modeling, analysis, and design of multi-tier and cognitive cellular wireless networks - prof ekram...
DESCRIPTION
Abstract: Multi-tier architecture with small cells such as femtocells, picocells, macrocells, and metrocells, overlaid with traditional macrocells is considered as a promising option to improve the network coverage and capacity of the next generation cellular wireless networks. Also, in such multi-tier networks, cognitive radio concepts will likely to be used by these small cells to improve the radio spectrum utilization. In this context, modeling, analysis, and design of multi-tier and cognitive cellular networks is increasingly attracting the attention of the research community. Recently, stochastic geometry models have been shown to provide tractable yet accurate performance bounds for multi-tier and cognitive cellular wireless networks. Given the need for interference characterization in multi-tier cellular networks, stochastic geometry models provide high potential to simplify their modeling and provide insights into their design. In this seminar, I will present a review of the stochastic geometry models for single-tier as well as multi-tier and cognitive cellular wireless networks. I will also present a taxonomy based on the target network model, the point process used, and the performance evaluation technique. To this end, I will discuss the open research challenges and future research directions. Prof Ekram HossainTRANSCRIPT
Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Modeling, Analysis, and Design of Multi-tier andCognitive Cellular Wireless Networks
Ekram Hossain
Department of Electrical and Computer EngineeringUniversity of Manitoba, Winnipeg, Canada
http://home.cc.umanitoba.ca/∼hossaina
Institut Technology Telcom (IT Telkom)27 August 2013
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
“Wireless Communications, Networks, and ServicesResearch Group” at U. of Manitoba
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
“Wireless Communications, Networks, and ServicesResearch Group” at U. of Manitoba
Current research interests:
Cognitive radio and dynamic spectrum access
Hierarchical cellular wireless networks (small cell networks)
Green cellular radio systems
Applied game theory and network economics
Current group members:
3 PDF, 8 Ph.D. students, 2 M.Sc. students
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Outline
1 Introduction
2 Challenges in Modeling, Analysis, and Design of HetNets
3 Preliminaries: Stochastic Geometry and Poisson Point Process
4 Categorization of Performance Evaluation Techniques
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Introduction
Evolution of the population of wireless devices
Nu
mb
er o
f co
nn
ecte
d d
evic
es
2020
10b
20b
30b
40b
50b
2015 2010
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Introduction
Evolution of the population of wireless devices
Global Mobile Data Traffic Forecast Report presented byCisco predicts 2.4 exabytes mobile data traffic per month forthe year 2013.
M2M communications and IoT (Internet of Things)
Three evolution phases of user population:1 connected consumer electronics phase (smart phones, tablets,
computers, IPTVs)2 connected industry phase (sensor networks, industry and
buildings automation, surveillance, and eHealth applications)3 connected everything phase (IoT phase)
A significant part of this traffic will be carried through cellularnetworks.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Introduction
Multi-tier cellular wireless networks
Improvement of cell coverage, network capacity, and betterquality-of-service (QoS) provisioning are some of the majorchallenges for next generation cellular networks.
Universal frequency reuse and make transmitters and receiverscloser
Hierarchical layering of cells (referred to as HetNets), anefficient solution to improve cell coverage and networkcapacity.
Adopted in the evolving Long Term Evolution(LTE)/LTE-Advanced (LTE-A) systems
3GPP Release-8 (LTE), 3GPP Release 10 onwards(LTE-Advanced)
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Introduction
LTE/LTE-A HetNet
Long-Term Evolution (LTE) and LTE-Advanced systems aredesigned to support high-speed packet-switched services in 4Gcellular wireless networks.
The cells or radio base stations in LTE/LTE-A can beclassified as: i) macrocell base station (referred as MeNB),and ii) small cells (e.g., microcells, picocells, femtocells).
“Small cell” is an umbrella term for low-power radio accessnodes that operate in both licensed and unlicensed spectrumand have a range of 10 meter to several hundred meters.
Small cells will improve the cell coverage and areaspectral-efficiency (capacity per unit area).
