Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
Asymptotics and Meta-Distribution of the
Signal-to-Interference Ratio in Wireless Networks
Part II
Martin Haenggi
University of Notre Dame, IN, USAEPFL, Switzerland
2015 Simons Conference on Networks and Stochastic Geometry
May 2015
Work supported by U.S. NSF (CCF 1216407)
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 1 / 47
Overview
Part II
Overview
Cellular networks and the HIP model
Standard analysis of some transmission techniques for the PPP
Non-Poisson network analysis using ASAPPPa
◮ The idea of the horizontal shift (gain) of SIR distributions◮ The relative distance process◮ The MISRb and the EFIRc
◮ Asymptotic gains at 0 and ∞◮ Examples
Concluding remarks
aApproximate SIR Analysis based on the PPP—or simply "as a PPP"bMean interference-to-signal ratiocExpected fading-to-interference ratio
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 2 / 47
Cellular networks models Setup
From bipolar to cellular networks
From yesterday: A generic cellular network (downlink)
Base stations form a stationaryand ergodic point process andall transmit at equal power.
Assume a user is located at o.Its serving base station is thenearest one (strongest onaverage).
The other base stations areinterferers (frequency reuse 1).
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 3 / 47
Cellular networks models Standard BS association
Single-tier cellular networks with reuse 1
SIR with strongest-base station (BS) association
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
SIR ,S
I
S = h‖x0‖−α
I =∑
x∈Φ\{x0}
hx‖x‖−α
Φ: point process of BSsx0: serving BSh, (hx): iid fading
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 4 / 47
Cellular networks models Standard BS association
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) Asymptotics of SIR Distributions May 2015 5 / 47
Cellular networks models Standard BS association
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) Asymptotics of SIR Distributions May 2015 6 / 47
Cellular networks models SIR analysis
Fact on SIR distributions
Only the PPP is tractable exactly—in some cases
If the base stations form a homogeneous Poisson point process (PPP):
ps(θ) , F̄SIR(θ) =1
2F1(1,−δ; 1 − δ;−θ), δ , 2/α.
For δ = 1/2, ps(θ) =(
1 +√θ arctan
√θ)−1
.
If the fading is not Rayleigh or if the point processis not Poisson, it gets hard very quickly.
So let us enjoy the beauty of Poissonia a littlelonger.
PoissoniaM. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 7 / 47
Cellular networks models The HIP model
The HIP baseline model for HetNets
The HIP (homogeneous independent Poisson) modela
aDhillon et al., “Modeling and Analysis of K-Tier Downlink
Heterogeneous Cellular Networks”. 2012.
−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 PPP.Here λ = 6.
Choose a number of tiers andintensities for each tier, sayλ1 = 1, λ2 = 2, and λ3 = 3.
Then randomly color the BSsaccording to the intensities toassign them to the different tiers:
P(tier = i) = λi/λ
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 8 / 47
Cellular networks models The HIP model
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
Here λi = 1, 2, 3. Assign power levels Pi to each tier.This model is doubly independent and thus highly tractable.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 9 / 47
Cellular networks models Equivalence to single-tier model
Equivalence of all HIP models
From the perspective of the typical user, this network is completelyequivalent to a single-tier Poisson model with unit power and unit density.
Hence for all HIP models (withRayleigh fading and power law pathloss), the SIR distribution is
ps(θ) , F̄SIR(θ) =1
2F1(1,−δ; 1 − δ;−θ).
In particular, for δ = 1/2:
ps(10) = 20.00%
The typical user is not impressed withthis performance.
−20 −10 0 10 200
0.2
0.4
0.6
0.8
1SIR CCDF
θ (dB)P
(SIR
>θ)
α = 4 (δ = 1/2).
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 10 / 47
Cellular networks models Equivalence to single-tier model
Explanation for equivalence
For a single tier with unit transmit power, let
Ξ = {ξi} , {x ∈ Φ: ‖x‖α/hx}.
