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Carrier to interference ratio, rate and coverage analysis in shotgun cellular systems over composite fading channels
Accepted Manuscript Carrier to interference ratio, rate and coverage analysis in shotgun cellular systems over composite fading channels Asma Bagheri, Ghosheh Abed Hodtani
PII: DOI: Reference:
S1874-4907(16)30156-2 http://dx.doi.org/10.1016/j.phycom.2017.07.005 PHYCOM 407
To appear in:
Physical Communication
Received date : 3 October 2016 Revised date : 26 May 2017 Accepted date : 24 July 2017 Please cite this article as: A. Bagheri, G.A. Hodtani, Carrier to interference ratio, rate and coverage analysis in shotgun cellular systems over composite fading channels, Physical Communication (2017), http://dx.doi.org/10.1016/j.phycom.2017.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Carrier to Interference Ratio, Rate and coverage Analysis in Shotgun Cellular Systems over Composite Fading Channels Asma Bagheri1 , Ghosheh Abed Hodtani2 1,2 2
Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran [email protected]; [email protected]
Abstract: Wireless communication performance in shotgun cellular system (SCS: wireless communication system with randomly placed base stations) is analysed. Aiming at this analysis, (i) the carrier to interference ratio (CIR), as an important measure in a cellular network, is considered in the downlink (path-loss, shadowing and multi-path) fading channel. Since direct calculating of the distribution of CIR is complicated, a related mathematical equation is determined through which the probability density function (PDF) of CIR can be calculated. And then, as a special case, the tail probability of CIR is obtained. (ii) A normal approximation for inverse of CIR is proposed which is applicable for calculating a tractable PDF for the downlink CIR. The analytically calculated and the approximate PDFs of CIR are compared with numerical PDFs. (iii) The distribution of the downlink rate and a lower bound for the average rate; the analytical expression for coverage of a user in an SCS based on its received CIR, and an average value for coverage are calculated. (iv) Simulation results show that the closed form and approximate PDFs over different models are close to numerical ones. Keywords- Carrier to interference ratio (CIR), Coverage, Multi-path fading effect, Rate and average rate, Shadowing effect, Shotgun cellular system, Tail probability of CIR. 1. Introduction Cellular communication consists of a set of radio base stations (BSs) distributed over a region that communicate with mobile stations (MSs). In contrast to hexagonal cellular systems (i.e. ideal systems) with regular BS placement in many wireless systems such as Local Area Networks (LANs) and femtocells [1], due to site acquisition difficulties, BSs are placed irregularly over the deployment region. These random systems called shotgun cellular systems (SCSs) can be described by BS density function
as a
function of the distances between BSs and the MS. In an SCS the signal propagation is affected by three phenomena. First is path-loss effect (here is random), the second is Log-normal shadowing, and third is multi-path fading effect. Hence many cellular deployments have significant randomness. So that, an SCS is a system affected by random phenomena. The SCS and its performance metrics have been studied under different channel models [2-10]. Introducing a new random substitute for hexagonal systems was started by [2]. In [3] the downlink communication in an SCS was compared with hexagonal system and it was shown that in Log-normal shadowing, SCS with fixed channel assignment has the signal level of about 4db lower than that of hexagonal cellular systems. So, in an SCS with random channel assignment the system performance needs
1
to be analysed. In [4] dynamic channel assignment (DCA) in an SCS was investigated, and the results were compared with hexagonal system. Also, downlink and uplink communications were both studied, and it was shown that performances of both links are the same, and system performance is improved to near ideal system performances, with self-organized DCA. In [5] CIR performance bounds for two-dimensional (2-D) interference-limited radio cellular systems with random channel assignment are introduced. It was calculated that an SCS provides lower bound for performance of cellular systems and hexagonal systems provide upper bound. They showed that the difference between upper and lower bounds is small under typical operating conditions, in modern TDMA and CDMA cellular systems. Furthermore, it was shown that under strong shadowing the bounds converge. Also, an SCS was studied in different scenarios such as different numbers of channel groups (CGs). In highly varying environment and low CIR levels, SCS is most similar to the hexagonal systems. In [6] CIR performance at an MS was investigated for one, two and three-dimensional SCS, and analytical tail probability of CIR was obtained. Also, the results of [1] were generalized to the multidimensional cases. Study of the CIR at an MS in an SCS with Poisson process is difficult and comparing of systems is not easy. Therefore, a simple analytical tool based on stochastic ordering was developed to compare the distributions of CIR at the MS in two SCSs [7], [8]. In [9] a multi-tier network composed of M tiers of homogenous N-dimensional systems was considered, and CIR performance at an MS for such a network was investigated. Rate and coverage are other important metrics of a wireless network. In [10], under a special channel model, the distribution of downlink rate at a user in a Heterogeneous cellular network was studied, where they assumed Log-normal shadowing and a special unit mean Rayleigh fading. In [11] the average rate of a cellular network with Poisson point process BSs placement was analysed, where only path-loss and unit mean Rayleigh fading with limited interference were assumed. Also, probability of coverage was defined as the probability that the Signal-to-Noise-Ratio (SINR) is upper than a threshold, which it was simulated based on the tail probability of SINR. The calculated expressions for distribution of SINR, although simple, are applicable in a special channel model. In [12], the coverage number for an MS in an SCS has been defined as the number of BSs which the received power of MS from them is more than a threshold. Assuming low power path-loss, limited shadowing with an arbitrary distribution, and Rayleigh fading, they computed numerically the probability that the coverage number is upper than a threshold k. Literature review shows that an SCS in addition to easiness and ability to be designed fast, when the noise power is noticeable or over shadowing environment, almost performs near to ideal system. On the
other hand, a general and tractable distribution for CIR is necessary to calculate quality metrics of network, which can be adopted with path-loss, shadowing and multi-path fading channels. Our work- In this paper, we analytically study 1-D SCS performances such as CIR, rate and coverage. Our contributions can be summarized as follows:
In previous works shadowing and path-loss effects on performance of an SCS were investigated, however, no exact probability distribution had been calculated for the CIR. Here, we first investigate an SCS over only path-loss channels and calculate an analytical probability distribution for CIR.
We study an SCS over multi-path fading channels with Nakagami-m random variables and determine expressions for the distribution of inverse CIR. Then, the probability density function of CIR is calculated in an analytical form. Finally, the results are generalized to composite fading channels with path-loss, Lognormal shadowing and Nakagami-m fading.
Because of complicated calculations, an approximation for the distribution of the inverse CIR over composite fading channels is proposed, and its parameters are determined. So, a clear-tractable expression for the distribution of CIR is approximated. While reducing the mathematical complexity, this approximation provides fairly accurate PDF for the CIR.
The second quality metric, i.e. the downlink rate distribution, is calculated based on the MS received CIR, over the composite fading model. A tail probability and an average for the downlink rate are calculated. Then, using the Markov's inequality, a lower bound for the average rate is proposed. Also, the approximate distribution of the downlink rate is calculated, in the SCS over composite fading channels.
Coverage metric for the MS in an SCS is defined as the number of BSs which the received power of the MS from them is more than a threshold. We analytically calculate the third quality metric, i.e. coverage. Then, we compare the metric for different channel models.
Finally, we simulate the performance metrics using the exact analytically calculated expression and the approximate
one. We also compare them with the PDF which is
calculated using Monte-Carlo method. The rest of the paper is organized as follows: Section 2 describes the SCS model and reviews previous related works. In section 3, we explain main results. In section 4, the details of simulation results are presented and lastly, in section 5, we conclude the paper.
2. System Model and Previous Works 2.1 The System Model
We assume a uniform 1-D SCS, which the M BSs are placed along a line, according to a BS density ~
function , where
λ . Since the focus is on downlink, i.e. BS-to-MS performance analysis,
we consider a single MS. This MS, without loss of generality, is assumed to be located at the origin. The received power at the MS from a BS, is given by
.
.
. .
