Tessellations in Wireless Communication
Networks: Voronoi and Beyond it
OUTLINE
I Introduction to Wireless Communication,
François Baccelli, Bartek Błaszczyszyn
The World a Jigsaw: Tessellations in the Sciences
Lorentz Center, Leiden University, 6–10 March 2006
INRIA-ENS / University of Wrocław
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
OUTLINE
2
OUTLINE
I Introduction to Wireless Communication,
I Introduction to Wireless Communication,
II Basic geometric models: Voronoi, Boolean, ...,
II Basic geometric models: Voronoi, Boolean, ...,
III Signal-to-Interference-and-Noise ratio (SINR) coverage model:
In between Voronoi and Boolean.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
2-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
2-b
OUTLINE
I INTRODUCTION TO WIRELESS COMMUNICATION
I Introduction to Wireless Communication,
II Basic geometric models: Voronoi, Boolean, ...,
• Physical layer,
III Signal-to-Interference-and-Noise ratio (SINR) coverage model:
• Multiple access,
In between Voronoi and Boolean.
• Network layer.
IV Power control in CDMA: Evaluating capacity of some Voronoi architecture.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
2-c
WIRELESS COMMUNICATION...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
3
WIRELESS COMMUNICATION...
Physical layer
Physical layer
• Emitter sends information emitting power Pe .
• Emitter sends information emitting power Pe .
• Transmission distance attenuates the
power; Pr = Pe × l(distance).
emitter used sign
al
receiver
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
emitter used sign
al
receiver
interferen
ce
4
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
interferen
ce
4-a
WIRELESS COMMUNICATION...
WIRELESS COMMUNICATION...
Physical layer
Physical layer
• Emitter sends information emitting power Pe .
• Transmission distance attenuates the
power; Pr = Pe × l(distance).
• Emitted powers cause interference at
• Emitter sends information emitting power Pe .
• Transmission distance attenuates the
power; Pr = Pe × l(distance).
• Emitted powers cause interference at
all receivers;
I=
P
i
all receivers;
Pei l(distancei ).
emitter used sign
al
receiver
I=
interferen
ce
P
i
Pei l(distancei ).
• Reception possible is if the Signal
to Interference (and Noise N ) Ratio
emitter used sign
al
receiver
interferen
ce
(SINR) at the receiver is large enough;
SINR = Pr /(N +I) > threshold T .
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
4-b
WIRELESS COMMUNICATION...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
4-c
WIRELESS COMMUNICATION...
Physical layer
Multiple access
• Emitter sends information emitting power Pe .
• Transmission distance attenuates the
power; Pr = Pe × l(distance).
• Emitted powers cause interference at
In case of simultaneous communications, how to distinguish who is talking to whom?
Lets separate communications. This will also reduce the interference. Three ways of
separation:
all receivers;
I=
P
i
Pei l(distancei ).
• Reception possible is if the Signal
to Interference (and Noise N ) Ratio
emitter used sign
al
receiver
interferen
ce
(SINR) at the receiver is large enough;
SINR = Pr /(N +I) > threshold T .
• T is related to the bitrate (amount of information transmitted per unit time); it
depends on coding used, exist information-theoretic bounds (Shannon theorem).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
4-d
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
5
WIRELESS COMMUNICATION...
WIRELESS COMMUNICATION...
Multiple access
Multiple access
In case of simultaneous communications, how to distinguish who is talking to whom?
In case of simultaneous communications, how to distinguish who is talking to whom?
Lets separate communications. This will also reduce the interference. Three ways of
Lets separate communications. This will also reduce the interference. Three ways of
separation:
separation:
• FDMA: Frequency division multiple access; different channels get different
• FDMA: Frequency division multiple access; different channels get different
sub-frequencies of the given bandwidth (technology of the first generation,
sub-frequencies of the given bandwidth (technology of the first generation,
analog wireless communication).
analog wireless communication).
• TDMA: Time division multiple access; different channels get different time-slots of
a given unit time interval (technology of the present GSM).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
5-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
5-b
WIRELESS COMMUNICATION...
Multiple access
In case of simultaneous communications, how to distinguish who is talking to whom?
Lets separate communications. This will also reduce the interference. Three ways of
CDMA basic principles
• Single channel with a bandwidth of 1.25 MHz; All users
transmit simultaneously on this single channel.
separation:
user i signal
↓ + signature ci
MODULATION
• FDMA: Frequency division multiple access; different channels get different
↓
adding users
EMISSION
sub-frequencies of the given bandwidth (technology of the first generation,
↓
analog wireless communication).
RECEPTION
↓
• TDMA: Time division multiple access; different channels get different time-slots of
+ signature ci
DEMODULATION
a given unit time interval (technology of the present GSM).
↓
user i signal + interference
• CDMA: Code division multiple access; different channels get different
(pseudo)-orthogonal codes to modulate their signals with (technology of the
arriving UMTS).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
5-c
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
6
CDMA basic principles
• Single channel with a bandwidth of 1.25 MHz; All users
transmit simultaneously on this single channel.
• User i uses signature process ci built from a pseudo
CDMA basic principles
• Single channel with a bandwidth of 1.25 MHz; All users
user i signal
↓ + signature ci
MODULATION
↓
random sequence to modulate its signal.
adding users
EMISSION
RECEPTION
+ signature ci
DEMODULATION
↓
user i signal + interference
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
• User i uses signature process ci built from a pseudo
6-a
WIRELESS COMMUNICATION...
user i signal
↓ + signature ci
MODULATION
↓
random sequence to modulate its signal.
• The signature process ci is used by the receiver to mod-
↓
↓
transmit simultaneously on this single channel.
adding users
EMISSION
↓
ulate the total received signal. This gives back the orig-
RECEPTION
i plus some (Gaussian) noise due to
the lack of perfect orthogonality between signatures c i ’s.
DEMODULATION
inal signal of user
This is the interferences.
↓
+ signature ci
↓
user i signal + interference
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
6-b
AD-HOC NETWORKS / Network layer / cellular networks ...
Network layer
Key issues concerning cellular networks:
Cellular networks: Infrastructure of base stations or access points provided by an
• How do the cells really look like?
operator. Individual users talk to these stations and listen to them.
regular
irregular
Voronoi is only a “protocol” model. It takes into account only locations of
antennas and ignores the physical aspect of the communication (emitted powers,
interference, etc.)
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
7
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
8
AD-HOC NETWORKS / Network layer / cellular networks ...
AD-HOC NETWORKS / Network layer / cellular networks ...
Key issues concerning cellular networks:
Key issues concerning cellular networks:
• How do the cells really look like?
• How do the cells really look like?
Voronoi is only a “protocol” model. It takes into account only locations of
Voronoi is only a “protocol” model. It takes into account only locations of
antennas and ignores the physical aspect of the communication (emitted powers,
antennas and ignores the physical aspect of the communication (emitted powers,
interference, etc.)
interference, etc.)
• How many users a given infrastructure can reliably serve?
• How many users a given infrastructure can reliably serve?
• Users send (uplink) and receive (downlink).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
8-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
AD-HOC NETWORKS / Network layer / cellular networks ...
AD-HOC NETWORKS / Network layer / cellular networks ...
Key issues concerning cellular networks:
Key issues concerning cellular networks:
• How do the cells really look like?
8-b
• How do the cells really look like?
