square grid [Baccelli, et al, 2010]

Remedy

• Add a regular square grid [Baccelli, et al, 2010]

A more practical solution

sourcedestination


sourcedestination

�


sourcedestination

dij�


sourcedestination

dij�


sourcedestination

dij�


sourcedestination

dij�

sij = hij`(dij) SINRij =sij

I +N


Power Control : power with prob.

sourcedestination

Pi = c`(dij)�1 pi = MP�1

i

dij�


I +N

Transmit with higher power Less frequently


Power Control : power with prob.

sourcedestination

Pi = c`(dij)�1 pi = MP�1

i

dij�


I +N

Results

• Expected delay to nearest neighbor is finite

• Information velocity is positive

Negative Result Primer

sij sij = Pihij`(dij)



Time to exit to any other nodeT



Drop the Interference SINRij =sij

Ij +NSNRij =

sijN





Ij +NSNRij =

sijN


P(T > k) = P(SNRij(t) < �, 8 j 2 �, t = 1, . . . , k)




Ij +NSNRij =

sijN


Conditioned over location of nodes, SIRs are independent

P(T > k) = P(SNRij(t) < �, 8 j 2 �, t = 1, . . . , k)




Ij +NSNRij =

sijN



P(T > k) = P(SNRij(t) < �, 8 j 2 �, t = 1, . . . , k)

P (T > k|�) =Y

j2�

P (SINRij(t) < �)k




Ij +NSNRij =

sijN



After unconditioning P(T > k) >1

k

P(T > k) = P(SNRij(t) < �, 8 j 2 �, t = 1, . . . , k)

P (T > k|�) =Y

j2�

P (SINRij(t) < �)k

Issues with New Policy

n(y)

n(x)

y

x

z

n(z)

Choice of cones at any time are correlated

Analysis Ideas - Power Control


P(T > k) = P(SINRon(o)(t) < �, t = 1, . . . , k)


Earlier it was sufficient to condition over �

P(T > k) = P(SINRon(o)(t) < �, t = 1, . . . , k)



With correlated choice of cones across time slots

Not any more

P(T > k) = P(SINRon(o)(t) < �, t = 1, . . . , k)



With correlated choice of cones across time slots condition over sigma field generated by and choice of cones �Gk

Not any more

P(T > k) = P(SINRon(o)(t) < �, t = 1, . . . , k)



With correlated choice of cones across time slots condition over sigma field generated by and choice of cones �Gk

Not any more

P(T > k) = E

(kY

t=1

P(SINRon(o)(t) < �|G

k

)

)

P(T > k) = P(SINRon(o)(t) < �, t = 1, . . . , k)

on(o)

x n1(x)


P(T > k) = E

(kY

t=1


k

)

)

dij

Pij

on(o)

x n1(x)


P(T > k) = E

(kY

t=1


k

)

)

dij

Pij

on(o)

x

n2(x)

n1(x)


P(T > k) = E

(kY

t=1


k

)

)

dij

Pij

on(o)

x

n2(x)

n1(x)Px

(t) changes


P(T > k) = E

(kY

t=1


k

)

)

for o, cone is fixed for k slots

dij

Pij

on(o)

x

n2(x)

n1(x)Px

(t) changes


For any interferer z, use the choice of cone that maximizes the interference seen at n(o) at time t=1,…,k

P(T > k) = E

(kY

t=1


k

)

)

for o, cone is fixed for k slots


E {T} =1X

k=0

P(T > k) P(T > k) = E

(kY

t=1


k

)

)


Power Control : power with prob.Pi = c`(dij)�1 pi = MP�1

i

E {T} =1X

k=0

P(T > k) P(T > k) = E

(kY

t=1


k

)

)


For any interferer z, the average interference is bounded


i

E {T} =1X

k=0

P(T > k) P(T > k) = E

(kY

t=1


k

)

)




i

pzPz M

E {T} =1X

k=0

P(T > k) P(T > k) = E

(kY

t=1


k

)

)




i

pzPz MZ

`(|x|)dx < 1Thus, if

E {T} =1X

k=0

P(T > k) P(T > k) = E

(kY

t=1


k

)

)




i

pzPz MZ

`(|x|)dx < 1Thus, if

using Campbell’s Theorem we can show that expected delay is finite

E {T} =1X

k=0

P(T > k) P(T > k) = E

(kY

t=1


k

)

)

Information Velocity

Look at tagged particle at origin

X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

o

V = limt!1

d(t)

t

dest



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

o

V = limt!1

d(t)

t

dest



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

o

V = limt!1

d(t)

t

dest

progress



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

T0

o

V = limt!1

d(t)

t

dest

progress



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

T0

T1

o

V = limt!1

d(t)

t

dest

progress



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

T0

T1

o

V = limt!1

d(t)

t

dest

progress

⌫ =

E{R cos(✓)}E{Ti}

First Guess



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

T0

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

T0

T1

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

T0

T1

T0,T1 are not identically distributed

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

T0

T1

T0,T1 are not identically distributed

vacant

o


X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

o


Dominate the delay by adding an infinite chain of points

X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

same distribution as R and ✓

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1

same distribution as R and ✓fill Poisson points

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1


T̂0

T̂1

o



X1 + C1

R2

θ−1

X−1

X2

φ

R1

R−2

R−1

X1

θ−2

θ1

θ2

X−2

C1


T̂0, T̂1, ... are identically distributed

T̂0

T̂1

o



ˆT0, ˆT1, ... are stationary

similar to before E{T̂i} < 1



Birkoff’s Ergodic Theorem limn!1

1

n

n�1X

k=0

T̂k = T̂

similar to before

T̂where is a rv with mean

E{T̂i} < 1

E{T̂i}



v � lim

t!1

PN(t)k=1 Rk cos(✓k)PN(t)+1

k=1ˆTk

=

E[R cos(✓)]ˆT

Birkoff’s Ergodic Theorem

effective distance progress

limn!1

1

n

n�1X

k=0

T̂k = T̂

similar to before

T̂where is a rv with mean

⌫

E{T̂i} < 1

E{T̂i}

Documents

Breaking the Poisson inﬁnite delay stranglehold Vaze.pdf · Rahul Vaze and Srikanth Iyer Breaking the Poisson inﬁnite delay stranglehold IISc, Bangalore