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The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control
Journal Pre-proof The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control Qiaoshou Liu, Zhongpei Zhang PII:
S2352-8648(19)30169-5
DOI:
https://doi.org/10.1016/j.dcan.2020.02.002
Reference:
DCAN 198
To appear in:
Digital Communications and Networks
Received Date: 14 June 2019 Revised Date:
6 January 2020
Accepted Date: 12 February 2020
Please cite this article as: Q. Liu, Z. Zhang, The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control, Digital Communications and Networks (2020), doi: https://doi.org/10.1016/j.dcan.2020.02.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2020 Chongqing University of Posts and Telecommunications. Production and hosting by Elsevier B.V. All rights reserved.
Digital Communications and Networks(DCN)
journal homepage: www.elsevier.com/locate/dcan
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control Qiaoshou Liu∗a,b , Zhongpei Zhanga a National
Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu, 611731, China b Key Laboratory of Optical Communication and Networks, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China
Abstract The ultra-dense network is a promising technology to increase the network capacity in the forthcoming fifth generation (5G) mobile communication networks by deploying lots of low power Small Base Stations (SBSs) which overlap with Macro Base Stations (MBSs). The interference and energy consumption increase rapidly with the number of SBSs although each SBS transmits with small power. In this paper, we model a downlink heterogeneous ultra-dense network where a lot of SBSs are randomly deployed with MBSs based on Poisson point process. We derive the coverage probability and its variance, and analyze the area spectral efficiency and energy efficiency of the network considering three Fractional Power Control (FPC) strategies. The numerical results and Monte Carlo simulation results show that power control can mitigate the interference and balance the performances of inner-user equipments and edge-user equipments. Especially, a great improvement of energy efficiency is archived with a little loss of area spectral efficiency when FPC is adopted. Finally, we analyze the effect of base stations’ (BSs’) sleeping on the performance of the network when it is partially loaded. c 2019 Published by Elsevier Ltd.
KEYWORDS: Heterogeneous ultra-dense network; Poisson point process; Coverage probability; Area spectral efficiency; Energy efficiency
1. Introduction To meet more and more User Equipments (UEs) accessing the networks at a fast speed, many key technologies, such as massive Multi-Input MultiOutput (MIMO), mm-Wave, and Ultra-Dense Network (UDN) are adopted in the forthcoming 5G wireless networks [1, 2, 3, 4]. As one of the key technologies of 5G, the UDN is a promising technology to cope with high capability requirements. the UDN can enhance the Signal-to-Interference-plus-Noise Ratio (SINR) of UEs and increase the space reuse efficiency of the spectrum by deploying more Small Base Stations (SBSs) to reduce the radius of coverage in a given area. Deploying a large number of SBSs will lead to a large amount of energy consumption although a SBS consumes less energy than a Macro ∗ (Corresponding
author) email: [email protected].
Base Station (MBS). In [5, 6], the statistical data indicates that the whole information and communication technology has been estimated to bring about 2% of global CO2 emissions. With the exponential growth of data traffic and the number of intelligent terminals, this figure about CO2 emissions is projected to increase significantly. It is estimated that the power consumption of Base Stations (BSs) accounts for about 57% of the total power consumption in wireless cellular networks. It is important to reduce the energy consumption of BSs in the UND. Recently, Poisson Point Process (PPP) as the most tractable model of stochastic geometry has been widely accepted to model and analyze the cellular networks due to the simple form of its Probability Generating FunctionaL (PGFL) model [7]. One of the most important contributions to the study of UDN using stochastic geometry can be found in [8], where the authors analyzed a one-tier UDN and developed closed
2 form expressions of coverage probability and ergodic rate under specific channels exploiting PPP. As a tractable tool for modelling and analyzing cellular networks, most of the prior research based on PPP focused on the standard success (coverage) probability as the performance metric [8, 9, 10]. The coverage probability is the spatial value averaging over the channel fading and the point process. This important value gives some basic information on the Signalto-Interference Ratio (SIR) performance but does not provide further information, such as the proportion of users in a Poisson cellular network achieving a desired link reliability with a given SIR threshold, or the differences between two networks that have the same coverage probability. In [11], the authors modelled and analyzed the meta distribution of SIR for cellular networks with power control based on a non-homogeneous PPP model. Utilizing the meta distribution of SIR, more fine-grained information on individual link reliability was revealed. In this paper, we focus on a two-tier Heterogeneous Ultra-Dense Network (HetUDN) in the downlink based on PPP, considering the network as fully loaded or partially loaded. To reduce energy consumption, we consider three power control strategies for SBSs and MBSs. We measure the network performance by coverage probability, variance, Area Spectral Efficiency (ASE), and Energy Efficiency (EE). Moreover, more information about the quality of link will be revealed utilizing meta distribution. The main contributions of this paper are summarized as follows: (a) We propose an HetUDN model consisting of an SBS-tier and an MBS-tier based on PPP. In our model, we consider that the number of SBSs is much greater than the number of MBSs, and that UEs openly access the network according to a positive bias of distance to offload UEs from the MBS-tier to the SBS-tier. We analyze the probability distribution of the transmit power with Fractional Power Control (FPC) to guide the selection of parameters to avoid exceeding the maximum transmit power limit and save power. (b) We derive the b-th moment of the conditional success probability for the Poisson downlink HetUDN with FPC considering three cases. First, the SBSs transmit with FPC but the MBSs transmit with fixed maximum power. Second, both the SBSs and the MBSs transmit with FPC under different parameters. Finally, in the special case, the SBSs and MBSs transmit with FPC under the same parameters, and the UEs are associated with their closest BSs without bias. So the two-tier HetUDN can be simplified as a one-tier UDN. (c) We analyze the ASE and EE of the whole HetUDN with three cases. The results show that power control greatly reduces the energy consumption of SBSs and MBSs. The remainder of this paper is organized as follows: we give our system model of the HetUDN based on
Qiaoshou Liu, Zhongpei Zhang. PPP in Section II. In Section III, we analyze the meta distribution of SIR in a two-tier downlink HetUDN considering three strategies of FPC. The analysis of ASE and EE are followed in Section IV. The conclusions are given in Section V. 2. System model We consider a two-tier heterogeneous ultral-dense cellular downlink network in a given area S ∈ R2 , as shown in Fig. 1. The network consists of an SBStier and an MBS-tier, and both BSs of the SBS-tier and the MBS-tier are deployed independently and randomly according to some homogeneous PPP Φm and Φ s with densities λm and λ s , i.e., Φ s ∼ PPP (λ s ) and Φm ∼ PPP (λm ), respectively. For the HetUDN, the number of SBSs is much larger than the number of MBSs, i.e., λ s λm . The UEs have certain temporal and spatial characteristics, which leads to the result that some SBs are fully loaded but others are partially loaded. In [12], the authors studied the heterogeneous cellular networks with spatio-temporal traffic. In our paper, for simplicity, we consider that all MBSs are fully loaded due to their particularities. Some SBSs in which no UE needs to be served are turned off to save energy. Assume that the active probability of SBSs is 0 < pa ≤ 1, where pa < 1 means that the SBS-tier is partially loaded, and pa = 1 means that the SBS-tier is fully loaded. For any UE, all active BSs act as interferers except the tagged BS, because all BSs reuse the same spectrum resources. All active SBSs are still a PPP Φa with density λa = pa λ s , according to Thinning theorem [13]. The UEs are modelled as a stationary point process Φu with density λu . According to Slivnyak-Mecke theorem [7], we consider a typical UE located at the origin. Without loss of generality, we denote SBSs, MBSs, n oand UEs by their locations, i.e., Φ s = {si }, Φm = m j , Φu = {uk } , and i, j, k ∈ N. Moreover, we denote the typical UE as uo , its closest SBS as so , and its closest n o MBS as mo , where uo ∈ {uk }, so ∈ {si }, and mo ∈ m j .
SBS
R
Dm
SBS
SBS UE
Rm
MBS UE
Dm
Typical MBS UE
MBS
Ds
MBS
SBS
Signal Interference
R
Ds
Rs SBS
Typical SBS UE
SBS
Fig. 1: The frame of heterogeneous ultra-dense networks.
