An overview of the transmission capacity of wireless networkspscn/slides/2011_02_29_weber.pdf · An overview of the transmission capacity of wireless networks Steven Weber Drexel

An overview of the transmission capacity ofwireless networks

Steven WeberDrexel University, Dept. of ECE

Philadelphia, PA [email protected]

joint work withJeffrey G. Andrews (The University of Texas at Austin)

Nihar Jindal (The University of Minnesota)

Steven Weber - Drexel University, Dept. of ECEUniversity of Pennsylvania - Philadelphia, PA - March 1, 2011

Transmission capacity and uncoordinated networks:setup

• Uncoordinated network: transmitters employ Aloha MAC

• Large ad hoc network with uniform (Poisson) distributed transmitternode locations at a snapshot in time

• Successful transmissions have a signal to interference ratio exceeding agiven SIR threshold (treat interference as noise)

• Aloha contention probability ⇒ density of transmissions ⇒ interferencelevel ⇒ outage probability.

• Transmission capacity: maximum spatial density of concurrent successfultransmissions subject to a bound on the outage probability


1

Increased density of interferers leads to increased outageprobability


2

Transmission capacity and uncoordinated networks:limitations

• Sub-optimal Aloha MAC: nodes transmit without the benefit of anytransmission coordination.

• Single hop: snapshot in time ignores the essential multi-hop nature of adhoc networks (routing not addressed).

• Independent transmitter locations: not true under any MAC withtransmission coordination.


3

Transmission capacity and uncoordinated networks:justification

• Sometimes the overhead of coordination is too high (high feedback cost,short channel coherence time, rapid topology changes): Aloha MAC maybe the only feasible choice.

• The benefit and cost of coordination must be gauged relative toperformance of uncoordinated network.

• Assumptions permit model tractability and yield crisp design insights forad hoc networks, not found in other setups.


4

Outline

• Background

• Basic model

• Applications

– Spread spectrum and multi-channel models– Interference cancellation– Fading and threshold scheduling– Fractional power control


5

Background: shot noise processes

• Model for the superposition of effects caused by a point process withimpulse response h(·) [Schottky 1918]:

I(t) =

∞∑j=−∞

h(t− tj), {tj} ∼ PPP.

• For power law impulse response [Lowen and Teich 1990]:

h(t) = Kt−β, A ≤ t ≤ B, 0 ≤ β ≤ 2,

the process {I(t)} is Levy stable random process for certain β,A,B.


6

Background: Levy stable distributions

• Stable distributions [Levy 1925] are closed under convolutions: X isstable iff for X1, X2 are iid copies of X there exist constants a, b, c, dsuch that

aX1 + bX2d= cX + d.

• No closed form CDF or PDF (in general), but the characteristic function(for symmetric case) is:

φ(t) = E[eitX

]= exp

{−γ|t|ν

},

for γ > 0 the dispersion parameter, and ν ∈ [0, 2] is stability exponent.

• Fractional order moments up to ν (ν < 2):

E[|X|p]{<∞, 0 ≤ p ≤ ν=∞, p > ν

.


7

Background: spatial shot noise processes


8

Background: spatial interference models

• Spatial models of co-channel interference in wireless networks first usedin [Musa and Wasylkiwskyj 1978].

• Power law attenuation linked with power law shot noise and Levy stabledistribution by [Sousa and Silvester 1990, 1992].

• Characterization of impact of random (distance independent) effects oninterference distribution in [Ilow and Hatzinakos 1998].

• Connections between stable shot noise processes and stochastic geometryin [Baccelli and Blaszczyszyn 2000].

• Recent developments: NOW monographs by Ganti/Haenggi and Baccelli,book by Franceschetti and Meester, SpaSWiN conference (since 2005),JSAC special issue in August, 2008, etc.


9

Outline

• Background

• Basic model

• Applications



10

Basic reception model: assumptions

• Pure pathloss attenuation: no shadowing or fading

• No thermal noise: SINR = SIR

• No scheduling: nodes make independent transmission decisions

• No power control: all nodes employ common power

• Fixed TX-RX distance: each TX has unique associated RX at fixeddistance

• No spreading: all nodes share same narrowband channel

Each of these assumptions can be relaxed.


