wireless communications: from simple stochastic geometry
TRANSCRIPT
![Page 1: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/1.jpg)
Wireless communications: from simplestochastic geometry models to practice
III CapacityB. Błaszczyszyn Inria/ENS
Workshop on Probabilistic Methods in TelecommunicationWIAS Berlin, November 14–16, 2016
– p. 1
![Page 2: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/2.jpg)
Outline
COVERAGE
CONNECTIVITY
CAPACITY
– p. 2
![Page 3: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/3.jpg)
CAPACITY
Ability to serve simultaneously many users. How many?Quality of service in function to the number of servedusers.
– p. 3
![Page 4: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/4.jpg)
CAPACITY
Ability to serve simultaneously many users. How many?Quality of service in function to the number of servedusers.
Queueing theory in association with stochastic geometry.
Space-time models. Simulations required for quantitativeresults.
We shall present some model capturing the dependencebetween the traffic demand and the quality of service inlarge cellular networks, validated w.r.t. some real data.
Fruit of long-standing collaboration with M.K. Karrayfrom Orange.
– p. 3
![Page 5: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/5.jpg)
Motivation
– p. 4
![Page 6: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/6.jpg)
Real data
0
2000
4000
6000
8000
10000
12000
0 500 1000 1500 2000 2500 3000 3500 4000
User
thro
ughput
[kbps]
Traffic demand per cell [kbps]
Throughput versus traffic per cell
Cellular network deployed in a big city. 9288 measurements (387stations during 24 hours) of some given day.
– p. 5
![Page 7: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/7.jpg)
Real data
0
2000
4000
6000
8000
10000
12000
0 500 1000 1500 2000 2500 3000 3500 4000
User
thro
ughput
[kbps]
Traffic demand per cell [kbps]
Throughput versus traffic per cell
Cellular network deployed in a big city. 9288 measurements (387stations during 24 hours) of some given day.
Some macroscopic law?
– p. 5
![Page 8: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/8.jpg)
A key QoS metric
Mean user throughput: average “speed” [bits/second] ofdata transfer during a typical data connection.More formally, the ratio of the average number of bitssent (or received) per data request to the averageduration of the data transfer.
– p. 6
![Page 9: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/9.jpg)
A key QoS metric
Mean user throughput: average “speed” [bits/second] ofdata transfer during a typical data connection.More formally, the ratio of the average number of bitssent (or received) per data request to the averageduration of the data transfer.
User-centric QoS metric.
– p. 6
![Page 10: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/10.jpg)
A key QoS metric
Mean user throughput: average “speed” [bits/second] ofdata transfer during a typical data connection.More formally, the ratio of the average number of bitssent (or received) per data request to the averageduration of the data transfer.
User-centric QoS metric.
Network heterogeneous in space and time. Appropriatetemporal and spatial averaging required.
– p. 6
![Page 11: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/11.jpg)
Various levels of averaging
Information theory (over bits processed in time)
Queueing theory (over users/calls served in time)
Stochastic geometry (over geometric patterns of cellsand users)
We are interested in the radio part of the problem.
– p. 7
![Page 12: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/12.jpg)
Outline
HOMOGENEOUS NETWORKS
TECHNOLOGY HETEROGENEOUS NETWORKS;micro/macro cells
SPATIALLY INHOMOGENEOUS NETWORKS; varyingdensity of BS deployment, frequency dimensioningproblem.
– p. 8
![Page 13: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/13.jpg)
HOMOGENEOUS NETWORKS
– p. 9
![Page 14: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/14.jpg)
Queuing theory
for one cell
– p. 10
![Page 15: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/15.jpg)
Little’s law
Consider a service system in its steady state. (Here onenetwork cell during a given hour). DenoteN — mean (stationary, time average) number of users(calls) served at a given time
λ — average number of call arrivals per unit of time [second]
T — average (Palm) call duration
– p. 11
![Page 16: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/16.jpg)
Little’s law
Consider a service system in its steady state. (Here onenetwork cell during a given hour). DenoteN — mean (stationary, time average) number of users(calls) served at a given time
λ — average number of call arrivals per unit of time [second]
T — average (Palm) call durationThe Little’s law:
N = λT
– p. 11
![Page 17: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/17.jpg)
Little’s law
Consider a service system in its steady state. (Here onenetwork cell during a given hour). DenoteN — mean (stationary, time average) number of users(calls) served at a given time
λ — average number of call arrivals per unit of time [second]
T — average (Palm) call durationThe Little’s law:
N = λT
Applies in to a very general system of service, production,
communication... No probabilistic assumptions regarding the
distribution of the arrivals, service times. Not related to a particular
service policy. Just stationarity!