The received powers from the nodes in Φ are {ξ−1}.If Φ ⊂ R
2 is Poisson with intensity λ, then Ξ is Poisson with intensityfunction µ(r) = λπδr δ−1
E(hδ).
For multiple independent Poisson tiers with transmit power Pk , the union
Ξ = {ξi} =⋃
k∈[K ]
{x ∈ Φk : ‖x‖α/(Pkhx)}.
is a PPP with intensity function
µ(r) =∑
k∈[K ]
πλkδPδk r
δ−1E(hδ) .
In any case, µ(r) ∝ r δ−1. The pre-constant does not matter for the SIR.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 11 / 47
Cellular networks models Equivalence to single-tier model
Path loss process
The point process Ξ = {ξi} ⊂ R+ where {ξ−1
i } are the receivedpowers (with or without fading) is called the path loss process orpropagation process.
It is a key ingredient in many proofs of many results for cellularnetworksa and HetNetsb.
The equivalence also holds for advanced transmission techniques, suchas BS cooperation and silencing.
Let us have a look at some of these advanced techniques.
aBlaszczyszyn, Karray, and Keeler, “Using Poisson Processes to
Model Lattice Cellular Networks”. 2013.bZhang and Haenggi, “A Stochastic Geometry Analysis of Inter-cell
Interference Coordination and Intra-cell Diversity”. 2014; Nigam,
Minero, and Haenggi, “Coordinated Multipoint Joint Transmission in
Heterogeneous Networks”. 2014.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 12 / 47
Advanced techniques for downlink BS silencing
BS silencing: neutralize nearby foes
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
The strongest BS (on average) is the serving BS, while the n − 1next-strongest ones are silenced. The model may include shadowing (whichstays constant over time).
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 13 / 47
Advanced techniques for downlink BS silencing
SIR distribution for silencing (ICIC)
With BS silencing (or inter-cell interference coordination, ICIC) of n − 1BSs, the SIR distribution isa
p(!n)s , P
(
power from serving BS
power from BSs beyond the nth> θ
)
= (n − 1)δ
∫ 1
0
(1 − xδ)n−2xδ−1
(C1(θx , 1))n dx ,
where C1(s,m) = 2F1(m,−δ; 1 − δ;−s).
This result does not depend on the shadowing distribution—as long as itsδ-th moment is finite.
aZhang and Haenggi, “A Stochastic Geometry Analysis of Inter-cell
Interference Coordination and Intra-cell Diversity”. 2014.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 14 / 47
Advanced techniques for downlink Intra-cell diversity
Intra-cell diversity from multiple resource blocks
Transmission over M resource blocks
Here, all base stations interfere, but the serving one uses M resource blocks(with independent fading) to serve the user.The success probability is the probability that the SIR in at least one ofthem exceeds θ:
p(∪M)s , P
(
M⋃
m=1
Sm
)
, where Sm = {SIRm > θ}.
For the joint success probability, we have
p(∩M)s , P
(
M⋂
m=1
Sm
)
=1
C1(θ,M)=
1
2F1(M,−δ; 1 − δ;−θ).
p(∪M)s follows from inclusion/exclusion.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 15 / 47
Advanced techniques for downlink Comparison of ICIC and ICD
n-BS silencing (ICIC) vs. transmission over M RBs (ICD)
−20 −15 −10 −5 0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ (dB)
ps(n)
κ = n
κ ≡ 1
✚✚✚✚✚❃
n = 1, 2, 3, 4, 5
n-BS ICIC. α = 4.
−20 −15 −10 −5 0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ (dB)
ps(M
) ✚✚✚✚❃
M = 1
M = 5
M-RB ICD. α = 4.
Observation
ICIC provides a gain in the SIR but no diversity. ICD has a diversity gain ofM. As a result, ICD is superior at small values of θ (θ < −5 dB).