We consider that the UEs openly access the network, i.e., any UE can be associated with an SBS or an MBS. In [14], a Reference Signal Received Power (RSRP)-based association scheme and its variant, i.e., Reference Signal Received Quality (RSRQ)-based association scheme are proposed for associations of UEs
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control in single-tier networks. But RSRP and RSRQ can not be directly used to the two-tier network because the transmit power of MBSs is much greater than that of SBSs, and most of UEs would choose MBSs to access. This leads to a huge traffic load imbalance between two tiers. A bias-based Cell Range Expansion (CRE) was exploited to offload UEs from the MBStier to the SBS-tier by adding a positive bias to their measured signal-strengths during cell association [15]. To offload UEs from the MBS-tier to the SBS-tier, we consider an open access policy based on distance [16], the UEs are associated with an SBS or an MBS by adding a positive bias to their measured distances from the closest BSs, which is more convenient for analysis than other methods. Let R s denote the distance from a UE to its closest SBS, and Rm denote the distance from a UE to its closest MBS. The open access policy is expressed as ( S BS → UE, i f R s ≤ κRm , κ ≥ 0, (1) MBS → UE, i f R s > κRm where κ is the offloading bias. So, the probability of a UE associated with an SBS is λ s /λ1 and the probability of a UE associated with an MBS is λm /λ2 , where λ1 = λ s + λm /κ2 and λ2 = λm + κ2 λ s . κ → 0 means that most of UEs are associated with MBSs; κ = 1 means that all UEs are associated with their closest BSs (SBSs or MBSs); κ → ∞ means that most of UEs are associated with SBSs, i.e., the number of UEs associated with SBSs increases with the increase of κ. Based on the characteristics of spatial PPP, we can easily get the Probability Density Function (PDF) of R s by conditioning on the open access policy as fRs |Rs ≤κRm (r) = 2λ1 πr · exp −λ1 πr2 , (2) and the PDF of Rm by conditioning on the open access policy as fRm |Rm
3
consumed and the downlink qualities of edge-UEs being worse than that of the inner-UEs. In [11], the author analyzed a one-tier cellular network with power control in the uplink and the downlink, not considering the multi-tier network. An Almost Blank Sub-frame (ABS) [19] was proposed to avoid the MBSs interfering the UEs of SBSs in a two-tier cellular downlink network. The ABS can eliminate the interference from MBSs to the UEs of the SBS-tier by sacrificing the spectrum resources of MBSs. As aforementioned, power control in the uplink benefits in two aspects: on the one hand, the UEs can save much power with power control, especially the inner-UEs, because they can transmit with very low power; on the other hand, power control will balance the performances of innerUEs and edge-UEs. In our downlink heterogeneous cellular model, we consider that the BSs transmit with FPC as the uplink. Adopting appropriate power control parameters can balance the performances of the inner-UEs and edge-UEs, and save power. Similar to the uplink, the BSs transmit data to the UEs with FPC and the transmit power of a BS is expressed as ( x Pt = t x Rα x , with FPC; (4) Ptx = T x , without FPC, where t x , x ∈ {s, m} is the minimum transmit power of BSs with FPC and is defined as the transmit power when R equals the unit distance. T x denotes the fixed transmit power of BSs without power control. We can get the distribution of transmit power Pts with FPC as 1 − exp −λ1 π Pts /t s 2/α , (5) as
and the distribution of transmit power Pm t with FPC 2/α 1 − exp −λ2 π Pm . (6) t /tm
Assume that the path loss between a UE and a BS is h x R−α x , x ∈ {s, m}, where h x is the small-scale fading, all h x are independent identical distribution (i.i.d.) and h x ∼ exp (1). R x ∈ {R s , Rm , D s , Dm } is the distance from a UE to the BSs. R s denotes the distance from a UE to its associated BS which belongs to the {si }, Rm denotes the distance from a UE to its associated BS which belongs to the {m j }, D s denotes the distance from a UE to one of its interfering SBSs which belongs to the {si }, and Dm denotes the distance from a UE to one of its interfering MBSs which belongs to the {m j }. α is the path loss exponent and α > 2 in the open air. In this case, the received signal of a UE from a BS is Ptx h x R−α x . For a quick reference, all of the notations and symbols used in this paper are summarized in Table 1. We assume there are 50 SBSs and 1 MBS in a 1 km2 area, i.e., λs = 1 · 10−6 pieces/m2 and λs = 5 · 10−5 pieces/m2 . We set t s = 1 mW, tm = 4 mW, 1/α and α = 4. In [16], let κ = Pts /Pm , then the intert tier interference from SBSs to macro-cell users is nullified. In this case, κ = 0.7 is adopted. Fig. 2 illustrates the Cumulative Distribution Function (CDF) of
4
Qiaoshou Liu, Zhongpei Zhang. Table 1: notations and symbols used in the paper
Notation λs pa λm ts tm Tm Rs Rm R
Description The density of the SBSs The probability of SBSs The density of the MBSs The minimum transmit power of the SBS The minimum transmit power of the MBS The fixed transmit power of the MBS Distance from a UE to its closest SBS Distance from a UE to its closest MBS Distance from the typical UE to its associated BS Distance from the typical UE to a interfering SBS Distance from the typical UE to a interfering MBS Offloading bias Path loss exponent Fractional power control efficient Probability operator The expectation over all random variables in [·] Euclidean distance
Ds Dm κ α P [·] E [·] k·k
BSs’ transmit power with FPC. In this paper, we limit the maximum transmit power of the SBSs and MBSs to 2 W and 5 W, receptively. As shown in Fig. 2, the transmit power curves will exceed the limit with a great probability if a larger is chosen. We choose = 0.3 as a middle value for simplicity in this paper if there is no special explanation.
openly access the network based on the policy defined in Eq. (1). Each BS serves at most one UE, and each UE is served by at most one BS at the same time. For the typical UE, we omit the subscripts of h x and R x . When the typical UE uo is associated with its closest SBS so according to the open access policy. In this case, R denotes the distance from uo to so , i.e., R = kso − uo k, where k·k is the Euclidean distance. The pa has no influence on the distribution of the distance from a UE to its closest BS but the the distribution of the interfering BSs. So, the SIR of the typical UE is P s hR−α ; (7) SIR s = t I s,s + I s,m if the typical UE is associated with its closest MBS mo according to the open access policy, R denotes the distance from uo to mo , i.e., R = kmo − uo k. The SIR of the typical UE is SIRm =
−α Pm t hR , Im,s + Im,m
(8)
P where I s,s = s∈Φa \{so } Pts h s D−α s is the cumulative interference from all other active SBSs except the so to P −α the typical UE uo ; I s,m = m∈Φm Pm t hm Dm is the cumulative interference from all MBSs to the typical UE P which associated with the so ; Im,s = s∈Φa Pts h s D−α s is the cumulative interference from all active SBSs to the typical UE which is associated with mo and P −α Im,m = m∈Φm \{mo } Pm t hm Dm is the cumulative interference from all MBSs except the mo to the typical UE which is associated with mo . Pts and Pm t are set based on Eq. (4).
1
3. Coverage probability
0.9
CDF of Tx_Power of BSs
0.8 0.7 0.6 0.5 0.4 0.3 SBS, MBS, SBS, MBS,
0.2 0.1
= 0.3 = 0.3 = 0.4 = 0.4
0 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Tx_Power of BSs
Fig. 2: The CDF of BSs’ transmit power with FPC, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, κ = 0.7, and pa = 1.