, where K,
,
and
are radio factor, transmitter power, shadowing factor, and multi-path fading factor, respectively. The path-loss is a function of the BS-MS separation R, and follows an inverse power law with as the pathloss exponent. Shadowing effect is usually modelled as a Log-normal random variable [3], [4]. Also, multi-path fading is generally modelled as a Nakagami-m random variable [13], which many other fading models are a special case of this general model. The MS communicates with only one BS. In a system with multiple channel reuse groups (CG), each BS is assigned to one CG. The channels are assumed to be perfectly orthogonal to each other. All the interferences are due to co-channel radios. One of the interest performance metrics in wireless systems is the signal quality at the MS. This metric is defined as the ratio of the received signal power to the sum of the interference powers, and it is denoted by CIR, when the interference power dominates the noise power. Within a CG, the BS with the strongest received signal power at the MS is chosen as the serving BS and all the remaining BSs will be co-channel interferers. The received power corresponding to the serving BS is denoted by interference power due to the other co-channel radios is denoted by
. Therefore, we have
Another performance metric is downlink rate which is denoted as
, and the total .
. This metric is defined as the
number of bits which an MS receives from the server BS in every second. We here calculate this metric as: 1
. Because of random nature of CIR in an SCS rate would be random, too.
Coverage is another important performance metric for a wireless system, which shows the ability of an MS to communicate with different BSs in a cellular system and it depends on the strength of the received signal. This metric also depends to BS placement in an SCS. Here we define coverage of an MS as number of BSs that their received CIR at the MS is upper than a threshold. We denote this random variable as
, and it is defined as:
∑
|
, that T denotes the
threshold level [11]. 2.2 Previous related works
In this subsection, we review the previous works. In [4] and [6], a 2-D SCS was considered, where the BSs were placed over the entire 2-D plane according to a 2-D Poisson process with a constant BS
density. They considered path-loss, while ignored shadowing, fading and noise. They showed that the CIR random variable only depends on the distances between BSs and MS, uniform ℓ-D SCS with a constant BS density BS density function
ℓ
.
ℓ
ℓ
,∀
.So, they concluded that a
is equivalent to a non-uniform one-sided 1-D SCS with a 0 , where
1,
2 ,and
4 . Therefore, all
analyses and results are sufficient to be stated in terms of a 1-D SCS. In [5], the performance of an SCS was compared with hexagonal cellular system over shadowing channels, and it was shown that the SCS is an useful system because its performance over very shadowed environment is close to hexagonal systems performance. In [1], the performance of a 1-D SCS with shadowing modelled by i.i.d Log-normal random variables has been studied. A semi-analytical expression for CIR and tail probability of CIR (it means the probability that CIR level is over a threshold level ,
1 ) obtained. Also, it was shown that an SCS
affected by Log-normal shadowing is equivalent to another SCS without shadowing with different BS density function. In [6], the performances of 1, 2 and 3-D SCSs with placing base stations as a non-homogenous Poisson distribution with random distances from origin have been investigated (r denotes the random distances). Also, a 1-D SCS with BS density function
were studied, where shadowing in the form of
i.i.d non-negative random factors, { }, is introduced. They showed that when the random variables
are
independent of the base stations placement, the resulting system is equivalent to another 1-D SCS with a different BS density as: , which
(1)
. denotes the statistical expectation with respect to . Also, it was shown that in a homogenous
ℓ-D SCS, shadowing does not affect the performance at the MS and in noisy shadowed channels this term is completely captured in the noise power. As mentioned, multi-path fading influences the performance of wireless systems, but as the best of our knowledge this term has not been studied in CIR performance of SCS. Also, the analytical distribution of CIR in an SCS has not been determined in a closed form for all channel models. The rate distribution and coverage of an MS in an SCS over multi-path general model have not been investigated, too. So, we analyse these metrics in section 3. 3. Main Results The received power at the MS from a BS is denoted with
. It can be expressed in a general form as:
.
.∏
.
and
where the shadowing and fading factors are introduced with
(2)
, respectively. The radio factor K
would be a random variable to capture the variations in antenna gains and orientations, but we just assume K as a unit constant. We model shadowing factor with variance
, which its distribution function is: P φ
, 1/
√
exp
.
(3)
≜ , is modelled as a Nakagami-m random variable denoted with
Also, multi-path fading factor ∼
≜ , as a zero mean Log-normal random variable
. It means that
is a gamma random variable with fading parameter m and the
distribution: f α
α
exp
α
(4)
In such a case, we can write the CIR at the MS in a general form as: ∏ ∑
∏
, ,
(5)
which subscripts ‘s’ and ‘i’ denote the serving BS and the co-channel interferer BSs, respectively. This expression has polynomial denominators, therefore, determining the distribution of CIR over this general channel model is complicated. So, we at first analyse the inverse CIR, and then by using the results, the PDF of CIR is calculated. 3.1 The CIR distribution for uniform 1-D SCS over channels with only path-loss
In this subsection, we obtain an expression for the distribution of inverse CIR, uniform 1-D SCS over channels with only path-loss
,
1 . From (5), we have:
,
, in the ∑
where M is the number of interferer BSs which is a Poisson random variable with mean . We denote , as the ratio of the received power at the MS from i-th interferer to the received power from server BS. The distribution of
is calculated as [14]: d
f
.
(6)
Proof: see appendix A. Now, assuming a non-random given M, the distribution of random variable f
/
x
where ‘*’ denotes convolution operator, and f to the M. Hence, the distribution of
f
x ∗ …∗ f
/
. denotes the conditional probability of x with respect
is written as:
x
would be: (7)
∑
f x
f
x .p M
|
η , (8)
which M is a Poisson random variable with parameter , so:
!
.
3.2 Multi-path fading effect on the CIR of the 1-D SCS
In this subsection, we calculate the performance of the SCS, assuming multi-path fading further path-loss channels
,
1,
,
1 . At first, the distribution of reverse CIR is calculated.
Then, the distribution and tail probability of CIR are calculated, in such a channel model. The expression (5), over this channel is rewritten as: . ∑
which
and
.
(9)
are Nakagami-m fading factors for the i-th interferer BS and the server BS channels to the
MS, respectively. The reverse of CIR is written as: ∑
≜
.
∑
.
(10)
In order to simplify, we define the tractable random variables " " and " " as: t ≜ which
v ≜ d .t
,
(11)
is a random variable introduced in section 3.1. So the inverse CIR is rewritten as:
∑
t
∑
d .t
∑
v .
(12)
and its distribution can be expressed as: ∑
f y
f
y ∗ …∗ f
y .p M
η
(13)
.
(14)
is calculated as:
The distribution of
f
!
v
.v
E
where "m" is the parameter of Nakagami-m distribution. Proof: see appendix B. Now, the analytical distribution of inverse CIR is calculated as following: f y
∑ ∑
f
y .p M
| !
.e
!
η .
y
E
∗ …∗ y
E
(15)
as:
We can analytically calculate the tail probability of CIR for an SCS over composite fading channels, P C/I
P
≜ , the analytical distribution of
If we define
f Ω
/
I/C
(16)
is calculated as follows: ∑
f
f y dy .
f
.p M
|
η.
(17)
3.3 Composite fading effect on the 1-D SCS
In this subsection, we show the effect of composite fading (path-loss, shadowing and multi-path fading) on a 1-D SCS, and find an analytical expression for probability density function of CIR. Moreover, the tail probability for this random variable is calculated. In [1], it has been proved that a shadowed SCS with BS density function
is equivalent to
another SCS with a different BS density. We use this result and determine the equivalent 1-D SCS BS density from (3): λ r
E
φ λ rφ
λ. E
φ
λ. e
≜λ
(18)
Now, the expressions (13) and (15) are expanded and the probability density function of inverse CIR over composite fading channels is concluded as: ∑
f y
f
|
y .p M
η λ
(19)
The tail probability of CIR over composite fading channels is calculated as: P
C I
θ !
∑
.e
!
y
E
∗ … ∗ y
E
dy λ . (20)
Shadowing affects the BS density function, therefore, the distributions of d and M change, respect to previous assumed channels modelled without shadowing. So we have: p M
ηλ
e
,
!
f
d
(21)
The distribution of CIR over this channel model is concluded by inserting (18) in (17), as: f Ω
f
∑
f
|
.p M
η λ r .