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 16 / 47
Advanced techniques for downlink BS cooperation (CoMP)
Cooperation by joint transmission
BS cooperation: turn nearby foes into friends
−4 −3 −2 −1 0 1 2 3 4−4
−3
−2
−1
0
1
2
3
4
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 17 / 47
Advanced techniques for downlink BS cooperation (CoMP)
SIR distribution with BS cooperation
In the HIP model, let the users receive combined signals from the n
strongest (on average) BSs, denoted by C.
Channels are Rayleigh fading, and BSs use non-coherent jointtransmission.
The amplitude fading coefficients (gx ) are zero-mean unit-variancecomplex Gaussian, and the signal power is
S =∣
∣
∣
∑
x∈C
gx√
Px‖x‖−α/2∣
∣
∣
2.
S is exponentially distributed with mean∑
Px‖x‖−α.
The interference stems from Φ \ C.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 18 / 47
Advanced techniques for downlink BS cooperation (CoMP)
BS cooperation with non-coherent JT
Letu = (u1, . . . , un)
~u = (un/u1, . . . , un/un)
Z (u) = ‖~u‖α/2θ−δ
F (x) =
∞∫
x
r
1 + rαdr
The success probability is independent
α = 4
−20 −15 −10 −5 0 5 100.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ (dB)
P(S
IR>
θ)
Single BSJT from 2 BSsJT from 3 BSs
of power levels and densitiesa
ps(θ) =
∫
0<u1<...<un<∞
exp
(
−un
(
1 + 2F (√
Z (u))
Z (u)
))
du.
aNigam, Minero, and Haenggi, “Coordinated Multipoint Joint
Transmission in Heterogeneous Networks”. 2014.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 19 / 47
ASAPPP Non-Poisson networks
What is possible outside Poissonia?
Ginibre point process (GPP)
For GPP with Rayleigh fadinga: ps(θ) =∫
∞
0
e−v
[
∞∏
j=0
1
j!
∫
∞
v
s je−s
1 + θ(v/s)α/2ds
][
∞∑
i=0
v i
(∫
∞
v
s ie−s
1 + θ(v/s)α/2ds
)−1 ]
dv
aMiyoshi and Shirai, “A Cellular Network Model with Ginibre
Configured Base Stations”. 2014.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 20 / 47
ASAPPP Shape of SIR distributions
Observation on SIR distributions
Shape of SIR distributions
In many cellular papers, we find figures like this:
−15 −10 −5 0 5 10 15 200
0.2
0.4
0.6
0.8
1SIR CCDF
θ (dB)
P(S
IR>
θ)
Scheme 1Scheme 2Scheme 3Scheme 4
It appears that: The curves all have the same shape—they are merelyshifted horizontally!
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 21 / 47
ASAPPP Shape of SIR distributions
Different BS point processes
−10 −5 0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
SINR Threshold (dB)
Cov
erag
e P
roba
bilit
y
Experimental dataPPPPoisson hard−core processStrauss process
Indeed—visually, the curves are shifts of each other. Since the shift (orgain) is due to the deployment, we call it deployment gaina.
aGuo and Haenggi, “Asymptotic Deployment Gain: A Simple
Approach to Characterize the SINR Distribution in General Cellular
Networks”. 2015.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 22 / 47
ASAPPP Horizontal gap (gain)
ASAPPP: Approximate SIR analysis based on the PPP
−15 −10 −5 0 5 10 15 200
0.2
0.4
0.6
0.8
1SIR CCDF
θ (dB)
P(S
IR>
θ)
G1
G2
G3
PPPBetter scheme 1Better scheme 2Better scheme 3
If the SIR ccdfs were indeed just shifted:
ps,PPP(θ) , P(SIRPPP > θ) ⇒ ps(θ) = ps,PPP(θ/G).