2.2. SIR In cellular networks, the thermal noise is always neglected because it hardly affects the network’s performance[8]. For a UE in a cellular downlink network, the interference comes from other SBSs and MBSs which reuse the same spectrum resources at the same time. In this paper, we consider that all SBSs and MBSs reuse the total spectrum resources and UEs
In the HetUDN, the network performance is mainly influenced by the SBS-tier because the density of SBSs is much greater than that of MBSs and most of UEs are offloaded from the MBS-tier to the SBStier. The UEs associated with SBSs will be strongly interfered by MBSs if the MBSs transmit with high power, which will lead to a drastic decline in the network’s performance. To mitigate the interference from the MBS-tier to the SBS-tier, the transmit power of MBSs should be reduced compared with the conventional MBSs. Moreover, we consider three FPC strategies to coordinate the interference and balance the performances of inner-UEs and edge-UEs. Based on Eq. (2), (3), (7), and (8), the conditional coverage probability of the SBS-tier is P ss (θ) = P {SIR s > θ|Φ s , Φm } n o −1 = P h > θ Ptso Rα I s,s + I s,m |Φa , Φm Y Y 1 1 (a) = a −α α(1−) · m −1 −α α(1−) , 1 + θR D R 1 + θP t Dm R s s s t Φ Φ \{s } a
o
m
(9)
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control and the conditional coverage probability of the MBS-tier is Pms (θ) = P {SIRm > θ|Φa , Φm } n o o −1 α = P h > θ Pm R Im,s + Im,m |Φa , Φm t Y Y 1 1 (a) = , mo −α α · m mo −1 −α α 1 + θ P /P D R m 1 + θ P t s Ra t t s Ds R Φa Φm \{mo } t
(10) where (a) follows the i.i.d. of h and its further independence from the point process Φ, θ is the predefined m o SIR threshold for demodulation, and Ptso , Pm t , Pt need to be set according to Eq. (4). Then, the conditional coverage of the whole network is P s (θ) =
pcs
·
P ss
(θ) +
pm c
·
Pms
(θ) ,
(11)
where pcs =λ s /λ1 is the probability of a UE’s being associated with an SBS, pm c =λm /λ2 is the probability of a UE’s being associated with an MBS based on the open access policy. 3.1. Meta distribution In order to get more fine-grained information for each individual link in the network, the meta distribution of the SIR is defined by [20] as F Ps (x) = P (P s (θ) > x) , x ∈ [0, 1] , o
(12)
∆
where P s (θ) = P (SIR > θ|Φ) is the conditional success probability (coverage probablity) taking over the fading and the random activities of the interferes, given the point process. Po (·) denotes the reduced Palm measure of the point process, given a receiver at the origin [7]. We focus on the moments of P s (θ) because it is most likely impossible to calculate the meta distribution directly from the definition in Eq. (12). The b-th moment of P s (θ) is denoted by Z 1 ∆ o b Mb (θ) = E P s (θ) = bxb−1 F Ps (x)dx, b ∈ C, 0
(13) where Eo (·) is the reduced Campbell measure [7]. Hence we easily have the standard success probabil∆ ity p s (θ) = P (SIR > θ) ≡ M1 (θ). And the variance of P s (θ) equals M2 (θ) − M12 (θ). The meta distribution of the uplink SIR can be obtained from Gil-Pelaez theorem [21] as Z − jtx 1 1 ∞ Im e M jt F Ps (x) = + dt, (14) 2 π 0 t where Im (z) denotes the imaginary of the complex ∆ √ number z and j = −1. As in [20], some classic bounds are given for the meta distribution of Eq. (14) and the best bound is the beta distribution approximation. We only consider the beta distribution approximation in this paper. Since P s (θ) is supported on [0,1], it is natural to approximate its distribution with the beta distribution
5
[20]. The PDF of a beta distributed random variable X with mean u is x 1−u (1 − x)β−1 , B (uβ/(1 − u), β) u(β+1)−1
fX (x) =
(15)
R1 where B (a, b) = 0 ta−1 (1 − t)b−1 dt is the beta function. The variance is given by var X =
u(1 − u)2 . β+1−u
(16)
Matching the mean u = M1 and the variance var X = M2 − M12 yields (1 − M2 )(1 − M1 ) . (M2 − M12 )
β=
(17)
3.2. Only SBSs transmit with FPC Firstly, we consider that MBSs transmit with fixed power T m and SBSs transmit with FPC, i.e., Pts = t s Rα s . Theorem 1. Considering that MBSs transmit with fixed power and SBSs transmit with FPC, the bmoment of the conditional success probability for the SBS-tier is given by Z ∞ Mbs = exp −z 1 + fb,1 (z) + fb,2 (z) dz, (18) 0
where
fb,1 (z) =
λa λ1
R∞R∞ −xz ze 1− 1 0
1
dxdy,
(1+θxα/2 y−α/2 )b
(19) and
fb,2 (z) =
λm λ1
R∞ 1/κ
! 1−
1 α/2 −α/2 −α/2 b y z ) (1+θTm t−1 s (λ1 π)
dy.
(20) The b-moment of the conditional success probability for the MBS-tier is given by Z ∞ Mbm = B · exp −Bz 1 + f3,b (z) + f4,b (z) dz, 0
(21)
where λm f3,b (z) = λ2
Z
∞
1
f4,b (z) R∞R∞ = λλ2a κ 0 ze−zx 1 −
1 −
1 1 + θy
dy, −α/2 b
(22)
!! 1
(1+θTm−1 ts (λ1
b π)−α/2 xα/2 y−α/2 zα/2
)
dxdy,
(23) where B = λ2 /λ1 . Proof of Theorem 1. The proof is given in Appendix 6.1.
6
Qiaoshou Liu, Zhongpei Zhang.
Compared with the conventional MBS, T m should be descended to reduce the energy consumption and mitigate the interference from MBSs to the UEs of SBSs because most of the edge-UEs of MBSs are offloaded to the SBS-tier with the bias κ. Let T m = 5000 mW, t s = 1 mW, κ = 0.7, = 0.3, and the network’s area is 5km × 5km for both numerical results and Monte Carlo simulation. Fig. 3 illustrates the coverage probability (the 1 st moment) and its variance when the network is fully loaded. The dotted lines with marks denote the Monte Carlo simulation results of coverage probabilities; the solid lines denote the numerical results of coverage probabilities; and the dotted lines with plus signs denote the numerical results of variances. As shown in Fig. 3, the coverage probability of the MBS-tier with a higher variance is larger than that of the SBS-tier, because MBSs transmit with fixed maximum power. The variance of the SBS-tier is less than that of the MBS-tier because FPC balances the performances of inner-UEs and edge-UEs on the SBS-tier. The whole network coverage probability and its variance are mainly dominated by the SBS-tier because the number of SBSs is much more than the number of MBSs. 1
0.2 P SBS - Num
0.9
P MBS - Num
0.18
0.8
P - Num P - Sim
0.16
0.7
P MBS - Sim
0.14
0.6
P - Sim V SBS - Num
0.12
0.5
V MBS - Num
0.1
V - Num
0.4
0.08
0.3
0.06
0.2
0.04
0.1
0.02
0 -20
-15
-10
-5
0
5
10
15
20
25
Variance
Coverage Probability
SBS
0 30
where
g1,b (z) = g2,b (z) =
λa λ1
λm λ1
R∞R∞ −xz ze 1− 1 0
R∞ R∞ 1/κ 0
−Bzx
Bze
1
(
dxdy,
b 1+θxα/2 y−α/2
)
(25) dxdy,
1 1− α/2 y−α/2 b (1+θtm t−1 ) s x
(26) the b-moment of the conditional success probability for the MBS-tier is given by Z ∞ Mbm = B · exp −Bz 1 + g3,b (z) + g4,b (z) dz, 0
(27)
where g3,b (z) =
λm λ2
R∞R∞
g4,b (z) =
λa λ2
R∞R∞ −zx ze 1− κ 0
1
0
Bze−Bzx 1 −
1
dxdy,
(1+θxα/2 y−α/2 )b
1 (1+θts tm−1 xα/2 y−α/2 )b
(28) dxdy,
(29) where B = λ2 /λ1 Proof of Theorem 2. The proof is given in Appendix 6.2. In this section, MBSs transmit with FPC. Let tm = 4 mW, and other parameters remain unchanged. Fig. 4 illustrates the coverage probability and variance when the network is fully loaded. The coverage probability of the MBS-tier descends and that of the SBS-tier rises a little compared with Fig. 3, because the transmit power of MBSs is reduced to mitigate the interference to thhe SBS-tier. Moreover, power control benefits the edge-UEs by sacrificing interests of the inner-UEs on the MBS-tier as well as the SBS-tier, which can be proved by the variance of the MBS-tier (green dotted line with plus signs), which is smaller than that in Fig. 3.
