Budapest University of Technology and Economics, Department of Networked Systems and Services,
Mobile Communication and Quantum Technologies Laboratory
Modelling Two-tier LTE-Advanced Small Cell Networks
with Stochastic Geometry
Zoltán Jakó Ph.D. Thesis book
Scientific supervisor:
Dr. Jeney Gábor, Ph.D.
2017, Budapest
Inroduction
Nowadays, under the definition of modern mobile communication system we usually mean LTE (Long-Term Evolution). Nevertheless, evolution and research process are incessantly runs in background, in order to make faster and better networks (based on principals of LTE). The updated version of LTE is the so-called LTE-Advanced (LTE-A) system. According to the liter- ature LTE-A is the first mobile network that really fulfils the requirements given for 4G mobile system. The standardization of LTE-A is running under the coordination of 3GPP, however standardization process is not closed yet. Besides, state-of-art research tasks focus on creating and specifying a new system, that fulfils the rigorous requirements (i. e. for throughput, latency etc.), that are already given for the next generation (so-called 5G) networks. The appearance of the first 5G capable networks are expected around 2020 [1].
One the mainstream research direction focuses on throughput enhancement based on in- creasing the level of coverage. The importance of coverage becomes a rather important issue, relying on the fact that the carrier frequency (and the frequency band also) used for communi- cation shifts for higher and higher values. However the value of path loss highly depends on the applied carrier frequency and the objects (e. g.buildings) located between the base station and mobile terminal. Furthermore, in densely populated areas, due to population, the number of (mobile) users are increasing rapidly. Therefore, the demand for mobile network services (especially high data rate or low latency services – such as data transfer, media stream) increase extremely. Due to the carrier frequency shifting phenomenon and the rapidly increasing user population, the size of the cells, (which is the access point for mobile devices) are reducing. For example in a densely populated city cores the average cell radius for a LTE cell is around (or less than) 500 m.
An essential solution to enhance the coverage and throughput is to extend the conventional,
“one-tier” macro cell structure to a multi-tier architecture. The macrocell layer can be aug- mented with second (underlaid) tier. This second tier is formed by the combination of several micro-, pico- and femtocells. A femtocell (or Home eNB) is a small base station, that are usu- ally deployed in a flat or an office in order to enhance indoor coverage and provide better QoE (Quality of Experience) to local users, by bringing the base station closer to the user. Home eNB concept differs from the previously used micro- or picocell concept because the data col- lected by a HeNB is delivered to the mobile operator via public internet. The data is brought to the mobile operators core network via wired techniques, such as copper cable- (e. g. DSL – Digital Subscriber Line) or fiber etc. This solution offloads macrocells [2]. At the mobile oper- ator a femtocell Gateway collects the incoming data and merges with the data stream collected from macro eNBs. Femtocells are low power emission base stations, thus the emitted power is typically between 10–100 mW and the radius of the covered area scales up to 30-100 meters.
It 4.5 G and 5 G systems these outdoor cells are deployed to for example lamp- or utility posts. Due to the lower distance to potential users utility post installed cells can offer better coverage (and QoS based services) for pedestrian users. In both cases (outdoor and indoor
solutions) goal is to bring the base station as close as possible to users, that can guarantee the good coverage (and high data rate). The literature uses the collecting term: “Small cells”for these low power emitting, underlaid base stations, therefore I refer them as small cells also hereafter in the thesis book.
In LTE-A and 5G systems small cells are denoted as primary access points [2]. Small cells are operating on licensed spectra. This is exactly the same spectra that the mobile service provider operates the macro e-NodeBs (in case of the frequency reuse is one). Thanks to the commonly used frequency band(s) the small cell users (users served by small cells), suffer from interference generated by the macro base stations and vice versa, from the point of view of macro users (the amount of users served by macrocells) the small cells are interference sources.
Due to the ad-hoc nature of femtocells and public Internet backhaul the central interfer- ence management (via femtocell gateway) is difficult to apply. Femtocells use public Internet to deliver data, which might suffer from significant latency (due to Internet’s best-effort na- ture). Furthermore, femtocells are deployed and operated by the users, thus the location and the
“uptime” is also non deterministic and mobile operators do not have influence on it.
One possible solution is to “upgrade” femtocells capabilities. In other words, femtocells have to have some kind of interference detection and mitigation techniques. For example the interference detector monitors the available (E-UTRA) frequency bands, that can be allocated to potential users (for communication purpose), and in case of high level interference on the investigated band, the scheduler do not use this band [3]. Thus for a small cell user the effect of macrocells interference can be reduced. However this dissertation investigates a worst case scenario, a heavily loaded system, where all resources are allocated continuously. Therefore all resources have been allocated independently from the interference.
Research Objectives
The dissertation (and the thesis book) is restricted to the access network, thus the small cell effects related to core network of an LTE-A system is not detailed in the dissertation. In this model the access network is separated into two tiers. The first tier is the macro tier and second tier is the so-called small cell tier. The dissertation analyses a two-tier LTE-A network with the aid ofStochastic geometry [4, 5, 6]. Stochastic geometry combines the theory of prob- ability and vector geometry. With the mathematical tools offered by Stochastic geometry we can analyse the average behaviour of a network, meanwhile the several input parameters are handled as random variable. In the dissertation for example the actual location of small cells is a random input variable. Therefore the location of the a small cell base station is modelled with two dimensional random point processes.