(22)
3.4 Approximate distribution of CIR over composite fading channel
In subsection 3.3, we determined the closed-analytical distributions of
≜
and
≜ , but the
results were complicated. Therefore, we introduce a novel approximate distribution for CIR in a dense SCS. Here, we use the Central Limit Theorem (CLT) to approximate the distribution of y. As we write
∑
≜
, which M and
are random variables, when the number of BSs over the region is
high (dense systems) according CLT, the approximate distribution can be written as: f y where σ and
exp
(23)
are variance and mean of random variable y, respectively. Since M and
are
independent, it can be easily proved that: μ σ
E∑
v
E M .σ
E M . E v , σ
. E v
which both mean and variance of M are . Mean and variance of μ ≜ E t .d σ ≜E t that mean of
(for m>2),
(for
(24) (25)
are calculated as:
E t .E d ,
.E d
(26)
Ev
(27)
∈ ) and square of them can be calculated as: !
!
Et
(28)
…
Ed
1 Et
Ed
∑
ε ln2
1 !
! …
1
2ε ln2
∑
(29)
(30)
1
(31)
Now, by using (22) and (23), we obtain an approximate expression for probability density function of CIR performance as: f Ω
/
exp
(32)
3.5 The downlink rate distribution of 1-D SCS over composite fading channel
In this section, we calculate the distribution of downlink rate, then average rate is defined, and a lower bound for average downlink rate is determined. We here define downlink rate as:
1
, where the strongest received signal at the MS is determining the server BS, and the other signals are interfering. So we use from relation between rate and CIR random variables. From (22), it is calculated as: f which
2e f e
1
. denotes the PDF of the downlink rate.
∑
f
|
.p M
η λ r
(33)
Proof: see appendix C. We analytically define the average rate as: μ ≜
.f
d
(34)
Since the expression for the downlink rate distribution is difficult to calculate, we define a lower bound for average downlink rate using Markov inequality as: ∗
μ which
∗
∗
.P
(35)
is a positive arbitrary rate. The tail probability of rate can be calculated as: ∗
P
P
e
∗
1
(36)
that the tail probability of CIR can analytically be determined from (20). 3.6 The coverage of an MS in a 1-D SCS over composite fading channels
Coverage of an MS in an SCS is analytically calculated in this subsection. Here every BS is chosen as a server, where other BSs are interfering. Then, the number of BSs that received power at the MS from them is upper than a threshold is defined as the coverage number. We calculate this random variable, i.e. , as [11]: ∑ The threshold
.∑
pM
|
(37)
1 ensures that only one BS can provide the CIR level. Since we assume that the MS
communicates with only one BS at a time, so the threshold is selected upper than one. In previous subsections, we determined the distribution and tail probability of CIR, so the coverage metric for the SCS with BS density ̅ M τ ∞ Pc ∑∞ τ 1 p M τ . ∑l 1 ∑η
is calculated using (37) and (22), as:
2m‐1 ! 1
Γ m
2
η
e‐λ
λη
1 T
η! 0
y m‐1 Ed1
d1 m y d1 2m
*…* y m‐1 Edη
dη m y dη
2m
dy
(38)
3.7 The approximate analyses of downlink rate and coverage of an MS
In section 3.4, we determined an approximate distribution of CIR in a 1-D SCS over composite fading channels. Now, we use the approximation to find the distribution of downlink rate and the coverage metric, which are easy to calculate. From (32) and (33), the approximate distribution of downlink rate is determined as: f
2e f e
1
exp
(39)
which mean and variance of y were introduced in (24) and (25). Also, the approximate coverage of the MS in an SCS over composite fading channels is concluded from (32) and (38), as:
∑
pM
.∑
|
exp
(40)
4. Simulation Results The details of simulations and comparisons between numerical, analytical and approximate results are presented. In every single trial, a random number
~
is generated for number of interferer
BSs. The region is a line with a length of 1000m, where a typical MS is assumed in the origin of this line, and the BSs are placed uniformly on the line. For calculating the numerical results, in every trial, the received power at the MS from each BS is computed by generating random variables for shadowing and multi-path fading. Monte-Carlo method is used with 1000000 number of iterations. In Fig. 1, the tail probability of CIR for a uniform 1-D SCS, over different channel models, at different BS densities are compared. Composite fading increases the tail probability, as was concluded in [1] with shadowing effect. Also, increasing the BS density, reduces the tail probability for all channel models; that it is because of increasing the interferer BSs. But when number of BSs goes up, the tail probability gets approximately fixed, because the ratios of interferer distances to server distance get very high; hence the path-loss gets approximately fix. It is the same as results in [1] and [6] , calculated for path-loss channels. Fig. 2 depicts the approximate distribution of CIR, the analytical distribution of CIR, and the numerical result which is calculated from Monte-Carlo method. This plot shows the ability of the approximation which obtains results close to the numerical results. We compare the approximate and numerical distributions of CIR for two different BS densities, in Fig. 3. It is concluded that the approximate results are fairly accurate. Also, it is obvious that increasing the BS density leads to upper interference, and reduces the average of CIR. The mean and median of CIR for case that =10, are 4.6 and 1.78, respectively. For =20 the measure of mean and median are 1.03 and 0.8, respectively. It shows that =20 for BS density is not proper for this network size and causes to low-quality communication with high interference, because of increasing interferers. The gap between the approximate and numerical results decreases when the BS density increases. We have calculated the changes of the gap in this figure as follows:
0.0049 0.0037 100 0.0049 0.0172 0.0131 100 20 0.0172
10
24.49% 23.8%
In Fig. 4, the tail probability of downlink rate for the SCS over different channel models are depicted at different BS densities. It is concluded that in this random system, downlink rate goes up over composite fading channels respect to the other models. Furthermore, increasing the BS density, reduces the downlink rate probability for all channel models. Fig. 5 plots the average rate simulations. Fig. 5a presents the lower bound of average rate at different BS densities, in the SCS over different channel models. Also, the composite fading model provides better lower bounds in low and medium BS sparseness. With the BS density increasing, the interference is increased and the order of the bounds change. Fig.5b depicts the average downlink rate, in the SCS over composite fading channels at different BS densities, for different path-loss exponents. In sparse SCS the path-loss exponent model has strong impact on the average rate. But in dense system the effect of pathloss exponent is reduced, because the ratios of the MS-interferer-BS distances to the MS-server distance goes up; hence the effect of ε in minus power reduces, so the curves get closer. Also, in sparse SCS, the greater exponent the higher the downlink rate. In Fig. 6 the approximate distribution of downlink rate is compared with Monte-Carlo result where BS density is ten, path-loss exponent is three and fading factor m is four. It is concluded that the approximate distribution provides results fairly accurate. Fig. 7 plots the coverage of the MS, in an SCS over different channel models. It is obvious that composite fading channel provides better coverage for the MS in this random system, too. 5. Conclusion In this paper, we studied the downlink CIR performance in a cellular system with randomly distributed BSs, i.e. SCS. We assumed path-loss, Lognormal shadowing and Nakagami-m fading to model the BSs-MS channels. The direct calculation of CIR distribution was complicated. So, we first analytically calculated the distribution of inverse CIR in a closed form. Then, the distribution of CIR was determined using the obtained results. Since the analysis of the analytical expression was very complex, we proposed a simple approximation for the distribution of CIR. We compared the approximate and the exact distributions, and also exploited the Monte-Carlo results by simulations, in order to show the ability of approximation and correctness of the analytical expression. We calculated the distributions of the downlink rate and the coverage in the SCS, too. Also, a closed form and a lower bound for the average rate were proposed. The proposed approximate distribution of CIR was used to approximately calculate the distribution of rate and coverage, as well. The approximate expressions were compared with numerical results by simulations. The comparisons showed the effectiveness of the approximations. Also, the simulations illustrated that the composite fading provides better metric performances in the SCS.