Voronoi is only a “protocol” model. It takes into account only locations of
Voronoi is only a “protocol” model. It takes into account only locations of
antennas and ignores the physical aspect of the communication (emitted powers,
antennas and ignores the physical aspect of the communication (emitted powers,
interference, etc.)
interference, etc.)
• How many users a given infrastructure can reliably serve?
• How many users a given infrastructure can reliably serve?
• Users send (uplink) and receive (downlink).
• Users send (uplink) and receive (downlink).
• Channels carry voice (fixed-birtate traffic) and/or data (elastic traffic).
• Channels carry voice (fixed-birtate traffic) and/or data (elastic traffic).
• Users arrive to the network, move and depart.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
8-c
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
8-d
AD-HOC NETWORKS / Network layer / cellular networks ...
WIRELESS COMMUNICATION / Network layer...
Key issues concerning cellular networks:
• How do the cells really look like?
Ad-hoc networks: No fixed infrastructure (no base stations, no access points, etc.)
Voronoi is only a “protocol” model. It takes into account only locations of
antennas and ignores the physical aspect of the communication (emitted powers,
interference, etc.)
• How many users a given infrastructure can reliably serve?
emitter
• Users send (uplink) and receive (downlink).
receiver
• Channels carry voice (fixed-birtate traffic) and/or data (elastic traffic).
• Users arrive to the network, move and depart.
• Evaluate Quality-of-Service characteristics of a “typical user” (e.g. call blocking
probability).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
8-e
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
9
WIRELESS COMMUNICATION / Network layer...
WIRELESS COMMUNICATION / Network layer...
Ad-hoc networks: No fixed infrastructure (no base stations, no access points, etc.)
Ad-hoc networks: No fixed infrastructure (no base stations, no access points, etc.)
• A random set of users distributed in
• A random set of users distributed in
space and sharing a common Hertzian
space and sharing a common Hertzian
medium.
medium.
• Users constitute ad-hoc network that is
emitter
in charge of transmitting information far
receiver
emitter
receiver
away via several hops.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
9-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
9-b
WIRELESS COMMUNICATION / Network layer...
AD-HOC NETWORKS / Network layer / Ad-hoc networks ...
Ad-hoc networks: No fixed infrastructure (no base stations, no access points, etc.)
• Emitter sends a packet in some given direction far away via several hops.
• A random set of users distributed in
space and sharing a common Hertzian
medium.
• Users constitute ad-hoc network that is
in charge of transmitting information far
towards destination
emitter
emitter
receiver
receiver
optimal receiver
away via several hops.
• Users switch between emitter and receiver modes.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
9-c
AD-HOC NETWORKS / Network layer / Ad-hoc networks ...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
10
AD-HOC NETWORKS / Network layer / Ad-hoc networks ...
• Emitter sends a packet in some given di-
• Emitter sends a packet in some given di-
rection far away via several hops.
rection far away via several hops.
• The packet is received by some number
• The packet is received by some number
(possibly 0) of neigbouring receivers.
(possibly 0) of neigbouring receivers.
emitter
receiver
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
towards destination
• An optimal receiver among them is in
emitter
charge of forwarding this packet in (one receiver
of) his next emission time-slots.
optimal receiver
10-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
towards destination
optimal receiver
10-b
AD-HOC NETWORKS / Network layer / Ad-hoc networks ...
AD-HOC NETWORKS / Network layer / ad-hoc networks ...
• Emitter sends a packet in some given direction far away via several hops.
Key issues concerning ad-hoc networks:
• The packet is received by some number
• Connectivity: Can every node be reached? No isolated (groups of) nodes?
(possibly 0) of neigbouring receivers.
• An optimal receiver among them is in
towards destination
emitter
charge of forwarding this packet in (one receiver
of) his next emission time-slots.
optimal receiver
• In the case of no reception, emitter reemits the packet next authorized time.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
10-c
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
AD-HOC NETWORKS / Network layer / ad-hoc networks ...
AD-HOC NETWORKS / Network layer / ad-hoc networks ...
Key issues concerning ad-hoc networks:
Key issues concerning ad-hoc networks:
• Connectivity: Can every node be reached? No isolated (groups of) nodes?
• Connectivity: Can every node be reached? No isolated (groups of) nodes?
• Protocols for routing.
• Protocols for routing.
11
“Protocol” models based on Delaunay graph. (Ignore the physical aspect of the
“Protocol” models based on Delaunay graph. (Ignore the physical aspect of the
communication).
communication).
• Capacity: How much own traffic every node can send, given it has to relay traffic
of other nodes?
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
11-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
11-b
WIRELESS COMMUNICATION / Network layer ...
WIRELESS COMMUNICATION / Network layer ...
Sensor networks: Variants of ad-hoc networks.
Sensor networks: Variants of ad-hoc networks.
• Nodes monitor some space (measuring temperature, detecting intruders, etc.)
• Nodes monitor some space (measuring temperature, detecting intruders, etc.)
• They send collected information in an ad-hoc manner to some “sink” locations.
• They send collected information in an ad-hoc manner to some “sink” locations.
• Issues: Coverage, connectivity, energy (batery) saving.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
12
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
12-a
BASIC MODELS...
Poisson Point Process
Planar Poisson point process (p.p.)
Φ of intensity λ:
• Number of Points Φ(B) of Φ in subset B of the plane is Poisson random variable
with parameter λ|B|, where | · | is the Lebesgue measure on the plane; i.e.,
II BASIC GEOMETRIC MODELS
• Poisson point process,
P{ Φ(B) = k } = e
• Voronoi tessellation and Delaunay graph,
−λ|B|
(λ|B|)k
,
k!
• Boolean model,
• Numbers of points of Φ in disjoint sets are independent.
• Shot-Noise model.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
13
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
14
BASIC MODELS...
BASIC MODELS/Poisson p.p. ...
Poisson Point Process
Planar Poisson point process (p.p.)
Φ of intensity λ:
Poisson p.p. is the basis of the stochastic-geometry modeling of communication
• Number of Points Φ(B) of Φ in subset B of the plane is Poisson random variable
with parameter λ|B|, where | · | is the Lebesgue measure on the plane; i.e.,
P{ Φ(B) = k } = e−λ|B|
k
(λ|B|)
,
k!
networks.
This modeling consist in treating the given architecture of the network as a snapshot
of a (homogeneous) random model, and analyzing it in a statistical way. In this
approach the physical meaning of the network elements is preserved and reflected in
the model, but their geographical locations are no longer fixed but modeled by
random points of, typically, homogeneous planar Poisson point processes.
• Numbers of points of Φ in disjoint sets are independent.
Consequently, any particular detailed pattern of locations is no longer of interest.
Laplace transform of the Poisson p.p.
LΦ (h) = E[e
R
h(x) Φ(dx)
Instead, the method allows for catching the essential spatial characteristics of the
R
−λ (1−eh(x) ),dx
]=e
,
R
P
where h(·) is a real function on the plane and h(x) Φ(dx) =
Xi ∈Φ h(Xi ).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
14-a
BASIC MODELS ...
network performance basically through the densities of these point processes (i.e.,
the densities of the network devices).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
15
BASIC MODELS ...