The goal of the dissertation (relaying on the state-of-art literature) is to propose a mathe- matical model for two-tier small cell network modelling. Some known values (e. g.the mean value of the user population) are given as input parameter, meanwhile other variables are re- main random. Thanks to Stochastic geometry, despite of several random parameters are given,
the model is traceable. As far as I know there is no such detailed downlink interference model for LTE-A small cells in the literature, that is proposed in the dissertation. The proposed in- terference model includes the effect of widely used fast fading types (Rayleigh, Nakagami-m, Rice and Weibull fading). Fast fading models are also applied for the calculation of the proba- bility service outage. The dissertation introduces two new two-dimensional point processes for modelling LTE-Advanced networks. This two introduced point process are belong to the family of Poisson cluster processes, namely: Matérn- and Thomas cluster process. Since small cell installation in a chosen area might not be homogeneous, therefore these cluster based models can model these scenarios in more accurate way compared to homogeneous point processes.
Research Methodology
In my research work I analyse a two-tier network and investigate important performance pa- rameters (related to QoS) with the tools offered by Stohastic geometry. The evaluation of the proposed formulas (given in the dissertation) provide results faster, than running tediously slow Monte-Carlo simulations.
In the first thesis group (Thesis group I.) the location of the small cells are modelled with homogeneous Poisson Point process (PPP). The downlinkinterference analysisfor a Poisson Point Process (PPP) modelled two-tier small cell network. The thesis group focuses on the interference caused by the small cells. The dissertation shows that the cumulative interference (from the small cells), has a closed form probability density function (p.d.f.) and cumulative distribution function (c.d.f.), furthermore it follows symmetric alpha-stable (sαs) distribution, more precisely Lévy distribution, regardless of the channel fast fading type. The effect of fast fading is included to the c.d.f and p.d.f. through the so-called fractional moment. Next, the outage probability for a macrocell user is investigated for various fading types such as:
log-normal fading, Rayleigh fading, Nakagami-mfading and Rice fading. The proposed forms are closed and evaluable. In the Rice fading case only a lower band forcoverage probability (or upper bound for outage probability) is given. Finally, in this thesis group the author has calculated the overall system capacity with the aid of the Signal-to-Interference Ratio (SIR) distribution.
The second thesis group (Thesis group II.) investigates the interference in case of cluster based small cell modelling. The investigated cluster processes are the so-calledThomas and Matérn cluster processes[9]. The thesis group gives formulas to calculate the outage proba- bility (or the complement event – the coverage probability) for a user served by macrocell. Due to the complexity of the forms, they are approximated. However, according to the simulation results, these approximations are still accurate. The cumulative interference and the overall system capacity are investigated with simulations, however these results are not included in this thesis book.
The results of the proposed formulas are compared with Monte-Carlo simulation results for validation purpose and in order to ensure the precision the proposed forms. The simulations are made with MathWorks MATLABr.
System model
Before presenting the new scientific results I briefly introduce the system model. In this model the access network is separated into two tiers. The first tier is the macro tier and second tier is the so-called small cell tier. The system model is documented in Chapter 2 in the dissertation.
The system model is limited to a finite field denoted byR. The area of this field is repre- sented by|R|and this parameter denotes the coverage area of a macro eNB. The macro eNB is located at the center ofR. This point is denoted as the origin point of the coordinate system. In other words, the macrocell is located at the origin of the coordinate system and covers the area
|R|. The small cells are deployed by the users on this finite area|R|. This finite area denotes a square with area of|R|=1000 m×1000 m.
Both eNB type (macro- and small eNB) operates on a fix, constant power. The emitted power of a macro eNB is denoted byPc, meanwhile the emitted power of the small cells are given by Ps. There are no power control applied for small cells. The emitted power of the small cells is given as an average value of the small cells market product data sheet [10]. It is assumed that there is no cooperation between small cells, thus they operate independently from each other. Moreover, there is no central coordination between small cells and macrocell(s). On the other hand small cells operate on the same licensed spectra as the macrocells. The location of the macrocell user (macro UE) is given by vector z. The distance between the macro UE and the macro eNB (from the origin) is the absolute value of the vectorkzk (in meter). The effect of the fast fading is also included in the system model. We include the fading effect in the received power. Parameterhdenotes the fast fading, andhis an independent and identically distributed random variable. The distribution ofh depends on the modelled fading type. The index of h refers to the type base station e. g. hc denotes the fast fading for a macro eNB.
Further information abouthis lavishly detailed in the dissertation (Chapter 2).
The applied path loss for small cells is based on Stanford University Interim (SUI) channel model [11]. The non-logarithmic version is given as a gain value:
g(z) = 1 Ki· 1
Ψlog· kzk−α,
where the constant parts (e. g. propagation loss due to carrier frequency) are merged intoKi. The outdoor path loss exponent is denoted byα. Nevertheless, for the sake of simplicity we chose Ki =1. Ψlog random variable denotes the shadowing (log-normal fading) component.
The modelled two-tier system is interference limited, thus the effect of the thermal noise at the receiver is ignored for the sake of simplicity.
The small cell base station generated interference at a macrocell user:
I(z) =
∑
x∈Φ
Pshxg(x−z). (1)
The vectorxrepresents the random location of the interference sources in the setΦand vector zgives the actual location of the macrocell user. The path-lossg(kx−zk)between the macro- user’s receiver and the interference source. The effect of the fast fading is included to the model byhx.
The outage probability is defined as the probability of the Signal-to-Interference ratio (SIR) being below the threshold value (T).