G is the SIR shift (in dB) or the SIR gain or gap.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 23 / 47
ASAPPP Horizontal gap (gain)
Horizontal gap and asymptotics
The shift at threshold θ is
G (θ) ,F̄−1
SIR(ps,PPP(θ))
θ,
hence we have ps(θ) = ps,PPP(θ/G (θ)).
The asymptotic gains are
G0 , limθ↓0
G (θ); G∞ , limθ↑∞
G (θ).
So (if G0 and G∞ exist),
ps(θ) ∼ ps,PPP(θ/G0), θ → 0 ; ps(θ) ∼ ps,PPP(θ/G∞), θ → ∞.
Observation: G (θ) ≈ G0 for all θ, i.e., a shift by G0 results in anapproximation that is quite accurate over the entire distribution.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 24 / 47
ASAPPP Examples for gains
Example 1: Deployment gain of square lattice
−20 −10 0 10 20 300
0.2
0.4
0.6
0.8
1SIR CCDFS for α= 3.0
θ (dB)
P(S
IR>θ
)
PPPSquare latticeShift by G
0
−20 −10 0 10 20 300
0.2
0.4
0.6
0.8
1SIR CCDFS for α= 4.0
θ (dB)
P(S
IR>θ
)
PPPSquare latticeShift by G
0
For the square lattice:
G0 = 3.19 dB for α = 3 and G0 = 3.14 dB for α = 4.
So applying a gain of 2 yields an accurate approximation. For α = 4,
psqs (θ) ≈ (1 +
√
θ/2 arctan√
θ/2)−1.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 25 / 47
ASAPPP Examples for gains
Example 2: Gain of joint transmission
−20 −15 −10 −5 0 5 100.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
θ (dB)
P(S
IR>
θ)
JT from 2 BSsSingle BSMISR−based gain
Again the ccdf for the cases without and with cooperation are very similarin shape.
The shift here is G0 = 2/(4 − π) ≈ 2.33.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 26 / 47
ASAPPP MISR
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.
Remarks
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 interferers are located at distances Rk ,
MISR , E(ISR) = E
(
Rα∑
hkR−αk
)
=∑
E
(
R
Rk
)α
.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 27 / 47
ASAPPP MISR
Relevance of the MISR for Rayleigh fading
pout(θ) = P(hR−α < θI ) = P(h < θ ISR)
Since h is exponential, letting θ → 0,
P(h < θ ISR | ISR) ∼ θ ISR ⇒ P(h < θ ISR) ∼ θMISR.
So the asymptotic gain at 0 is the ratio of the two MISRsa:
G0 =MISRPPP
MISR
The MISR for the PPP is easily calculated to be
MISRPPP =2
α− 2=
δ
1 − δ= δ + δ2 + δ3 + . . . .
aHaenggi, “The Mean Interference-to-Signal Ratio and its Key Role
in Cellular and Amorphous Networks”. 2014.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 28 / 47
ASAPPP MISR
ASAPPP
The method of approximating the SIR ccdf by shifting the PPP’s ccdf iscalled ASAPPP—"Approximate SIR analysis based on the PPP".
Can we explain the unreasonable effectiveness of ASAPPP?
Can we calculate G0 and G∞? How close are they?
Can we show that the shape of the SIR distributions are similar bycomparing the asymptotics?
How sensitive are the gains to the path loss exponent and the fadingmodel?
Some of these question are addressed in (very) recent work with RadhaK. Gantia.
aGanti and Haenggi, “Asymptotics and Approximation of the SIR
Distribution in General Cellular Networks”. 2015, arXiv.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 29 / 47
ASAPPP RDP
RDP and MISR
Definition (The relative distance process (RDP))
For a stationary point process Φ with x0 = arg min{x ∈ Φ: ‖x‖}, let
R , {x ∈ Φ \ {x0} : ‖x0‖/‖x‖} ⊂ (0, 1).