0.9 0.85
Tail Probability of CIR
0.8 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
0.75 0.7 0.65 0.6 0.55 0.5 0.45
2
4
6
8
10
12
14
16
18
20
BS density
Fig. 1 The CIR tail probability at different BS densities, in an uniform 1-D SCS over different channel models.(
,
0.035 Mont-Carlo Approximated pdf Analytical expression pdf
0.03
CIR distribution
0.025
0.02
0.015
0.01
0.005
0 0.5
1
1.5
2
2.5
3
CIR
Fig. 2 The comparison of analytical, approximate and numerical distributions of CIR in an uniform 1-D SCS.(
, m=4)
)
0.018 Numerical, BS density=20 Analytical, BS density=20 Numerical, BS density=10 Analytical, BS density=10
0.016
Distribution of CIR
0.014 0.012 0.01 0.008 0.006 0.004 0.002 0
0
0.5
1
1.5
2
2.5 CIR
3
3.5
4
4.5
5
Fig. 3 The comparison of numerical and approximate distributions of CIR at different BS densities.(
, m=4)
0.75 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
Tail Probability of Rate P(R>1)
0.7
0.65
0.6
0.55
0.5
0.45
2
4
6
8
10
12
14
16
18
20
BS density
Fig. 4 Tail probability of the downlink rate at different BS densities, for the uniform 1-D SCS over different channel models.( ,m=2)
1
4 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
3
0.9
A v e ra g e R a te
L o w e r b o u n d o f A v e r a g e R a te
0.95
Path-loss exponent=2 Path-loss exponent=3 Path-loss exponent=4
3.5
0.85
2.5 2 1.5
0.8
1
0.75
0.5 0.7
2
4
6
8
10
12
BS density
a
14
16
18
20
0
0
2
4
6
8
10
12
14
16
18
20
BS density
b
Fig. 5 a) The lower bound of the average rate at different BS densities, for the uniform 1-D SCSs over different channel models.( =4,m=2) b) The average downlink rate at different BS densities, for different path-loss models.(m=4)
0.035 Numerical Results Approximated Results
0.03
Rate Distribution
0.025
0.02
0.015
0.01
0.005
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Rate Fig. 6 The approximate and numerical distribution of downlink rate. (
,
)
0.95 0.9
Average Coverage Number
0.85 0.8 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
0.75 0.7 0.65 0.6 0.55 0.5 0.45
1
2
3
4
5
6
7
8
9
10
BS density
Fig. 7 The average of coverage at different BS densities, for the uniform 1-D SCSs over different channel models.( =3, m=4)
6. Appendices
In the following mathematical derivations, we have used the usual relations for the joint and marginal distribution functions from [14].
Appendix A
The random variable
is defined as a function of random variables
the 1-D homogenous SCS, the random variables we define
≜
≜
and
and
and
as:
. In
are independent exponential with mean . So,
to use the joint distribution, and write: R ,R
,
(41)
. , . denotes the Jacobian matrix. Then, the joint density function of random variables
where
and
is
written as: f
,
So the distribution of random variable
Now, we have:
f
d
exp
.
c
(42)
is calculated as: f
.
, so f
,
g
f
,
,
g ,c
g , c dc
,
(43)
. The distribution of d
/
.d
is calculated as:
/
/
(44)
Appendix B
To calculate the distribution of , we define random variable " ≜
" to use the joint distribution,
and write: α ,α
4
4√t ,
(45)
. , . denotes the Jacobian matrix. Then, the joint density function for random variables
where
and u is
written as: f
,
t ,u
,
,
f
,
So the distribution of random variable f t
f
,
t , u du
√u. t . f
√u
u
t
exp
1
t u
(46)
is calculated as: t
u
. exp
1
t u du
t
B (47)
which the integral B is defined as
≜
.
1
, and it is calculated as [15]:
B
exp
1
∑
t u .
Therefore, we have the distribution of
∏
1
distribution of
j
!
(48)
as: !
f t with the distributions of
2m
(49)
in (49), d in (6), and another use of Jacobian matrix we determine the
as follows : f
v
!
v
f
d . dd
(50)
Comparing this expression with statistical expectation definition we write: f
!
v
.v
E
.
(51)
Appendix C
We define random variable
1
as a function of CIR, as :
1, and therefore,
is CIR and it reversely is calculated as:
, which the random variable 2
. Using the relations for
the distribution of function of a random variable, it is concluded: f
2e f e
1 .
(52)
7. References
[1] P. Madhusudhanan, J. G. Restrepo, Y. E. Liu and T. X. Brown, "Carrier to interference ratio analysis for the shotgun cellular system," in IEEE Global Telecommunications Conference (GLOBECOM), Honolulu, 2009. [2] V. Macdonald, "The Cellular Concept," The Bell SystemTechnical Journal, 1979, 58, pp. 15-41. [3] T. Brown, "Analysis and Coloring of a Shotgun Cellular System," in IEEE Radio and Wireless Conference (RAWCON), Colorado Springs, 1998. [4] T. Brown, "Dynamic channel assignment in shotgun cellular systems," in IEEE Radio and Wireless Conference, Denver, CO, 1999. [5] T. Brown, "Cellular performance bounds via shotgun cellular systems," IEEE Journal on Selected Areas in Communications (JSAC), 2000, 18, (11), pp. 2443 - 2455. [6] P. Madhusudhanan, J. Restrepo, Y. Liu, T. Brown and K. Baker, "Generalized carrier to interference ratio analysis for the shotgun cellular system in multiple dimensions," CORR, Vol. abs/1002,3943, 2011. [7] C. Tepedelenlioğlu, A. Rajan and Y. Zhang, "Applications of stochastic ordering to wireless communications," IEEE Transactions on Wireless Communications, 2011, 10, (12), pp. 4249-4257. [8] P. Madhusudhanan, J. G. Restrepo, Y. Liu, T. X. Brown and K. R. Baker, "Stochastic ordering based carrier-to-interference ratio analysis for the shotgun cellular systems," IEEE Wireless Communications Letters, 2012, 1, (6), pp. 565-568. [9] P. Madhusudhanan, J. G. Restrepo, Y. Liu, T. X. Brown and K. R. Baker, "Multi-tier network performance analysis using a shotgun cellular system," in IEEE Global Telecommunications Conference
(GLOBECOM), Houston, 2011. [10] H. S. Dhillon and J. G. Andrews, "Downlink rate distribution in heterogeneous cellular networks under generalized cell selection," IEEE Wireless Communications Letters, 2014, 3, (1), pp. 42-45. [11] J. G. Andrews, F. Baccelli and R. K. Ganti, "A tractable approach to coverage and rate in cellular networks," IEEE Transactions on Communications, 2011, 59, (11), pp. 3122-3134. [12] H. P. Keeler, B. Blaszczyszyn and M. K. Karray, "SINR-based k-coverage probability in cellular networks with arbitrary shadowing," in IEEE International Symposium on Information Theory Proceedings (ISIT), Istanbul, 2013. [13] P. C. Pinto and M. Z. Win, "A Unifying Framework for Local Throughput in Wireless Networks," arXiv:1007.2814., 1, 2010. [14] A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic processes, McGraw-Hill, 1985. [15] I. Gradshteyn and I. Ryzhik, Table of integrals, series, and products, Elsevier , 2007.