IP{out}(z) = IP
Pchcg(z) I(z) ≤T
, (2)
where the fast fading between the receiver and the macro eNB is given by parameterhc. The threshold value (T) is the minimal required Signal-to-interference (SIR) value. The comple- menter event of the outage probability is the so-called coverage probability:
IP{cov}(z) =1−IP{out}(z) = IP
Pchcg(z) I(z) ≥T
. (3)
Location of Small Cells
The dissertation introduces three random processes, that can be used for two-tier small cell modelling. These random processes models the planar locations of the small cells on finite area R. The first one is the widely used two dimensional homogeneous Poisson point process (PPP).
According to the homogeneous PPP model the small cells are scatter in|R|uniformly. The actual number of small cells follows Poisson distribution with densityλ. The mean value of the small cells, thereforeNs =λ· |R|. In this model we assume that the process is homogeneous, thus the points are scattered uniformly in the planar. In this case the intensity parameterλ is constant in every part ofR. The actual location of the small cells is given by vectorx. The distance between a small cell and the macro eNB (located at the origin) is given by the absolute value of the vectorkxk. Furthermore the distance between a small cell and the user is given by kx−zk. The homogeneous version of Poisson point process is isotropic and stationary, which makes the process traceable. An illustrative figure is presented in Figure 1a.
The Poisson cluster process (PCP) based model breaks the homogeneous nature and groups the small cells into clusters. If one takes a photo about a realization, in some parts of the field
|R|the density of small cell is higher than others. Furthermore in some fraction of|R|is free from small cells. This cluster based process provides an accurate model, rather than PPP. Since in a real environment some parts the small cell intensity is higher (e. g. block of flats), than others (e. g. public parks). Nevertheless, the introduced cluster models remains traceable due to Stochastic geometry.
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−300
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x (m)
y(m)
x
z x−z
Macro UE
Small cell Macro Base Station
(a) Illustration of Homogeneous Poisson point process
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x (m)
y(m)
Parent point Small cell
Cluster
(b) Illustration of Thomas cluster process
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x (m)
y (m)
Parent point
Cluster Small cell
(c) Illustration of Matérn cluster process
Figure 1. Illustration of Small cells system models
The dissertation introduces two Poisson cluster processes that can be used for small cell modelling, namely the Thomas cluster process and Matérn cluster process [4]. Both pro- cesses are belong to the family of Neyman-Scott point process [4] and composed with superpo- sition of simple Poisson point processes The base for Thomas and Matérn cluster process is a Poisson point process with parameterλp. The literature calls this points as parent points. The set of parent points is denoted by(Φp={x1,x2, . . .}). In our model the actual location of the parent points is given by vectorx. Around the parent points scattered the daughter points (or the name offspring points is also used by the literature). The daughter points are independent from each other and from the other parent points. Furthermore, they are identically distributed around
Figure 2. PPP based Random graph model
the parent point. The set of cluster is represented byNxi ={Ni,xi}, whereNi={y1,y2, . . .}
denotes the i.i.d. set of daughter points. Note thatNiis independent from the parent process, in other words, the parent points are not members of the cluster. The whole cluster therefore can be written as follows:
Φ=[
i
(Ni∪xi) = [
x∈Φp
Nx.
The number of daughter points is random, and follows Poisson distribution. The mean value of the daughter points in a cluster is represented byc. The distance between daughter points and the cluster centres (parent points) is given bykyk. In PCP the daughter points represents the small cells. Their actual position is given by vectorxi+y. The density of the cluster process can be calculated withλ =λp·c, where c represents the average number of small cells in a cluster.
The main difference betweenThomas andMatérn claster is the scattering of the daughter points. In case ofThomas cluster process the daughter points scatter around the parent points according to a symmetrical normal distribution (zero mean andδ2variance) with the following density function:
f(y) = 1 2π δ2exp
−kyk2 2δ2
. (4)
However, inMatérn cluster processthey are scatter around the parent points uniformly in a circle with radiusR. The density function is given by [13]:
f(y) = ( 1
πR2, ifkyk ≤R
0, oherwise (5)
Further details of cluster process is given in the dissertation (Chapter 2.).
In Thesis I.5. we apply PPP model in order to calculate the system level throughput in down- town. Road structure can be represented with a graph, where the vertices are crossings and the edges are roads. In this graph, shortest paths exist and can be found between two nodes, if the graph is connected. One possible graph structure represented in Figure 2, where the macrocell and small cells are denoted by blue and green rectangles, respectively. In this case the model should take into account the road structure of a district, therefore the following restrictions are given extending the general PPP model:
• The location of the users is modelled with PPP. The actual number of user is given byNc.
• In general case it is expected that small cells are covering a circle areas with radius Rs (thus the area covered by one small cell isR2sπ). The whole system area is|R|.
• On the other hand in random graph model (city road structure) small cells are covering only road segments with distance 2Rs. The roads segments are denoted bye1,e2,· · ·,en. The total length of the road segments is given by ∑i|ei|. Figure 2 gives an illustration of the proposed random graph model. Small cells are installed along the edges to provide extended coverage for the users i. e. small cells are located on the edges of the graph (e. g.
installed on a street light). The second tier base stations (i. e. small cells) are assumed to be uniformly distributed along the edges. The distribution of the small cells thus follow a Poisson Point Process. However the area where small cells possibly occur is restricted to the road structure, following [5], PPP makes possible to analyse this wireless network.
The number of users in theith small cell is denoted byUi.Ui’s are i.i.d. random variables which follow Poisson distribution with parameterλ.
Research objectives
Thesis group I. Thesis group II.