MISR using the RDP
We haveISR =
∑
y∈R
hyyα
and
MISR = E
∑
y∈R
yα =
∫ 1
0rαΛ(dr).
For the stationary PPP, Λ(dr) = 2r−3dr .
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 30 / 47
ASAPPP RDP
Pgfl and moment densities of the RDP of the PPP
For the PPP, the probability generating functional (pgfl) of the RDP is
GR[f ] , E
∏
x∈R
f (x) =1
1 + 2∫ 10 (1 − f (x))x−3dx
,
and the moment densities are
ρ(n)(t1, t2, . . . , tn) = n! 2nn∏
i=1
t−3i .
Pgfl for general BS processes
For a general stationary process Φ, the pgfl can be expressed as
GR[f ] = λ
∫
R2
G!o
[
f
( ‖x‖‖ ·+x‖
)
1(·+ x ∈ b(o, ‖x‖)c)]
dx .
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 31 / 47
ASAPPP Generalized MISR
Generalized MISR
We defineMISRn , (E(ISR
n))1/n.
For a Poisson cellular network with arbitrary fading,
E(ISRn) =
n∑
k=1
k!Bn,k
(
δ
1 − δ, . . . ,
δE(hn−k+1)
n− k + 1 − δ
)
,
where Bn,k are the Bell polynomials. A good lower bound on MISRn isobtained by only considering the term n = k in the sum:
MISRn ≥ MISR1(n!)1/n =
δ
1 − δ(n!)1/n
The bound does not depend on the fading. For δ → 1 (α → 2), it isasymptotically tight.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 32 / 47
ASAPPP Generalized MISR
Generalized MISR for PPP with Rayleigh fading
2.5 3 3.5 4 4.5 50
2
4
6
8
10
n=1
n=2
n=5
α
MIS
Rn
ExactLower bound 1Lower bound 2
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 33 / 47
ASAPPP Generalized MISR
Generalized MISR for PPP with Nakagami-m fading
2.5 3 3.5 4 4.5 50
2
4
6
8
10
n=1
n=2
n=5
α
MIS
Rn
n=1
n=2
n=5
n=1
n=2
n=5
m=1m=2m=10
MISRn as a function of α
0 5 10 15 200
2
4
6
8
10
12α=4
n
MIS
Rn
m=1m=2m=10Lower bound
MISRn as a function of n for α = 4
For the PPP, MISRn is essentially proportional to n. For Rayleigh fading,MISRn ∼ (n/e)MISR1, n → ∞.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 34 / 47
ASAPPP Generalized MISR
Gain G0 for general fading
If Fh(x) ∼ cmxm, x → 0,
ps(θ) ∼ 1 − cm E[(θ ISR)m], θ → 0,
and thus
G(m)0 =
(
E(ISRm
PPP)
E(ISRm)
)1/m
=MISRm,PPP
MISRm
.
The ASAPPP approximation follows as
p(m)s (θ) ≈ p
(m)s,PPP(θ/G
(m)0 ).
This applies more generally to any transmission scheme with diversity m.
If MISRm grows roughly in proportion to MISR1, G(m)0 ≈ G0, and G0 is
insensitive to the fading statistics.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 35 / 47
ASAPPP Generalized MISR
Deployment gain for lattice with Nakagami fading
−15 −10 −5 0 5 100.3
0.4
0.5
0.6
0.7
0.8
0.9
1SIR CCDF, α=4, Nakagami fading, m= 2
θ (dB)
P(S
IR>
θ)
PPPsquare latticeISR−based approx
−15 −10 −5 0 5 100.3
0.4
0.5
0.6
0.7
0.8
0.9
1SIR CCDF, α=4, Nakagami fading, m= 5
θ (dB)
P(S
IR>
θ)
PPPsquare latticeISR−based approx
Here the gain for m = 1 (Rayleigh fading) is applied, which is 3 dB. Indeed
G(m)0 ≈ G0 in this case.