Voronoi Tessellation (VT) and Delaunay graph
Voronoi Tessellation (VT) and Delaunay graph
= {Xi } on the plane and a given point x, we define
the Voronoi cell of this point Cx = Cx (Φ) as the subset of the plane of all locations
that are closer to x than to any point of Φ; i.e.,
Given a collection of points Φ
= {Xi } on the plane and a given point x, we define
the Voronoi cell of this point Cx = Cx (Φ) as the subset of the plane of all locations
that are closer to x than to any point of Φ; i.e.,
Given a collection of points Φ
Cx (Φ) = {y ∈ R2 : |y − x| ≤ |y − Xi | ∀Xi ∈ Φ} .
Cx (Φ) = {y ∈ R2 : |y − x| ≤ |y − Xi | ∀Xi ∈ Φ} .
= {Xi } is a Poisson p.p. we call
the (random) collection of cells {CXi (Φ)}
When Φ
Borders of Voronoi Cells
Borders of Voronoi Cells
the Poisson-Voronoi tessellation (PVT).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
16
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
16-a
BASIC MODELS ...
BASIC MODELS/VT ...
Voronoi Tessellation (VT) and Delaunay graph
= {Xi } on the plane and a given point x, we define
the Voronoi cell of this point Cx = Cx (Φ) as the subset of the plane of all locations
that are closer to x than to any point of Φ; i.e.,
Given a collection of points Φ
VT is a frequently used generic model of tessellation of the plane.
Points denote locations of various structural elements (devices) of the network (base
Cx (Φ) = {y ∈ R2 : |y − x| ≤ |y − Xi | ∀Xi ∈ Φ} .
station antennas and/or network controllers in cellular networks, concentrators in
fixed telephony, access nodes in ad hoc networks, etc.).
Cells denote mutually disjoint regions of the plane served in some sense by these
= {Xi } is a Poisson p.p. we call
the (random) collection of cells {CXi (Φ)}
When Φ
devices.
Borders of Voronoi Cells
Delaunay graph is a “protocol” model of the neighbourhood.
the Poisson-Voronoi tessellation (PVT).
Edges of the Delaunay graph connect nuclei of the adjacent cells.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
16-b
BASIC MODELS...
BASIC MODELS...
Boolean Model (BM)
Let Φ̃
Boolean Model (BM)
= {(Xi , Gi )} be a marked Poisson p.p., where {Xi } are points and {Gi }
Let Φ̃
are iid random closed stets (grains). We define the Boolean Model (BM) as the union
Ξ=
17
[
Xi ⊕ Gi
where x ⊕ G
= {x + y : y ∈ G}.
= {(Xi , Gi )} be a marked Poisson p.p., where {Xi } are points and {Gi }
are iid random closed stets (grains). We define the Boolean Model (BM) as the union
Ξ=
i
[
Xi ⊕ Gi
where x ⊕ G
= {x + y : y ∈ G}.
i
10
10
8
8
6
6
4
2
0
2
4
6
8
10
Known:
• Poisson distribution of the
10
10
8
8
6
6
4
4
4
2
2
2
0
number of grains intersecting
any given set.
2
4
6
8
10
BM with spherical grains of random radii
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
0
2
4
6
8
10
0
2
4
6
8
BM with spherical grains of random radii
18
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
18-a
10
BASIC MODELS...
BASIC MODELS/BM ...
Boolean Model (BM)
Let Φ̃
= {(Xi , Gi )} be a marked Poisson p.p., where {Xi } are points and {Gi }
are iid random closed stets (grains). We define the Boolean Model (BM) as the union
Ξ=
[
Xi ⊕ Gi
where x ⊕ G
= {x + y : y ∈ G}.
BM is a generic coverage model.
i
Known:
• Poisson distribution of the
number of grains intersecting
any given set.
• Asymptotic results (λ → ∞)
for the probability of complete
Points denote locations of various structural elements (devices) of the network.
10
10
8
8
6
6
4
4
2
2
Granis denote independent regions of the plane served these devices .
In wireless networks it is a simplified model for the study of coverage and
connectivity. It takes into account transmission regions, but it ignores the interference
effect.
covering of a given set.
0
2
4
6
8
10
0
2
4
6
8
10
BM with spherical grains of random radii
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
18-b
BASIC MODELS...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
19
BASIC MODELS...
Shot-Noise (SN) model
Shot-Noise (SN) model
Let Φ̃
= {(Xi , Si )} be a marked p.p., where {Xi } are points and {Si } are iid
random variables. Given a real response function L(·) of the distance on the plane
Let Φ̃
we define the Shot-Noise field
we define the Shot-Noise field
IΦ̃ (y) =
X
Si L(y − Xi ) .
= {(Xi , Si )} be a marked p.p., where {Xi } are points and {Si } are iid
random variables. Given a real response function L(·) of the distance on the plane
IΦ̃ (y) =
i
X
Si L(y − Xi ) .
i
When Φ̃ is a marked Poisson p.p. then we call IΦ̃ the Poisson SN.
For the Poisson SN, the Laplace transform of the vector (I Φ (y1 ), . . . , IΦ (yn )) is
known for any y1 , . . . , yn
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20
∈ R2 (via Laplace transform of the Poisson p.p.).
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F. Baccelli, B. Błaszczyszyn ;
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20-a
BASIC MODELS/SN ...
III SINR COVERAGE MODEL
In between Voronoi and Boolean
Φ = {Xi , (Si , Ti )} marked point process (Poisson)
{Xi } points of the p.p. on R2 — antenna locations,
(Si , Ti ) ∈ (R+ )2 possibly random mark of point Xi — (power,threshold)
½
¾
Si l(y − Xi )
y:
cell attached to point Xi : Ci (Φ, W ) =
≥ Ti
W + κIΦ (y)
P
where Iφ (y) =
i6=0 Si l(y − Xi ) shot noise process, κ interference
factor, W ≥ 0 external noise, l(·) attenuation (response) function.
SN is a good model for interference in wireless networks.
Marks Si correspond to emitted powers.
Response function correspond to attenuation function.
Ci is the region where the SINR from Xi is bigger than the threshold Ti .
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
21
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
22
SINR COVERAGE MODEL ...
III SINR COVERAGE MODEL
In between Voronoi and Boolean
Φ = {Xi , (Si , Ti )} marked point process (Poisson)
• Motivations,
{Xi } points of the p.p. on R2 — antenna locations,
(Si , Ti ) ∈ (R+ )2 possibly random mark of point Xi — (power,threshold)
½
¾
Si l(y − Xi )
y:
cell attached to point Xi : Ci (Φ, W ) =
≥ Ti
W + κIΦ (y)
P
where Iφ (y) =
i6=0 Si l(y − Xi ) shot noise process, κ interference
factor, W ≥ 0 external noise, l(·) attenuation (response) function.
Ci is the region where the SINR from Xi is bigger than the threshold Ti .
Coverage PROCESS:
Ξ(Φ; W ) =
[
Ci (Φ, W ).
i∈N
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
22-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
23
SINR COVERAGE MODEL ...
SINR COVERAGE MODEL ...
• Motivations,
• Motivations,
• Snapshots and qualitative results,
• Snapshots and qualitative results,
• Typical cell study
(coverage probability for the point is some distance to the antenna, simultaneous
coverage of several points, mean area of the cell),
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
23-a
SINR COVERAGE MODEL ...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
23-b
SINR COVERAGE MODEL ...