11

Basic reception model: random ISR at the origin

! = {Xi}

r

• Π = {Xi, i ∈ N}: PPP interfererlocations, intensity λ

• WLOG: add reference TX-RXpair at origin

• |Xi|−α: interference power atorigin from i, α > 2

• r−α: RX signal power at origin

• ISR: Y = 1r−α

∑i∈Π |Xi|−α

• Y is stable, no closed-form CDF,resort to bounds


12

Outage probability and Transmission capacity

The outage probability is the probability the SIR at the reference receiverat the origin is below the SIR threshold β: (equiv: ISR > y ≡ 1/β)

q(λ) ≡ P(SIR < β) = P

(r−α∑

i∈Π(λ) |Xi|−α< β

)

= P(ISR > y) = P

1

r−α

∑i∈Π(λ)

|Xi|−α > y

= P(Y > y).

The transmission capacity is the maximum intensity of successfultransmissions subject to an outage probability of ε ∈ [0, 1]:

c(ε) ≡ q−1(ε)(1− ε), ε ∈ (0, 1).

Think of ε as a QoS constraint.


13

Outage probability q(λ) and transmission capacity c(ε)

0.0001 0.001 0.01Λ

0.01

0.1

1.qHΛL

0.01 0.1 1.Ε

0.00001

0.0001

0.001cHΕL

Note an upper (lower) bound on the OP yields a lower (upper) bound onthe TC.


14

Dominant and non-dominant interferers

• An interferer i is dominant if 1r−α|Xi|−α > y (⇔ |Xi| < ry−

1α).

• Interferers may be split into dominant and non-dominant subsets:

Πy = {i : |Xi| < ry−1α} ⊂ Π, Πc

y = Π \Πy.

• Dominant and non-dominant ISR is

Yy =1

r−α

∑i∈Πy

|Xi|−α, Y cy =1

r−α

∑i∈Πcy

|Xi|−α, Y = Yy + Y cy .


15

Lower bound on outage probability

Probability of outage lower bounded by dropping non-dominant interferers:

q(λ) = P(Y > y) = P(Yy + Y cy > y) > P(Yy > y) = 1− P(Yy ≤ y).

Low dominant interference same as void probability in a disk

{Yy ≤ y} = {Yy = 0} = {Πy = ∅}.

Void probability for a Poisson process is known explicitly:

P(Πy = ∅) = P(Π ∩ b(o, ry− 1α) = ∅) = exp

{−λπ

(ry−

1α

)2}.

Final lower bound on OP and upper bound on TC:

q(λ) > 1− exp{−λπr2y−

2α

}, c(ε) <

(1− ε) log(1− ε)−1

πr2y−2α

.


16

Upper bound on outage probability

Condition on the dominant interferers:

P(Y > y) = P(Y > y|Yy > y)P(Yy > y) + P(Y > y|Yy ≤ y)P(Yy ≤ y)

= P(Yy > y) + P(Y cy > y)P(Yy ≤ y)

= 1−(1− P(Y cy > y)

)exp

{−λπr2y−

2α

}An upper bound on the non-dominant interferers is achieved by standardtail probability bounding techniques (Markov, Chebychev, Chernoff, etc.):

P(Y cy > y) = P

1

r−α

∑i∈Πcy

|Xi|−α > y

≤ . . . .


17

Outage probability q(λ) and transmission capacity c(ε)

0.0001 0.001 0.01Λ

0.01

0.1

1.qHΛL

0.01 0.1 1.Ε

0.00001

0.0001

0.001cHΕL


18

Tightness of the dominant interferer bound and thesubexponential property

Fix n interferers. The aggregate ISR may be decomposed as

Y =1

r−α

n∑i=1

|Xi|−α = V1 + · · ·+ Vn

where the ISR contribution of each node i is:

Vi =1

r−α|Xi|−α.

Then (V1, . . . , Vn) are n iid subexponential rvs meaning

limy→∞

P(V1 + · · ·+ Vn > y)

P(max{V1, . . . , Vn} > y)= 1, n ≥ 2.

The numerator and denominator are the aggregate OP and the dominantinterferer OP.


19

Spatial throughput and transmission capacity

• Spatial throughput (attempted transmission intensity × success prob.):

τ(λ) = λ(1− q(λ)) < λ exp{−λπr2y−

2α

}.

• Same form as throughput under (non-spatial) Aloha: λe−gλ: λ∗ = 1g,

τ∗ = ge .

• Success probability under throughput optimal density of contention is1 − q(λ∗) = e−gλ

∗= e−1 ≈ 0.36: incur 64% outage to maximize

throughput!

• Energy efficiency concerns of ad hoc networks means spatial throughputmay not be the correct performance metric.

• Transmission capacity maximizes throughput subject to a bound on themaximum admissible outage probability (energy inefficiency).


20

Transmission capacity as sphere packing

• First order Taylor series approximation of cu(ε) in ε around ε = 0 yields:

cu(ε) =−(1− ε) log(1− ε)

πr2y−2α

=1

πr(ε)2+ Θ(ε2), r(ε) =

β1αr√ε.