Poisson Point process based model Poisson cluster process based model Stochastic
Geometry based investigation
Simulation
Stochastic Geometry based
investigation
Simulation Interference
investigation: 3I. 1. Thesis 3 - 3
Outage/Coverage probability investigation:
3 I. 2. Thesis, 3 I. 3. Thesis, 3 I. 4. Thesis, 3 I. 5. Thesis
3
3 II. 1a. Thesis, 3 II. 1b. Thesis, 3 II. 2a. Thesis, 3 II. 2b. Thesis
3
Average System Throughput investigation:
3I. 6. Thesis 3 - 3
Table 1. Structure of theses
New Scientific Results
The research objectives can be categorized into three categories. These are the investigation of interference, the outage/coverage probability and the average system throughput. The theses are collected in Table 1. In Poisson cluster case the results are given only for outage/coverage prob- ability. For the sake of completeness simulation results are provided for cumulative interference and the average system throughput. Nevertheless, these simulation results are represented only in the dissertation.
Small Cell modelling with Homogeneous Poisson Point Process
In this subsection the first thesis group is proposed. This thesis group contains six theses. All of the theses investigate two-tier small cell networks with stochastic geometry. The location of small cells are modelled with homogeneous Poisson point process.
Firstly, I give a closed form for the cumulative interference caused by small cells. Af- terwards, I define the macro users outage probability at positionz. The outage probability is
Fading Type Fading moment
Without fading: E { √
h} = 1,
Rayleigh fading: E { √
h} = Γ 1 +
2α
= Γ
32=
√ π 2
,
Nakagami-m fading: E { √
h} =
(2m−1)!!2m(m−1)!√ m
√ π , Rice fading: E { √
h} = q
11+K
· e
−K2(1 + K) I
0 K2+ K · I
1 K2√
π 2
,
Weibull fading: E { √
h} = √
γ · Γ 1 +
2n1,
Lognormal fading: E { p
1/Ψ} = e
(−1/2·µ+σ2·1/8).
Table 2. E{√
h}Fading moments calculated for slow fading and multiple fast fading scenarios.
I propose two ways to calculate the service outage probability. The first method is based on the interference distribution, meanwhile the second method is based on the PGFL (Probability Generating Functional), which is provided by Stochastic geometry. The detailed explanation of PGFL is given in Section 2.2.5 in my dissertation.
Finally, in this thesis group, I propose a form that allows to calculate the average system capacity of a two-tier small cell network.
Before introducing the theses, let us propose the PGFL for a homogeneous Poisson Point process (for details see Section 2.2.5 in the dissertation):
GN(v) =exp
−λ Z
R[1−ν(x)]dx
. (6)
THESIS I.1. [J1], [J5], [C6] I have proved, that in a two-tier OFDMA based system, the cumulative small cell interference I(z) follows Lévy distribution, if the (interference source) small cell locations are modelled with homogeneous Poisson Point process, and the outdoor path loss exponent (α) equals 4. The c.d.f and p.d.f. are given as follows:
fI(x) = r c
2π
exp −2xc
x3/2 , (7)
FI(x) = IP{I≤x}=erfc r c
2x
. (8)
The distributions location parameter (µ) is 0, meanwhile the scale parameter is:
c=π3λt2Ki−1
E{√ h}2
Ps/2. (9)
Regardless of the channel fast fading type (i. e. Rayleigh, Nakagami-m, Rician or Weibull faded) the interference distribution remains Lévy distribution.
The function erfc(x) =1−erf(x) = √2
π
R∞
x e−t2dt is the complementary Gauss error func- tion [14]. The intensity parameter of the Poisson distribution is λ. For homogeneous Point process we can apply Slivnyak’s theorem [5]. According to this theorem the cumulative in- terference distribution is the same regardless of the receiver position. Thus, for the sake of simplicity we chose the origin of the coordinate system (I(0)) for the observation point. The fast fading is included to the form as the non-integer order moments (E{√
h}). The values for the 1/2 order moments of slow- and fast fading types are given in Table 2. For further details please read Section 2.2.4 of the dissertation. C.D.F.s are calculated with (8) to several fading types (by substituting the correctE{√
h}). The stochastic geometry based forms are validated with MATLAB simulations (Figure 3).
The simulations are evaluated with the same values (λ,Ps etc.), that used in (8) and (9) to gain the results. Thus the results from the proposed forms and the simulation results are compa- rable. The simulation results are constructed from the evaluation of 104experiments. In every iteration the actual location of the small cells are given as a PPP and thehis a random param- eter, which obeys to the currently applied fast fading type (see Section 2.2.4 of the dissertation for details). We illustrated also the results related to the case without fading in every figure as a reference point. The c.d.f curves are calculated for average small cells numbersNs=50 and Ns=100.