– p. 11
![Page 18: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/18.jpg)
Mean user throughput via Little’s law
Denote:1µ — average data volume [bits] transmitted during one call
ρ := 1µ× λ mean traffic demand [bits/second]
r := 1µ/T mean user throughput [bits/second]
– p. 12
![Page 19: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/19.jpg)
Mean user throughput via Little’s law
Denote:1µ — average data volume [bits] transmitted during one call
ρ := 1µ× λ mean traffic demand [bits/second]
r := 1µ/T mean user throughput [bits/second]
From Little’s law
N = λT ⇒ 1
T=
λ
N⇒ 1
µT=
λ
µN
r =ρ
N=
mean traffic demand
average number of users served at a given time
– p. 12
![Page 20: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/20.jpg)
Mean user throughput via Little’s law
Denote:1µ — average data volume [bits] transmitted during one call
ρ := 1µ× λ mean traffic demand [bits/second]
r := 1µ/T mean user throughput [bits/second]
From Little’s law
N = λT ⇒ 1
T=
λ
N⇒ 1
µT=
λ
µN
r =ρ
N=
mean traffic demand
average number of users served at a given time
But N depends on ρ. What is the relation between N and ρ?
– p. 12
![Page 21: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/21.jpg)
Processor sharing (PS) queue
Poisson process of call arrivals, iid data volume requests.
– p. 13
![Page 22: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/22.jpg)
Processor sharing (PS) queue
Poisson process of call arrivals, iid data volume requests.
Denote: R — total service (transmission) rate [bits/second].
PS: server resources shared equally among all calls; whenn user served simultaneously, one user gets the rate R/n.
– p. 13
![Page 23: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/23.jpg)
Processor sharing (PS) queue
Poisson process of call arrivals, iid data volume requests.
Denote: R — total service (transmission) rate [bits/second].
PS: server resources shared equally among all calls; whenn user served simultaneously, one user gets the rate R/n.
Recall: ρ — traffic demand [bits/second]
– p. 13
![Page 24: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/24.jpg)
Processor sharing (PS) queue
Poisson process of call arrivals, iid data volume requests.
Denote: R — total service (transmission) rate [bits/second].
PS: server resources shared equally among all calls; whenn user served simultaneously, one user gets the rate R/n.
Recall: ρ — traffic demand [bits/second]
θ :=ρ
R— system load
– p. 13
![Page 25: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/25.jpg)
Processor sharing (PS) queue
Poisson process of call arrivals, iid data volume requests.
Denote: R — total service (transmission) rate [bits/second].
PS: server resources shared equally among all calls; whenn user served simultaneously, one user gets the rate R/n.
Recall: ρ — traffic demand [bits/second]
θ :=ρ
R— system load
If θ < 1 the system stable, otherwise unstable.
– p. 13
![Page 26: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/26.jpg)
Processor sharing (PS) queue
Poisson process of call arrivals, iid data volume requests.
Denote: R — total service (transmission) rate [bits/second].
PS: server resources shared equally among all calls; whenn user served simultaneously, one user gets the rate R/n.
Recall: ρ — traffic demand [bits/second]
θ :=ρ
R— system load
If θ < 1 the system stable, otherwise unstable.Number of users in the system has a geometric distributionwith the mean
N =
{
θ1−θ when θ < 1
∞ when θ ≥ 1(∗)
– p. 13
![Page 27: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/27.jpg)
Spatial processor sharing queue
Service (transmission) rate R(y) depends on user location y
(because of the distance to transmitting station, etc...)
Denote: V — the service zone (cell)
– p. 14
![Page 28: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/28.jpg)
Spatial processor sharing queue
Service (transmission) rate R(y) depends on user location y
(because of the distance to transmitting station, etc...)
Denote: V — the service zone (cell)
Equation (∗) applies with R being the harmonic average ofthe local service rates R(y)
R =|V |
∫
V1/R(y) dy
as often in the case of rate averaging...
– p. 14
![Page 29: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/29.jpg)
Spatial processor sharing queue
Service (transmission) rate R(y) depends on user location y
(because of the distance to transmitting station, etc...)
Denote: V — the service zone (cell)
Equation (∗) applies with R being the harmonic average ofthe local service rates R(y)
R =|V |
∫
V1/R(y) dy
as often in the case of rate averaging...
How to model service rates R(y)?
– p. 14
![Page 30: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/30.jpg)
Information theory
for the link quality
– p. 15
![Page 31: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/31.jpg)
Transmissin rate as channel capacity
Transmission rate R(y) is technology dependent(information coding, simple or multipletransmission/reception antenna systems, etc)
– p. 16
![Page 32: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/32.jpg)
Transmissin rate as channel capacity
Transmission rate R(y) is technology dependent(information coding, simple or multipletransmission/reception antenna systems, etc)
We express available R(y) with respect to theinformation-theoretic capacity bounds for appropriatechannel models.
– p. 16
![Page 33: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/33.jpg)
Transmissin rate as channel capacity
Transmission rate R(y) is technology dependent(information coding, simple or multipletransmission/reception antenna systems, etc)
We express available R(y) with respect to theinformation-theoretic capacity bounds for appropriatechannel models.