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Introduction
LTE/LTE-A HetNet
Macrocell Base Station A (MeNB-A)
MeNB-A UE 1
HeNB-A1 UE 1MeNB-A UE 2
Picocell
PC-A1RN-A1 UE 1
Relay Node
RN-A1
MeNB-A UE 3
PC-A1 UE 1
PC-A1 UE 2
HeNB-A1
HeNB-A2 UE 1HeNB-A2
HeNB-A3 UE 1 HeNB-A3
X2Un
MeNB-B
HeNB-B1 UE 1
MeNB-B UE 2
PC-B1
RN-B1 UE 1
Relay Node
RN-B1
MeNB-B UE 1
PC-B1 UE 1
PC-B1 UE 2
HeNB-B1
HeNB-B2 UE 1
HeNB-B2
X2
Un
MeNB-C
MeNB-C UE 1
HeNB-C3 UE 1
MeNB-C UE 2
MeNB-C UE 3
RN-C1 UE 1
Relay Node
RN-C1
MeNB-C UE 4
HeNB-C3 HeNB-C2 UE 1
HeNB-C2
HeNB-C1 UE 1
HeNB-C1
X2
UnPicocell
PC-C1PC-C1 UE 1
PC-C1 UE 2
LTE Evolved Packet Core
HeNB Gateway
MME / S-GW
HeNB Gateway
X2
X2
X2
S1
S1
S1
S1 S1
S1
Internet
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Introduction
Comparison among different radio base stations inLTE/LTE-A
Attributes MeNB Picocell HeNB Wi-Fi
BS Installation Mobile Operator Mobile Operator Customer Customer
Site Acquisition
Mobile Operator Mobile Operator Customer Customer
Transmission Range
300-2000 m 40-100 m 10-30 m 100-200 m
Transmission Power
40 W (approx.) 200 mW- 2 W 10-100 mW 100-200 mW
Band License Licensed band Licensed band Licensed band Unlicensed band
System Bandwidth
5, 10, 15, 20 MHz (with CA up to 100 MHz)
5, 10, 15, 20 MHz (with CA up to 100 MHz)
5, 10, 15, 20 MHz (with CA up to 100 MHz)
5, 10, 20 MHz
Transmission Rate
up to 1 Gbps up to 300 Mbps 100 Mbps-1 Gbps
up to 600 Mbps
Cost $ 60,000/yr $ 10,000/yr $ 200/yr $ 100-200/yr
Power Consumption
High Moderate Low Low
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Introduction
Motivations for small cells
High data rate and improved quality-of-services to subscribers
Eliminate coverage holes in macrocell footprint
Extended battery life of mobile phones
Macrocell load reduced (hence more resources for macrocellusers)
Mitigate spectrum underutilization problem
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
HetNet characteristics
Introduction of small cells results in a substantial shift in thecellular network architecture with features such as
topological randomnesshigh variability in the specifications of the network elementsunbalanced uplink-downlink associationtraffic offloading and load balancingvery dense deployment
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
Topological randomness
!
5
(a) (b) (c)
Fig. 2: Example of different macrocell only models. Traditional grid networks remain the most popular, but 4G systems havesmaller and more irregular cell sizes, and perhaps are just as well modeled by a totally random BS placement.
to the femtocell user is assumed to be only from the variousmacrocells, which in a fairly sparse femtocell deployment, isprobably accurate. In the uplink as well, the strong interferenceis bound to come from nearby mobiles transmitting at highpower up to the macro base station, so the model may bereasonable. The main limitation of this model vs. Model 1 isthat the performance of downlink macrocell users – who mayexperience strong femtocell interference depending on theirposition – cannot be accurately characterized.
The third model, which appears to be the most recent, isto allow both the macrocells and femtocells to be randomlyplaced. This is the approach of three papers in this specialissue [61]–[63], and to the best of our knowledge, theseare the first full-length works to propose such an approach(earlier versions being [64], [65]. Both of these papers arefor the downlink only and an extension to the uplink wouldbe desirable. An appealing aspect of this approach is that therandomness actually allows significantly improved tractabilityand the SINR distribution can be found explicitly. This mayallow the fundamental impact of different PHY and MACdesigns to be evaluated theoretically in the future.
A fourth model is simply to keep all the channel gains(including interfering channels) and possibly even the variousper-user capacities general, without specifying the precisespatial model for the various base stations, e.g. [66], [67]. Thiscan be used in many higher-level formulations, e.g. for gametheory [59], power control, and resource allocation, althoughultimately some distribution of these channel gains must beassumed in order to do any simulation, and the gains areto a first order determined by the locations of the varioustransmitting sources. So ultimately, this fourth model typicallywill conform to one of the above three models.