10−9 10−8 10−7 10−6 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x[W]
F(x)=Pr{I≤x}
Without Fast Fading (analytic c.d.f) Without Fast Fading (empirical c.d.f) Rayleigh fading (analytic c.d.f) Rayleigh Fading (empirical c.d.f) Nakagami−4 fading (analytic c.d.f) Nakagami−4 Fading (empirical c.d.f)
Ns= 100 Ns=50
10−9 10−8 10−7
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
x[W]
F(x)=Pr{I≤x}
Without Fast Fading (analytic c.d.f) Without Fast Fading (empirical c.d.f) Rayleigh fading (analytic c.d.f) Rayleigh Fading (empirical c.d.f) Nakagami−4 fading (analytic c.d.f) Nakagami−4 Fading (empirical c.d.f)
Ns= 100 Ns=50
(a) Nakagami-mfaded channel
10−9 10−8 10−7 10−6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x[W]
F(x)=Pr{I≤x}
Without Fast Fading (analytic c.d.f) Without Fast Fading (empirical c.d.f) Rice fading K=1/2 (analytic c.d.f) Rice fading K=1/2 (empirical c.d.f) Rice fading K=2 (analytic c.d.f) Rice fading K=2 (empirical c.d.f)
Ns=50 Ns= 100
10−9 10−8 10−7
0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
x[W]
F(x)=Pr{I≤x}
Without Fast Fading (analytic c.d.f) Without Fast Fading (empirical c.d.f) Rice fading K=1/2 (analytic c.d.f) Rice fading K=1/2 (empirical c.d.f) Rice fading K=2 (analytic c.d.f) Rice fading K=2 (empirical c.d.f)
Ns= 100 Ns=50
(b) Rician faded channel
10−9 10−8 10−7 10−6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x[W]
F(x)=Pr{I≤x}
Without Fast Fading (analytic c.d.f) Without Fast Fading (empirical c.d.f) Weibull Fading n=2 (analytic c.d.f) Weibull Fading n=2 (empirical c.d.f) Weibull Fading n=10 (analytic c.d.f) Weibull Fading n=10 (empirical c.d.f)
Ns= 100 Ns=50
10−9 10−8 10−7
0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85
x[W]
F(x)=Pr{I≤x}
Without Fast Fading (analytic c.d.f) Without Fast Fading (empirical c.d.f) Weibull Fading n=2 (analytic c.d.f) Weibull Fading n=2 (empirical c.d.f) Weibull Fading n=10 (analytic c.d.f) Weibull Fading n=10 (empirical c.d.f)
Ns=50
Ns= 100
(c) Weibull faded channel
Figure 3. Validation of the proposed formulas with Monte-Carlo simulations
THESIS I.2. [C6] Relying onTHESIS I.1. I have introduced a form, that allows the cal- culation of the (users’) probability of service outage in an OFDMA based two-tier small cell system, if the channel is infected independent Lognormal fading. The proposed form relies on the theory that the interference obeys Lévy distribution:
IP{out}(z) = Z ∞
0
1−erfc
Nsπ3/2
q
PsKi−1E np
1/Ψ o 2· |R|
s t
T Pcg(z)
fΨ(t)dt, (10) where fΨ(t)the p.d.f of a zero mean and 10 dB variance Lognormal fading is:
fΨ(t) = 1
√2πln(10)σ10 t
exp − (ln(t))2 2(ln(10)σ10 )2
!
. (11)
Applying the definition of the service outage probability (3) we get:
IP{out}(z) = IP
Pc/Ψg(z)
I(z) ≤T
=1−IP
I(z)≤ Pcg(z) TΨ
. (12)
With the Law of total probability the equation can be written as follows:
IP{out}(z) = Z ∞
0
1−IP
I(z)≤ Pcg(z) T t
| {z }
Lévy distribution c.d.f
·fΨlog(t)dt. (13)
Finally substituting the c.d.f of the Lévy distribution given in (8) with some algebraic manipu- lation yields (10).E
np 1/Ψo
is given in Table 2.
1 10 100 600
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
Ns Prout( z) = Pr{SIR < T}
|| z || =100 m, w/o lognormal fading
|| z || = 200 m, w/o lognormal fading
|| z || =100 m, lognormal fading (0,10dB)
|| z || = 200 m, lognormal fading (0,10dB)
|| z || = 100 m, Simulation w/o lognormal fading
|| z || = 200 m, Simulation w/o lognormal fading
|| z || = 100 m, Simulation lognormal fading (0,10dB)
|| z || = 200 m, Simulation lognormal fading (0,10dB)
Figure 4. Probability of service outage from (10)
THESIS I.3. [C6] Service outage probability based on Lévy distribution: I have modified the previously proposed form (that allows the calculation of the users’ probability of service outage in a slow faded environment), in order to calculate the users’ probability of service outage, when the channel is Rayleigh or Nakagami-m faded:
IP{out}= Z ∞
0
1−erfc
Nsπ3/2 q
PsKi−1E√ hc 2· |R| ·
s h
T Pckzk−α)
·hm−1e−hm m−mΓ(m)dh.
(14) The proof of (14) is given in the dissertation Section 3.2.2.1. The main steps are similar to the Lognormal fading case.
The fast fading values (E√
hc ) can be calculated from Table 2.
THESIS I.4. [J2], [J3] Service outage probability based on PGFL:I have proposed a math- ematically form in order to evaluate outage/coverage probability for a user at location z, if the channel is Nakagami-m faded, and the actual location of small cells are modelled with homogeneous Poisson Point process:
IP{cov}(z) =
m−1 k=0
∑
(−1)k k!
sk dk
dskLI(z)(s)
| {z }
G(ν(x,z))
s= T m
Pcg(z)
. (15)
The proof of (15) is given in the dissertation Section 3.2.2.2.
The proposed form can be interpreted as a general form, due to the fact that in this case we use the PGFL to calculate the outage probability. In the previous (Lévy based) form for outage probability we had a restriction. The Lévy distribution is valid only if the outdoor path loss exponent isα =4. However the PGFL based form does not depend fromα, the only restriction is, that α should be non-negative. If we chose m=1 for Nakagami-m (thus the channel is Rayliegh faded), then the form modifies to:
IP{cov}(z) =LI(z)(s) s= T
Pcg(z)
,
which corresponds to the results in [12, 13].