Asm ma Bagherri received the B.Sc. and a the M.Sc. degree in electriccal engineerring from m the Ferdo owsi Univerrsity of Masshhad, Iran,, in 2010 annd 2013, resspectively. She is currently c pu ursuing the P Ph.D. degreee in commu unications aat the Babol Universityy of Tecchnology. Her H current research in nterests include mobilee networks and cognittive radios.
hosheh Abeed Hodtanii received the t B.Sc. degree in eleectronics en ngineering and Gh the M.Sc. deg gree in com mmunication ns engineerin ng, both froom Isfahan n Universityy of Tecchnology, Isfahan, Irran ,in 198 85, 1987, respectivelyy. He join ned Electrical Enggineering Dept., D at Ferrdowsi Univ versity of Mashhad, M M Mashhad, Iraan, in 1987. He deccided to purrsue his studdies in 2005 5 and receiv ved the Ph..D. degree (with ( excelllent grade) from Sharif S Univeersity of Teechnology, Tehran, Iraan, in 2008 8. His reseaarch inteerests are in multi-uuser inform mation theorry, commuunication theory, wirelless com mmunicatioons and signnal processing. Dr. Hoddtani is the author of a textbook oon electricall circuits and is thee winner of the best papper award at IEEE ICT T -2010
PII: DOI: Reference:
S1874-4907(16)30156-2 http://dx.doi.org/10.1016/j.phycom.2017.07.005 PHYCOM 407
To appear in:
Physical Communication
Received date : 3 October 2016 Revised date : 26 May 2017 Accepted date : 24 July 2017 Please cite this article as: A. Bagheri, G.A. Hodtani, Carrier to interference ratio, rate and coverage analysis in shotgun cellular systems over composite fading channels, Physical Communication (2017), http://dx.doi.org/10.1016/j.phycom.2017.07.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Carrier to Interference Ratio, Rate and coverage Analysis in Shotgun Cellular Systems over Composite Fading Channels Asma Bagheri1 , Ghosheh Abed Hodtani2 1,2 2
Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran [email protected]; [email protected]
Abstract: Wireless communication performance in shotgun cellular system (SCS: wireless communication system with randomly placed base stations) is analysed. Aiming at this analysis, (i) the carrier to interference ratio (CIR), as an important measure in a cellular network, is considered in the downlink (path-loss, shadowing and multi-path) fading channel. Since direct calculating of the distribution of CIR is complicated, a related mathematical equation is determined through which the probability density function (PDF) of CIR can be calculated. And then, as a special case, the tail probability of CIR is obtained. (ii) A normal approximation for inverse of CIR is proposed which is applicable for calculating a tractable PDF for the downlink CIR. The analytically calculated and the approximate PDFs of CIR are compared with numerical PDFs. (iii) The distribution of the downlink rate and a lower bound for the average rate; the analytical expression for coverage of a user in an SCS based on its received CIR, and an average value for coverage are calculated. (iv) Simulation results show that the closed form and approximate PDFs over different models are close to numerical ones. Keywords- Carrier to interference ratio (CIR), Coverage, Multi-path fading effect, Rate and average rate, Shadowing effect, Shotgun cellular system, Tail probability of CIR. 1. Introduction Cellular communication consists of a set of radio base stations (BSs) distributed over a region that communicate with mobile stations (MSs). In contrast to hexagonal cellular systems (i.e. ideal systems) with regular BS placement in many wireless systems such as Local Area Networks (LANs) and femtocells [1], due to site acquisition difficulties, BSs are placed irregularly over the deployment region. These random systems called shotgun cellular systems (SCSs) can be described by BS density function
as a
function of the distances between BSs and the MS. In an SCS the signal propagation is affected by three phenomena. First is path-loss effect (here is random), the second is Log-normal shadowing, and third is multi-path fading effect. Hence many cellular deployments have significant randomness. So that, an SCS is a system affected by random phenomena. The SCS and its performance metrics have been studied under different channel models [2-10]. Introducing a new random substitute for hexagonal systems was started by [2]. In [3] the downlink communication in an SCS was compared with hexagonal system and it was shown that in Log-normal shadowing, SCS with fixed channel assignment has the signal level of about 4db lower than that of hexagonal cellular systems. So, in an SCS with random channel assignment the system performance needs
1
to be analysed. In [4] dynamic channel assignment (DCA) in an SCS was investigated, and the results were compared with hexagonal system. Also, downlink and uplink communications were both studied, and it was shown that performances of both links are the same, and system performance is improved to near ideal system performances, with self-organized DCA. In [5] CIR performance bounds for two-dimensional (2-D) interference-limited radio cellular systems with random channel assignment are introduced. It was calculated that an SCS provides lower bound for performance of cellular systems and hexagonal systems provide upper bound. They showed that the difference between upper and lower bounds is small under typical operating conditions, in modern TDMA and CDMA cellular systems. Furthermore, it was shown that under strong shadowing the bounds converge. Also, an SCS was studied in different scenarios such as different numbers of channel groups (CGs). In highly varying environment and low CIR levels, SCS is most similar to the hexagonal systems. In [6] CIR performance at an MS was investigated for one, two and three-dimensional SCS, and analytical tail probability of CIR was obtained. Also, the results of [1] were generalized to the multidimensional cases. Study of the CIR at an MS in an SCS with Poisson process is difficult and comparing of systems is not easy. Therefore, a simple analytical tool based on stochastic ordering was developed to compare the distributions of CIR at the MS in two SCSs [7], [8]. In [9] a multi-tier network composed of M tiers of homogenous N-dimensional systems was considered, and CIR performance at an MS for such a network was investigated. Rate and coverage are other important metrics of a wireless network. In [10], under a special channel model, the distribution of downlink rate at a user in a Heterogeneous cellular network was studied, where they assumed Log-normal shadowing and a special unit mean Rayleigh fading. In [11] the average rate of a cellular network with Poisson point process BSs placement was analysed, where only path-loss and unit mean Rayleigh fading with limited interference were assumed. Also, probability of coverage was defined as the probability that the Signal-to-Noise-Ratio (SINR) is upper than a threshold, which it was simulated based on the tail probability of SINR. The calculated expressions for distribution of SINR, although simple, are applicable in a special channel model. In [12], the coverage number for an MS in an SCS has been defined as the number of BSs which the received power of MS from them is more than a threshold. Assuming low power path-loss, limited shadowing with an arbitrary distribution, and Rayleigh fading, they computed numerically the probability that the coverage number is upper than a threshold k. Literature review shows that an SCS in addition to easiness and ability to be designed fast, when the noise power is noticeable or over shadowing environment, almost performs near to ideal system. On the
other hand, a general and tractable distribution for CIR is necessary to calculate quality metrics of network, which can be adopted with path-loss, shadowing and multi-path fading channels. Our work- In this paper, we analytically study 1-D SCS performances such as CIR, rate and coverage. Our contributions can be summarized as follows:
In previous works shadowing and path-loss effects on performance of an SCS were investigated, however, no exact probability distribution had been calculated for the CIR. Here, we first investigate an SCS over only path-loss channels and calculate an analytical probability distribution for CIR.
We study an SCS over multi-path fading channels with Nakagami-m random variables and determine expressions for the distribution of inverse CIR. Then, the probability density function of CIR is calculated in an analytical form. Finally, the results are generalized to composite fading channels with path-loss, Lognormal shadowing and Nakagami-m fading.
Because of complicated calculations, an approximation for the distribution of the inverse CIR over composite fading channels is proposed, and its parameters are determined. So, a clear-tractable expression for the distribution of CIR is approximated. While reducing the mathematical complexity, this approximation provides fairly accurate PDF for the CIR.
The second quality metric, i.e. the downlink rate distribution, is calculated based on the MS received CIR, over the composite fading model. A tail probability and an average for the downlink rate are calculated. Then, using the Markov's inequality, a lower bound for the average rate is proposed. Also, the approximate distribution of the downlink rate is calculated, in the SCS over composite fading channels.
Coverage metric for the MS in an SCS is defined as the number of BSs which the received power of the MS from them is more than a threshold. We analytically calculate the third quality metric, i.e. coverage. Then, we compare the metric for different channel models.
Finally, we simulate the performance metrics using the exact analytically calculated expression and the approximate
one. We also compare them with the PDF which is
calculated using Monte-Carlo method. The rest of the paper is organized as follows: Section 2 describes the SCS model and reviews previous related works. In section 3, we explain main results. In section 4, the details of simulation results are presented and lastly, in section 5, we conclude the paper.
2. System Model and Previous Works 2.1 The System Model
We assume a uniform 1-D SCS, which the M BSs are placed along a line, according to a BS density ~
function , where
λ . Since the focus is on downlink, i.e. BS-to-MS performance analysis,
we consider a single MS. This MS, without loss of generality, is assumed to be located at the origin. The received power at the MS from a BS, is given by
.
.
. .