Required SIR (dB)
Due to the special status of MBSs, it is impossible for MBSs to transmit to all UEs with vary small fixed power, although it will reduce the interference to the UEs on the SBS-tier. So, we consider that both MBSs α and SBSs transmit with FPC, i.e., Pm t = tm Rm and s α Pt = t s R s , respectively. In this case, the performance of edge-UEs of MBSs can be guaranteed. Theorem 2. Considering that both MBSs and SBSs transmit with FPC, the b-moment of the conditional success probability for the SBS-tier is given by Mbs =
Z 0
∞
exp −z 1 + g1,b (z) + g2,b (z) dz,
(24)
Coverage Probability
3.3. Both BSs of the SBS-tier and MBS-tier transmit with FPC
0.2 P SBS - Num
0.9
P
MBS
- Num
0.18
0.8
P - Num P SBS - Sim
0.16
0.7
P MBS - Sim
0.14
0.6
P - Sim V - Num
0.12
0.5
V
0.1
SBS
MBS
- Num
V - Num
0.4
0.08
0.3
0.06
0.2
0.04
0.1
0.02
0 -20
-15
-10
-5
0
5
10
15
20
25
Variance
1
Fig. 3: The coverage probability and its variance in the case of only SBSs transmitting with FPC, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, and pa = 1.
0 30
Required SIR (dB)
Fig. 4: The coverage probability and variance in the case of both SBSs and MBSs transmitting with FPC, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, and pa = 1.
3.4. Special case In this section, we consider a special case, i.e., letting t s = tm , κ = 1 when SBSs and MBSs both transmit with FPC. So, the two-tier heterogeneous cellular network will become a homogeneous network with density λ = λa + λm .
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control
0
1 Case 1, pa = 0.5 Case 2, pa = 0.5 Case 3, pa = 0.5 nPC, pa = 0.5 Case 1, pa = 1 Case 2, pa = 1 Case 3, pa = 1 nPC, pa = 1
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Coverage Probability
Corollary 1. From a mathematical point of view, when tm = t s , κ = 1, we can get λ1 = λ2 = λ s + λm = λ, B = 1. So, the b-moment of the conditional success probability for the whole network is Z ∞ Mb = exp (−z (1 + fb (z)))dz, (30)
7
where
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Z Z pa λ s + λm ∞ ∞ −xz 1 dxdy ze 1 − fb (z) = b α/2 −α/2 λ s + λm 1 0 1 + θx y Z ∞Z ∞ −xz (b) 1 dxdy. ≈ pa ze 1 − b α/2 −α/2 1 0 1 + θx y
(31)
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Fig. 6: The coverage probabilities in three cases vary with pa , where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
where (b) follows λ s λm . 0.16
Proof of Corollary 1. The proof is given in Appendix 6.3.
0.12 0.1
Var
Eq. (30) is consistent with Eq. (25) in [11] when pa = 1. It is the moment of the conditional success probability for a one-tier cellular network with FPC.
Case 1, pa = 0.5 Case 2, pa = 0.5 Case 3, pa = 0.5 nPC, pa = 0.5 Case 1, pa = 1 Case 2, pa = 1 Case 3, pa = 1 nPC, pa = 1
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Fig. 5: The coverage probability and variance of the special case, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, and pa = 1.
Fig. 7: The variance of coverage probabilities in three cases vary with pa , where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
lower when the network is partially loaded. The performance of cases 3 is the best among the three cases, but lower than the case of nPC in the high-θ regime. This is because power control benefits the edge-UEs by sacrificing interests of the inner-UEs. 1 Case 1, pa = 0.5 Case 2, pa = 0.5 Case 3, pa = 0.5 nPC, pa = 0.5 Case 1, pa = 1 Case 2, pa = 1 Case 3, pa = 1 nPC, pa = 1
0.8 0.7 0.6
1-F(x)
Fig. 5 illustrates the coverage probability and its variance of the special case when the network is fully loaded. The black solid line is the coverage probability when all SBSs transmit with a fixed maximum power 2000 mW (nPC) but other parameters remain unchanged in a one-tier UDN. As shown in Fig. 5, the coverage probability with FPC (red solid line) is greater than that of nPC in the low-θ regime but lower than that of nPC in the high-θ regime. This is because FPC can balance the performance of the innerUEs and edge-UEs. This can be proved from the variance where the variance of nPC (black dotted line with plus) is greater than that with FPC (red dotted line with plus). Fig. 6 and Fig. 7 illustrate the coverage probabilities and their variances of the SBS-tier in three cases when the network is fully loaded or partially loaded. We only consider the SBS-tier due to the fact that λ s λm . We add the nPC as the baseline. Obviously, the coverage probabilities are higher and variances are
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Fig. 8: The meta distribution by beta distribution approximation varies with pa , where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4.
Fig. 8 illustrates the beta distribution approximation of the meta distribution through matching the mean and variance with the required SIR threshold θ = 5 dB for the three cases and the nPC case considering that the network is either fully loaded or partially loaded. It can reveal the percentage (y-axis) of UEs whose suc-
8
Qiaoshou Liu, Zhongpei Zhang.
cess probability is greater than some values (x-axis) with a given SIR threshold θ. As shown in Fig. 8, a larger proportion of UEs have a larger probability to achieve the SIR requirement in case 3 among the three cases. For example, there are about 10 percent of UEs whose success probability is greater than 0.8 (x-axis) in case 3 when the the network is fully loaded. Meanwhile, the UEs in case 1 and case 2 are less than 10 percent. The proportions of case 1, case 2, and case 3 increase to 21%, 23%, and 25% when the network is partially loaded with pa = 0.5 for the same x. The UEs’ proportion of nPC is highest in the high-x regime (means high-θ regime), but lower than those of case 2 and 3 in low-x regime (means low-θ regime). This further proves that power control can benefit the edgeUEs by sacrificing interests of the inner-UEs.
SBSs is much more than that of MBSs. Moreover, case 3 achieves the highest ASE among the three FPC cases. The ASE of nPC is greater than the ASE of the three FPC cases in the high-θ regime by guaranteeing the inner-UEs’ link quality at the expense of more energy consumption. Obviously, when the network is partially loaded, the spatial reuse efficiency of spectrum resources reduces due to BSs’ sleeping although the coverage probability increases. It leads to the descending of all ASE (line curves). The ASE of the network being fully loaded is the maximum ASE that the network can reach and the ASE of the network being partially loaded can be regarded as the real-time ASE of the network.
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4. ASE and EE
Case 2 ,p a = 0.5 Case 3 ,p a = 0.5
In our HetUND model, MBSs and SBSs are randomly deployed in a given area and all base stations totally reuse the same spectrum resources. There are not only intra-tier but also inter-tier interference in the HetUDN. In the previous section, we have evaluated the network performance in three FPC cases by coverage probability, its variance, and the meta distribution of SIR to reveal the distribution of UEs whose success probability is greater than some values with a given SIR threhold. In fact, power control not only mitigates the interference, balances the performances between inner-UEs and edge-UEs in the cell, but also reduces the energy consumption of BSs. In this section, we further measure the network performance by the ASE and EE. Definition 1. In this section, the ASE is the spectral efficiency in a unit area, and the ASE of the SBS-Tier is defined as ASE s =pa λ s Pcs (θ) log2 1 + SIRreq , (32) the ASE of the MBS-tier is defined as ASEm =λm Pm c (θ) log2 1 + SIRreq ,
(33)
and the ASE of the whole network is ASE = ASE s +ASEm ,
(34)
m where Pcs (θ) = M1s and Pm c (θ) = M1 , i.e., the coverage probability of SBS and MBS, respectively; SIRreq = θ is the required SIR threshold for demodulation.
Fig. 9 illustrates the ASE of the three FPC cases in the previous section compared with the case of nPC. As shown in Fig. 9, the three colour solid lines of FPC with pa = 1 basically coincide and the three colour dotted lines of FPC with pa = 0.5 basically coincide too. This is because the SBS-tier is set with same parameters in the three FPC cases and the number of
ASE (bps/Hz/m 2 )
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Fig. 9: The area spectral efficiency of three cases, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
Let BW denote the total bandwidth which is reused by all SBSs and MBSs, S denote the area of the network, and τ denote the running time. The EE is defined as the number of bits sent at unit bandwidth by consuming one Joule energy.
Definition 2. The EE of SBS-tier is defined as total capacity o f S BS − tier (total energy o f S BS − tier) · BW ASE s · BW · τ · S = s E Pt · BW · τ · S · λa ASE s = s , E Pt · λa
EE s =
(35)
the EE of MBS-tier is defined as total capacity o f MBS − tier (total energy o f MBS − tier) · BW ASEm · BW · τ · S = m E Pt · BW · τ · S · λm ASEm , = m E Pt · λm
EEm =
(36)
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control the EE of the whole network is defined as total capacity (total energy) · BW ASE · BW · τ · S = s E Pt · τ · S · λa + E Pm t · τ · S · λm · BW ASE = s . E Pt · λa + E Pm t · λm (37)
EE =
Based on Eq. (5) and (6), we can get α ts α E Pts = · ·Γ , α/2 2 2 (λ1 π)
(38)
and tm α α α/2 · 2 · Γ m 2 , with FPC (λ π) 2 E Pt = T m , without FPC.