V. OVERVIEW OF KEY CHALLENGES
Building on the models developed in last section, as well asthe preceding discussions on standards and historical trends,
in this section we turn our attention to some of the newchallenges that arise in femtocell deployments. To motivatefuture research and an appreciation for the disruptive potentialof femtocells, we now overview the broader challenges of fem-tocells, focusing on both technical and economic/regulatoryissues.
A. Technical Challenges
1) Interference Coordination: Perhaps the most significantand widely-discussed challenge for femtocell deployments isthe possibility of stronger, less predictable, and more variedinterference, as shown in Fig 3. This occurs predominantlywhen femtocells are deployed in the same spectrum as thelegacy (outdoor) wireless network, but can also occur evenwhen femtocells are in a different but adjacent frequency banddue to out-of-band radiation, particularly in dense deploy-ments. As discussed in the previous section, the introductionof femtocells fundamentally alters the cellular topology bycreating an underlay of small cells, with largely randomplacements and possible restrictions on access to certain BSs.Precise characterizations of the interference conditions in suchheterogeneous and multi-tier networks has been the subject ofextensive study [68], [69]. One of the important and perhapssurprising results shown in [61] is that in principle, with open-access and strongest cell selection, heterogeneous, multi-tierdeployments do not worsen the overall interference conditionsor even change the SINR statistics. This “invariance prop-erty” has also been observed in real-world systems by NokiaSiemens [70] and Qualcomm [71], and provides optimism thatfemtocell deployments need not compromise the integrity ofthe existing macrocell network.
However, in practice, at least two aspects of femtocellnetworks can degrade the interference significantly. First,under closed access, unregistered mobiles cannot connect toa femtocell even if they are close by. As noted in Section
J. G. Andrews, H. Claussen, M. Dohler, S. Rangan, and M. C. Reed, “Femtocells: Past, present, and future,” IEEEJournal on Selected Areas in Communications, Special Issue on “Femtocell Networks”, April 2012.
!
Modeling a Heterogeneous Cellular Network (HCN)
20
25
10
15
T diti l id d l0 5 10 15 20 250
5
A t l 4G ll t dTraditional grid model Completely random BSsActual 4G macrocells today
cells
Zoo
m
w/ f
emto
c
m w
/ pico
c
Zo
om
cells
too
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
Effect of network geometry
SINR is one of the main performance metrics in wirelesscommunications:
SINR(y) =Pt(x0)Ah0 ‖x0 − y‖−η
N +∑
xi∈ΨI
Pt(xi )Ahi ‖xi − y‖−η
Network geometry along with propagation environment affectsSINR.
SINR impacts network performance metrics such as
outage probability, Pout = P(SINR < β)coverage probability, Pc = 1− Pout
bandwidth normalized average rate, E[ln(1 + SINR)]network capacity (or throughput), C = λ(1− Pout), subject toPout < ε, λ = no. of active links per unit area
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
User association
User association, spectrum access methods, etc. affectnetwork geometry (and hence SINR) and performance ofresource allocation methodsIn a single-tier network with all BSs having the same transmitpower, a user associates to the nearest BS (for which theaverage RSS is also the highest in the downlink).
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
User association
Different BSs having different transmit powers.With the strongest RSS or SINR-based association, the BSmay not necessarily be the closest one.Distance to the BS depends on relative transmit powers andpropagation conditions.Example: In first fig., r is larger than rs , butr × (Ps/Pm)1/η < rs .
rs > r (Ps/Pm)1/ƞ
r
rm>r
r
rs>r
rm > r (Pm/Ps)1/ƞ
Highest RSS Connectivity
Macro-cell User Scenario Small-cell User Scenario
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
Unbalanced uplink-downlink association
In downlink, a user may associate with a macro BS, while inthe uplink, it may associate with a small cell BS.
Downlink
Uplink
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
Traffic offloading and load balancing
Biasing can be used in multi-tier cellular networks to offload usersfrom one network tier to another tier.
Biasing is known as range extension in the 3GPP standard.
Instead of associating to the network entity offering the highestsignal power, a user associates to a small cell if
PsTr−ηs > Pmr
−ηm , where T ≥ 1.
i.e., if rm >(
Pm
PsT
) 1η
rs .
Without biasing, rm >(
Pm
Ps
) 1η
rs , that is, biasing will decrease the
minimum distance between a small cell user and interfering MBSs.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
Multi-tier cognitive cellular network
Each network element performs spectrum sensing to accessthe spectrum.