, where K,
,
and
are radio factor, transmitter power, shadowing factor, and multi-path fading factor, respectively. The path-loss is a function of the BS-MS separation R, and follows an inverse power law with as the pathloss exponent. Shadowing effect is usually modelled as a Log-normal random variable [3], [4]. Also, multi-path fading is generally modelled as a Nakagami-m random variable [13], which many other fading models are a special case of this general model. The MS communicates with only one BS. In a system with multiple channel reuse groups (CG), each BS is assigned to one CG. The channels are assumed to be perfectly orthogonal to each other. All the interferences are due to co-channel radios. One of the interest performance metrics in wireless systems is the signal quality at the MS. This metric is defined as the ratio of the received signal power to the sum of the interference powers, and it is denoted by CIR, when the interference power dominates the noise power. Within a CG, the BS with the strongest received signal power at the MS is chosen as the serving BS and all the remaining BSs will be co-channel interferers. The received power corresponding to the serving BS is denoted by interference power due to the other co-channel radios is denoted by
. Therefore, we have
Another performance metric is downlink rate which is denoted as
, and the total .
. This metric is defined as the
number of bits which an MS receives from the server BS in every second. We here calculate this metric as: 1
. Because of random nature of CIR in an SCS rate would be random, too.
Coverage is another important performance metric for a wireless system, which shows the ability of an MS to communicate with different BSs in a cellular system and it depends on the strength of the received signal. This metric also depends to BS placement in an SCS. Here we define coverage of an MS as number of BSs that their received CIR at the MS is upper than a threshold. We denote this random variable as
, and it is defined as:
∑
|
, that T denotes the
threshold level [11]. 2.2 Previous related works
In this subsection, we review the previous works. In [4] and [6], a 2-D SCS was considered, where the BSs were placed over the entire 2-D plane according to a 2-D Poisson process with a constant BS
density. They considered path-loss, while ignored shadowing, fading and noise. They showed that the CIR random variable only depends on the distances between BSs and MS, uniform ℓ-D SCS with a constant BS density BS density function
ℓ
.
ℓ
ℓ
,∀
.So, they concluded that a
is equivalent to a non-uniform one-sided 1-D SCS with a 0 , where
1,
2 ,and
4 . Therefore, all
analyses and results are sufficient to be stated in terms of a 1-D SCS. In [5], the performance of an SCS was compared with hexagonal cellular system over shadowing channels, and it was shown that the SCS is an useful system because its performance over very shadowed environment is close to hexagonal systems performance. In [1], the performance of a 1-D SCS with shadowing modelled by i.i.d Log-normal random variables has been studied. A semi-analytical expression for CIR and tail probability of CIR (it means the probability that CIR level is over a threshold level ,
1 ) obtained. Also, it was shown that an SCS
affected by Log-normal shadowing is equivalent to another SCS without shadowing with different BS density function. In [6], the performances of 1, 2 and 3-D SCSs with placing base stations as a non-homogenous Poisson distribution with random distances from origin have been investigated (r denotes the random distances). Also, a 1-D SCS with BS density function
were studied, where shadowing in the form of
i.i.d non-negative random factors, { }, is introduced. They showed that when the random variables
are
independent of the base stations placement, the resulting system is equivalent to another 1-D SCS with a different BS density as: , which
(1)
. denotes the statistical expectation with respect to . Also, it was shown that in a homogenous
ℓ-D SCS, shadowing does not affect the performance at the MS and in noisy shadowed channels this term is completely captured in the noise power. As mentioned, multi-path fading influences the performance of wireless systems, but as the best of our knowledge this term has not been studied in CIR performance of SCS. Also, the analytical distribution of CIR in an SCS has not been determined in a closed form for all channel models. The rate distribution and coverage of an MS in an SCS over multi-path general model have not been investigated, too. So, we analyse these metrics in section 3. 3. Main Results The received power at the MS from a BS is denoted with
. It can be expressed in a general form as:
.
.∏
.
and
where the shadowing and fading factors are introduced with
(2)
, respectively. The radio factor K
would be a random variable to capture the variations in antenna gains and orientations, but we just assume K as a unit constant. We model shadowing factor with variance
, which its distribution function is: P φ
, 1/
√
exp
.
(3)
≜ , is modelled as a Nakagami-m random variable denoted with
Also, multi-path fading factor ∼
≜ , as a zero mean Log-normal random variable
. It means that
is a gamma random variable with fading parameter m and the
distribution: f α
α
exp
α
(4)
In such a case, we can write the CIR at the MS in a general form as: ∏ ∑
∏
, ,
(5)
which subscripts ‘s’ and ‘i’ denote the serving BS and the co-channel interferer BSs, respectively. This expression has polynomial denominators, therefore, determining the distribution of CIR over this general channel model is complicated. So, we at first analyse the inverse CIR, and then by using the results, the PDF of CIR is calculated. 3.1 The CIR distribution for uniform 1-D SCS over channels with only path-loss
In this subsection, we obtain an expression for the distribution of inverse CIR, uniform 1-D SCS over channels with only path-loss
,
1 . From (5), we have:
,
, in the ∑
where M is the number of interferer BSs which is a Poisson random variable with mean . We denote , as the ratio of the received power at the MS from i-th interferer to the received power from server BS. The distribution of
is calculated as [14]: d
f
.
(6)
Proof: see appendix A. Now, assuming a non-random given M, the distribution of random variable f
/
x
where ‘*’ denotes convolution operator, and f to the M. Hence, the distribution of
f
x ∗ …∗ f
/
. denotes the conditional probability of x with respect
is written as:
x
would be: (7)
∑
f x
f
x .p M
|
η , (8)
which M is a Poisson random variable with parameter , so:
!
.
3.2 Multi-path fading effect on the CIR of the 1-D SCS
In this subsection, we calculate the performance of the SCS, assuming multi-path fading further path-loss channels
,
1,
,
1 . At first, the distribution of reverse CIR is calculated.
Then, the distribution and tail probability of CIR are calculated, in such a channel model. The expression (5), over this channel is rewritten as: . ∑
which
and
.
(9)
are Nakagami-m fading factors for the i-th interferer BS and the server BS channels to the
MS, respectively. The reverse of CIR is written as: ∑
≜
.
∑
.
(10)
In order to simplify, we define the tractable random variables " " and " " as: t ≜ which
v ≜ d .t
,
(11)
is a random variable introduced in section 3.1. So the inverse CIR is rewritten as:
∑
t
∑
d .t
∑
v .
(12)
and its distribution can be expressed as: ∑
f y
f
y ∗ …∗ f
y .p M
η
(13)
.
(14)
is calculated as:
The distribution of
f
!
v
.v
E
where "m" is the parameter of Nakagami-m distribution. Proof: see appendix B. Now, the analytical distribution of inverse CIR is calculated as following: f y
∑ ∑
f
y .p M
| !
.e
!
η .
y
E
∗ …∗ y
E
(15)
as:
We can analytically calculate the tail probability of CIR for an SCS over composite fading channels, P C/I
P
≜ , the analytical distribution of
If we define
f Ω
/
I/C
(16)
is calculated as follows: ∑
f
f y dy .
f
.p M
|
η.
(17)
3.3 Composite fading effect on the 1-D SCS
In this subsection, we show the effect of composite fading (path-loss, shadowing and multi-path fading) on a 1-D SCS, and find an analytical expression for probability density function of CIR. Moreover, the tail probability for this random variable is calculated. In [1], it has been proved that a shadowed SCS with BS density function
is equivalent to
another SCS with a different BS density. We use this result and determine the equivalent 1-D SCS BS density from (3): λ r
E
φ λ rφ
λ. E
φ
λ. e
≜λ
(18)
Now, the expressions (13) and (15) are expanded and the probability density function of inverse CIR over composite fading channels is concluded as: ∑
f y
f
|
y .p M
η λ
(19)
The tail probability of CIR over composite fading channels is calculated as: P
C I
θ !
∑
.e
!
y
E
∗ … ∗ y
E
dy λ . (20)
Shadowing affects the BS density function, therefore, the distributions of d and M change, respect to previous assumed channels modelled without shadowing. So we have: p M
ηλ
e
,
!
f
d
(21)
The distribution of CIR over this channel model is concluded by inserting (18) in (17), as: f Ω
f
∑
f
|
.p M
η λ r .