Instead of the Laplace form of the interference, we can substitute the PGFL belongs to the homogeneous PPP (6):
IP{cov}(z) =
m−1
∑
k=0
(−1)k k!
"
sk dk dskG(ν)
# s=Pcg(z)T m
. (16)
1 10 50 10−3
10−2 10−1 100
Ns Outage Probability [Prout]
Analyitic || z || = 100 m (PGFL) Analytic || z || = 200 m (PGFL) Simulation || z || = 100 m Simulation || z || = 200 m Analyitic || z || = 100 m (Lévy) Analyitic || z || = 200 m (Lévy)
(a) PPPm=1 (Rayleigh fading) (b) PPPm=4 (Nakagami fading)
10 20 30 40 50
10−3 10−2 10−1 100
Ns Outage Probability [Prout]
Analytic ||z||=100 m Analytic ||z||=200 m Simulation ||z||= 100 m Simulation ||z||= 200 m
(c) PPPK=0.5 (Rice fading)
10 20 30 40 50
10−3 10−2 10−1 100
Ns Outage Probability [Prout]
Analytic ||z||=100 m Analytic ||z||=200 m Simulation ||z||= 100 m Simulation ||z||= 200 m
(d) PPPK=1 (Rician fading)
Figure 5. Outage probability in PPP with Rayleigh, Nakagami-mand Rician fading In case of Rayleigh fading (m=1) the sum disappears:
IP{out}(z) =1−exp
−λp Z
R2
1− 1
1+PcPkzksT−αkx−zk−αdx
. (17)
The Lévy based result are calculated with (14), meanwhile the PGFL based results are cal- culated with (16) and (17). Both forms are validated with simulations. The figures are given in Figure 5a and Figure 5b. The dashed lines are the PGFL based results, and the markers de- notes the Lévy based results, finally the "full" markers are the simulation results. Both Rayleigh and Nakagami-m (m=4) are illustrated. From the results it is visible that the Lévy- and the PGFL based method are equivalent, ifα =4. Furthermore the simulation results validates the accuracy of the forms.
THESIS I.5. [J4] I have proposed an mathematical form that allows to calculate coverage probability for a macrocell user in a two-tier interference limited network, where the small cell positions obey a Poisson Point Process and the channel is Rician faded:
IP{cov}(z)=e−K
∞
∑
m=0
Km m!
m
∑
k=0
(−1)ksk k!
dk
dskLI(z)(s)
s=(K+1)TPcg(z)
. (18)
From this formula we get a lower bound:
IP{cov}(z)≥e−K
N
∑
m=0
Km m!
m
∑
k=0
(−1)ksk k!
dk
dskGN(ν)
s=(K+1)·TPcg(z)
. (19)
The detailed proof is shown in the dissertation Section 3.2.3. If the Rician fading parameter (K) is chosen forK=0 (Rayleigh faded channel), then the form changes to:
IP{cov}(z) =LI(z)(s) s=Pcg(z)T
=GN(ν),
which corresponds with the results of [12, 13]. With reference to the relationship between the Laplace transform of the interference and the PGFL we can substitute the PPP’s PGFL into (18). Furthermore, since the odd-order derivatives of the PGFL are negative values, the sum in (18) contains only non-negative elements. Hence we can define a lower bound for the coverage probability (or upper bound to the service outage probability) by limiting the number of terms to a finite numberN. The results are shown in Figure 5c and Figure 5d for K=0.5 andK=1, respectively. The results are gained from (18) and represented as dashed lines, while the markers represent the MATLAB simulation results. Although (18) is just an upper bound results the simulation results validate the proposed formulas (approximated) results are close to the ones from the simulator for bothK values.
I have compared the coverage probability with different path loss exponent values for three fast fading types (Figure 6). I investigated how the coverage probability changes for different threshold values (T), ifNs=50 andkzk=100 m. The results are compared with simulations.
−10 −8 −6 −4 −2 0 2 4 6 8 10 0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T [dB]
Coverage probability [Prcov]
α=3 Analytic α=4 Analytic α=5 Analytic α=3 Simulation α=4 Simulation α=5 Simulation
(a) Rayleigh fading (m=1)
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T [dB]
Coverage probability [Prcov]
α=3 Analytic α=4 Analytic α=5 Analytic α=3 Simulation α=4 Simulation α=5 Simulation
(b) Nakagami fading (m=4)
−10 −8 −6 −4 −2 0 2 4 6 8 10
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T [dB]
Coverage Probability Prcov
α = 3 α = 4 α = 5 α = 3 Simulation α = 4 Simulation α = 5 Simulation
(c) Rice fadingK=0.5
Figure 6. The coverage probability for different fast fading types with various path loss expo- nents (α)
SIR[dB] System Throughput[kbit/s] SIR[dB] System Throughput[kbit/s]
-12 dB 0 kbit/s 9 dB 340 kbit/s
-9 dB 15 kbit/s 12 dB 480 kbit/s
-6 dB 38 kbit/s 15 dB 620 kbit/s
-3 dB 70 kbit/s 18 dB 740 kbit/s
0 dB 110 kbit/s 21 dB 850 kbit/s
3 dB 170 kbit/s 24 dB 910 kbit/s
6 dB 250 kbit/s 27 dB 930 kbit/s
Table 3. Throughputs used for Average System capacity calculation in MIMO 2×2 configura- tion [15]
THESIS I.6. [C3], [C4] I have proposed a form to calculate the overall two-tier system ca- pacity in a two-tier small cell network, if the small cell are scattered according to homogeneous Poisson point process. Given the average number of active small cells, the overall two-tier system capacity can calculated with
Cfull=
∑
U1,...,UN f
IP
U1, . . . ,UNf
U1
∑
j1=1
Csj
1+· · ·+
UN f
∑
jN f=1
Csj
N f +
Nc−∑iUi k=1
∑
Ckm
=
=NPRB·
e−λ·Ns·Csfull+Cmfull
, (20)
whereNf denotes the actual (random) number of small cells in the area. The first sum collects the combination of user distribution along small cells, IP
U1, . . . ,UNf represents the corre- sponding probability (which is actually the product of Poisson probabilities) and finally the terms in brackets show the capacities which are available for the users – they are summed. The last term stands for the macrocell. Note, that the sums go from one: if there is no user under a base station (the zero case), there is nothing to sum up. Note that the average number of active small cells equalse−λNs (the details are given in the dissertation Section 3.4.2), where:
λ = (N
cR2sπ
|R| two-dimensional PPP space width circular coverage of small cells,
Nc2Rs
∑iei graph model of PPP width line coverage of small cells.