• Motivations,
• Motivations,
• Snapshots and qualitative results,
• Snapshots and qualitative results,
• Typical cell study
• Typical cell study
(coverage probability for the point is some distance to the antenna, simultaneous
(coverage probability for the point is some distance to the antenna, simultaneous
coverage of several points, mean area of the cell),
coverage of several points, mean area of the cell),
• Handoff study
• Handoff study
(overlapping of cells, coverage probability for a typical point, distance to different
(overlapping of cells, coverage probability for a typical point, distance to different
handoff states),
handoff states),
• Macroeconomic optimization example,
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
23-c
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
23-d
SINR COVERAGE MODEL ...
SINR COVERAGE MODEL ...
• Motivations,
Motivation I: CDMA handoff cells
• Snapshots and qualitative results,
• Typical cell study
x0 — a point in R2 (location of an antenna),
s0 ≥ 0 and t0 ≥ 0 — (pilot signal power of
(coverage probability for the point is some distance to the antenna, simultaneous
the antenna and SINR threshold (bit energy-to-
coverage of several points, mean area of the cell),
noise spectral power density
w
Eb /NO ) for the
pilot signal),
• Handoff study
(overlapping of cells, coverage probability for a typical point, distance to different
handoff states),
• Macroeconomic optimization example,
φ = {xi , (si , ti )} — pattern of antennas,
w ≥ 0 — external noise,
0 ≤ κ ≤ 1 — orthogonality factor,
l(·) — attenuation function
(x 2 , (s2 , t 2))
(x 1 , (s1 , t1 ))
y
(x 3 , (s3 , t3 ))
(x 0 , (s0 , t0 ))
(x 4 , (s4 , t4 ))
s0 l(y − x0 )
≥ t0
w + Iφ (y)
• References.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
23-e
SINR COVERAGE MODEL / CDMA motivation ...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
24
SINR COVERAGE MODEL / CDMA motivation ...
Parameter values
Intensity of Poisson process of base stations
λBS ∼ 0.2 BS/km2 .
Pilot signal power s0
∼ 30 mW
SINR threshold (bit energy-to-noise spectral
power density
−14 dB
Eb /NO ) for the pilot t0
External noise w
∼
This is a relatively simple model, which takes into account only locations of the Base
Stations, their pilot signal powers and SIR’s for the pilots.
∼ −105 dB
In particular there is no any pattern of mobiles assumed yet and it does not take into
account power control issues.
Interference factor for pilots from different BS’s
κ=1
Attenuation function
l(x) = A max(|x|, r0 )−α or
l(x) = (1 + A|x|)−α with α ∼ 3 − 6.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
25
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
26
SINR COVERAGE MODEL / Motivations II
SINR COVERAGE MODEL ...
Snapshots and qualitative results
Aplications to ad-hoc networks
• Gupta & Kummar (2000) studied the capacity of ad-hoc networks under similar
model.
• Percolation in a variant of this model was studied by Douse et al. (2003, 2006) to
20000
20000
20000
20000
18000
18000
18000
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6000
6000
6000
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2000
0
2000
4000
6000
8000 10000 12000 14000 16000 18000 20000
κ = 0.5
address connectivity issues of large ad-hoc networks.
0
2000
4000
6000
8000 10000 12000 14000 16000 18000 20000
0
2000
2000
κ = 0.1
4000
6000
8000 10000 12000 14000 16000 18000 20000
0
κ = 0.01
2000
4000
6000
8000 10000 12000 14000 16000 18000 20000
κ=0
≡ 1, T = 0.4 and
interference factor κ → 0.
Constant emitted powers Si , ei
(BTW, percolation of the classical Boolean model was proposed as a connectivity
model for wireless communication networks by Gilbert back in 1961!)
Small interference factor allows one to approximate SINR cells by a Boolean model
(quantitative results via perturbation methods).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
27
SINR COVERAGE MODEL / Snapshots ...
500
400
300
200
100
0
100
200
300
400
500
20000
18000
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16000
14000
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10000
10000
8000
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6000
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4000
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28
SINR COVERAGE MODEL / Snapshots ...
20000
2000
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
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2000
2000
4000
6000
8000 10000 12000 14000 16000 18000 20000
0
2000
4000
6000
8000 10000 12000 14000 16000 18000 20000
≡ 1, T = 0.4, W = 0, l(r) = (Ar)−β and
attenuation exponent β → ∞.
Constant emitted powers Si , ei
Cells without and with point dependent fading.
Fading reflects variations in time and space of the channel quality about its average
SIR cells tend to Voronoi cells whenever attenuation is stronger, e.g. in urban areas.
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F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
29
state.
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F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
30
SINR COVERAGE MODEL / Snapshots ...
SINR COVERAGE MODEL / Typical cell study ...
Probability for a typical cell to cover a point
Φ — marked Poisson point process representing antennas in R2 ,
(0, (S, T )) — additional antenna located at fixed point 0 with random (S, T )
distributed as any mark of Φ, independent of it (thus Φ ∪ {(0, (S, T )} has
2
Poisson Palm distribution), y — location (of a mobile) in R .
Given:
Probability for C0 to cover a given point y located at the distance R to the origin:
Cells with macrodiversity K
= 1 and the gain of the macrodiversity K = 2.
Macrodiversity K : possibility of being connected simultaneously to K stations and to
combine signals from them.
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31
³
´
pR = P y ∈ C0
³
´
= P S(1/T − 1)l(R) − W − IΦ (y) > 0 .
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32
SINR COVERAGE MODEL / Typical cell study ...
SINR COVERAGE MODEL / Typical cell study ...
Res. For M/G case (general distribution of (S, T )) the coverage probability p R can
Res. For M/G case (general distribution of (S, T )) the coverage probability p R can
be given via Laplace transforms of S(1/T
IΦ (y) that is
− 1), W
and the Laplace transform of
IΦ (y) that is
· Z ³
´
i
1 − LS (ξl(y − z)) µ(dz) ,
E[exp(−ξIΦ (y))] = exp −
Rd
where LS (ξ)
be given via Laplace transforms of S(1/T
= E[e−ξS ] is the Laplace transform of S .
− 1), W
and the Laplace transform of
· Z ³
´
i
1 − LS (ξl(y − z)) µ(dz) ,
E[exp(−ξIΦ (y))] = exp −
Rd
where LS (ξ)
= E[e−ξS ] is the Laplace transform of S .
Cor. Fourier transform of the Poisson shot-noise variable I φ (y) →
Rieman Boundary Problem
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F. Baccelli, B. Błaszczyszyn ;
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33
→ probability of coverage by the typical cell.
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33-a
SINR COVERAGE MODEL / Typical cell study ...
SINR COVERAGE MODEL / Typical cell study ...
Special M/M case
Example
Fourier transform FIΦ (ξ) of the homogeneous Poisson (intensity λ) shot noise with
exponential S (parameter m) and attenuation l(x)
for ξ
−4
= A max(|x|, r 0 )
£
¤
FIΦ (ξ) = E e−iξIΦ
" r
r
µ r
¶
λ
iAξ
m
iAξ
= exp λπ
arctan r02
− π2
m
iAξ
2
m
¸
4
4
r
−
iAξ
−
r
m
0
+ λπr02 0
,
4
iAξ + r0 m
Res. [Baccelli&BB&Muhlethaler (2004)] Assume that {Si } are exponential r.vs. with
µ, Ti = T are constant and denote LW the Laplace transform of W . Then the
probability for C0 to cover a given point located at the distance R is equal to
½
¾
Z ∞
u
pR = exp − 2πλ
du LW (µT /l(R)) .