How about G∞? Is it close to G0?
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 36 / 47
ASAPPP EFIR
EFIR
Definition (Expected fading-to-interference ratio (EFIR))
Let I∞ ,∑
x∈Φ hx‖x‖−α and let h be a fading random variableindependent of all (hx). The expected fading-to-interference ratio (EFIR) isdefined as
EFIR ,
(
λπE!o
[
(
h
I∞
)δ])1/δ
, δ , 2/α,
where E!o is the expectation w.r.t. the reduced Palm measure of Φ.
EFIR properties
The EFIR does not depend on λ, since E!o(I
−δ∞ ) ∝ 1/λ. It does not depend
on the distribution of the distance to the serving BS, either.
For the PPP with arbitrary fading:
EFIRPPP = (sinc δ)1/δ = (sinc(2/α))α/2 / 1 − δ.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 37 / 47
ASAPPP EFIR
SIR tail and G∞
Theorem (SIR tail)
For all stationary BS point processes Φ, where the typical user is served by
the nearest BS, with arbitrary fading,
ps(θ) ∼(
θ
EFIR
)−δ
, θ → ∞.
Corollary
G∞ =EFIR
EFIRPPP=
(
λπE!o(I
−δ∞ )E(hδ)
sinc δ
)1/δ
.
Implication on tail of SIR distribution
The asymptotic behavior ps(θ) = Θ(θ−δ) is unavoidable for the singularpath loss law and stationary BS deployment.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 38 / 47
ASAPPP Example: Square lattice
Scaled success probability ps(θ)θδ for square lattice
5 10 15 20 25 300.5
1
1.5
2
2.5
θ (dB)
P(S
IR>
θ)*θ
δ
Square lattice with Rayleigh fading for α=4
SimulationEFIR−based AsymptoteAnalytical upper boundAnalytical lower bound
The curve approaches EFIRδ. The EFIR is bounded as
(πΓ(1 + δ))1/δ
Z (2/δ)≤ EFIRsq ≤
( π
sinc δ
)1/δ 1
Z (2/δ),
where Z is the Epstein zeta function. The asymptote is at√
EFIR ≈ 1.19.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 39 / 47
ASAPPP Summary
Summary: MISR and EFIR
For θ → 0 and Rayleigh fading:
ps(θ) ∼ 1 − θMISR ; G0 =MISRPPP
MISR
For θ → ∞ and arbitrary fading:
ps(θ) ∼(
θ
EFIR
)−δ
; G∞ =EFIR
EFIRPPP
The reference MISR and EFIR for the PPP have very simple expressions:
MISRPPP =δ
1 − δ; EFIRPPP = (sinc δ)1/δ
They are efficiently obtained by simulation for arbitrary point processes.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 40 / 47
ASAPPP Comparison of gains
Asymptotic gains for lattices
3 3.2 3.4 3.6 3.8 41
1.5
2
2.5
3
3.5
4
4.5
5
α
Gai
n
G0 and G∞ for square and triangular lattices
G0 square
G0 tri
G∞ square
G∞ tri
G0 barely depends on α, while G∞ slightly increases.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 41 / 47
ASAPPP Comparison of gains
Insensitivity of G0 to α
Recall: MISR =∫ 10 rαλ(r)dr , where λ is the intensity function of the RDP.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Relative density of RDP for square lattice
r
λ(r)
*r3 /2
λsq(r)/λPPP(r)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1Relative density of RDP for triangular lattice
r
λ(r)
*r3 /2
λtri(r)/λPPP(r)
Relative intensity of RDPs of square and triangular lattices.
The straight line corresponds to 1/G0,sq and 1/G0,tri. It is essentially theaverage of the relative densities.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 42 / 47
ASAPPP Comparison of gains
The gains for the β-Ginibre process
0 0.2 0.4 0.6 0.8 11
1.2
1.4
1.6
1.8
2
β
Gai
n
G0 and G∞ for β−GPP for α=4
G0
G0 approx.