(22)
3.4 Approximate distribution of CIR over composite fading channel
In subsection 3.3, we determined the closed-analytical distributions of
≜
and
≜ , but the
results were complicated. Therefore, we introduce a novel approximate distribution for CIR in a dense SCS. Here, we use the Central Limit Theorem (CLT) to approximate the distribution of y. As we write
∑
≜
, which M and
are random variables, when the number of BSs over the region is
high (dense systems) according CLT, the approximate distribution can be written as: f y where σ and
exp
(23)
are variance and mean of random variable y, respectively. Since M and
are
independent, it can be easily proved that: μ σ
E∑
v
E M .σ
E M . E v , σ
. E v
which both mean and variance of M are . Mean and variance of μ ≜ E t .d σ ≜E t that mean of
(for m>2),
(for
(24) (25)
are calculated as:
E t .E d ,
.E d
(26)
Ev
(27)
∈ ) and square of them can be calculated as: !
!
Et
(28)
…
Ed
1 Et
Ed
∑
ε ln2
1 !
! …
1
2ε ln2
∑
(29)
(30)
1
(31)
Now, by using (22) and (23), we obtain an approximate expression for probability density function of CIR performance as: f Ω
/
exp
(32)
3.5 The downlink rate distribution of 1-D SCS over composite fading channel
In this section, we calculate the distribution of downlink rate, then average rate is defined, and a lower bound for average downlink rate is determined. We here define downlink rate as:
1
, where the strongest received signal at the MS is determining the server BS, and the other signals are interfering. So we use from relation between rate and CIR random variables. From (22), it is calculated as: f which
2e f e
1
. denotes the PDF of the downlink rate.
∑
f
|
.p M
η λ r
(33)
Proof: see appendix C. We analytically define the average rate as: μ ≜
.f
d
(34)
Since the expression for the downlink rate distribution is difficult to calculate, we define a lower bound for average downlink rate using Markov inequality as: ∗
μ which
∗
∗
.P
(35)
is a positive arbitrary rate. The tail probability of rate can be calculated as: ∗
P
P
e
∗
1
(36)
that the tail probability of CIR can analytically be determined from (20). 3.6 The coverage of an MS in a 1-D SCS over composite fading channels
Coverage of an MS in an SCS is analytically calculated in this subsection. Here every BS is chosen as a server, where other BSs are interfering. Then, the number of BSs that received power at the MS from them is upper than a threshold is defined as the coverage number. We calculate this random variable, i.e. , as [11]: ∑ The threshold
.∑
pM
|
(37)
1 ensures that only one BS can provide the CIR level. Since we assume that the MS
communicates with only one BS at a time, so the threshold is selected upper than one. In previous subsections, we determined the distribution and tail probability of CIR, so the coverage metric for the SCS with BS density ̅ M τ ∞ Pc ∑∞ τ 1 p M τ . ∑l 1 ∑η
is calculated using (37) and (22), as:
2m‐1 ! 1
Γ m
2
η
e‐λ
λη
1 T
η! 0
y m‐1 Ed1
d1 m y d1 2m
*…* y m‐1 Edη
dη m y dη
2m
dy
(38)
3.7 The approximate analyses of downlink rate and coverage of an MS
In section 3.4, we determined an approximate distribution of CIR in a 1-D SCS over composite fading channels. Now, we use the approximation to find the distribution of downlink rate and the coverage metric, which are easy to calculate. From (32) and (33), the approximate distribution of downlink rate is determined as: f
2e f e
1
exp
(39)
which mean and variance of y were introduced in (24) and (25). Also, the approximate coverage of the MS in an SCS over composite fading channels is concluded from (32) and (38), as:
∑
pM
.∑
|
exp
(40)
4. Simulation Results The details of simulations and comparisons between numerical, analytical and approximate results are presented. In every single trial, a random number
~
is generated for number of interferer
BSs. The region is a line with a length of 1000m, where a typical MS is assumed in the origin of this line, and the BSs are placed uniformly on the line. For calculating the numerical results, in every trial, the received power at the MS from each BS is computed by generating random variables for shadowing and multi-path fading. Monte-Carlo method is used with 1000000 number of iterations. In Fig. 1, the tail probability of CIR for a uniform 1-D SCS, over different channel models, at different BS densities are compared. Composite fading increases the tail probability, as was concluded in [1] with shadowing effect. Also, increasing the BS density, reduces the tail probability for all channel models; that it is because of increasing the interferer BSs. But when number of BSs goes up, the tail probability gets approximately fixed, because the ratios of interferer distances to server distance get very high; hence the path-loss gets approximately fix. It is the same as results in [1] and [6] , calculated for path-loss channels. Fig. 2 depicts the approximate distribution of CIR, the analytical distribution of CIR, and the numerical result which is calculated from Monte-Carlo method. This plot shows the ability of the approximation which obtains results close to the numerical results. We compare the approximate and numerical distributions of CIR for two different BS densities, in Fig. 3. It is concluded that the approximate results are fairly accurate. Also, it is obvious that increasing the BS density leads to upper interference, and reduces the average of CIR. The mean and median of CIR for case that =10, are 4.6 and 1.78, respectively. For =20 the measure of mean and median are 1.03 and 0.8, respectively. It shows that =20 for BS density is not proper for this network size and causes to low-quality communication with high interference, because of increasing interferers. The gap between the approximate and numerical results decreases when the BS density increases. We have calculated the changes of the gap in this figure as follows:
0.0049 0.0037 100 0.0049 0.0172 0.0131 100 20 0.0172
10
24.49% 23.8%
In Fig. 4, the tail probability of downlink rate for the SCS over different channel models are depicted at different BS densities. It is concluded that in this random system, downlink rate goes up over composite fading channels respect to the other models. Furthermore, increasing the BS density, reduces the downlink rate probability for all channel models. Fig. 5 plots the average rate simulations. Fig. 5a presents the lower bound of average rate at different BS densities, in the SCS over different channel models. Also, the composite fading model provides better lower bounds in low and medium BS sparseness. With the BS density increasing, the interference is increased and the order of the bounds change. Fig.5b depicts the average downlink rate, in the SCS over composite fading channels at different BS densities, for different path-loss exponents. In sparse SCS the path-loss exponent model has strong impact on the average rate. But in dense system the effect of pathloss exponent is reduced, because the ratios of the MS-interferer-BS distances to the MS-server distance goes up; hence the effect of ε in minus power reduces, so the curves get closer. Also, in sparse SCS, the greater exponent the higher the downlink rate. In Fig. 6 the approximate distribution of downlink rate is compared with Monte-Carlo result where BS density is ten, path-loss exponent is three and fading factor m is four. It is concluded that the approximate distribution provides results fairly accurate. Fig. 7 plots the coverage of the MS, in an SCS over different channel models. It is obvious that composite fading channel provides better coverage for the MS in this random system, too. 5. Conclusion In this paper, we studied the downlink CIR performance in a cellular system with randomly distributed BSs, i.e. SCS. We assumed path-loss, Lognormal shadowing and Nakagami-m fading to model the BSs-MS channels. The direct calculation of CIR distribution was complicated. So, we first analytically calculated the distribution of inverse CIR in a closed form. Then, the distribution of CIR was determined using the obtained results. Since the analysis of the analytical expression was very complex, we proposed a simple approximation for the distribution of CIR. We compared the approximate and the exact distributions, and also exploited the Monte-Carlo results by simulations, in order to show the ability of approximation and correctness of the analytical expression. We calculated the distributions of the downlink rate and the coverage in the SCS, too. Also, a closed form and a lower bound for the average rate were proposed. The proposed approximate distribution of CIR was used to approximately calculate the distribution of rate and coverage, as well. The approximate expressions were compared with numerical results by simulations. The comparisons showed the effectiveness of the approximations. Also, the simulations illustrated that the composite fading provides better metric performances in the SCS.
