New ASAPPP: A Simple Approximative Analysis Framework for …mhaenggi/talks/hetsnets14.pdf · 2014....

ASAPPP: A Simple Approximative AnalysisFramework for Heterogeneous Cellular Networks

Martin Haenggi

University of Notre Dame, IN, USAEPFL, Switzerland

HetSNets Keynote

Dec. 12, 2014

M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 1 / 38

Overview

Bird’s view of cellular networks: The SIR walk

The HIP model and a key result for the downlink

ASAPPP: Approximate SIR Analysis based on the PPP

How much better is BS cooperation than BS silencing?

Back to modeling: Inter- and intra-tier dependence

Conclusions

Cellular networks The big picture

Big picture in cellular networks

Frequency reuse 1: A single friend, many foes

−4 −3 −2 −1 0 1 2 3 4−4

The SIR walk and coverage at 0 dB

−4 −3 −2 −1 0 1 2 3 4−4

−3 −2 −1 0 1 2 3−6

12SIR (no fading)

SIR distribution

−3 −2 −1 0 1 2 3−25

15SIR (with fading)

The fraction of a long curve (or large region) that is above the threshold θis the ccdf of the SIR at θ:

ps(θ) , F̄SIR(θ) , P(SIR > θ)

It is the fraction of the users with SIR > θ for each realization of the BSand user processes.

HIP model Definition

The HIP baseline model for HetNets

The HIP (homogeneous independent Poisson) model [DGBA12]

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−0.5

Start with a homogeneous Poisson point process (PPP). Here λ = 6.Then randomly color them to assign them to the different tiers.

HIP model Definition

The HIP (homogeneous independent Poisson) model

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−0.5

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−0.5

Randomly assign BS to each tier according to the relative densities. Hereλi = 1, 2, 3. Assign power levels Pi to each tier.This model is doubly independent and thus highly tractable.

HIP model Basic result

Basic result for downlink [NMH14]

Assumptions:

A user connects to the BS that is strongest on average, while allothers interfere.

Homogeneous path loss law ℓ(r) = r−α and Rayleigh fading.

Remarkably, the SIR distribution is independent of the number of tiers,their densities, and their power levels:

ps(θ) , F̄SIR(θ) =1

2F1(1,−δ; 1 − δ;−θ)

In particular, for α = 4,

ps(θ) = P(SIR > θ) = F̄SIR(θ) =1

1 +√θ arctan

√θ.

So as far as the SIR is concerned, we can replace the multi-tier HIP modelby an equivalent single-tier model.

HIP model Basic result

Conclusions from HIP SIR distribution

The SIR does not improve with small cells (but the per-user capacity does).

If there are enough BSs so that many of them are inactive densification provides an SIR

For θ = 1, ps = 56%.

Question: How to improve the SIR distribution?⇒ Non-Poisson deployment⇒ BS silencing⇒ BS cooperation

How to quantify the improvement in the SIR distribution?

Comparing SIR Distributions

Comparing SIR distributions

Two distributions

−15 −10 −5 0 5 10 15 20 250

1SIR CCDF

θ (dB)

baseline schemeimproved scheme

How to quantify the improvement?

Comparing SIR Distributions Vertical comparison

The standard comparison: vertical

−15 −10 −5 0 5 10 15 20 250

1SIR CCDF

θ (dB)

At -10 dB, the gap is 0.058. Or 6.4%.

At 0 dB, the gap is 0.22. Or 39%.

Comparing SIR Distributions Horizontal comparison

A better choice: horizontal

−15 −10 −5 0 5 10 15 20 250

1SIR CCDF

θ (dB)

Gbaseline schemeimproved scheme

Use the horizontal gap instead.

This SIR gain is nearly constantover θ in many cases.

ps(θ) = P(SIR > θ) ⇒ ps(θ) = P(SIR > θ/G ).

Can we quantify this gain?

Comparing SIR Distributions The horizontal gap

Horizontal gap at probability p

The horizontal gap between two SIR ccdfs is

G (p) ,F̄−1

SIR2(p)

F̄−1SIR1

(p), p ∈ (0, 1),

where F̄−1SIR is the inverse of the ccdf of the SIR, and p is the target success

probability.We also define the asymptotic gain (whenever the limit exists) as

G = limp→1

G (p).

Relevance

We will show that

G is relatively easy to determine.

G (p) ≈ G for all practical p (which is p ' 3/4).

Comparing SIR Distributions ISR

The ISR and the MISR

Definition (ISR)

The interference-to-average-signal ratio is

ISR ,I

Eh(S),

where Eh(S) is the desired signal power averaged over the fading.