Cognitive spectrum access affects the locations and density ofinterferers.
re
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
Challenges in HetNet modeling, analysis, and design
Traditional grid-based model fails to capture the basic HetNetcharacteristics.
New modeling/design paradigms are required.
Need design methods that account for the topological randomness
Consider universal frequency reuse (which is essential for highspectral efficiency).
Network functionalities and their optimization techniques have to berevisited and adapted to the HetNet characteristics.
Centralized control for HetNets is infeasible.
Innovative distributed network management is required.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Challenges in Modeling, Analysis, and Design of HetNets
Stochastic geometry for modeling HetNets
Stochastic geometry is a powerful tool used to study and analyzenetworks with random topologies.
Stochastic geometry has been successfully adapted to model ad hocwireless networks from more than three decades.
Stochastic geometry has recently been used to model and analyzesingle-tier cellular networks and HetNets.
Stochastic point process is used to abstract the network model.
Stochastic geometry analysis provides statistical and spatialaverages for the performance metrics.
Stochastic geometry analysis
distribution
Physical layer characteristics
MAC layer behavior
Distribution of simultaneous active nodes
Network Capacity
Outage Probability
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Point process
A stochastic point process is a type of random process for whichany one realization consists of a set of isolated points either intime or geographical space, or in even more general spaces.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Applications of point processes
Modeling and analysis of spatial/temporal data
Diverse disciplines: forestry, plant ecology, epidemiology,geography, seismology, materials science, astronomy,economics
Frequently used as models for random events in time, e.g.,arrival of customers in a queue (queueing theory), impulses ina neuron (computational neuroscience), particles in a Geigercounter, location of users in a wireless/mobile network,searches on the web
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Formulation of a point process
by observing the arrival or inter-arrival timeby counting the number of pointsby counting the number of points within a specific interval orregion
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Constructing a new point process
By transforming or changing an existing point processTransformation operations include:
mapping (scaling, translation, rotation, projection, etc. )superpositionclusteringthinningmarking
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Major point processes
Stochastic point process is used to abstract the networktopology.
Four main point processes used in the literature for modelingwireless networks:
Poisson point process (PPP)binomial point process (BPP)hard core point process (HCPP)Poisson cluster process (PCP)
PPP is the simplest and most widely used point process.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
PPP, HCPP, and PCP
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Poison Point Process (PPP)
PPP provides tractable results that help understanding therelationship between the performance metrics and the designparameters.
PPP can model random network with randomized channel access.
Provides tight bound for networks with planned deployment andnetworks with coordinated spectrum access
Most of the available literature assume that the nodes aredistributed according to a PPP.
Results obtained using PPP are accurate (within 1-2 dB) with thoseobtained (by measurements) for legacy cellular networks as well asHetNets.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
PPP
Let Ψ = {xi ; i = 1, 2, 3, ...} be a point process in Rd withintensity λd , then Ψ is a PPP iff
for any compact set A ⊂ Rd , the number of points in A is aPoisson random variablenumbers of points existing within disjoint sets are independent.
Number of points inside any bounded region A ∈ Rd is givenby
P{N(A) = k
}=
(λd A
)ke−λd A
k!
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
PPP
Slivnyak’s theorem: the statistics seen from a PPP is independentof the test location.
Campbell’s theorem (valid for a general point process): Letf : Rd → [ 0,∞) be a function over a PP Ψ and Λ(B) is theintensity of the PP. Then the average of the sum of the function
E
[∑xi∈Ψ
f (xi )
]=
∫Rd
f (x)Λ(dx)
i.e., when the transmitting nodes form a point process Ψ and f (x)represents path-loss, the average interference seen at the origin.
Example: For a PPP with density λ, E
[ ∑xi∈Ψ
f (xi )
]= λ
∫Rd f (x)dx ,
and when f (x) = ||x ||−η (i.e., singular path-loss model), for
d = 2, E(I) = E
[ ∑xi∈Ψ
f (xi )
]= 2πλ
∫∞0
r−ηrdr = 2πλ[r2−η
2−η
]∞0
=
∞. 30/48
Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
PPP
E[I] =∞ for a PPP with a singular path-loss model.