Ifksmall cells are active (they have at least one active user), then the bracket term in (20) yieldsk·Cfulls +Cfullm , for each PRB. Though inactive small cells also have free capacities, they are not added: they do not raise the system capacity due to their inactivity.
We assume ON/OFF model, therefore if there is one user on the small cell, then all resources are allocated by the eNB. Thus using the form for active small cells:
Cfull=NPRB·
e−λ·Ns·Cfulls +Cfullm ,
0 2 4 6 8 10 12 14 16 18 20 101
102 103 104 105 106
log2(N
f): Logarithm of the number of Small cell base stations
Total system throughput [Mbps] in logarithmic scale
PPP model
Random graph PPP model
(a) Average System throughput vs. number of Small cells
−1 0 1 2 3 4 5
105 106
log2(Nf): Logarithm of the number of Small cell base stations
Total system throughput [kbps] in logarithmic scale
Simulation results (PPP) Analytic results (PPP)
(b) Comparing simulation and form results
Figure 7. Throughput enhancement with Small Cells where the multiplication byNPRB represents the number of PRB’s in the system.
The results are calculated for NPRB =100. The numerical results are gained from the SIR distribution and throughputs from Table reftab:throughput. The average system capacity for two small cell deployment model is shown in Figure 7a.
Small Cell modelling with Poisson cluster Process
The second thesis group deals with the Poisson cluser process based small cell modelling. The model is discussed in details in the dissertation (Chapter 4.). The thesis group contains four theses.
Before describing the theses, let us introduce the PGFL for the Poisson cluster processes:
GN(ν) =exp
−λp Z
R
1−exp
c Z
Rν(x+y)f(y)dy−1
dx
, (21)
where f(y)can be (4) or (5). Parametercdenotes the average small cells in a cluster.
Thomas cluster process
THESIS II.1a. [J2] I have developed a method, that allows to calculate the macrocell users service outage probability, if the interference source small cells are modelled with Thomas cluster process and the channel is Rayleigh faded:
IP{out}(z) =1−exp
−λp Z
R1−exp c(κray−1) dx
s= T
Pcg(z)
, (22)
where
κray= 1 2π δ2
Z
R
exp
−kyk2
2δ2
1+sPsg(x+y−z)dy. (23)
Furthermore an approximated form is given:
IP{out}(z)≈1−exp
−λp Z
R1−exp c(κˆray−1) dx
. (24)
Let us apply the Jensen’s inequality forκray, then ˆκraycan be substituted into (22). Further information is given in Subsection 4.1.2.1.
κˆray= 1
1+ PsT
Pckzk−α (δ2+kx−zk2)2+2δ4+4δ2kx−zk2
!−α4 . (25)
Results: The results are calculated with (24) and the results are compared with simulation results are given in Figure 9a. The results shows that the proposed approximated form is a tight upper bound for service outage probability.
In Poisson cluster processes the outage probability highly depends from the number of clus- ters and the number of small cells in a cluster. Compared to PPP based small cell modelling in the PCP based model the average number of parent points and cluster size is also a rele- vant input parameter. ParameterNs does not sufficient information in this case. We know that Ns =λp· |R| ·c. Let us explain through an example, where Ns =50. In the first scenario the average number of parent points is 50. This implies that the average number of small cells in a cluster (c) equals 1. In the second scenario the mean number of parent points was 10, therefore the average number of small cells in a cluster 5 (c=5). In both scenariosNs=50, however the two cases are very different.
10−6 10−4 10−2 100 102
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
β Outage Probability [Prout]
Ns = 100 Ns= 200 Ns = 300 Ns = 400
(a)T =0 dB
10−6 10−4 10−2 100 102
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
β Outage Probability [Prout]
Ns = 100 Ns = 200 Ns = 300 Ns = 400
(b)T =−5 dB
Figure 8. Outage probability for differentβ values in Thomas cluster process (Rayleigh fading)
Let us introduce a new variable for the sake of convenience:
β = c λp· |R|.
If the value ofβ is small, then we have a lot of clusters with low number small cells in it. On the other hand, ifβ is high, then we have a few, but large clusters. The results for different SIR threshold values (T) are given in Figure 8. The horizontal axis denotes the service outage probability, meanwhile the vertical axis represents the values ofβ in logarithmic scale.