1 + l(R)/(T l(u))
0
par.
∈ R, where the branch of the complex square root function is chosen with
positive real part.
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F. Baccelli, B. Błaszczyszyn ;
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34
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
35
SINR COVERAGE MODEL / Typical cell study / M/M case ...
SINR COVERAGE MODEL / Typical cell study ...
Special M/M case
Res. [Baccelli&BB&Muhlethaler (2004)] Assume that {Si } are exponential r.vs. with
µ, Ti = T are constant and denote LW the Laplace transform of W . Then the
probability for C0 to cover a given point located at the distance R is equal to
¾
½
Z ∞
u
du LW (µT /l(R)) .
pR = exp − 2πλ
1 + l(R)/(T l(u))
0
par.
Cor. For the attenuation function l(u)
pR (λ) = e−λR
proof: Say the emitter is at the origin and consider the corresp. Palm distribution P;
pR = P(S ≥ T (W + IΦ1 /l(R)))
Z ∞
=
e−µsT /l(R) dP(W + IΦ ≤ s)
= (Au)−β and W = 0
where C
2 T 2/β C
,
³
´
= C(β) = 2πΓ(2/β)Γ(1 − 2/β) /β .
0
= LIΦ (µT /l(R))LW (µT /l(R)) ,
where LIΦ (·) is the Laplace transform of the value of the hom. Poisson SN I Φ .
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F. Baccelli, B. Błaszczyszyn ;
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35-a
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F. Baccelli, B. Błaszczyszyn ;
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36
SINR COVERAGE MODEL / Typical cell study / M/M case ...
SINR COVERAGE MODEL / Typical cell study / M/M case optimization
Some optimizations
One can study the following optimization problems for the expected effective
transmission range r
• given the targeted range R find the density of emitters λ that optimize the spatial
× pr :
density of successful transmission λ × pR :
• given the density of stations λ find the targeted range r that optimizes the
expected effective transmission range
1
R2 T 2/β C
1
max{λpR (λ)} = 2 2/β
λ≥0
R T eC
λmax = λmax (R) =argmaxλ≥0 {λpR (λ)} =
1
√
r≥0
T 1/β 2λpC
1
√
= rmax (p) = argmaxr≥0 {rpr (λ)} =
T 1/β 2λC
ρ = ρ(p) = max{rpr (p)} =
rmax
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37
SINR COVERAGE MODEL / Typical cell study ...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
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SINR COVERAGE MODEL / Typical cell study ...
Probability for a typical cell to cover two points
Coverage probability via perturbation of Boolean model
valid for small interference factor κ
(y1 , y2 ) — two point to be covered by a given cell C0 (Φ, W ) under Palm
distribution of Φ ∪ {(0, (S, T )}
(κ)
(κ)
Denote pR
(κ)
n = P(x ∈ C0 ), where |x| = R ando
= y ∈ R2 : Sl(y) ≥ κIΦ (y) + W .
We need the joint Laplace transform of
C0
(IΦ (y1 ), IΦ (y2 )) that is given by
´¤
h ³
E exp −ξ1 IΦ (y1 ) − ξ2 IΦ (y2 )
· Z ³
´
i
= exp −
1 − LS (ξ1 l(y1 − z) + ξ2 l(y2 − z)) µ(dz) .
Assume F∗ (u)
Rd
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38
39
= P((Sl(x) − W ) ≤ u) admits Taylor approximation at 0:
F∗ (u) = F∗ (0) +
h
(k)
X
F∗ (0)
k=1
and R∗ (u)
k!
uk + R∗ (u)
= o(uh ) u ց 0.
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40
SINR COVERAGE MODEL / Typical cell study / Perturbation of Boolean model ...
SINR COVERAGE MODEL / Typical cell study ...
Mean cell area formula
= E[|C0 |].
Recall that pR is the coverage probability for location at distance R.
Denote the mean area of the cell of the BS located at 0 by v 0
Res.
(κ)
pR =
Res. We have
correcting terms
value for the Boolean model
z³
}|
´{
P Sl(x) ≥ W
provided E[(IΦ (x))
2h
−
z
h
X
k=1
] < ∞.
v0 =
}|
{ error
(k)
£
¤ z }|h{
k
k F∗ (0)
E (IΦ (y)) + o(κ ) ,
κ
k!
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41
SINR COVERAGE MODEL / Typical cell study ...
Z
R2
p|y| dy.
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42
SINR COVERAGE MODEL / Typical cell study ...
Numerical examples
Mean cell area formula
= E[|C0 |].
Recall that pR is the coverage probability for location at distance R.
p(κ) (1)
Denote the mean area of the cell of the BS located at 0 by v 0
p(0.1) (y)
1
v (κ) (1)
1
2
1.8
0.8
0.8
1.6
1.4
Res. We have
v0 =
Z
R2
0.6
0.6
1.2
0.4
0.4
0.8
0.2
0.2
0.4
1
p|y| dy.
0.6
0.2
0
0.2
0.4
0.6
1
0.8
κ
proof:
v0 = E
·Z
R2
¸
1(y ∈ C0 ) dy =
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Z
R2
0 1
1.2
1.4
1.6
1.8
0
2
function of the interference
0.06
0.08
function of the distance.
0.1
Mean cell area as a
function of the interference
coefficient
42-a
0.04
κ
Probability of coverage as a Probability of coverage as a
p|y| dy.
0.02
y
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coefficient.
43
SINR COVERAGE MODEL / Handoff study ...
SINR COVERAGE MODEL / Handoff study ...
Overlapping of cells
Overlapping of cells
Deterministic scenario: given n cells C(xi , si , ti ; φ, w), i = 1, . . . , n
Pn
Res. The inequality
i=1 ti /(1 + ti ) < 1 is a necessary condition for
Tn
i=1 C(xi , si , ti ; φ, w) 6= ∅
Deterministic scenario: given n cells C(xi , si , ti ; φ, w), i = 1, . . . , n
Pn
Res. The inequality
i=1 ti /(1 + ti ) < 1 is a necessary condition for
Tn
i=1 C(xi , si , ti ; φ, w) 6= ∅
Random scenario:
Cor. If the distribution of the ratio T is such that T
≥τ
for some τ
> 0, then the
number Ky of cells of the coverage process Ξ covering any given point y is a.s.
bounded
1+τ
.
τ
(Given point cannot be covered by (1 + τ )/τ or more cells, no matter how close
Ky <
they are located and how their signal is strong — “pole handoff number”.)
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44
SINR COVERAGE MODEL / Handoff study / Overlapping of cells...
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44-a
SINR COVERAGE MODEL / Handoff study ...
Moment expansion of the number of cells Ky covering y
Res. The factorial moment of Ky is given by
(n)
E[Ky
Example: For the maximal pilot’s bit energy-to-noise spectral power density
τ = Eb /NO = −14 dB the pole handoff number (theoretical maximal handoff
number) K ≤ 26.