A consequence of the path-loss law and the property of the PPPthat nodes can be arbitrarily close
Probability generating functional (PGFL): the average of aproduct of a function over the point process
PGFL for PPP:
E
[∏xi∈Ψ
f (xi )
]= exp
{−∫Rd
(1− f (x)) Λ(dx)
}.
A PGFL is very useful to evaluate the Laplace transform of the sum∑x∈Ψ f (x).
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Poisson field of interferers and aggregate interference
Laplace transform of the pdf of interference for a PPPnetwork:
LI(s) =e−λπ
((Ps)
2η E[h
2η ΓL(1− 2
η ,Pshr−ηe )
]−r2
e E[
1−e−Pshr−ηe
])
where h can follow any distribution.
For Rayleigh fading,
LI(t) =e−πλ
((Ps)
2η Eh
[h
2η ΓL(1− 2
η ,sPhr−ηe )
]− Psr2
ePs+µr
ηe
).
For η = 4,
LI(s) =e−πλ
√Psµ arctan
(√Psµ
r2e
).
In general, the Laplace transform cannot be inverted toobtain the pdf of the interference.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Preliminaries: Stochastic Geometry and Poisson Point Process
Modeling steps
1 Abstract the network into a convenient point process.
2 Identify the network geometry w.r.t. the test receiver basedon the network characteristics.
3 Identify the point process for the interference sources andderive its parameters.
4 Derive the Laplace transform (LT), moment generatingfunction (MGF) or the characteristic function (CF) of the pdfaggregate interference.
Note that MGF of I(t) = LI(−s) and CF of I(t) =LI(−jω), where j =
√−1 and ω is a real-valued parameter.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Five techniques for performance evaluation
There are FIVE main techniques to overcome the problem dueto the non-existence of pdf of the aggregate interference.
1 Assume Rayleigh fading and obtain the exact SINR statistics2 Obtain tight bounds3 Generate moments/cumulants and approximate the pdf4 Plancherel-Parseval theorem5 Inversion
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #1: Rayleigh fading assumption
With Rayleigh fading on the useful link, forinterference-limited networks, the cdf of SINR (i.e., outageprobability) for a receiver at a distance r from its transmitteris evaluated as FSINR(θ) = 1− LI(s)|s=cθ
FSINR(θ) = P {SINR ≤ θ}
= P{
PtAh0r−η
N + I≤ θ
}
= 1− P{h0 >
(N + I)θrη
PtA
}
= 1− E[
exp
(−
(N + I)µθrη
PtA
)]
= 1− exp
(−
Nµθrη
PtA
)E[
exp
(−Iµθrη
PtA
)]
= 1− exp (−Ncθ) LI (s)|s=cθ , where c =µrη
PtA. (1)
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #1: Rayleigh fading assumption
Coverage probability, Pc = 1− FSINR(θ)= exp (−Ncθ) LI(s)|s=cθ.
With Rayleigh fading, the coverage probability in aninterference-limited network is
Pc =
∫rfR(r)LI(cθ)dr . (2)
The average transmission rate is
E[ln (1 + SINR)] =
∫ ∞0
P {ln (1 + SINR) > t} dt
=
∫ ∞0
P{
SINR >(et − 1
)}dt
=
∫ ∞0
e−Nc(et−1)LI(c(et − 1
))dt. (3)
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #2: Dominant interferers by region bounds ornearest n interferers
Tight lower bound on the cdf of SINRcan be obtained by looking at thevulnerability region.
Laplace transform of theinterference distribution is notrequired.
For deterministic channel gains, thevulnerability radius is given by
rv = β1η r .
Vulnerability Circle Bx(rv)
rv
r
x
Tight lower bound on the cdf of SINR can also be obtained byconsidering the strongest n interferers.
Upper bound can be obtained by Markov inequality, Chebyshev’sinequality, Jensen’s inequality, or Chernoff bound.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #2: Dominant interferers by region bounds
For example, in CSMA networks, transmitters contend toaccess the spectrum.
Contention-based access creates protection regions for thereceivers.
Protection regions are centred around transmitters.
Vulnerability region is crescent shaped.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #2: Dominant interferers by region bounds
Divide the aggregate interference I into interferences from twodisjoint regions I1 and I2.