(39)
Fig. 10 illustrates the EE in the three cases with FPC compared with the case of nPC. As shown in Fig. 10, power control greatly improves the EE, especially in case 3. Power control reduces the energy consumption of BSs but almost does not affect the ASE. In Fig. 9, the ASE of nPC is greater than that of the three cases with FPC in the high-θ regime, but the EE of nPC is the lowest among all cases in Fig. 10. This is because power control sacrifices the performance of the inner-UEs to guarantee the performance of the edge-UEs and mitigate the interference to improve the network’s EE. The EE of all cases increases when the network is partially loaded because its coverage probability is greater than that of the full loaded network. This means the BSs can send more bits with the same energy consumption.
Case ,p = 0.5 a
Case ,p = 0.5
6
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EE, will increase if we reasonably offload UEs from the MBS-tier to the SBS-tier, especially in case 3 by changing the HetUDN into a one-tier UDN.
5. Conclusion In this paper, we model a heterogeneous ultra-dense cellular network based on spatial PPP. In our model, we consider three cases of FPC strategies, and derive the probability and variance of the UEs’ successfully being associated with BSs utilizing the meta distribution. The numerical results and Monte Carlo simulation results show that the network performance is mainly influenced by the SBS-tier in the HetUDN because the number of SBSs is much larger than that of the MBSs and most edge-UEs of MBSs are offload to the SBS-tier. Power control can mitigate the interference by sacrificing the performance of inner-UEs to guarantee the performance of edge-UEs. Furthermore, power control can balance the performances of two tiers if both tiers transmit with power control. Especially, in case 3, we change the two-tier HetUDN into a one-tier homogeneous network by setting appropriate parameters. And the numerical results show it can greatly improve the EE with little loss of the network performance. Finally, in the HetUDN, some BSs which no UE needs to be served can be turned off to save energy and to increase the coverage probability and EE. But BSs’ sleeping will reduce the ASE due to the decrease of active BSs.
Acknowledgement This work was supported in part by the Major Program of the National Nature Science Foundation of China (Grant No. 61831004).
7 1
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6. Appendix
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6.1. Proof of theorem 1
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Fig. 10: The energy efficiency of three cases, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
In the HetUDN, the inter-cell interference exists on both the same tier and different tiers. It is very important to balance loads and jointly optimize resources [22, 23]. The numerical and simulation results show the network’s performance, such as the coverage probability, the proportion of UEs achieving some success probabilities with a given SIR threshold, ASE, and
SBSs transmit with FPC, and MBSs transmit with m fixed power, i.e., Pts = t x Rα s , and Pt = T m , respecmo m tively. Then Pt /Pt = 1. From Eq. (9), the Mbs follows as
Mbs
Y =ER ERs ,Ds
Φa \{so }
Y EDm Φm
1 1+
−α α(1−) b θRa s Ds R
. b −α α(1−) 1 + θT m t−1 D R m s 1
· (40)
10
Qiaoshou Liu, Zhongpei Zhang. Let x = t/r, y = v/r, z = λ1 πr2 . We can get
and "
1 a −α α(1−) b Φa \{so } 1 + θR s D s R Z ∞ Z ∞ 2 (c) 2λ1 πte−λ1 πt 1 − = exp −2λa π dt vdv b r 0 1 + θtα v−a rα(1−) Z Z ∞ λa ∞ e−u 1 − dw = exp − du b α/2 −α/2 α(1−)/2 λ1 z 0 1 + θu w z Z ∞ Z ∞ −u λa e−u du dw = exp − e − b α/2 −α/2 α(1−)/2 λ1 z 0 1 + θu w z Z ∞Z ∞ −xz λa 1 ze 1 − , = exp − z dxdy b α/2 −α/2 λ1 1 0 1 + θx y
Y ERs ,Ds
ERs ,Ds
# 1
Q
−α α b 1+θT m−1 t s Rα s Ds R
) Φa ( R R∞ 2 ∞ 2λ1 πte−λ1 πt = exp −2λa π κr 1 − 0 b dt vdv −1 α −a α (1+θTm ts t v r ) ! ! R∞ R∞ e−u = exp − λλa1 κz 1 − 0 du dw (1+θTm−1 ts (λ1 π)−α/2 uα/2 w−α/2 zα/2 )b ! ! R∞R∞ e−u = exp − λλa1 κz 0 e−u − dudw (1+θTm−1 ts (λ1 π)−α/2 uα/2 w−α/2 zα/2 )b ! ! R R ∞ ∞ ze−zx = exp − λλa2 Bz k 0 ze−zx − dxdy −α/2 α/2 −α/2 α/2 b −1 (1+θTm ts (λ1 π) x y z )
(45) where B = λ2 /λ1 , s = Bz. Combining Eq. (3), (43), (44), and (45) yields Eq. (21).
(41) 6.2. Proof of theorem 2
and
SBSs and MBSs transmit with FPC, Pts = t s Rα s , and mo m α −α = tm Rα , respectively, then P /P = R . m mR t t The rest of the proof is similar to Appendix 6.1. Pm t
Y 1 EDm −1 D−α Rα(1−) 1 + θP P m s m Φm Z ∞ (c) 1 vdv = exp −2λm π 1 − b −1 −a α(1−) r/κ 1 + θT m t s v r λm Z ∞ 1 1 − = exp − b dw λ1 z/κ α/2 −α/2 α(1−)/2 (λ w z 1 + θT m t−1 π) 1 s Z ∞ 1 λm 1 − = exp − z dy , b λ1 1/κ α/2 −α/2 −α/2 −1 z 1 + θT m t s (λ1 π) y
6.3. Proof of corollary 1 mo SBSs and MBSs transmit with FPC, then Pm t /Pt = Moreover, tm = t s , κ = 1, and we can get λ1 = λ2 = λ s + λm = λ, B = 1. The rest of the proof is similar to Appendix 6.1. −α . Rα mR
References
(42) where (c) follows the PGFL. Combining Eq. (2), (40), (41), and (42) yields Eq. (18). From Eq. (10), the Mbm follows as
Mbm
Y 1 =ER EDm · −α α b Φm \{mo } (1 + θDm R ) Y ERs ,Ds Φa
(43) b . α −α α m −1 1 + θ Pt ts Rs Ds R 1
Let x = t/r, y = v/r, s = λ2 πr2 = Bz, and we can get Y 1 EDm −α α b (1 ) + θD R m Φm \{mo } ! ! Z ∞ 1 vdv = exp −2λm π 1− (1 + θv−a rα )b r Z ∞ λm 1 1 − = exp − b dw (44) −α/2 α/2 λ2 s 1 + θw s Z ∞ λm 1 1 − dy = exp − s b λ2 1 1 + θy−α/2 Z ∞ λm 1 = exp − Bz 1 − b dy , −α/2 λ2 1 1 + θy
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Conflict of interest statement
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service or company that could be construed as influencing the position presented in the manuscript entitled “The analysis of coverage probability, ASE and EE in heterogeneous ultradense networks with power control”
Yours Sincerely, Qiaoshou Liu, Zhongpei Zhang 07/01/2020
S2352-8648(19)30169-5
DOI:
https://doi.org/10.1016/j.dcan.2020.02.002
Reference:
DCAN 198
To appear in:
Digital Communications and Networks
Received Date: 14 June 2019 Revised Date:
6 January 2020
Accepted Date: 12 February 2020
Please cite this article as: Q. Liu, Z. Zhang, The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control, Digital Communications and Networks (2020), doi: https://doi.org/10.1016/j.dcan.2020.02.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2020 Chongqing University of Posts and Telecommunications. Production and hosting by Elsevier B.V. All rights reserved.