0.9 0.85
Tail Probability of CIR
0.8 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
0.75 0.7 0.65 0.6 0.55 0.5 0.45
2
4
6
8
10
12
14
16
18
20
BS density
Fig. 1 The CIR tail probability at different BS densities, in an uniform 1-D SCS over different channel models.(
,
0.035 Mont-Carlo Approximated pdf Analytical expression pdf
0.03
CIR distribution
0.025
0.02
0.015
0.01
0.005
0 0.5
1
1.5
2
2.5
3
CIR
Fig. 2 The comparison of analytical, approximate and numerical distributions of CIR in an uniform 1-D SCS.(
, m=4)
)
0.018 Numerical, BS density=20 Analytical, BS density=20 Numerical, BS density=10 Analytical, BS density=10
0.016
Distribution of CIR
0.014 0.012 0.01 0.008 0.006 0.004 0.002 0
0
0.5
1
1.5
2
2.5 CIR
3
3.5
4
4.5
5
Fig. 3 The comparison of numerical and approximate distributions of CIR at different BS densities.(
, m=4)
0.75 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
Tail Probability of Rate P(R>1)
0.7
0.65
0.6
0.55
0.5
0.45
2
4
6
8
10
12
14
16
18
20
BS density
Fig. 4 Tail probability of the downlink rate at different BS densities, for the uniform 1-D SCS over different channel models.( ,m=2)
1
4 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
3
0.9
A v e ra g e R a te
L o w e r b o u n d o f A v e r a g e R a te
0.95
Path-loss exponent=2 Path-loss exponent=3 Path-loss exponent=4
3.5
0.85
2.5 2 1.5
0.8
1
0.75
0.5 0.7
2
4
6
8
10
12
BS density
a
14
16
18
20
0
0
2
4
6
8
10
12
14
16
18
20
BS density
b
Fig. 5 a) The lower bound of the average rate at different BS densities, for the uniform 1-D SCSs over different channel models.( =4,m=2) b) The average downlink rate at different BS densities, for different path-loss models.(m=4)
0.035 Numerical Results Approximated Results
0.03
Rate Distribution
0.025
0.02
0.015
0.01
0.005
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Rate Fig. 6 The approximate and numerical distribution of downlink rate. (
,
)
0.95 0.9
Average Coverage Number
0.85 0.8 Pathloss Shadowing-Pathloss Multipath-Pathloss Composite Fading
0.75 0.7 0.65 0.6 0.55 0.5 0.45
1
2
3
4
5
6
7
8
9
10
BS density
Fig. 7 The average of coverage at different BS densities, for the uniform 1-D SCSs over different channel models.( =3, m=4)
6. Appendices
In the following mathematical derivations, we have used the usual relations for the joint and marginal distribution functions from [14].
Appendix A
The random variable
is defined as a function of random variables
the 1-D homogenous SCS, the random variables we define
≜
≜
and
and
and
as:
. In
are independent exponential with mean . So,
to use the joint distribution, and write: R ,R
,
(41)
. , . denotes the Jacobian matrix. Then, the joint density function of random variables
where
and
is
written as: f
,
So the distribution of random variable
Now, we have:
f
d
exp
.
c
(42)
is calculated as: f
.
, so f
,
g
f
,
,
g ,c
g , c dc
,
(43)
. The distribution of d
/
.d
is calculated as:
/
/
(44)
Appendix B
To calculate the distribution of , we define random variable " ≜
" to use the joint distribution,
and write: α ,α
4
4√t ,
(45)
. , . denotes the Jacobian matrix. Then, the joint density function for random variables
where
and u is
written as: f
,
t ,u
,
,
f
,
So the distribution of random variable f t
f
,
t , u du
√u. t . f
√u
u
t
exp
1
t u
(46)
is calculated as: t
u
. exp
1
t u du
t
B (47)
which the integral B is defined as
≜
.
1
, and it is calculated as [15]:
B
exp
1
∑
t u .
Therefore, we have the distribution of
∏
1
distribution of
j
!
(48)
as: !
f t with the distributions of
2m
(49)
in (49), d in (6), and another use of Jacobian matrix we determine the
as follows : f
v
!
v
f
d . dd
(50)
Comparing this expression with statistical expectation definition we write: f
!
v
.v
E
.
(51)
Appendix C
We define random variable
1
as a function of CIR, as :
1, and therefore,
is CIR and it reversely is calculated as:
, which the random variable 2
. Using the relations for
the distribution of function of a random variable, it is concluded: f
2e f e
1 .
(52)
7. References
[1] P. Madhusudhanan, J. G. Restrepo, Y. E. Liu and T. X. Brown, "Carrier to interference ratio analysis for the shotgun cellular system," in IEEE Global Telecommunications Conference (GLOBECOM), Honolulu, 2009. [2] V. Macdonald, "The Cellular Concept," The Bell SystemTechnical Journal, 1979, 58, pp. 15-41. [3] T. Brown, "Analysis and Coloring of a Shotgun Cellular System," in IEEE Radio and Wireless Conference (RAWCON), Colorado Springs, 1998. [4] T. Brown, "Dynamic channel assignment in shotgun cellular systems," in IEEE Radio and Wireless Conference, Denver, CO, 1999. [5] T. Brown, "Cellular performance bounds via shotgun cellular systems," IEEE Journal on Selected Areas in Communications (JSAC), 2000, 18, (11), pp. 2443 - 2455. [6] P. Madhusudhanan, J. Restrepo, Y. Liu, T. Brown and K. Baker, "Generalized carrier to interference ratio analysis for the shotgun cellular system in multiple dimensions," CORR, Vol. abs/1002,3943, 2011. [7] C. Tepedelenlioğlu, A. Rajan and Y. Zhang, "Applications of stochastic ordering to wireless communications," IEEE Transactions on Wireless Communications, 2011, 10, (12), pp. 4249-4257. [8] P. Madhusudhanan, J. G. Restrepo, Y. Liu, T. X. Brown and K. R. Baker, "Stochastic ordering based carrier-to-interference ratio analysis for the shotgun cellular systems," IEEE Wireless Communications Letters, 2012, 1, (6), pp. 565-568. [9] P. Madhusudhanan, J. G. Restrepo, Y. Liu, T. X. Brown and K. R. Baker, "Multi-tier network performance analysis using a shotgun cellular system," in IEEE Global Telecommunications Conference
(GLOBECOM), Houston, 2011. [10] H. S. Dhillon and J. G. Andrews, "Downlink rate distribution in heterogeneous cellular networks under generalized cell selection," IEEE Wireless Communications Letters, 2014, 3, (1), pp. 42-45. [11] J. G. Andrews, F. Baccelli and R. K. Ganti, "A tractable approach to coverage and rate in cellular networks," IEEE Transactions on Communications, 2011, 59, (11), pp. 3122-3134. [12] H. P. Keeler, B. Blaszczyszyn and M. K. Karray, "SINR-based k-coverage probability in cellular networks with arbitrary shadowing," in IEEE International Symposium on Information Theory Proceedings (ISIT), Istanbul, 2013. [13] P. C. Pinto and M. Z. Win, "A Unifying Framework for Local Throughput in Wireless Networks," arXiv:1007.2814., 1, 2010. [14] A. Papoulis and S. U. Pillai, Probability, random variables, and stochastic processes, McGraw-Hill, 1985. [15] I. Gradshteyn and I. Ryzhik, Table of integrals, series, and products, Elsevier , 2007.
Asm ma Bagherri received the B.Sc. and a the M.Sc. degree in electriccal engineerring from m the Ferdo owsi Univerrsity of Masshhad, Iran,, in 2010 annd 2013, resspectively. She is currently c pu ursuing the P Ph.D. degreee in commu unications aat the Babol Universityy of Tecchnology. Her H current research in nterests include mobilee networks and cognittive radios.
hosheh Abeed Hodtanii received the t B.Sc. degree in eleectronics en ngineering and Gh the M.Sc. deg gree in com mmunication ns engineerin ng, both froom Isfahan n Universityy of Tecchnology, Isfahan, Irran ,in 198 85, 1987, respectivelyy. He join ned Electrical Enggineering Dept., D at Ferrdowsi Univ versity of Mashhad, M M Mashhad, Iraan, in 1987. He deccided to purrsue his studdies in 2005 5 and receiv ved the Ph..D. degree (with ( excelllent grade) from Sharif S Univeersity of Teechnology, Tehran, Iraan, in 2008 8. His reseaarch inteerests are in multi-uuser inform mation theorry, commuunication theory, wirelless com mmunicatioons and signnal processing. Dr. Hoddtani is the author of a textbook oon electricall circuits and is thee winner of the best papper award at IEEE ICT T -2010