Comments

The ISR is a random variable due to the random positions of BSs andusers. Its mean MISR is a function of the network geometry only.

If the desired signal comes from a BS at distance R , (Eh(S))−1 = Rα.

If the interferers are located at distances Rk ,

MISR , E(ISR) = E

Rα∑

hkR−αk

Relevance of the MISR [Hae14]

−30 −25 −20 −15 −10 −5 0

10−3

10−2

10−1

SIR CDF (outage probability)

θ (dB)

Outage probability:

pout(θ) = P(hR−α < θI )

= P(h < θ ISR)

For exponential h:

= 1 − e−θ ISR

∼ θMISR, θ → 0.

So FSIR(θ) ∼ θMISR =⇒ F̄−1SIR(p) ∼ (1 − p)/MISR, (p → 1).

So the asymptotic gain is the ratio of the two MISRs: G =MISR1

We need to find a reference MISR1 that is easy to calculate...

The MISR for the HIP model

For the (single-tier) HIP model,

MISR = E

∞∑

R−αk

∞∑

where Rk is the distance to the k-th nearest BS.The distribution of νk = R1/Rk is

Fνk (x) = 1 − (1 − x2)k−1, x ∈ [0, 1].

Summing up the α-th moments E(ναk ), we obtain (remarkably) [Hae14]

MISR =2

α− 2.

This is the baseline MISR relative to which we can measure the gain G .For α = 4, it is 1.

The ASAPPP approach

Gain relative to HIP

We can approximate the SIR distribution of arbitrary point processes andtransmission schemes by shifting the Poisson curve:

ps(θ) = pHIPs

θMISR

MISRHIP

Acronym

ASAPPP stands for "approximate SIR analysis based on the PPP".

It also can be read as "as a PPP", since we are (first) treating thenetwork as if it was based on a PPP.

Thirdly, it is a strong ASAP, which means that it can be very efficientin obtaining a good approximation on the SIR distribution.

Gains due to Deployment and Cooperation Deployment gain

Deployment gain [GH13, GH14]

−20 −15 −10 −5 0 5 100.3

1PPP and square lattice, α= 3.0

θ (dB)

PPPsquare latticeISR−based gain

−20 −15 −10 −5 0 5 100.3

1PPP and square lattice, α= 4.0

θ (dB)P

PPPsquare latticeISR−based gain

For the square lattice, the gap (deployment gain) is quite exactly 3dB—irrespective of α! For α = 4, psq

s = (1 +√

θ/2 arctan√

θ/2)−1.

For the triangular lattice, it is 3.4 dB. This is the maximum achievable.

Gains due to Deployment and Cooperation Deployment gain

The bandgap of SIR distributions

−15 −10 −5 0 5 10 15 200

1SIR CCDF

θ (dB)

Poissontriangular lattice

All (repulsive) deployments have SIRs that fall into this thin green region.

Higher gains can only be achieved using interference-mitigating and/orsignal-boosting schemes.

Gains due to Deployment and Cooperation BS silencing

BS silencing: neutralize nearby foes [ZH14]

−4 −3 −2 −1 0 1 2 3 4−4

Gains due to Deployment and Cooperation BS silencing

Gain due to BS silencing for HIP model [ZH14, Hae14]

Let ISR(!n)

be the ISR obtained when the n − 1 strongest (on average)interferers are silenced. All other BSs are still interfering.So there is cooperation from n BSs, with the strongest one transmittingand the other n − 1 being silent.For HIP,

E(ISR(!n)

α− 2

Γ(1 + α/2)Γ(n + 1)

Γ(n + α/2).

Gsilence =Γ(n + α/2)

Γ(1 + α/2)Γ(n + 1)∼ (n + 1)α/2−1

Γ(1 + α/2).

For α = 4, in particular,

Gsilence =n + 1

Gains due to Deployment and Cooperation BS cooperation

Cooperation by joint transmission

BS cooperation: turn nearby foes into friends

−4 −3 −2 −1 0 1 2 3 4−4

SIR with joint transmission from n BS [NMH14]

For the analysis of JT, we introduce the function

ψn(α) ,

· · ·1∫

1 +∑n−1

k=1 t−α/2k

dt1 · · · dtn−1.

This is the expected value

1 + ‖t‖−α/2−α/2

, where t = (t1, . . . , tn−1) ∼ U([0, 1]n−1).

Since t−α/2k ≥ 1, certainly ψn(α) < 1. We have

ψ2(4) = 2− π

2≈ 0.43 ; ψ3(4) =

4π−3π−2+

2arcsin

≈ 0.264.