Digital Communications and Networks(DCN)
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The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control Qiaoshou Liu∗a,b , Zhongpei Zhanga a National
Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu, 611731, China b Key Laboratory of Optical Communication and Networks, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China
Abstract The ultra-dense network is a promising technology to increase the network capacity in the forthcoming fifth generation (5G) mobile communication networks by deploying lots of low power Small Base Stations (SBSs) which overlap with Macro Base Stations (MBSs). The interference and energy consumption increase rapidly with the number of SBSs although each SBS transmits with small power. In this paper, we model a downlink heterogeneous ultra-dense network where a lot of SBSs are randomly deployed with MBSs based on Poisson point process. We derive the coverage probability and its variance, and analyze the area spectral efficiency and energy efficiency of the network considering three Fractional Power Control (FPC) strategies. The numerical results and Monte Carlo simulation results show that power control can mitigate the interference and balance the performances of inner-user equipments and edge-user equipments. Especially, a great improvement of energy efficiency is archived with a little loss of area spectral efficiency when FPC is adopted. Finally, we analyze the effect of base stations’ (BSs’) sleeping on the performance of the network when it is partially loaded. c 2019 Published by Elsevier Ltd.
KEYWORDS: Heterogeneous ultra-dense network; Poisson point process; Coverage probability; Area spectral efficiency; Energy efficiency
1. Introduction To meet more and more User Equipments (UEs) accessing the networks at a fast speed, many key technologies, such as massive Multi-Input MultiOutput (MIMO), mm-Wave, and Ultra-Dense Network (UDN) are adopted in the forthcoming 5G wireless networks [1, 2, 3, 4]. As one of the key technologies of 5G, the UDN is a promising technology to cope with high capability requirements. the UDN can enhance the Signal-to-Interference-plus-Noise Ratio (SINR) of UEs and increase the space reuse efficiency of the spectrum by deploying more Small Base Stations (SBSs) to reduce the radius of coverage in a given area. Deploying a large number of SBSs will lead to a large amount of energy consumption although a SBS consumes less energy than a Macro ∗ (Corresponding
author) email: [email protected].
Base Station (MBS). In [5, 6], the statistical data indicates that the whole information and communication technology has been estimated to bring about 2% of global CO2 emissions. With the exponential growth of data traffic and the number of intelligent terminals, this figure about CO2 emissions is projected to increase significantly. It is estimated that the power consumption of Base Stations (BSs) accounts for about 57% of the total power consumption in wireless cellular networks. It is important to reduce the energy consumption of BSs in the UND. Recently, Poisson Point Process (PPP) as the most tractable model of stochastic geometry has been widely accepted to model and analyze the cellular networks due to the simple form of its Probability Generating FunctionaL (PGFL) model [7]. One of the most important contributions to the study of UDN using stochastic geometry can be found in [8], where the authors analyzed a one-tier UDN and developed closed
2 form expressions of coverage probability and ergodic rate under specific channels exploiting PPP. As a tractable tool for modelling and analyzing cellular networks, most of the prior research based on PPP focused on the standard success (coverage) probability as the performance metric [8, 9, 10]. The coverage probability is the spatial value averaging over the channel fading and the point process. This important value gives some basic information on the Signalto-Interference Ratio (SIR) performance but does not provide further information, such as the proportion of users in a Poisson cellular network achieving a desired link reliability with a given SIR threshold, or the differences between two networks that have the same coverage probability. In [11], the authors modelled and analyzed the meta distribution of SIR for cellular networks with power control based on a non-homogeneous PPP model. Utilizing the meta distribution of SIR, more fine-grained information on individual link reliability was revealed. In this paper, we focus on a two-tier Heterogeneous Ultra-Dense Network (HetUDN) in the downlink based on PPP, considering the network as fully loaded or partially loaded. To reduce energy consumption, we consider three power control strategies for SBSs and MBSs. We measure the network performance by coverage probability, variance, Area Spectral Efficiency (ASE), and Energy Efficiency (EE). Moreover, more information about the quality of link will be revealed utilizing meta distribution. The main contributions of this paper are summarized as follows: (a) We propose an HetUDN model consisting of an SBS-tier and an MBS-tier based on PPP. In our model, we consider that the number of SBSs is much greater than the number of MBSs, and that UEs openly access the network according to a positive bias of distance to offload UEs from the MBS-tier to the SBS-tier. We analyze the probability distribution of the transmit power with Fractional Power Control (FPC) to guide the selection of parameters to avoid exceeding the maximum transmit power limit and save power. (b) We derive the b-th moment of the conditional success probability for the Poisson downlink HetUDN with FPC considering three cases. First, the SBSs transmit with FPC but the MBSs transmit with fixed maximum power. Second, both the SBSs and the MBSs transmit with FPC under different parameters. Finally, in the special case, the SBSs and MBSs transmit with FPC under the same parameters, and the UEs are associated with their closest BSs without bias. So the two-tier HetUDN can be simplified as a one-tier UDN. (c) We analyze the ASE and EE of the whole HetUDN with three cases. The results show that power control greatly reduces the energy consumption of SBSs and MBSs. The remainder of this paper is organized as follows: we give our system model of the HetUDN based on
Qiaoshou Liu, Zhongpei Zhang. PPP in Section II. In Section III, we analyze the meta distribution of SIR in a two-tier downlink HetUDN considering three strategies of FPC. The analysis of ASE and EE are followed in Section IV. The conclusions are given in Section V. 2. System model We consider a two-tier heterogeneous ultral-dense cellular downlink network in a given area S ∈ R2 , as shown in Fig. 1. The network consists of an SBStier and an MBS-tier, and both BSs of the SBS-tier and the MBS-tier are deployed independently and randomly according to some homogeneous PPP Φm and Φ s with densities λm and λ s , i.e., Φ s ∼ PPP (λ s ) and Φm ∼ PPP (λm ), respectively. For the HetUDN, the number of SBSs is much larger than the number of MBSs, i.e., λ s λm . The UEs have certain temporal and spatial characteristics, which leads to the result that some SBs are fully loaded but others are partially loaded. In [12], the authors studied the heterogeneous cellular networks with spatio-temporal traffic. In our paper, for simplicity, we consider that all MBSs are fully loaded due to their particularities. Some SBSs in which no UE needs to be served are turned off to save energy. Assume that the active probability of SBSs is 0 < pa ≤ 1, where pa < 1 means that the SBS-tier is partially loaded, and pa = 1 means that the SBS-tier is fully loaded. For any UE, all active BSs act as interferers except the tagged BS, because all BSs reuse the same spectrum resources. All active SBSs are still a PPP Φa with density λa = pa λ s , according to Thinning theorem [13]. The UEs are modelled as a stationary point process Φu with density λu . According to Slivnyak-Mecke theorem [7], we consider a typical UE located at the origin. Without loss of generality, we denote SBSs, MBSs, n oand UEs by their locations, i.e., Φ s = {si }, Φm = m j , Φu = {uk } , and i, j, k ∈ N. Moreover, we denote the typical UE as uo , its closest SBS as so , and its closest n o MBS as mo , where uo ∈ {uk }, so ∈ {si }, and mo ∈ m j .
SBS
R
Dm
SBS
SBS UE
Rm
MBS UE
Dm
Typical MBS UE
MBS
Ds
MBS
SBS
Signal Interference
R
Ds
Rs SBS
Typical SBS UE
SBS
Fig. 1: The frame of heterogeneous ultra-dense networks.