Z
=
] = E[Ky (Ky − 1) . . . (Ky − n + 1)+ ]
µ
n
n
³
´¶
\
X
P y∈
C xk , S k , Tk ; Φ +
ε(xi ,(Si ,Ti )) , W
(Rd )n
i=1
k=1
i6=k
×µ(dx1 ) . . . µ(dxn ).
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45
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F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
46
SINR COVERAGE MODEL / Handoff study ...
SINR COVERAGE MODEL / Handoff study ...
Contact distribution functions
Little law
Example of contact d.f.’s estimation
p (fraction of the space covered by Ξ) is given by
p=
k=1
k!
6
2
0
0.0
where |C0 | is the area of the typical cell. Moreover, in this case the volume fraction
∞
X
(−1)k+1
4
Density
1.5
1.0
Density
2.0
2.5
= λE[|C0 |] ,
0.5
E[K0 ]
8
In particular, for a homogeneous Poisson point process with intensity λ
0
1
2
3
4
0.0
0.2
0.4
L
0.6
0.8
1.0
R
Histograms of linear L and spherical R
contact d.f. given the point is not covered.
(k)
E[(K0 )
].
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F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
47
SINR COVERAGE MODEL / Handoff study / Contact distribution functions ...
EL
varL
ER
varR
0.423 km
0.191 km2
0.121 km
0.013 km2
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F. Baccelli, B. Błaszczyszyn ;
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48
SINR COVERAGE MODEL / Handoff study / Contact distribution functions ...
Almost exact simulation of the shot-noise
Conditional distribution of the model
Two finite sets of points:
For a given size of observation window (ra-
z1 , . . . , zn and z1′ , . . . , zp′ .
dius R) one selects a larger influence win′
dow (radius R ) in order to get good es-
Condition:
timate of the shot-noise term
points zi are covered by at least ni cells
Iφ
in the
smaller observation window.
and
′
′
points zi are covered by at most ni cells,
′
′
for some given numbers n1 , . . . , nn and n1 , . . . , np .
This type of conditions allows one to consider cases where the exact number of cells
covering a point is specified.
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F. Baccelli, B. Błaszczyszyn ;
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49
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
50
SINR COVERAGE MODEL / Handoff study / Contact distribution functions ...
SINR COVERAGE MODEL / Handoff study / Contact distribution functions ...
Almost exact simulation of the shot-noise
Perfect simulation in the observation window
For a given size of observation window (ra-
One constructs a Markov process (Z̃t ) of patterns of points that has for its
dius R) one selects a larger influence win′
dow (radius R ) in order to get good estimate of the shot-noise term
Iφ
stationary distribution the conditional distribution.
Points are generated at exponential periods and located in the window but only
in the
smaller observation window.
Th. If the attenuation functions is of the form l(x, y)
constants C >
1/(β/2−δ)
< C/|x − y|β for some
0, β > 0 and if the distribution of S has finite moment
E[S
] < ∞ for some δ ∈ [1, β/2], then one can show that for any
R, ε, α > 0³, there exists R′ > 0 such that
´
P
P sup|y|<R |Xi |>R′ Si l(y, Xi ) < ε > 1 − α .
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50-a
SINR COVERAGE MODEL ...
Macroeconomic optimization example:
if their presence does not violate conditions of maximal coverage of the points
zi′ . Points located in the window stay there for exponential times and are
removed, but only if their absence does not violate the conditions of maximal
′
coverage of the points zi . If a particular removal would lead to the violation,
then the point are exponentially perpetuated.
The exact stationary distribution of the Markov process ( Z̃t ) is obtained using
backward simulation (coupling from the past) similar to that proposed by Kendall.
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51
SINR COVERAGE MODEL ...
densification / magnification
Macroeconomic optimization example:
densification / magnification
Increase the mean power m of existing antennas or
Increase the mean power m of existing antennas or
increase the density λ of antennas?
increase the density λ of antennas?
C total budget of an operator per km2 ,
Cλ cost of one antenna,
Cm cost of increasing the power of one antenna by 1W.
λCλ + Cm λm = C.
constraint:
C total budget of an operator per km2 ,
Cλ cost of one antenna,
Cm cost of increasing the power of one antenna by 1W.
λCλ + Cm λm = C.
constraint:
mean handoff level EN0
10
8
Plots of mean handoff as a functions of mean antenna power
under budget constraint with
top: Cm
m
C = 1000, Cλ = 500 and from the
= 1, 2, 5.
Solution: Plot maximum = Optimal configuration.
6
4
2
0
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52
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2
4
6
8
10
12
mean antenna power m
52-a
SINR COVERAGE MODEL ...
IV POWER CONTROL IN CDMA
References
Evaluating capacity of the Voronoi architecture
1. Baccelli & Błaszczyszyn (2001), On a coverage process ranging from the
• Voronoi network architecture,
Boolean model to the Poisson Voronoi tessellation, with applications to wireless
communications Adv. Appl. Probab. 33
2. Tournois (2002), Perfect Simulation of a Stochastic Model for CDMA coverage
INRIA report 4348,
3. Baccelli, Błaszczyszyn & Tournois (2002), Spatial averages of downlink coverage
characteristics in CDMA networks (INFOCOM ).
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
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53
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
IV POWER CONTROL IN CDMA
IV POWER CONTROL IN CDMA
Evaluating capacity of the Voronoi architecture
Evaluating capacity of the Voronoi architecture
• Voronoi network architecture,
• Voronoi network architecture,
• CDMA Power allocation algebra,
• CDMA Power allocation algebra,
54
• Feasibility probabilities and maximal load estimates,
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54-a
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F. Baccelli, B. Błaszczyszyn ;
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54-b
IV POWER CONTROL IN CDMA
IV POWER CONTROL IN CDMA
Evaluating capacity of the Voronoi architecture
Evaluating capacity of the Voronoi architecture
• Voronoi network architecture,
• Voronoi network architecture,
• CDMA Power allocation algebra,
• CDMA Power allocation algebra,
• Feasibility probabilities and maximal load estimates,
• Feasibility probabilities and maximal load estimates,
• Dynamic study: blocking rates via a spatial Erlang formula,
• Dynamic study: blocking rates via a spatial Erlang formula,
• References.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
54-c
POWER CONTROL ...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
POWER CONTROL / Network architectures ...
Network architecture models
Poisson-Voronoi (P-V) model (“too random”)
• BS’s {Yj } located according to Poisson p.p,
Hexagonal (Hex) model (“too regular”)
with intensity λBS .
• BS’s {Yj } located according to hexagonal
• All antenna parameters are i.i.d. marks.
• All antenna parameters are i.i.d. marks.
• All mobiles form independent Poisson p.p.
grid, p.p, with spatial density λBS .
NM
• All mobiles form independent Poisson p.p.
NM
54-d
• Each mobile is served by the nearest BS.
with spatial density λM .
j
(Equivalently: Each BS j serves mobiles NM
• Each mobile is served by the nearest BS.
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with intensity λM .
in its Voronoi cell.)
55
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56
POWER CONTROL ...
POWER CONTROL ...
CDMA Power allocation algebra
NBS = {Yj }j : locations of base-stations (BS’s) in R2 ,
NM = {Xij }i : locations of mobiles served by BS No. j ,
Cij : SINR required for mobile Xij ,
Wij : total non-traffic noise at Xij (from common overhead channels, thermal noise),
κj , γ : orthogonality factors,
CDMA: Interference limited radio channel
(Y2 , S 2 )
(Y5 , S5 )
W
l(x, y): path-loss from y to x.