E.g. the Shannons law for the Gaussian channel saysR(y) = aW log(1 + SNR(y)),
whereW — channel bandwidth [Hertz]SNR — signal-to-noise ratioa (0 < a < 1) — calibration coefficient (real v/s theoretical rate)
– p. 16
![Page 34: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/34.jpg)
Transmissin rate as channel capacity
Transmission rate R(y) is technology dependent(information coding, simple or multipletransmission/reception antenna systems, etc)
We express available R(y) with respect to theinformation-theoretic capacity bounds for appropriatechannel models.
E.g. the Shannons law for the Gaussian channel saysR(y) = aW log(1 + SNR(y)),
whereW — channel bandwidth [Hertz]SNR — signal-to-noise ratioa (0 < a < 1) — calibration coefficient (real v/s theoretical rate)
More specific expressions for MMSE, MMSE-SIC, MIMO, etc...– p. 16
![Page 35: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/35.jpg)
Concluding for one cell
– p. 17
![Page 36: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/36.jpg)
Throughput r v/s traffic demandρ
Putting together previously explained relations for one cell Vr = ρ/N , N = θ/(1 − θ), θ = ρ/R one obtains
r = (ρc − ρ)+,where
ρc := R =|V |
∫
V1/R(SINR(y)) dy
can be interpreted as the critical traffic demand for cell Vand R(SINR(y)) are location dependent user peak servicerates, which depend on the SINR experienced at y.
– p. 18
![Page 37: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/37.jpg)
Throughput r v/s traffic demandρ
Putting together previously explained relations for one cell Vr = ρ/N , N = θ/(1 − θ), θ = ρ/R one obtains
r = (ρc − ρ)+,where
ρc := R =|V |
∫
V1/R(SINR(y)) dy
can be interpreted as the critical traffic demand for cell Vand R(SINR(y)) are location dependent user peak servicerates, which depend on the SINR experienced at y.
Adequate model for the spatial distribution of the SINR incellular networks is required!
– p. 18
![Page 38: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/38.jpg)
r = (ρc − ρ)+ cell by cell?
– p. 19
![Page 39: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/39.jpg)
Stochastic geometry
for a large multi-cell network
– p. 20
![Page 40: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/40.jpg)
Network of interacting cells
Vi — network cells on the plane (i = 1, . . .)
– p. 21
![Page 41: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/41.jpg)
Network of interacting cells
Vi — network cells on the plane (i = 1, . . .)
ρi := ρ × |Vi| — traffic demand to cell i
ρci =
|Vi|∫Vi
1/R(SINRi(y)) dy— critical traffic demand for cell i
θi = ρi/ρci = ρ
∫
Vi1/R(SINRi(y)) dy — load of cell i
Ni = θi/(1 − θi) — mean number of users in cell i
ri = (ρci − ρi)
+ — throughput in cell i.
– p. 21
![Page 42: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/42.jpg)
Network of interacting cells
Vi — network cells on the plane (i = 1, . . .)
ρi := ρ × |Vi| — traffic demand to cell i
ρci =
|Vi|∫Vi
1/R(SINRi(y)) dy— critical traffic demand for cell i
θi = ρi/ρci = ρ
∫
Vi1/R(SINRi(y)) dy — load of cell i
Ni = θi/(1 − θi) — mean number of users in cell i
ri = (ρci − ρi)
+ — throughput in cell i.
However, SINRi(y) depends on the extra-cell interference.Study of such dependent PS-queues is impossible!
– p. 21
![Page 43: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/43.jpg)
Decoupling of cells
Simplifying idea:“decoupling cells in time”, keeping only spatial dependence
– p. 22
![Page 44: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/44.jpg)
Decoupling of cells
Simplifying idea:“decoupling cells in time”, keeping only spatial dependence
Come up with a model in which stochastic processesdescribing the evolution of PS-queues at different cells areconditionally independent, given locations of network BS,which will be assumed (random) point process.
– p. 22
![Page 45: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/45.jpg)
Cell load equations
Assume that the cell loads θi i = i, . . . satisfy the system offixed-point equations
θi = ρ
∫
Vi
1
R
(
P/l(|y−Xi|)N+P
∑
j 6=i
min(θj,1)/l(|y−Xj|)
) dy
where Xi is the location of the BS i, l(·) is the path lossfunction, N external noise, P BS transmit power.
– p. 23
![Page 46: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/46.jpg)
Cell load equations
Assume that the cell loads θi i = i, . . . satisfy the system offixed-point equations
θi = ρ
∫
Vi
1
R
(
P/l(|y−Xi|)N+P
∑
j 6=i
min(θj,1)/l(|y−Xj|)
) dy
where Xi is the location of the BS i, l(·) is the path lossfunction, N external noise, P BS transmit power.
Recall: θi (provided θi < 1) is the probability that the PSqueue of cell i is not idle.