We consider that the UEs openly access the network, i.e., any UE can be associated with an SBS or an MBS. In [14], a Reference Signal Received Power (RSRP)-based association scheme and its variant, i.e., Reference Signal Received Quality (RSRQ)-based association scheme are proposed for associations of UEs
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control in single-tier networks. But RSRP and RSRQ can not be directly used to the two-tier network because the transmit power of MBSs is much greater than that of SBSs, and most of UEs would choose MBSs to access. This leads to a huge traffic load imbalance between two tiers. A bias-based Cell Range Expansion (CRE) was exploited to offload UEs from the MBStier to the SBS-tier by adding a positive bias to their measured signal-strengths during cell association [15]. To offload UEs from the MBS-tier to the SBS-tier, we consider an open access policy based on distance [16], the UEs are associated with an SBS or an MBS by adding a positive bias to their measured distances from the closest BSs, which is more convenient for analysis than other methods. Let R s denote the distance from a UE to its closest SBS, and Rm denote the distance from a UE to its closest MBS. The open access policy is expressed as ( S BS → UE, i f R s ≤ κRm , κ ≥ 0, (1) MBS → UE, i f R s > κRm where κ is the offloading bias. So, the probability of a UE associated with an SBS is λ s /λ1 and the probability of a UE associated with an MBS is λm /λ2 , where λ1 = λ s + λm /κ2 and λ2 = λm + κ2 λ s . κ → 0 means that most of UEs are associated with MBSs; κ = 1 means that all UEs are associated with their closest BSs (SBSs or MBSs); κ → ∞ means that most of UEs are associated with SBSs, i.e., the number of UEs associated with SBSs increases with the increase of κ. Based on the characteristics of spatial PPP, we can easily get the Probability Density Function (PDF) of R s by conditioning on the open access policy as fRs |Rs ≤κRm (r) = 2λ1 πr · exp −λ1 πr2 , (2) and the PDF of Rm by conditioning on the open access policy as fRm |Rm
3
consumed and the downlink qualities of edge-UEs being worse than that of the inner-UEs. In [11], the author analyzed a one-tier cellular network with power control in the uplink and the downlink, not considering the multi-tier network. An Almost Blank Sub-frame (ABS) [19] was proposed to avoid the MBSs interfering the UEs of SBSs in a two-tier cellular downlink network. The ABS can eliminate the interference from MBSs to the UEs of the SBS-tier by sacrificing the spectrum resources of MBSs. As aforementioned, power control in the uplink benefits in two aspects: on the one hand, the UEs can save much power with power control, especially the inner-UEs, because they can transmit with very low power; on the other hand, power control will balance the performances of innerUEs and edge-UEs. In our downlink heterogeneous cellular model, we consider that the BSs transmit with FPC as the uplink. Adopting appropriate power control parameters can balance the performances of the inner-UEs and edge-UEs, and save power. Similar to the uplink, the BSs transmit data to the UEs with FPC and the transmit power of a BS is expressed as ( x Pt = t x Rα x , with FPC; (4) Ptx = T x , without FPC, where t x , x ∈ {s, m} is the minimum transmit power of BSs with FPC and is defined as the transmit power when R equals the unit distance. T x denotes the fixed transmit power of BSs without power control. We can get the distribution of transmit power Pts with FPC as 1 − exp −λ1 π Pts /t s 2/α , (5) as
and the distribution of transmit power Pm t with FPC 2/α 1 − exp −λ2 π Pm . (6) t /tm
Assume that the path loss between a UE and a BS is h x R−α x , x ∈ {s, m}, where h x is the small-scale fading, all h x are independent identical distribution (i.i.d.) and h x ∼ exp (1). R x ∈ {R s , Rm , D s , Dm } is the distance from a UE to the BSs. R s denotes the distance from a UE to its associated BS which belongs to the {si }, Rm denotes the distance from a UE to its associated BS which belongs to the {m j }, D s denotes the distance from a UE to one of its interfering SBSs which belongs to the {si }, and Dm denotes the distance from a UE to one of its interfering MBSs which belongs to the {m j }. α is the path loss exponent and α > 2 in the open air. In this case, the received signal of a UE from a BS is Ptx h x R−α x . For a quick reference, all of the notations and symbols used in this paper are summarized in Table 1. We assume there are 50 SBSs and 1 MBS in a 1 km2 area, i.e., λs = 1 · 10−6 pieces/m2 and λs = 5 · 10−5 pieces/m2 . We set t s = 1 mW, tm = 4 mW, 1/α and α = 4. In [16], let κ = Pts /Pm , then the intert tier interference from SBSs to macro-cell users is nullified. In this case, κ = 0.7 is adopted. Fig. 2 illustrates the Cumulative Distribution Function (CDF) of
4
Qiaoshou Liu, Zhongpei Zhang. Table 1: notations and symbols used in the paper
Notation λs pa λm ts tm Tm Rs Rm R
Description The density of the SBSs The probability of SBSs The density of the MBSs The minimum transmit power of the SBS The minimum transmit power of the MBS The fixed transmit power of the MBS Distance from a UE to its closest SBS Distance from a UE to its closest MBS Distance from the typical UE to its associated BS Distance from the typical UE to a interfering SBS Distance from the typical UE to a interfering MBS Offloading bias Path loss exponent Fractional power control efficient Probability operator The expectation over all random variables in [·] Euclidean distance
Ds Dm κ α P [·] E [·] k·k
BSs’ transmit power with FPC. In this paper, we limit the maximum transmit power of the SBSs and MBSs to 2 W and 5 W, receptively. As shown in Fig. 2, the transmit power curves will exceed the limit with a great probability if a larger is chosen. We choose = 0.3 as a middle value for simplicity in this paper if there is no special explanation.
openly access the network based on the policy defined in Eq. (1). Each BS serves at most one UE, and each UE is served by at most one BS at the same time. For the typical UE, we omit the subscripts of h x and R x . When the typical UE uo is associated with its closest SBS so according to the open access policy. In this case, R denotes the distance from uo to so , i.e., R = kso − uo k, where k·k is the Euclidean distance. The pa has no influence on the distribution of the distance from a UE to its closest BS but the the distribution of the interfering BSs. So, the SIR of the typical UE is P s hR−α ; (7) SIR s = t I s,s + I s,m if the typical UE is associated with its closest MBS mo according to the open access policy, R denotes the distance from uo to mo , i.e., R = kmo − uo k. The SIR of the typical UE is SIRm =
−α Pm t hR , Im,s + Im,m
(8)
P where I s,s = s∈Φa \{so } Pts h s D−α s is the cumulative interference from all other active SBSs except the so to P −α the typical UE uo ; I s,m = m∈Φm Pm t hm Dm is the cumulative interference from all MBSs to the typical UE P which associated with the so ; Im,s = s∈Φa Pts h s D−α s is the cumulative interference from all active SBSs to the typical UE which is associated with mo and P −α Im,m = m∈Φm \{mo } Pm t hm Dm is the cumulative interference from all MBSs except the mo to the typical UE which is associated with mo . Pts and Pm t are set based on Eq. (4).
1
3. Coverage probability
0.9
CDF of Tx_Power of BSs
0.8 0.7 0.6 0.5 0.4 0.3 SBS, MBS, SBS, MBS,
0.2 0.1
= 0.3 = 0.3 = 0.4 = 0.4
0 0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Tx_Power of BSs
Fig. 2: The CDF of BSs’ transmit power with FPC, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, κ = 0.7, and pa = 1.
