New ASAPPP: A Simple Approximative Analysis Framework for …mhaenggi/talks/hetsnets14.pdf · 2014....
Transcript of 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
Menu
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
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 2 / 38
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
−3
−2
−1
0
1
2
3
4
SIR =
S
I
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 3 / 38
Cellular networks The big picture
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
4
−3 −2 −1 0 1 2 3−6
−4
−2
0
2
4
6
8
10
12SIR (no fading)
x
SIR
(dB
)
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 4 / 38
Cellular networks The big picture
SIR distribution
−3 −2 −1 0 1 2 3−25
−20
−15
−10
−5
0
5
10
15SIR (with fading)
x
SIR
(dB
)
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 5 / 38
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
−1
−0.5
0
0.5
1
1.5
2
Start with a homogeneous Poisson point process (PPP). Here λ = 6.Then randomly color them to assign them to the different tiers.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 6 / 38
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
−1
−0.5
0
0.5
1
1.5
2
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 7 / 38
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 8 / 38
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
gain.
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?
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 9 / 38
Comparing SIR Distributions
Comparing SIR distributions
Two distributions
−15 −10 −5 0 5 10 15 20 250
0.2
0.4
0.6
0.8
1SIR CCDF
θ (dB)
P(S
IR>
θ)
baseline schemeimproved scheme
How to quantify the improvement?
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 10 / 38
Comparing SIR Distributions Vertical comparison
The standard comparison: vertical
−15 −10 −5 0 5 10 15 20 250
0.2
0.4
0.6
0.8
1SIR CCDF
θ (dB)
P(S
IR>
θ)
baseline schemeimproved scheme
At -10 dB, the gap is 0.058. Or 6.4%.
At 0 dB, the gap is 0.22. Or 39%.
At 10 dB, the gap is 0.15. Or 73%.
At 20 dB, the gap is 0.05. Or 78%.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 11 / 38
Comparing SIR Distributions Horizontal comparison
A better choice: horizontal
−15 −10 −5 0 5 10 15 20 250
0.2
0.4
0.6
0.8
1SIR CCDF
θ (dB)
P(S
IR>
θ)
G
G
G
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?
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 12 / 38
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).
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 13 / 38
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
)
=∑
E
(
R
Rk
)α
.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 14 / 38
Comparing SIR Distributions ISR
Relevance of the MISR [Hae14]
−30 −25 −20 −15 −10 −5 0
10−3
10−2
10−1
100
SIR CDF (outage probability)
θ (dB)
P(S
IR<
θ)
Gasym
baseline schemeimproved scheme
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
MISR2
We need to find a reference MISR1 that is easy to calculate...
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 15 / 38
Comparing SIR Distributions ISR
The MISR for the HIP model
For the (single-tier) HIP model,
MISR = E
(
Rα1
∞∑
k=2
R−αk
)
=
∞∑
k=2
E
(
R1
Rk
)α
,
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 16 / 38
Comparing SIR Distributions ISR
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 17 / 38
Gains due to Deployment and Cooperation Deployment gain
Deployment gain [GH13, GH14]
−20 −15 −10 −5 0 5 100.3
0.4
0.5
0.6
0.7
0.8
0.9
1PPP and square lattice, α= 3.0
θ (dB)
P(S
IR>
θ)
PPPsquare latticeISR−based gain
−20 −15 −10 −5 0 5 100.3
0.4
0.5
0.6
0.7
0.8
0.9
1PPP and square lattice, α= 4.0
θ (dB)P
(SIR
>θ)
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 18 / 38
Gains due to Deployment and Cooperation Deployment gain
The bandgap of SIR distributions
−15 −10 −5 0 5 10 15 200
0.2
0.4
0.6
0.8
1SIR CCDF
θ (dB)
P(S
IR>
θ)
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 19 / 38
Gains due to Deployment and Cooperation BS silencing
BS silencing: neutralize nearby foes [ZH14]
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−2
−1
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4
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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)
) =2
α− 2
Γ(1 + α/2)Γ(n + 1)
Γ(n + α/2).
So
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
2.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 21 / 38
Gains due to Deployment and Cooperation BS cooperation
Cooperation by joint transmission
BS cooperation: turn nearby foes into friends
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Gains due to Deployment and Cooperation BS cooperation
SIR with joint transmission from n BS [NMH14]
For the analysis of JT, we introduce the function
ψn(α) ,
1∫
0
· · ·1∫
0
n
1 +∑n−1
k=1 t−α/2k
dt1 · · · dtn−1.
This is the expected value
E
(
n
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) =
9√
2
4π−3π−2+
9
2arcsin
(
1
3
)
≈ 0.264.
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Gains due to Deployment and Cooperation BS cooperation
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 =
1
ψn(α).
For α = 4, n = 2:
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0.4
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0.6
0.7
0.8
0.9
1
θ (dB)
P(S
IR>
θ)
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.
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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
5
10
15
20
25
n
SIR
gai
n
Joint transmission from n BSsSilencing n−1 BSs
α = 3
0 10 20 30 40 500
10
20
30
40
50
60
70
80
n
SIR
gai
n
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 → ∞.
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Gains due to Deployment and Cooperation Cooperation for worst-case users
Cooperation for worst-case users
SIR at Voronoi vertices with cooperation
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At these locations (×), the user isfar away from any BS, and thereare two interfering BS at thesame distance.
In the Poisson model,
F̄×
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))
−2.
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Gains due to Deployment and Cooperation Cooperation for worst-case users
With ASAPPP
−20 −15 −10 −5 0 5 100
0.2
0.4
0.6
0.8
1SIR CCDF for PPP, α=4
θ (dB)
P(S
IR>
θ)
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 +
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 27 / 38
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 28 / 38
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 29 / 38
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
−5
0
5
PPP with n=65
−5 0 5−8
−6
−4
−2
0
2
4
6
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 .
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 30 / 38
HetNet modeling The Ginibre point process
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 31 / 38
HetNet modeling The Ginibre point process
The Ginibre point process in action
We would like to model these two deployments:
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Data taken from Ofcom website.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 32 / 38
HetNet modeling The Ginibre point process
The Ginibre point process in action
SIR distributions for different path loss exponents:
−10 −5 0 5 10 15 200
0.2
0.4
0.6
0.8
1
SIR Threshold (dB)
Cov
erag
e P
roba
bilit
y
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
0.2
0.4
0.6
0.8
1
SIR Threshold (dB)C
over
age
Pro
babi
lity
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 33 / 38
HetNet modeling The Ginibre point process
The Ginibre process and the MISR
0 0.2 0.4 0.6 0.8 11
1.1
1.2
1.3
1.4
1.5
β
Gai
n
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 34 / 38
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 35 / 38
Conclusions
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 36 / 38
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.
M. Haenggi (ND & EPFL) ASAPPP 12/12/2014 37 / 38
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.
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