• Interference radius of sphere depends upon four critical parameters:α, β, ε, r.

D|D| = A

rε

nε = Aλεspheres

rε =1!εβ1αrtx

disk

interference radius

rtx


21

Outline

• Background

• Basic model

• Applications



22

Spread spectrum: direct sequence

• Relax the narrowband assumption: large BW is now available.

• Signals multiplied by a “spreading sequence” with BW M times largerthan NB

• Used in IS-95, 3G cellular networks, 802.11b LANs

• Interference suppression reduces effective SINR req. by M : β → β/M :

c(ε) =F−1Z (ε)(1− ε)πr2β

2α

→ cDS(ε) =F−1Z (ε)(1− ε)

πr2(βM

) 2α

= c(ε)M2α.

• Suppresses interference


23

Spread spectrum: frequency hopping

• Each user’s code dictates a pseudo-random sequence of hops across Mnarrow bands

• Used in Bluetooth

• Reduces intensity of interferers on each band by M : λ→ λ/M :

c(ε) =F−1Z (ε)(1− ε)πr2β

2α

→ cFH(ε) = MF−1Z (ε)(1− ε)πr2β

2α

= c(ε)M.

• Avoids interference

Design insight: better to avoid interference than suppress it:

cFH(ε)

cDS(ε)= M1− 2

α =√M (for α = 4).


24

How to divide spectrum into sub-bands to maximize TC?

Given a total BW of W (Hz) and a data rate requirement of R (bps), what# of sub-bands M , each W/M (Hz), maximizes the TC?

W

1236

Competing effects:

• Increasing M reduces the intensity of interferers: λ/M per band

• Increasing M increases the required SINR:

R =W

Mlog2(1 + β) requires an SINR of β(M) = 2

MRW − 1.


25

How to divide spectrum into sub-bands to maximize TC?

The SINR requirement can also be written as a band requirement:

R =W

Mlog2(1 + β) requires M =

W

Rlog2(1 + β).

Substituting into the expression for the TC under FH:

cFH(ε) =F−1Z (ε)(1− ε)πr2β

2α

M =F−1Z (ε)(1− ε)πr2β

2α

W

Rlog2(1 + β).

Maximize cFH(ε) over all β:

arg maxβ≥0

F−1Z (ε)(1− ε)πr2β

2α

W

Rlog2(1 + β) = arg max

β≥0β−

2α log2(1 + β).

has solutionβ∗(α) = exp

{α2

+W(−α

2e−

α2

)}− 1,

where W(z) solves W(z)eW(z) = z.


26

Optimal SINR threshold β∗(α) versus pathloss exp. α

2 2.5 3 3.5 4 4.5 5−10

−8

−6

−4

−2

0

2

4

6

8

10

Path Loss Exponent (α)

SIN

R T

hres

hold

(dB

)


27

Capacity vs. M for varying SNR (30, 20, 5, 0) dB

0 20 40 60 80 100 120 1400

0.5

1

1.5

2

2.5

3

3.5

4x 10−3

N (# sub−bands)

Dens

ity (1

/m2 )

Design insight: M should be be chosen so that β(M) = β∗(α). Choosingβ too small means high interference per band and thus sub-optimalparallelization of FH. Choosing β too large means low interference perband and thus sub-optimal TC per band.


28

Outline

• Background

• Basic model

• Applications



29

Successive interference cancellation

Signals are decoded sequentially, with the receiver cancelling interferenceafter each user.

• Restrict cancellation to nodes with interference power exceeding receivedsignal power

• At most K nodes may be decoded due to delay constraints (modeled bya cancellation region)

• Cancellation only partially effective:

– cancelled interference power reduced from I to zI– z ∈ [0, 1] is the cancellation effectiveness parameter– z = 0 gives perfect cancellation, z = 1 recovers no cancellation.


30


R2intended

transmitter uncancelable interferer

cancelable interferers

K = 2

R2

potentially cancelable

nodes

intended transmitter

uncancelable nodes


31


For β > 1, the UB on TC is

cu(ε) =

(1−ε) log(1−ε)−1

πr2(β2α(1+z

2α)−1)

,(

0 ≤ z ≤ 1β , λ <

Kπr2

)(1−ε)(log(1−ε)−1+K)

πr2β2α(1+z

2α)

,

(0 ≤ z ≤ 1

β ,Kπr2 ≤ λ < K

π(zβ)2αr2

)(1−ε) log(1−ε)−1

πr2β2α

,

(0 ≤ z ≤ 1

β , λ ≥K

π(zβ)2αr2

)or 1

β ≤ z ≤ 1

• Case 1: TC depends upon quality of cancellation (z) not on quantity(K)