– p. 23
![Page 47: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/47.jpg)
Cell load equations
Assume that the cell loads θi i = i, . . . satisfy the system offixed-point equations
θi = ρ
∫
Vi
1
R
(
P/l(|y−Xi|)N+P
∑
j 6=i
min(θj,1)/l(|y−Xj|)
) dy
where Xi is the location of the BS i, l(·) is the path lossfunction, N external noise, P BS transmit power.
Recall: θi (provided θi < 1) is the probability that the PSqueue of cell i is not idle.
Existence of a solution! We assume uniqueness; partially supported by
[Siomina&Yuan, “Analysis of cell load coupling for LTE network ...” IEEE
TWC 2012].– p. 23
![Page 48: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/48.jpg)
Stable fraction of the network
There is no one global network stability condition.Recall: for a given traffic demand ρ per unit of surface, cell iis stable provided ρi = ρ|Vi| < ρc
i .
– p. 24
![Page 49: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/49.jpg)
Stable fraction of the network
There is no one global network stability condition.Recall: for a given traffic demand ρ per unit of surface, cell iis stable provided ρi = ρ|Vi| < ρc
i .
Denote:S =
⋃
i:ρi<ρciVi — union of all stable cells
πS — fraction of the surface covered by S; equivalently:probability that the typical user is covered by a stable cell.
– p. 24
![Page 50: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/50.jpg)
Mean user throughput in large network
Define the mean user throughput in the network as the ratio
r =average number of bits per data request
average duration of the data transfer in the stable part Sin the stable part S of the network;(“ratio of averages” not the “average of ratios”).
– p. 25
![Page 51: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/51.jpg)
Mean user throughput in large network
Define the mean user throughput in the network as the ratio
r =average number of bits per data request
average duration of the data transfer in the stable part Sin the stable part S of the network;(“ratio of averages” not the “average of ratios”).
We have
r :=ρπS
λBSN0
whereλBS is the density of BS deployment (stationary, ergodic)N0 := 1/n
∑ni:ρi<ρc
iNi is the spatial average of the
(steady-state mean) number of users per stable cell;(typical (stable) network cell interpretation).
– p. 25
![Page 52: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/52.jpg)
“Mean cell” approach
Mean cell load: constant θ̄ satisfying
θ̄ =ρ
λBSE
[
1/R
(
P/l (|X∗|)N + P
∑
Xj 6=X∗ θ̄/l (|y − Z|)
)]
, where
X∗ BS serving the typical user (located at 0).
– p. 26
![Page 53: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/53.jpg)
“Mean cell” approach
Mean cell load: constant θ̄ satisfying
θ̄ =ρ
λBSE
[
1/R
(
P/l (|X∗|)N + P
∑
Xj 6=X∗ θ̄/l (|y − Z|)
)]
, where
X∗ BS serving the typical user (located at 0).
Mean traffic demand (to the typical cell): ρ̄ =ρ
λBS.
– p. 26
![Page 54: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/54.jpg)
“Mean cell” approach
Mean cell load: constant θ̄ satisfying
θ̄ =ρ
λBSE
[
1/R
(
P/l (|X∗|)N + P
∑
Xj 6=X∗ θ̄/l (|y − Z|)
)]
, where
X∗ BS serving the typical user (located at 0).
Mean traffic demand (to the typical cell): ρ̄ =ρ
λBS.
Other “mean cell” characteristics calculated from θ̄ and ρ̄ asin the case of a single (isolated) cell:
N̄ =θ̄
1 − θ̄— mean number of users
r̄ = ρ̄(1/θ̄ − 1) — mean user throughput.
– p. 26
![Page 55: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/55.jpg)
Numerical results
for some real network
(a homogeneous BS deployment region)
– p. 27
![Page 56: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/56.jpg)
Mean cell load and the stable fraction
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200
Cell load
S
table
fra
ction
Traffic demand per cell [kbps]
Typical cell loadStable fraction
Mean cellField measurements
– p. 28
![Page 57: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/57.jpg)
Mean number of users per cell
0
2
4
6
8
10
0 200 400 600 800 1000 1200
Num
ber
of
users
Traffic demand per cell [kbps]
Typical cellMean cell
Field measurements
– p. 29
![Page 58: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/58.jpg)
Mean user throughput
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 200 400 600 800 1000 1200
User
thro
ughput
[kbps]
Traffic demand per cell [kbps]
Typical cellMean cell
Field measurements
– p. 30
![Page 59: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/59.jpg)
Mean user throughput, another example
– p. 31
![Page 60: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/60.jpg)
Conclusions
– p. 32
![Page 61: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/61.jpg)
Conclusions
QoS in large irregular multi-cellular networks usinginformation theory (for link quality), processor sharingqueues (traffic demand and service model, cell by cell),stochastic geometry (to handle a spatially distributednetwork).
The mutual-dependence of the cells (due to theextra-cell interference) is captured via some system ofcell-load equations accounting for the spatial distributionof the SINR.