THESIS II.1b. [J3] I have generalizedTHESIS II.1a. in order to calculate the macrocell users service outage probability, if the interference source small cells are modelled with Thomas cluster process and the channel is Nakagami-m faded:
IP{out}(z) =1−
m−1 k=0
∑
(−1)k k! sk dk
dsk
exp
−λp Z
R1−exp(c(κnaka−1))dx
s= T m
Pcg(z)
, (26)
where
κnaka= 1 2π δ2
Z
R
exp
−kyk2
2δ2
1+sPmsg(x+y−z)mdy.
Furthermore an approximation is given:
IP{out}(z)≈1−
m−1
∑
k=0
(−1)k k!
"
sk dk dskexp
−λp Z
R1−exp(c(κˆnaka−1))dx #
s=Pcg(z)T m
(27)
and
κˆnaka= 1
1+PcPkzksT−α
(δ2+kx−zk2)2+2δ4+4δ2kx−zk2−α4!m. (28) Similarly to (23) vectorxdenotes the coordinates of the parent points and vectoryrepresents the coordinates around vectorx.
THESIS II.1b. is a general form, thus in Rayleigh fading case (m=1) (24) simplifies to:
IP{out}(z)≈1−exp
−λp Z
R2
1−exp c(κˆray−1) dx
, which corresponds withTHESIS II.1a..
Results: I only calculated the approximated versions from THESIS II.1a. and THE- SIS II.1b.. These approximate forms and validated with simulations. Nevertheless, it is clearly visible that the forms provide an accurate upper bound for service outage for both fading types.
Due to the nature of Thomas cluster I calculated with numerouscvalues. The results are given in Figure 9a and Figure 9b. In this case I investigated the service outage probability compared to the mean value of small cells. It is visible in Figure 9b, that the mean number of small cells
1 10 50
10−3 10−2 10−1
Ns Outage Probability [Prout]
Analyticc= 5 Analyticc= 10 Simulationc= 5 Simulationc= 10
(a) Rayleigh fading (m=1) andkzk=100 m
1 10 50
10−2 10−1
Ns Outage Probability [Prout]
Analyticc= 5 Simulationc= 5 Analyticc= 10 Simulationc= 10
(b) Nakagami-m(m=4) andkzk=100 m
Figure 9. Service outage probability for Thomas klaszter in Nakagami-mfaded channel (T =1)
0 1 2 3 4 5 6 7 8 9 10 1E−2
1E−3 1E−4
1E−05 1E−06 0
0.2 0.4 0.6 0.8 1
c γ
Proutage
(a)λp=4·10−4
0 1 2 3 4 5 6 7 8 9 10
1E−2 1E−3
1E−4 1E−5
1E−06 0
0.2 0.4 0.6 0.8 1
γ c Prout
(b)λp=2·10−4 Figure 10. Service outage probability vs. parameterγ
−10 −5 0 5 10
0.2 0.4 0.6 0.8 1
T [dB]
Prout
c= 1 c= 2 c= 4 c= 6
(a) Service outage probability vs.T,λp=4·10−4
−10 −5 0 5 10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T [dB]
Prout
c= 1 c= 2 c= 4 c= 6
(b) Service outage probability vs.T,λp=2·10−4
Figure 11. Service outage probability vs. T ,kzk=100 m,m=4 in a cluster has a great impact to the service outage probability.
Next, the cluster density is changed to λp=4·10−4 and λp=2·10−4. In this case c is investigated in contrast of parameterT. If the finite area isR =500 m×500 m, then the mean number of parent points are 100 and 50, respectively. The Nakagami-mfadingmparameter is 4. The results are visible in Figure 10.
Untill now, I investigated Thomas clusters with constant macro- and small cell emitted pow- ers (20 W and 20 mW), meanwhile the path loss constant parameter equals 1 (Ki=1). I intro- duce a new variable:
γ = PsKi PcKcT.
With this new variable we can calculate the service outage probability with several emitted power and path loss constant. The service outage probability versusγ is given in Figure 10.
Matérn Cluser process
THESIS II.2a. [C5] I have given a formula to calculate the the macro users’ probability of service outage in case of the small cell (interference source) locations are modelled with Matérn cluster process and the channel is Rayleigh faded:
IP{cov}(z) =exp
−λp
Z
R1−exp c(κray−1) dx
# s=Pcg(z)T
, (29)
ahol
κray= 1 πR2
Z
R
1
1+sPsg(x+y−z)dy. (30)
Similarly to the Thomas cluster process case we can use the Jensen’s inequality in order to get an approximation forκray:
κray≤κˆray= 1 1+ PsT
Pckzk−α kx−zk4+2·R2kx−zk2+25R4
!−α4 . (31)
If we substitute this parameter to (29), then we get an approximation for the coverage prob- ability in case of Matérn cluster process:
IP{cov}(z)≈exp
−λp· Z
R1−exp c(κˆray−1) dx
| {z }
G(ν)
s=Pcg(z)T
. (32)
THESIS II.2b. [C5] I generalized the form given inTHESIS II.2a.in order to calculate the coverage probability in a Matérn cluster modelled small cell environment in case of Nakagami- m faded channel:
IP{cov}(z) =
m−1
∑
k=0
(−1)k k! sk dk
dsk
"
exp
−λp· Z
R1−exp c(κnaka−1) dx
| {z }
G(ν)
# s=Pcg(z)T m
, (33)
where
κnaka= 1 πR2
Z
R
1
1+sPmsg(x+y−z)mdy. (34)