SIR with joint transmission from n BS

Let the desired signal consist of the superposition of the signals from the n

strongest (on average) BS in the HIP model. We have [NMH14]

E(ISR(+n)

) =2ψn(α)

α− 2=⇒ Gcoop =

ψn(α).

For α = 4, n = 2:

−20 −15 −10 −5 0 5 100.2

θ (dB)

JT from 2 BSsSingle BSMISR−based gain

Gcoop =2

4 − π≈ 2.33.

The red curve is the exact ccdf(given by an n-dim. integral).

So JT from 2 BSs in the HIP modelis better than even a triangular lat-tice without JT.

Gains due to Deployment and Cooperation Comparison of silencing and JT

SIR with cooperation from n BS: Silencing vs. joint transmission

0 10 20 30 40 500

Joint transmission from n BSsSilencing n−1 BSs

α = 3

0 10 20 30 40 500

Joint transmission from n BSsSilencing n−1 BSs3/2*n

α = 4

For α = 3, the ratio approaches 4, while for α = 4, the ratio approaches 3.

Conjecture:Gcoop

Gsilence∼ 2 − δ

1 − δ=

2 − 2/α

1 − 2/α, n → ∞.

Gains due to Deployment and Cooperation Cooperation for worst-case users

Cooperation for worst-case users

SIR at Voronoi vertices with cooperation

−4 −3 −2 −1 0 1 2 3 4−4

At these locations (×), the user isfar away from any BS, and thereare two interfering BS at thesame distance.

In the Poisson model,

SIR(θ) =F̄ 2

SIR(θ)

(1 + θ)2.

With BS cooperation from the 3equidistant BSs, for α = 4,

F̄×,coop

SIR (θ) = F̄ 2SIR(θ/3) =

1 +√

θ/3 arctan(√

θ/3))

Gains due to Deployment and Cooperation Cooperation for worst-case users

With ASAPPP

−20 −15 −10 −5 0 5 100

1SIR CCDF for PPP, α=4

θ (dB)

general userworst−case, no coop.worst−case, 3−BS coop.shifted by E(ISR) ratio

For worst-case users withn ∈ {1, 2, 3} BSs cooperating,

E(ISR) =4 + (3 − n)(α− 2)

n(α− 2).

So for n = 3, the ratio of the twoMISRs is

Gcoop =MISR

MISRcoop= 3 +

2(α− 2).

The shape of the curve does not change, it is merely shifted.

This indicates that BS cooperation of k BSs (non-coherent jointtransmission) does not provide a diversity gain.

Effectiveness of ASAPPP

The unreasonable effectiveness of ASAPPP

Why is shifting so accurate?

Not fully clear (yet).Intuition: cdfs all have the same shape (single inflection point), and in thecase of the SIR, both tails can only differ in the pre-constant.

For small θ, by definition of the diversity gain d ,

1 − ps(θ) = Θ(θd), θ → 0.

Theorem (Tail of SIR distribution)

For all stationary point processes and all fading distributions,

ps(θ) = Θ(θ−δ), θ → ∞.

For HIP with Rayleigh fading, ps(θ) ∼ sinc δ θ−δ.

HetNet modeling Dependencies

Back to modeling

Introducing dependencies

Intra-tier dependence: BS of one tier are not placed independently.

Inter-tier dependence: BS of different tiers are not placedindependently.

Intra-tier dependence

The HIP model is conservative since it may place BSs arbitrarily closeto each other. The lattice model is extreme.

Current and future real-world deployments fall in between. It isunlikely to have two BSs very close, so the BSs form a soft-core orhard-core process. In other words, the BSs process is repulsive.

HetNet modeling The Ginibre point process

The Ginibre model [DZH14]

Realizations of PPP and the Ginibre point process (GPP) on b(o, 8)

−5 0 5

PPP with n=65

−5 0 5−8

81−GPP with n=64

The GPP exhibits repulsion—just as BSs in a cellular network.Its pair correlation function is g(r) = 1 − e−r2 .

The Ginibre point process

The GPP is a motion-invariant determinantal point process.

Remarkable property: If Φ = {x1, x2, . . .} ⊂ R2 is a GPP, then

{‖x1‖2, ‖x2‖2, . . .} d= {y1, y2, . . .}

where (yk) are independent gamma distributed random variables with pdf

fyk (x) =xk−1e−x

Γ(k); E(yk) = k .

The intensity is 1/π but can be adjusted by scaling.

The GPP can be made less repulsive by independently deleting pointswith probability 1 − β and re-scaling. This β-GPP approaches thePPP in the limit as β → 0.