Identify macroscopic laws regarding networkperformance metrics involving averaging both over timeand the network geometry.
Validated against real field measurement in anoperational network.
– p. 33
![Page 62: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/62.jpg)
What next?
Heterogeneous networks; micro/macro cells.
– p. 34
![Page 63: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/63.jpg)
What next?
Heterogeneous networks; micro/macro cells.
Spatially inhomogeneous networks; varying density ofBS deployment, as observed at the level of a wholecountry; useful for macroscopic network planning anddimensioning.
– p. 34
![Page 64: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/64.jpg)
TECHNOLOGY HETEROGENEOUSNETWORKS
– p. 35
![Page 65: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/65.jpg)
Motivation
– p. 36
![Page 66: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/66.jpg)
Variable BS transmission power
30 35 40 450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Emitted power [dBm] without antenna gain
Cum
ulat
ive
dist
ribut
ion
func
tion
Data from a commercial network in a big European city.
– p. 37
![Page 67: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/67.jpg)
Multi-tier network approach
micro cells macro cells
30 35 40 450
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Emitted power [dBm] without antenna gain
Cum
ulat
ive
dist
ribut
ion
func
tion
– p. 38
![Page 68: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/68.jpg)
Network of interacting cells; recap.
Vi — network cells on the plane (i = 1, . . .), each cellserving its users according to the processor sharing (PS)model,
– p. 39
![Page 69: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/69.jpg)
Network of interacting cells; recap.
Vi — network cells on the plane (i = 1, . . .), each cellserving its users according to the processor sharing (PS)model,
ρi := ρ × |Vi| — traffic demand to cell i
ρci =
|Vi|∫Vi
1/R(SINRi(y)) dy— critical traffic demand for cell i
– p. 39
![Page 70: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/70.jpg)
Network of interacting cells; recap.
Vi — network cells on the plane (i = 1, . . .), each cellserving its users according to the processor sharing (PS)model,
ρi := ρ × |Vi| — traffic demand to cell i
ρci =
|Vi|∫Vi
1/R(SINRi(y)) dy— critical traffic demand for cell i
θi = ρi/ρci = ρ
∫
Vi1/R(SINRi(y)) dy — load of cell i; if
θi < 1, it is equal to the probability that the PS queue of celli is not idle,
– p. 39
![Page 71: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/71.jpg)
Network of interacting cells; recap.
Vi — network cells on the plane (i = 1, . . .), each cellserving its users according to the processor sharing (PS)model,
ρi := ρ × |Vi| — traffic demand to cell i
ρci =
|Vi|∫Vi
1/R(SINRi(y)) dy— critical traffic demand for cell i
θi = ρi/ρci = ρ
∫
Vi1/R(SINRi(y)) dy — load of cell i; if
θi < 1, it is equal to the probability that the PS queue of celli is not idle,
Ni = θi/(1 − θi) — mean number of users in cell i
ri = (ρci − ρi)
+ — throughput in cell i.
– p. 39
![Page 72: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/72.jpg)
Network of interacting cells; recap.
Vi — network cells on the plane (i = 1, . . .), each cellserving its users according to the processor sharing (PS)model,
ρi := ρ × |Vi| — traffic demand to cell i
ρci =
|Vi|∫Vi
1/R(SINRi(y)) dy— critical traffic demand for cell i
θi = ρi/ρci = ρ
∫
Vi1/R(SINRi(y)) dy — load of cell i; if
θi < 1, it is equal to the probability that the PS queue of celli is not idle,
Ni = θi/(1 − θi) — mean number of users in cell i
ri = (ρci − ρi)
+ — throughput in cell i.
Cell loads θi mutually dependent θi = θi({θj : j 6= i}) viathe extra-cell interference (cell load equations).
– p. 39
![Page 73: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/73.jpg)
More specifically...
Base stations (BS) locations modeled by a point processΦ = {Xi} on the plane R2, assumed stationary, simpleand ergodic, with intensity parameter λ := λBS > 0,
– p. 40
![Page 74: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/74.jpg)
More specifically...
Base stations (BS) locations modeled by a point processΦ = {Xi} on the plane R2, assumed stationary, simpleand ergodic, with intensity parameter λ := λBS > 0,
BS Xi transmission power Pi > 0.
– p. 40
![Page 75: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/75.jpg)
More specifically...
Base stations (BS) locations modeled by a point processΦ = {Xi} on the plane R2, assumed stationary, simpleand ergodic, with intensity parameter λ := λBS > 0,
BS Xi transmission power Pi > 0.
Propagation loss: deterministic path-loss functionl : R2 → R+ of the transmitter-receiver vector y − Xi,and random shadowing Si (y − Xi) (random fields{Si (·)} are i.i.d. marks of Φ).
– p. 40
![Page 76: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/76.jpg)
More specifically...