G∞G∞ approx.
So quite exactly (and almost independently of α):
G0(β) ≈ 1 + β/2; G∞(β) ≈ 1 + β.
The square lattice has gains of 2 and 3.5, so the 1-GPP falls quite exactlyin between the PPP and the square lattice, both for G0 and G∞.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 43 / 47
ASAPPP Gain trajectories
Gain trajectories G (θ) and asymptotics for lattices
−20 −10 0 10 20 302
2.5
3
3.5
4
4.5
5
5.5
6
θ [dB]
Gap
[dB
]
Horizontal gap for lattices, α=3
squaretriangularG
0, G∞ square
G0, G∞ triang.
−20 −10 0 10 20 302
2.5
3
3.5
4
4.5
5
5.5
6
θ [dB]G
ap [d
B]
Horizontal gap for lattices, α=4
squaretriangularG
0, G∞ square
G0, G∞ triang.
The gap is relatively constant over more than 5 orders of magnitude for θ.
It is not monotonic, but probably G (θ) ≤ max{G0,G∞}.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 44 / 47
Conclusions
Conclusions
The world outside Poissonia is harsh. Even for the PPP, the SIR ccdfsfor advanced transmission techniques (including MIMO) are unwieldy.
To explain the unreasonable effectiveness of the ASAPPP method
ps(θ) ≈ ps,PPP(θ/G0),
we have compared G0 with G∞, which is the gap at θ → ∞.
The asymptotic gains G0 and G∞ are given by the MISR and theEFIR, respectively. The MISR is closely related to the relative distanceprocess and can be generalized for different types of fading.
G0 and G∞ are insensitive to fading, and G0 is insensitive to α.
The ASAPPP method is relatively accurate over the entire range of θand highly accurate for ps(θ) > 3/4 (or θ < 10).
A lot more work can and needs to be done.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 45 / 47
References
References I
B. Blaszczyszyn, M. K. Karray, and H.-P. Keeler, “Using Poisson Processes to ModelLattice Cellular Networks”. In: INFOCOM’13. Turin, Italy, Apr. 2013.
H. S. Dhillon et al., “Modeling and Analysis of K-Tier Downlink Heterogeneous CellularNetworks”. In: IEEE Journal on Selected Areas in Communications 30.3 (Mar. 2012),pp. 550–560.
R. K. Ganti and M. Haenggi, “Asymptotics and Approximation of the SIR Distribution inGeneral Cellular Networks”. ArXiv, http://arxiv.org/abs/1505.02310 . May 2015.
A. Guo and M. Haenggi, “Asymptotic Deployment Gain: A Simple Approach toCharacterize the SINR Distribution in General Cellular Networks”. In: IEEE Transactions
on Communications 63.3 (Mar. 2015), pp. 962–976.
M. Haenggi, “The Mean Interference-to-Signal Ratio and its Key Role in Cellular andAmorphous Networks”. In: IEEE Wireless Communications Letters 3.6 (Dec. 2014),pp. 597–600.
N. Miyoshi and T. Shirai, “A Cellular Network Model with Ginibre Configured BaseStations”. In: Advances in Applied Probability 46.3 (Aug. 2014), pp. 832–845.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 46 / 47
References
References II
G. Nigam, P. Minero, and M. Haenggi, “Coordinated Multipoint Joint Transmission inHeterogeneous Networks”. In: IEEE Transactions on Communications 62.11 (Nov. 2014),pp. 4134–4146.
X. Zhang and M. Haenggi, “A Stochastic Geometry Analysis of Inter-cell InterferenceCoordination and Intra-cell Diversity”. In: IEEE Transactions on Wireless
Communications 13.12 (Dec. 2014), pp. 6655–6669.
M. Haenggi (ND & EPFL) Asymptotics of SIR Distributions May 2015 47 / 47