According to the PPP, I1 and I2 are independent, and the outageprobability is obtained as
P{S
I< θ
}= P
{I >
S
θ
}= P
{I1 + I2 >
S
θ
}= P
{I1 >
S
θ
}+ P
{I2 >
S
θ
}+ P
{I1 + I2 >
S
θ|I1 <
S
θ, I2 <
S
θ
}︸ ︷︷ ︸
=0
= P{I1 >
S
θ
}+ P
{I2 >
S
θ
}> P
{I1 >
S
θ
}= P {Ψ ∩ Bx (rv ) 6= φ}
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #3: Approximation of the pdf of interference
We can resort to approximate the pdf of the aggregateinterference using the moments generated from LT, MGF orCF
There is no fixed criterion how to choose the approximate pdffor the interference.
Accuracy can only be validated via simulations.
The aggregate interference has be approximated by using theGaussian distribution, complex Gaussian distribution,truncated alpha-stable distribution, and log-normaldistribution.
Moments or cumulants are generated using the LT as follows:
E[Inagg] = (−1)ndn
dsnLIagg(s)
∣∣∣∣s=0
(4)
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #4: Plancherel-Parseval theorem (Fourierintegration technique)
Plancherel-Parseval theorem states that if f1(t) and f2(t) aresquare integrable complex functions, then
∫Rf1(t)f ∗2 (t)dt =
∫RF1(ω)F∗2 (ω)dω
The Fourier transform of a function is equivalent to the CF ofthat function, which can be obtained from its Laplacetransform.
Plancherel-Parseval theorem precludes the need of invertingthe Laplace transform of pdf of interference (i.e., Laplacetransform itself can be used for performance evaluation).
Results for general fading environment can be obtained.
The integrals are quite involved due to the complex nature ofthe characteristic functions.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #4: Example
Suppose we want to calculate the coverage probability
P{S
I> θ
}= P
{I < S
θ
}=
∫y
1{y< Sθ}fI(y)dy
The indicator function has the Fourier transform
1{y<θ}FT
=⇒ 1− e−2πiωθ
2πiω, and (5)
fI(y)FT
=⇒ F(ω)
Fourier transform can be directly obtained from the Laplacetransform, i.e., FI(ω) = LI(−2πiω)
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #4: Example
Then using Plancherel-Parseval theorem, we have
P{S
I< θ
}=
∫y
1{y< Sθ}fI(y)dy
=
∫ω
LI(−2πiω)1− e−
2πiωSθ
2πi Sθdω
If S is random, then the unconditional coverage probability isobtained as
P{S
I< θ
}=
∫s
fS(s)
∫ω
LI(−2πiω)1− e−
2πiωsθ
2πi sθdωds
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Technique #5: Inversion
The LT, CF, or MGF is inverted to obtain the pdf of theinterference.
Due to the complex nature of the expressions for LT, CF, orMGF, generally we are unable to find the pdf in closed form.
This technique is only useful for very special cases of the PPP,where the expressions for LT, CF, or MGF are invertible ormatch the LT, CF, or MGF of a known distribution.
Otherwise, inversion is done numerically.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Summary
Stochastic geometry modeling provides tractable yet accurateexpressions for several important performance metrics in terms ofthe design parameters.
Generally, the interference cannot be characterized via its pdf or cdf.
However, the LT (or CF, or MGF) of the pdf of interference can beobtained for any fading scenarios.
The cdf of SIR or the lower/upper bounds on the cdf of SIR can beobtained for any fading scenarios (in both useful and interferencelinks).
Technique #1 and technique #2 are the most popular performanceevaluation techniques due to their simplicity and tractability.
Technique #4 provides a potential to obtain exact general resultsvia stochastic geometry modeling, but at the expense of reducedtractability.
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Summary
Stochastic geometry is an elegant mathematical technique to modelcellular networks.
Under certain assumptions stochastic geometry gives simpleclosed-form expressions for the performance metrics.
PPP gives accurate lower bound for the coverage probability andachievable data rate.
The baseline models can be extended and capture more realisticcellular network characteristics.
The different techniques in the literature can be exploited forrelaxing some of the simplifying assumptions (e.g., Rayleigh fadingassumption).
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Future research directions
More accurate/general point processes
New performance metrics
More practical system model
cognitioncooperative multipoint transmission (COMP)MIMOmobilitymultiple channelsdifferent channel allocation strategies,power control
Uplink modeling
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Modeling, Analysis, and Design of Multi-tier and Cognitive Cellular Wireless Networks
Categorization of Performance Evaluation Techniques
Thank you!
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