Base stations (BS) locations modeled by a point processΦ = {Xi} on the plane R2, assumed stationary, simpleand ergodic, with intensity parameter λ := λBS > 0,
BS Xi transmission power Pi > 0.
Propagation loss: deterministic path-loss functionl : R2 → R+ of the transmitter-receiver vector y − Xi,and random shadowing Si (y − Xi) (random fields{Si (·)} are i.i.d. marks of Φ).
The power received at location y from BS Xi isPXi
(y) =PiSi(y−Xi)l(y−Xi)
.
– p. 40
![Page 77: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/77.jpg)
More specifically...
Base stations (BS) locations modeled by a point processΦ = {Xi} on the plane R2, assumed stationary, simpleand ergodic, with intensity parameter λ := λBS > 0,
BS Xi transmission power Pi > 0.
Propagation loss: deterministic path-loss functionl : R2 → R+ of the transmitter-receiver vector y − Xi,and random shadowing Si (y − Xi) (random fields{Si (·)} are i.i.d. marks of Φ).
The power received at location y from BS Xi isPXi
(y) =PiSi(y−Xi)l(y−Xi)
.
Cell Vi of BS Xi ∈ Φ is the strongest received signal zoneVi = {y : PXn
(y) ≥ PY (y) for all Y ∈ Φ}.
– p. 40
![Page 78: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/78.jpg)
Cell load equations
Cell loads {θi : i = i, . . .} is the (minimal) solution of thesystem of fixed-point equations
θi = ρ
∫
Vi
1
R
(
PXi(y)
N+∑
j 6=i
min(θj,1)PXj(y)
) dy for all i,
where N is the external noise power.
– p. 41
![Page 79: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/79.jpg)
Multi-tier network, basic facts
Consider J types (tiers) of BS characterized by different(constant) transmitting powers Pj, j = 1, . . . , J , modeled byindependent homogeneous Poisson point processes Φj ofintensity λj.
– p. 42
![Page 80: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/80.jpg)
Multi-tier network, basic facts
Consider J types (tiers) of BS characterized by different(constant) transmitting powers Pj, j = 1, . . . , J , modeled byindependent homogeneous Poisson point processes Φj ofintensity λj.FACT 1: Probability that the typical cell is of type j is equalto λj/λ, where λ =
∑
j λj.
– p. 42
![Page 81: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/81.jpg)
Multi-tier network, basic facts
Consider J types (tiers) of BS characterized by different(constant) transmitting powers Pj, j = 1, . . . , J , modeled byindependent homogeneous Poisson point processes Φj ofintensity λj.FACT 1: Probability that the typical cell is of type j is equalto λj/λ, where λ =
∑
j λj.FACT 2: Probability that the cell covering the typical user isof type j is equal to aj/a, where a =
∑
j aj and
aj :=πE[
S2/β]
K2λjP
2/βj .
– p. 42
![Page 82: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/82.jpg)
Network equivalence
FACT 3: The distribution of the signal powers received bythe typical user in the multi-tier network is the same as in thehomogeneous network with all emitted powers equal to
P =
J∑
j=1
λj
λP
2/βj
β/2
.
– p. 43
![Page 83: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/83.jpg)
Network equivalence
FACT 3: The distribution of the signal powers received bythe typical user in the multi-tier network is the same as in thehomogeneous network with all emitted powers equal to
P =
J∑
j=1
λj
λP
2/βj
β/2
.
Moreover, probability that a given received power is emittedby a station of type j does not depend on the value of thereceived power (and is equal to aj).
– p. 43
![Page 84: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/84.jpg)
Network equivalence
FACT 3: The distribution of the signal powers received bythe typical user in the multi-tier network is the same as in thehomogeneous network with all emitted powers equal to
P =
J∑
j=1
λj
λP
2/βj
β/2
.
Moreover, probability that a given received power is emittedby a station of type j does not depend on the value of thereceived power (and is equal to aj).FACT 4: The mean load of the typical cell of type j is
θ̄j = θ̄λaj
λja= θ̄
P2/βj
P 2/β,
where θ̄ is the load of the typical cell in the equivalenthomogeneous network.
– p. 43
![Page 85: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/85.jpg)
Cell load per cell type
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800 1000 1200
Cel
l loa
d
S
tabl
e fr
actio
n
Traffic demand per cell [kbps]
Mean cell loadMacroMicro
Typical cell loadMacroMicro
Stable fractionMacroMicro
Field measurementsMacroMicro
Real data for micro/macro cells fit the analytical prediction.( Data from a commercial network in a big European city.)
– p. 44
![Page 86: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/86.jpg)
Mean user throughput prediction per cell type
0
1000
2000
3000
4000
5000
6000
7000
0 200 400 600 800 1000 1200
Use
r th
roug
hput
[kbp
s]
Traffic demand per cell [kbps]
Mean cellMacroMicro
Typical cellMacroMicro
Field measurementsMacroMicro
Real data for micro/macro cells (quite) fit the analytical prediction.( Data from a commercial network in a big European city.)