(Y1 ,S1 )
X
(Y3 ,S3 )
(Y4 ,S4 )
S1 l(X − Y1 )
≥C
W + I(X)
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57
POWER CONTROL ...
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F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
POWER CONTROL / Power allocation algebra...
CDMA Power allocation algebra
NBS = {Yj }j : locations of base-stations (BS’s) in R2 ,
NM = {Xij }i : locations of mobiles served by BS No. j ,
Cij : SINR required for mobile Xij ,
Wij : total non-traffic noise at Xij (from common overhead channels, thermal noise),
κj , γ : orthogonality factors,
Local and global problem
l(x, y): path-loss from y to x.
Power allocation feasible
Power allocation feasible if exist antenna powers 0
≤ Sij < ∞ such that
Sij l(Yj − Xij )
j
X
X
X
≥
C
i
Wij + κj l(Yj , Xij )
Sij′ + γ
l(Yk , Xij )
Sik′
i′ 6=i
|
{z
58
own-cell interference
|
Local power allocation feasible
all
i, j .
for each BS
and
Global power allocation feasible
i′
k6=j
}
m
{z
other-cell interference
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}
58-a
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59
POWER CONTROL / Power allocation algebra...
POWER CONTROL / Power allocation algebra...
Local problem
Local problem
Fix BS j .
Fix total powers emitted on traffic channels by other BS’s:
Sk =
Power allocation is locally feasible in cell j if exist powers 0
Fix BS j .
P
k
i′ Si′ (k
≤ Sij < ∞ s.t.
6= j ).
Sij l(Yj − Xij )
X
X
≥ Cij
j
j P
j
j
k
Wi + κj l(Yj , Xi ) i′ 6=i Si′ + γ
l(Yk , Xi )
Si′
i′
k6=j
|
i
all .
fixed=γ
P
{z
j
k6=j l(Yk ,Xi )Sk
Fix total powers emitted on traffic channels by other BS’s:
Power allocation is locally feasible in cell j if exist powers 0
Res. Local power allocation is feasible iff
κj
X
i
P
j
k6=j l(Yk ,Xi )Sk
Cij
1 + κj Cij
Global problem
Define:
Define:
ajk = γ
i
bj =
l(Yj , Xij )
X HjW j
i
i
i
l(Yj , Xij )
Denote the matrix (ajk )
,
for j
6= k
< 1.
POWER CONTROL / Power allocation algebra...
Suppose for each BS local power allocation is feasible.
i
}
60-a
Suppose for each BS local power allocation is feasible.
i
{z
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Global problem
X H j l(Yk , X j )
i
all .
i′
fixed=γ
(local) pole capacity condition
POWER CONTROL / Power allocation algebra...
Sik′ (k 6= j ).
i′
Sij l(Yj − Xij )
X
X
≥ Cij
j
j P
j
j
k
Wi + κj l(Yj , Xi ) i′ 6=i Si′ + γ
l(Yk , Xi )
Si′
|
60
P
≤ Sij < ∞ s.t.
k6=j
}
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Sk =
and
j
where Hi
ajj =
X
κj Hij ,
ajk = γ
i
=
Cij
1 + κj Cij
X H j l(Yk , X j )
i
i
bj =
.
l(Yj , Xij )
X HjW j
i
i
i
= A, the vector (bj ) = b and (Sj ) = S.
i
l(Yj , Xij )
Denote the matrix (ajk )
,
for j
6= k
and
j
where Hi
ajj =
X
κj Hij ,
i
=
Cij
1 + κj Cij
.
= A, the vector (bj ) = b and (Sj ) = S.
Global power allocation is feasible if exist antenna powers 0
powers emitted on traffic channels) such that
≤ Sj < ∞ (total
S ≥ b + AS
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61
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61-a
POWER CONTROL / Power allocation algebra...
POWER CONTROL / Power allocation algebra...
Res.
Res.
• Global power allocation feasible iff the spectral radius of the matrix A is less
• Global power allocation feasible iff the spectral radius of the matrix A is less
than 1.
than 1.
• The minimal solution S is equal to
X
An b.
n
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62
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POWER CONTROL / Power allocation algebra...
POWER CONTROL / Power allocation algebra...
Res.
Res.
• Global power allocation feasible iff the spectral radius of the matrix A is less
than 1.
62-a
• Global power allocation feasible iff the spectral radius of the matrix A is less
than 1.
• The minimal solution S is equal to
X
An b.
• The minimal solution S is equal to
n
X
An b.
n
• The minimal solution can be obtained as the limit of the iteration
• The minimal solution can be obtained as the limit of the iteration
2
Ab, A2 b, . . .
Ab, A b, . . .
• A sufficient condition for the spectral radius to be less than one is that A is
substochastic (has row-sums less then 1).
Principle
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62-b
⇒ Decentralized Power Allocation
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62-c
POWER CONTROL ...
POWER CONTROL ...
Maximal load estimations of P-V and Hex model
Decentralized Power Allocation Principle (DPAP)
(Wrong) Idea: Given density of BS’s λBS find maximal density of mobiles λM , such
j
Each BS j verifies for the pattern NM of the mobiles it controls if
that power allocation is feasible with probability 1.
total path loss of user i
< 1.
Hij
own-BS path loss of user i
j |
{z
}
Xij ∈NM
X
user’s i weight
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63
POWER CONTROL ...
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64
POWER CONTROL ...
Maximal load estimations of P-V and Hex model
Maximal load estimations of P-V and Hex model
(Wrong) Idea: Given density of BS’s λBS find maximal density of mobiles λM , such
(Wrong) Idea: Given density of BS’s λBS find maximal density of mobiles λM , such
that power allocation is feasible with probability 1.
that power allocation is feasible with probability 1.
Res. Given density of BS’s λBS , for any λM
Res. Given density of BS’s λBS , for any λM
> 0 in both P-V and Hex model, the
> 0 in both P-V and Hex model, the
spectral radius of A is equal ∞ with probability 1, and thus power allocation in not
spectral radius of A is equal ∞ with probability 1, and thus power allocation in not
feasible!
feasible!
Conclusion: A reduction of mobiles (admission control) is necessary for any λ M
> 0.
Calculate blocking probabilities.
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64-a
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64-b
POWER CONTROL ...
POWER CONTROL / Feasibility probabilities ...
Pr.
of
1
DPAP
Feasibility probabilities for DPAP
0.8
¶
µ X
j total path loss of user i
<1 .
P
Hi
own-BS path loss of user i
j
j
X ∈NM
|i
{z
DPAP satisfied
0.6
0.4
}
0.2
It says how often an non-constrained Poisson configuration of users in a given cell
0
10
20
30
40
50
λM
λBS =0.18 BS/km2
C = 0.011797
γ=1
κ = 0.2
α=3
Simulated DPAP failure probability for P-V model (more flat
cannot be entirely accepted by the admission scheme DPAP.
curve) and Hex model (more steep) curve.
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65
POWER CONTROL ...
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66
POWER CONTROL ...
Blocking rates under DPAP — spatial dymamic modeling
Blocking rates under DPAP — spatial dymamic modeling
• Fix one BS, say Y 0 = 0. Denote its cell, considered as a subset of R2 , by C0 .