The Ginibre point process in action

We would like to model these two deployments:

360000 380000 400000 420000

Rural Region (75000m x 65000m)

528000 529000 53000018

Urban Region (2500m x 1800m)

Data taken from Ofcom website.

The Ginibre point process in action

SIR distributions for different path loss exponents:

−10 −5 0 5 10 15 200

SIR Threshold (dB)

Rural Region

Experimental Data w. α=4Experimental Data w. α=3.5Experimental Data w. α=3Experimental Data w. α=2.5The fitted β−GPP

−10 −5 0 5 10 15 200

SIR Threshold (dB)C

Urban Region

Experimental Data w. α=4Experimental Data w. α=3.5Experimental Data w. α=3Experimental Data w. α=2.5The fitted β−GPP

For the rural region, β = 0.2. For the urban region, β = 0.9.

Other models that fit well are the Strauss process and the perturbed lattice[GH13]. They are less tractable, though.

The Ginibre process and the MISR

0 0.2 0.4 0.6 0.8 11

MISR−based gain for β−GPP

So quite exactly Gβ ≈ 1 + β/2 (barely depends on α).The square lattice has a gain of 2, so the 1-GPP falls exactly in betweenthe PPP and the lattice.

Also: The 1-GPP provides the same SIR distribution as a PPP with 1-BSsilencing.

HetNet modeling Dependence

Two-tier models with intra- and inter-tier dependence

Intra-tier dependence

For a single tier, the Ginibre process models the repulsion.For a capacity-oriented deployment, a cluster process can be used for thesmall cells [DZH15].In this case, the users do not form a PPP—a Cox process with higherdensities in the hotspots may be suitable as a model.

Inter-tier dependence

Since small cells are not deployed close to macro-BSs, they can be assumedto form a Poisson hole process [DZH15].The macro-BSs may still be assumed to form a PPP.

Conclusions

The SIR distribution is a key metric from which other metric can bederived (spectral efficiency, rate, throughput, delay, reliability).

ASAPPP: The SIR gain of a deployment/architecture/scheme is bestmeasured as the horizontal gap relative to the Poisson model.

For α = 4 : ps(θ) ≈1

1 +√θMISR arctan

√θMISR

The MISR is easy to obtain by simulation if it cannot be calculated.

Joint transmission provides a fixed gain over BS silencing as thenumber of cooperating BSs increases.

Future work should also include models with intra- and inter-tierdependence. The Ginibre point process is promising as a repulsivemodel due to its tractability.Having data to verify the models would be very helpful.

References

References I

H. S. Dhillon, R. K. Ganti, F. Baccelli, and J. G. Andrews.Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks.IEEE Journal on Selected Areas in Communications, 30(3):550–560, March 2012.

N. Deng, W. Zhou, and M. Haenggi.The Ginibre Point Process as a Model for Wireless Networks with Repulsion.IEEE Transactions on Wireless Communications, 2014.Accepted. Available at http://www.nd.edu/~mhaenggi/pubs/twc14c.pdf .

N. Deng, W. Zhou, and M. Haenggi.Heterogeneous Cellular Network Models with Dependence.IEEE Journal on Selected Areas in Communications, 2015.Submitted. Available at http://www.nd.edu/~mhaenggi/pubs/jsac15b.pdf .

A. Guo and M. Haenggi.Spatial Stochastic Models and Metrics for the Structure of Base Stations in CellularNetworks.IEEE Transactions on Wireless Communications, 12(11):5800–5812, November 2013.

References

References II

A. Guo and M. Haenggi.Asymptotic Deployment Gain: A Simple Approach to Characterize the SINR Distributionin General Cellular Networks.IEEE Transactions on Communications, 2014.Accepted. Available at http://www.nd.edu/~mhaenggi/pubs/tcom14b.pdf .

M. Haenggi.The Mean Interference-to-Signal Ratio and its Key Role in Cellular and AmorphousNetworks.IEEE Wireless Communications Letters, 3(6):597–600, December 2014.

G. Nigam, P. Minero, and M. Haenggi.Coordinated Multipoint Joint Transmission in Heterogeneous Networks.IEEE Transactions on Communications, 62(11):4134–4146, November 2014.

X. Zhang and M. Haenggi.A Stochastic Geometry Analysis of Inter-cell Interference Coordination and Intra-cellDiversity.IEEE Transactions on Wireless Communications, 13(12):6655–6669, December 2014.

New ASAPPP: A Simple Approximative Analysis Framework for …mhaenggi/talks/hetsnets14.pdf · 2014....

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