– p. 45
![Page 87: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/87.jpg)
SPATIALLY INHOMOGENEOUSNETWORKS
– p. 46
![Page 88: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/88.jpg)
Motivation
– p. 47
![Page 89: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/89.jpg)
Frequency dimensioning problem
What frequency bandwidth required for a network deployedacross the whole country?
– p. 48
![Page 90: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/90.jpg)
Scaling laws for homogeneous networks
Assume path-loss function l (x) = (K |x|)β , x ∈ R2,where K > 0 and β > 2.
For α > 0 consider a network obtained from the originalone by the following dilation:
base station locations Φ′ = {X′ = αX}X∈Φ
intensity of traffic demand ρ′ = ρ/α2
distance coefficient K′ = K/α
shadowing processes S′i (y) = Si
(
yα
)
while preserving the original powers P ′i = Pi
Consider the cells V ′i of the rescaled network and their
respective characteristics their ρ′i, ρ
′ci , r′i, N
′i , θ
′i.
– p. 49
![Page 91: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/91.jpg)
Scaling laws for homogeneous networks
FACT: For α > 0 consider the homogeneous networkscaling as above. We have V ′
i = αVi. Moreover, theminimal solution of the system of load equations {θi} of theoriginal network is the minimal solution of the scaled one
θ′i = θi for all i.
– p. 50
![Page 92: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/92.jpg)
Scaling laws for homogeneous networks
FACT: For α > 0 consider the homogeneous networkscaling as above. We have V ′
i = αVi. Moreover, theminimal solution of the system of load equations {θi} of theoriginal network is the minimal solution of the scaled one
θ′i = θi for all i.
Consequently ρ′ci = ρc
i , r′i = ri, N ′
i = Ni.
– p. 50
![Page 93: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/93.jpg)
Scaling laws for homogeneous networks
FACT: For α > 0 consider the homogeneous networkscaling as above. We have V ′
i = αVi. Moreover, theminimal solution of the system of load equations {θi} of theoriginal network is the minimal solution of the scaled one
θ′i = θi for all i.
Consequently ρ′ci = ρc
i , r′i = ri, N ′
i = Ni.And thus, the typical cell of the scaled networks has thesame mean characteristics as the typical cell of the originalnetwork. In particular E0
[θ′0] = E0
[θ0].
– p. 50
![Page 94: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/94.jpg)
Networks with homogeneous QoS response
Assume a non-homogeneous network deploymentcovering urban, suburban and rural areas.
– p. 51
![Page 95: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/95.jpg)
Networks with homogeneous QoS response
Assume a non-homogeneous network deploymentcovering urban, suburban and rural areas.
Propagation-losses differ depending on these networkares. Specifically, assume for i ∈ {urban, suburban, rural}
Ki/√λi = const,
where Ki is the path-loss distance factor and λi localdensity of BS.
– p. 51
![Page 96: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/96.jpg)
Networks with homogeneous QoS response
Assume a non-homogeneous network deploymentcovering urban, suburban and rural areas.
Propagation-losses differ depending on these networkares. Specifically, assume for i ∈ {urban, suburban, rural}
Ki/√λi = const,
where Ki is the path-loss distance factor and λi localdensity of BS.
The scaling laws: locally, in urban, suburban and ruralareas, the same relations between the meanperformance metrics and the (per-cell) traffic demand.
– p. 51
![Page 97: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/97.jpg)
Networks with homogeneous QoS response
Assume a non-homogeneous network deploymentcovering urban, suburban and rural areas.
Propagation-losses differ depending on these networkares. Specifically, assume for i ∈ {urban, suburban, rural}
Ki/√λi = const,
where Ki is the path-loss distance factor and λi localdensity of BS.
The scaling laws: locally, in urban, suburban and ruralareas, the same relations between the meanperformance metrics and the (per-cell) traffic demand.
One relation is enough to capture the key dependenciesfor heterogeneous network dimensioning!
– p. 51
![Page 98: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/98.jpg)
Justifying assumptions
Assumption Ki/√λi = const means that the average
distance D between neighbouring base stations is inverselyproportional to the distance coefficient of the path-lossfunction: D × K = const.
– p. 52
![Page 99: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/99.jpg)
Justifying assumptions
Assumption Ki/√λi = const means that the average
distance D between neighbouring base stations is inverselyproportional to the distance coefficient of the path-lossfunction: D × K = const.May be justified by the fact that operators aim to assuresome coverage condition of the form (D × K)β = const.
– p. 52
![Page 100: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/100.jpg)
Justifying assumptions
Assumption Ki/√λi = const means that the average
distance D between neighbouring base stations is inverselyproportional to the distance coefficient of the path-lossfunction: D × K = const.May be justified by the fact that operators aim to assuresome coverage condition of the form (D × K)β = const.
Example: propagation parameters for carrier frequency 1795MHz.