• Fix one BS, say Y 0 = 0. Denote its cell, considered as a subset of R2 , by C0 .
• Spatial Birth-and-Death (SBD) process of call arrivals to C0 :
⊂ C0 , call inter-arrival times to A are independent
exponential random variables with mean 1/λ(A), where λ(·) is some given
intensity measure of arrivals to C0 unit of time,
– for a given subset A
– call holding times are independent exponential random variables with mean τ .
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67
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67-a
POWER CONTROL / Blocking rates ...
POWER CONTROL ...
Define blocking rate associated with a given location in the cell as the fraction of
Blocking rates under DPAP — spatial dymamic modeling
users arriving according to the SBD process at this location that are rejected.
• Fix one BS, say Y 0 = 0. Denote its cell, considered as a subset of R2 , by C0 .
• Spatial Birth-and-Death (SBD) process of call arrivals to C0 :
⊂ C0 , call inter-arrival times to A are independent
exponential random variables with mean 1/λ(A), where λ(·) is some given
intensity measure of arrivals to C0 unit of time,
– for a given subset A
– call holding times are independent exponential random variables with mean τ .
• Call acceptance/rejection: given some configuration of calls in progress
P
{Xm ∈ C0 }, accept a new call at x if f (x) + m f (Xm ) < 1, where f (·) is
the call weight function defined on C0 , and reject otherwise.
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67-b
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68
POWER CONTROL / Blocking rates ...
POWER CONTROL / Blocking rates ...
Define blocking rate associated with a given location in the cell as the fraction of
Define blocking rate associated with a given location in the cell as the fraction of
users arriving according to the SBD process at this location that are rejected.
users arriving according to the SBD process at this location that are rejected.
Res.
Res.
• The stationary distribution (time-limit) of the (non-constrained) SBD process of
call arrivals is the distribution of a (spatial) Poisson process Π with density λ(·).
• The stationary distribution (time-limit) of the (non-constrained) SBD process of
call arrivals is the distribution of a (spatial) Poisson process Π with density λ(·).
• The stationary distribution of calls accepted is the distribution of a Poisson
P
process truncated to the state space { m f (Xm ) < 1}.
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68-a
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
68-b
POWER CONTROL / Blocking rates ...
POWER CONTROL / Blocking rates ...
Define blocking rate associated with a given location in the cell as the fraction of
Define blocking rate associated with a given location in the cell as the fraction of
users arriving according to the SBD process at this location that are rejected.
users arriving according to the SBD process at this location that are rejected.
Res.
Res.
• The stationary distribution (time-limit) of the (non-constrained) SBD process of
call arrivals is the distribution of a (spatial) Poisson process Π with density λ(·).
• The stationary distribution (time-limit) of the (non-constrained) SBD process of
call arrivals is the distribution of a (spatial) Poisson process Π with density λ(·).
• The stationary distribution of calls accepted is the distribution of a Poisson
P
process truncated to the state space { m f (Xm ) < 1}.
• The stationary distribution of calls accepted is the distribution of a Poisson
P
process truncated to the state space { m f (Xm ) < 1}.
• Blocking rate bx at x ∈ C0 is given by the spatial Erlang formula
P
Π{1 − f (x) ≤ m f (Xm ) < 1}
P
bx =
.
Π{ m f (Xm ) < 1}
• Blocking rate bx at x ∈ C0 is given by the spatial Erlang formula
P
Π{1 − f (x) ≤ m f (Xm ) < 1}
P
bx =
.
Π{ m f (Xm ) < 1}
P
Rem. Note that m f (Xm ) is a compound Poisson r.v., whose distribution can be
effectively approximated by Gaussian distribution.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
68-c
POWER CONTROL / Blocking rates ...
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
68-d
POWER CONTROL ...
Numerical results (assumptions correspond to UMTS)
Blocking rate
0.055
approximation I
approximation II
0.05
References
0.045
0.04
• Baccelli & Blaszczyszyn & Tournois (2003) Downlink admission/congestion
0.035
control and maximal load in CDMA networks, IEEE INFOCOM
0.03
0.025
• Baccelli & Blaszczyszyn & Karray (2004) Up and Downlink
0.02
Admission/Congestion Control and Maximal Load in Large Homogeneous CDMA
0.015
0.01
Networks, Mobile Networks 9(6),
0.005
0
0.1
0.2
0.3
0.4
0.5
0.6
• Baccelli & Blaszczyszyn & Karray (2005) Blocking Rates in Large CDMA
normalized distance
Approximations of the blocking probability as functions of the distance to BS for the
mean number M̄
Networks via a Spatial Erlang Formula, IEEE INFOCOM.
= 27 of users per cell.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
69
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
70
SUMMARY
SUMMARY
• In wireless communication we deal with emitters and their coverage regions. This
• In wireless communication we deal with emitters and their coverage regions. This
regions sometimes look like (“quasi” tessellations). Cellular networks are here
regions sometimes look like (“quasi” tessellations). Cellular networks are here
particular examples.
particular examples.
• Voronoi tessellation is often a good, first approximation of what is going on.
However more sophisticated models (taking into account the physical aspect of
wireless communication) should and can be developed.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
71
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
71-a
SUMMARY
SUMMARY
• In wireless communication we deal with emitters and their coverage regions. This
• In wireless communication we deal with emitters and their coverage regions. This
regions sometimes look like (“quasi” tessellations). Cellular networks are here
regions sometimes look like (“quasi” tessellations). Cellular networks are here
particular examples.
particular examples.
• Voronoi tessellation is often a good, first approximation of what is going on.
• Voronoi tessellation is often a good, first approximation of what is going on.
However more sophisticated models (taking into account the physical aspect of
However more sophisticated models (taking into account the physical aspect of
wireless communication) should and can be developed.
wireless communication) should and can be developed.
• One particular “physical” model of coverage regions in cellular network was
• One particular “physical” model of coverage regions in cellular network was
proposed and analyzed. Depending on the choice of parameters it can resemble
proposed and analyzed. Depending on the choice of parameters it can resemble
the Voronoi or the Boolean model.
the Voronoi or the Boolean model.
• Going more deep into the engineering details of the performance of the CDMA
cellular network (power control aspect) we evaluated capacity of a large such
network under simplified (Voronoi) model of its architecture.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
71-b
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
71-c
What I have not time to talk about:
What I have not time to talk about:
• Many interesting problems related to geometry arise when analyzing so called
• Many interesting problems related to geometry arise when analyzing so called
ad-hoc networks (without a fixed cellular structure), but I do not have time to
ad-hoc networks (without a fixed cellular structure), but I do not have time to
address this subject.
address this subject.
• I do not have time either to present our (more mathematical) work on
Cox-Voronoi tessellations that can addresses the issue of non-homogeneity of
communication networks.
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
72
What I have not time to talk about:
• Many interesting problems related to geometry arise when analyzing so called
ad-hoc networks (without a fixed cellular structure), but I do not have time to
address this subject.
• I do not have time either to present our (more mathematical) work on
Cox-Voronoi tessellations that can addresses the issue of non-homogeneity of
communication networks.
THANK YOU
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
72-b
Tessellations in Wireless Communication Networks: Voronoi and Beyond it
F. Baccelli, B. Błaszczyszyn ;
The World a Jigsaw: TIS, Lorentz Center, Leiden University, March 6–10, 2006
72-a