• Case 3: SIC can’t suppress all dominant interferers causing outage when:

– high SINR requirement (large β), or,– poor cancellation effectiveness (high z), or,– high density of interferers (high λ)


32


0.0001

0.001

0.01

0.1

1

0 5 10 15 20

trans

miss

ion

capa

city

(c)

max. # of cancelable nodes (K)

TC vs. max. # of cancelable nodes

no SIC, UBperfect SIC, UB

imperfect SIC, UB, z = 1/100imperfect SIC, UB, z = 1/10

imperfect SIC, UB, z = 1/2

Design insight: Quality of cancellation (low z) more important than quantityof cancellation (high K).


33

Outline

• Background

• Basic model

• Applications



34

Transmission capacity under fading

Channel model without fading is

r−α, |Xi|−α

Channel model with fading is

H00r−α, Hi0|Xi|−α

for {Hij} iid. E.g., Hij ∼ Exp(1) for Rayleigh fading. The SIR is

SIR =H00r

−α∑i∈ΠHi0|Xi|−α

.

Overall effect of fading unclear:

• Good: strong channels for signal, poor channels for interference

• Bad: weak channels for signal, strong channels for interference


35

Fading has an overall negative effect on TC

Upper bound on the TC without fading:

cu(ε) =(1− ε) log(1− ε)−1

πr2β2α

.

Approximate (for ε small) upper bound on the TC with fading:

cu(ε) =(1− ε) log(1− ε)−1

πr2β2αE[H

2α]E[H−

2α].

Ratio of the TC UB without fading over the approximate TC UB withfading is

cu(ε)

cu(ε)= E[H

2α]E[H−

2α] ≥ 1.

E.g., Rayleigh fading with α = 4 reduces TC by 57%.


36

Exact OP and TC for Rayleigh fading (Baccelli 2006)

Define the aggregate interference seen at the origin as

Z =∑

i∈Π(λ)

Hi0|Xi|−α,

The success probability under Rayleigh fading is the Laplace transform of Zevaluated at s = βrα:

P(SIR > β) = P(H00 > βrαZ) =

∫ ∞0

e−βrαzfZ(z)dz = E

[e−sZ

]∣∣∣∣s=βrα

.

This transform can be computed explicitly, yielding exact OP and TCexpressions:

q(λ) = 1− exp

{−λπr2β

2α

2π

αcsc

(2π

α

)}, c(ε) =

(1− ε) log(1− ε)−1

πr2β2α 2πα csc

(2πα

) .


37

Exploiting fading through threshold scheduling

• Fading is a source of diversity: TC increases if we exploit fading(encourage use of strong channels and avoid weak ones)

• Uncoordinated networks: TX decisions independent, but TX-RXcoordination okay

• Threshold scheduling rule: each transmitter elects to transmit only if itschannel gain to its receiver is above a threshold t

Intuitive tradeoff for threshold t:

• t small: not adequately discriminate between strong and weak signals(limited effectiveness)

• t large: risk being over-selective, and thereby under-utilize the network


38

Exploiting fading through threshold scheduling

• Spatial intensity of attempted transmissions for threshold t is µ(t) ≡λP(H > t).

• Outage probability with threshold t:

q(ν, t) = P

(H00r

−α∑i∈Π(ν)Hi0|Xi|−α

< β

∣∣∣∣∣ H00 ≥ t

).

• Transmission capacity approximation is given by:

c(ε) ≈ (1− ε) log(1− ε)−1

πr2β2αE[H

2α]E[H−

2α|H ≥ t]

.

• Ratio of TC with and without threshold scheduling under Rayleigh fading:

cTS(ε)

cRA(ε)=

E[H−2α]

E[H−2α|H ≥ t]

=Γ(1− 2

α)

etΓ(1− 2α, t)

.


39

Multiplicative TC gain of threshold scheduling inRayleigh fading vs t for α = 2.5, 3, 4

0 0.5 1 1.5 21

2

3

4

5

6

7

8

9

10

11

Threshold (t)

Mul

tiplic

ativ

e T

C G

ain

α=2.5

α=3

α=4

Design insight: opportunistic scheduling can be tuned to achieve very highspatial throughputs at fixed outage at the expense of scheduling delays.