Environment A B K = 10A/B Kurban/K
Urban 133.1 33.8 8667 1
Suburban 102.0 31.8 1612 5
Rural 97.0 31.8 1123 8
Suburban and rural BS distance D should be, respectively, 5 and 8 timeslarger than in the urban scenario. Realistic?
– p. 52
![Page 101: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/101.jpg)
Cell load in different network-density zones
0
0.05
0.1
0.15
0.2
0.25
0 100 200 300 400 500
Cell
load
Traffic demand per cell [kbit/s]
Network 3G, Carrier frequency 2.1GHz
Analytical modelInter-BS distance in [6,8[ km
[4,6[ km[2,4[ km[0,2[ km
Real data in different zones fit the same analytical prediction.(Data from a commercial network of an international operatorin a big European country — a “reference network”)
– p. 53
![Page 102: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/102.jpg)
Cell load with regular network decomposition
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 100 200 300 400 500 600 700 800
Cell
load
Traffic demand per cell [kbit/s]
Network 3G, Carrier frequency 2.1GHz
Analytical modelMesh size=3km
10km30km
100km
The analytical prediction fits the real data regardless of thenetwork decomposition scale. The “reference network”.
– p. 54
![Page 103: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/103.jpg)
Networks in two other countries
0
0.2
0.4
0.6
0.8
1
0 200 400 600 800 1000 1200 1400
Cell
load
Traffic demand per cell [kbit/s]
Network 3G, Carrier frequency 2.1GHz
Analytical modelMeasurements, Country 2
Country 3
The analytical prediction fits the real data. In Country 2 (blue points),
for the traffic > 600 kbit/s per cell, an admission control is applied.
– p. 55
![Page 104: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/104.jpg)
Bandwidth dimensioning
What frequency bandwidth to run cellular network at agiven QoS?
– p. 56
![Page 105: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/105.jpg)
Bandwidth dimensioning
What frequency bandwidth to run cellular network at agiven QoS?
Conjecturing the homogeneous QoS response networkmodel focus on urban zones.
– p. 56
![Page 106: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/106.jpg)
Bandwidth dimensioning
What frequency bandwidth to run cellular network at agiven QoS?
Conjecturing the homogeneous QoS response networkmodel focus on urban zones.
Use the mean-cell model to predict the mean userthroughput for increasing traffic demand given thenetwork density and the frequency bandwidth.
– p. 56
![Page 107: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/107.jpg)
Bandwidth dimensioning
What frequency bandwidth to run cellular network at agiven QoS?
Conjecturing the homogeneous QoS response networkmodel focus on urban zones.
Use the mean-cell model to predict the mean userthroughput for increasing traffic demand given thenetwork density and the frequency bandwidth.
Find the minimal frequency bandwidth for which theprediction of the mean user throughput reaches a giventarget value.
– p. 56
![Page 108: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/108.jpg)
Bandwidth dimensioning solution
0
10000
20000
30000
40000
50000
60000
70000
0 2000 4000 6000 8000 10000 12000 14000
Fre
quency b
andw
idth
[kH
z]
Traffic demand per cell [kbit/s]
Mean user throughput in the network=5 Mbit/s
3G, Carrier freq=2.1GHz, Du=1km4G, Carrier freq=2.6GHz, Du=1km
4G, Carrier freq=800MHz, Du=1.5km4G, Carrier freq=800MHz, Du=1km
Frequency bandwidth required to provide 5 MBit/s of mean userthroughput; given the technology, carrier frequency and network density.
– p. 57
![Page 109: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/109.jpg)
Conclusions
We have presented macroscopic laws regarding networkperformance metrics involving averaging both over timeand the network geometry.
We are able to consider bothlocal network heterogeneity (e.g. micro/macro cells)andspatially inhomogeneity of network deployment(varying density of BS)
This latter extension is useful for macroscopic networkplanning and dimensioning.
– p. 58
![Page 110: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/110.jpg)
More details in
BB., Jovanovic, Karray, M. K. How user throughput depends on the
traffic demand in large cellular networks. In Proc. of
WiOpt/SpaSWiN 2014 (arxiv:1307.8409)
Jovanovic, Karray, BB. QoS and network performance estimation in
heterogeneous cellular networks validated by real-field
measurements. In Proc. of ACM PM2HW2N 2014 (hal-01064472)
BB, Jovanovic, Karray, Performance laws of large heterogeneous
cellular networks In Proc. of WiOpt/SpaSWiN 2015
(arXiv:1411.7785)
BB., Karray, What frequency bandwidth to run cellular network in a
given country? - a downlink dimensioning problem. In Proc. of
WiOpt/SpaSWiN 2015(arxiv:1410.0033)
– p. 59
![Page 111: Wireless communications: from simple stochastic geometry](https://reader031.vdocuments.us/reader031/viewer/2022022112/62121b4e05760e583f621aa9/html5/thumbnails/111.jpg)
Thank you!
– p. 60