40

Outline

• Background

• Basic model

• Applications



41

Power control: constant power and channel inversion

Two common Tx power control policies:

• Constant power (CP): simple, but may use excessive power on goodchannels and insufficient power on bad channels

• Channel inversion (CI):

– Tx compensates for the channel to ensure a constant Rx signal power– More “fair” than CP: bad, good channels have equal chance


42

Channel inversion lowers TC relative to constant power

The outage probability under constant power is approximately

qCP(λ) ≈ 1− E[exp

{−λπr2β

2αE[H

2α]H−

2α

}].

The outage probability under channel inversion is approximately

qCI(λ) ≈ 1− exp{−λπr2β

2αE[H

2α]E[H−

2α]}.

By Jensen’s inequality for the convex function ex:

E[exp

{−λπr2β

2αE[I

2α]S−

2α

}]> exp

{−λπr2β

2αE[H

2α]E[H−

2α]}.

It follows that:

qCP(λ) < qCI(λ) ⇒ cCP(ε) > cCI(ε).


43

Fractional power control

The transmit and receive powers for each node i, under both CP and CI,are

P tx,cpi = ρ =

ρ

E[H−0ii ]

H−0ii P rx,cp

i = ρHiir−α

P tx,cii =

ρ

E[H−1]H−1ii P rx,ci

i =ρ

E[H−1]r−α.

Tx power expressions suggest class of FPC policies with exponents s ∈ [0, 1]:

P tx,fpci =

ρ

E[H−sii ]H−sii P rx,fpc

i =ρ

E[H−sii ]H1−sii r−α.

Note s = 0 is CP, s = 1 is CI.


44


Approximate (small ε) upper bound on the TC under FPC is:

cu(ε, s) ≈ (1− ε) log(1− ε)−1

πr2β2αE[H

2α

]E[H−s

2α

]E[H−(1−s) 2

α

].Bound-optimal FPC exponent solves:

s∗ = arg mins∈[0,1]

E[H−s

2α

]E[H−(1−s) 2

α

]Holder’s inequality yields s∗ = 1/2. Design insight:

• s � 12: over-compensates for signal fading, leads to excessively high

interference levels

• s � 12: small interference levels but an under-compensation for signal

fading


45


0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Out

age

prob

abilit

y (q

)

Fractional power control parameter (s)

simulationlower bound

Jensen approx

Design insight: the benefit of compensating for low signal power mustbe appropriately balanced with the additional interference it generates fornearby receivers. The power control exponent s helps optimize this tradeoff.


46

Practical engineering design insights

• Spread spectrum: better to avoid interference (FH) than suppress it(DS)

• Multiple sub-bands: choose SINR threshold to generate optimalinterference per band β(M) = β∗(α)

• SIC: quality of cancellation (low z) more important than quantity ofcancellation (high K)

• Threshold scheduling: opportunistic scheduling achieves throughput atexpense of delay

• FPC: parameterizes throughput/fairness tradeoff and tradeoff betweensignal compensation and additional interference.


47

References

1. Steven Weber, Xiangying Yang, Jeffrey G. Andrews and Gustavo de Veciana, “Transmission capacity ofwireless ad hoc networks with outage constraints,” IEEE Transactions on Information Theory Vol. 51,No. 12, pp. 4091–4102, December, 2005.

2. Steven Weber, Jeffrey G. Andrews, Xiangying Yang and Gustavo de Veciana, “Transmission capacity ofwireless ad hoc networks with successive interference cancellation,” IEEE Transactions on InformationTheory Vol. 53, No. 8, pp. 2799–2814, August, 2007.

3. Steven Weber, Jeffrey G. Andrews, and Nihar Jindal, “The effect of fading, channel inversion, andthreshold scheduling on ad hoc networks”, IEEE Transactions on Information Theory, Vol. 53, No. 11,pp. 4127–4149, November, 2007.

4. Andrew Hunter, Jeffrey G. Andrews, and Steven Weber, “Capacity scaling of ad hoc networks withspatial diversity”, IEEE Transactions on Wireless Communications, December, 2008.

5. Nihar Jindal, Steven Weber, and Jeffrey G. Andrews, “Fractional power control for decentralized wirelessnetworks”, IEEE Transactions on Wireless Communications, December, 2008.

6. Nihar Jindal, Jeffrey G. Andrews, and Steven Weber, “Bandwidth partitioning in decentralized wirelessnetworks”, IEEE Transactions on Wireless Communications, December, 2008.

7. Steven Weber, Jeffrey G. Andrews, and Nihar Jindal. ”An overview of the transmission capacity ofwireless networks”. IEEE Transactions on Communications, vol. 58, no. 12, December 2010.


48

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An overview of the transmission capacity of wireless networkspscn/slides/2011_02_29_weber.pdf · An overview of the transmission capacity of wireless networks Steven Weber Drexel