2.2. SIR In cellular networks, the thermal noise is always neglected because it hardly affects the network’s performance[8]. For a UE in a cellular downlink network, the interference comes from other SBSs and MBSs which reuse the same spectrum resources at the same time. In this paper, we consider that all SBSs and MBSs reuse the total spectrum resources and UEs
In the HetUDN, the network performance is mainly influenced by the SBS-tier because the density of SBSs is much greater than that of MBSs and most of UEs are offloaded from the MBS-tier to the SBStier. The UEs associated with SBSs will be strongly interfered by MBSs if the MBSs transmit with high power, which will lead to a drastic decline in the network’s performance. To mitigate the interference from the MBS-tier to the SBS-tier, the transmit power of MBSs should be reduced compared with the conventional MBSs. Moreover, we consider three FPC strategies to coordinate the interference and balance the performances of inner-UEs and edge-UEs. Based on Eq. (2), (3), (7), and (8), the conditional coverage probability of the SBS-tier is P ss (θ) = P {SIR s > θ|Φ s , Φm } n o −1 = P h > θ Ptso Rα I s,s + I s,m |Φa , Φm Y Y 1 1 (a) = a −α α(1−) · m −1 −α α(1−) , 1 + θR D R 1 + θP t Dm R s s s t Φ Φ \{s } a
o
m
(9)
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control and the conditional coverage probability of the MBS-tier is Pms (θ) = P {SIRm > θ|Φa , Φm } n o o −1 α = P h > θ Pm R Im,s + Im,m |Φa , Φm t Y Y 1 1 (a) = , mo −α α · m mo −1 −α α 1 + θ P /P D R m 1 + θ P t s Ra t t s Ds R Φa Φm \{mo } t
(10) where (a) follows the i.i.d. of h and its further independence from the point process Φ, θ is the predefined m o SIR threshold for demodulation, and Ptso , Pm t , Pt need to be set according to Eq. (4). Then, the conditional coverage of the whole network is P s (θ) =
pcs
·
P ss
(θ) +
pm c
·
Pms
(θ) ,
(11)
where pcs =λ s /λ1 is the probability of a UE’s being associated with an SBS, pm c =λm /λ2 is the probability of a UE’s being associated with an MBS based on the open access policy. 3.1. Meta distribution In order to get more fine-grained information for each individual link in the network, the meta distribution of the SIR is defined by [20] as F Ps (x) = P (P s (θ) > x) , x ∈ [0, 1] , o
(12)
∆
where P s (θ) = P (SIR > θ|Φ) is the conditional success probability (coverage probablity) taking over the fading and the random activities of the interferes, given the point process. Po (·) denotes the reduced Palm measure of the point process, given a receiver at the origin [7]. We focus on the moments of P s (θ) because it is most likely impossible to calculate the meta distribution directly from the definition in Eq. (12). The b-th moment of P s (θ) is denoted by Z 1 ∆ o b Mb (θ) = E P s (θ) = bxb−1 F Ps (x)dx, b ∈ C, 0
(13) where Eo (·) is the reduced Campbell measure [7]. Hence we easily have the standard success probabil∆ ity p s (θ) = P (SIR > θ) ≡ M1 (θ). And the variance of P s (θ) equals M2 (θ) − M12 (θ). The meta distribution of the uplink SIR can be obtained from Gil-Pelaez theorem [21] as Z − jtx 1 1 ∞ Im e M jt F Ps (x) = + dt, (14) 2 π 0 t where Im (z) denotes the imaginary of the complex ∆ √ number z and j = −1. As in [20], some classic bounds are given for the meta distribution of Eq. (14) and the best bound is the beta distribution approximation. We only consider the beta distribution approximation in this paper. Since P s (θ) is supported on [0,1], it is natural to approximate its distribution with the beta distribution
5
[20]. The PDF of a beta distributed random variable X with mean u is x 1−u (1 − x)β−1 , B (uβ/(1 − u), β) u(β+1)−1
fX (x) =
(15)
R1 where B (a, b) = 0 ta−1 (1 − t)b−1 dt is the beta function. The variance is given by var X =
u(1 − u)2 . β+1−u
(16)
Matching the mean u = M1 and the variance var X = M2 − M12 yields (1 − M2 )(1 − M1 ) . (M2 − M12 )
β=
(17)
3.2. Only SBSs transmit with FPC Firstly, we consider that MBSs transmit with fixed power T m and SBSs transmit with FPC, i.e., Pts = t s Rα s . Theorem 1. Considering that MBSs transmit with fixed power and SBSs transmit with FPC, the bmoment of the conditional success probability for the SBS-tier is given by Z ∞ Mbs = exp −z 1 + fb,1 (z) + fb,2 (z) dz, (18) 0
where
fb,1 (z) =
λa λ1
R∞R∞ −xz ze 1− 1 0
1
dxdy,
(1+θxα/2 y−α/2 )b
(19) and
fb,2 (z) =
λm λ1
R∞ 1/κ
! 1−
1 α/2 −α/2 −α/2 b y z ) (1+θTm t−1 s (λ1 π)
dy.
(20) The b-moment of the conditional success probability for the MBS-tier is given by Z ∞ Mbm = B · exp −Bz 1 + f3,b (z) + f4,b (z) dz, 0
(21)
where λm f3,b (z) = λ2
Z
∞
1
f4,b (z) R∞R∞ = λλ2a κ 0 ze−zx 1 −
1 −
1 1 + θy
dy, −α/2 b
(22)
!! 1
(1+θTm−1 ts (λ1
b π)−α/2 xα/2 y−α/2 zα/2
)
dxdy,
(23) where B = λ2 /λ1 . Proof of Theorem 1. The proof is given in Appendix 6.1.
6
Qiaoshou Liu, Zhongpei Zhang.
Compared with the conventional MBS, T m should be descended to reduce the energy consumption and mitigate the interference from MBSs to the UEs of SBSs because most of the edge-UEs of MBSs are offloaded to the SBS-tier with the bias κ. Let T m = 5000 mW, t s = 1 mW, κ = 0.7, = 0.3, and the network’s area is 5km × 5km for both numerical results and Monte Carlo simulation. Fig. 3 illustrates the coverage probability (the 1 st moment) and its variance when the network is fully loaded. The dotted lines with marks denote the Monte Carlo simulation results of coverage probabilities; the solid lines denote the numerical results of coverage probabilities; and the dotted lines with plus signs denote the numerical results of variances. As shown in Fig. 3, the coverage probability of the MBS-tier with a higher variance is larger than that of the SBS-tier, because MBSs transmit with fixed maximum power. The variance of the SBS-tier is less than that of the MBS-tier because FPC balances the performances of inner-UEs and edge-UEs on the SBS-tier. The whole network coverage probability and its variance are mainly dominated by the SBS-tier because the number of SBSs is much more than the number of MBSs. 1
0.2 P SBS - Num
0.9
P MBS - Num
0.18
0.8
P - Num P - Sim
0.16
0.7
P MBS - Sim
0.14
0.6
P - Sim V SBS - Num
0.12
0.5
V MBS - Num
0.1
V - Num
0.4
0.08
0.3
0.06
0.2
0.04
0.1
0.02
0 -20
-15
-10
-5
0
5
10
15
20
25
Variance
Coverage Probability
SBS
0 30
where
g1,b (z) = g2,b (z) =
λa λ1
λm λ1
R∞R∞ −xz ze 1− 1 0
R∞ R∞ 1/κ 0
−Bzx
Bze
1
(
dxdy,
b 1+θxα/2 y−α/2
)
(25) dxdy,
1 1− α/2 y−α/2 b (1+θtm t−1 ) s x
(26) the b-moment of the conditional success probability for the MBS-tier is given by Z ∞ Mbm = B · exp −Bz 1 + g3,b (z) + g4,b (z) dz, 0
(27)
where g3,b (z) =
λm λ2
R∞R∞
g4,b (z) =
λa λ2
R∞R∞ −zx ze 1− κ 0
1
0
Bze−Bzx 1 −
1
dxdy,
(1+θxα/2 y−α/2 )b
1 (1+θts tm−1 xα/2 y−α/2 )b
(28) dxdy,
(29) where B = λ2 /λ1 Proof of Theorem 2. The proof is given in Appendix 6.2. In this section, MBSs transmit with FPC. Let tm = 4 mW, and other parameters remain unchanged. Fig. 4 illustrates the coverage probability and variance when the network is fully loaded. The coverage probability of the MBS-tier descends and that of the SBS-tier rises a little compared with Fig. 3, because the transmit power of MBSs is reduced to mitigate the interference to thhe SBS-tier. Moreover, power control benefits the edge-UEs by sacrificing interests of the inner-UEs on the MBS-tier as well as the SBS-tier, which can be proved by the variance of the MBS-tier (green dotted line with plus signs), which is smaller than that in Fig. 3.
Required SIR (dB)
Due to the special status of MBSs, it is impossible for MBSs to transmit to all UEs with vary small fixed power, although it will reduce the interference to the UEs on the SBS-tier. So, we consider that both MBSs α and SBSs transmit with FPC, i.e., Pm t = tm Rm and s α Pt = t s R s , respectively. In this case, the performance of edge-UEs of MBSs can be guaranteed. Theorem 2. Considering that both MBSs and SBSs transmit with FPC, the b-moment of the conditional success probability for the SBS-tier is given by Mbs =
Z 0
∞
exp −z 1 + g1,b (z) + g2,b (z) dz,
(24)
Coverage Probability
3.3. Both BSs of the SBS-tier and MBS-tier transmit with FPC
0.2 P SBS - Num
0.9
P
MBS
- Num
0.18
0.8
P - Num P SBS - Sim
0.16
0.7
P MBS - Sim
0.14
0.6
P - Sim V - Num
0.12
0.5
V
0.1
SBS
MBS
- Num
V - Num
0.4
0.08
0.3
0.06
0.2
0.04
0.1
0.02
0 -20
-15
-10
-5
0
5
10
15
20
25
Variance
1
Fig. 3: The coverage probability and its variance in the case of only SBSs transmitting with FPC, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, and pa = 1.
0 30
Required SIR (dB)
Fig. 4: The coverage probability and variance in the case of both SBSs and MBSs transmitting with FPC, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, and pa = 1.
3.4. Special case In this section, we consider a special case, i.e., letting t s = tm , κ = 1 when SBSs and MBSs both transmit with FPC. So, the two-tier heterogeneous cellular network will become a homogeneous network with density λ = λa + λm .
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control
0
1 Case 1, pa = 0.5 Case 2, pa = 0.5 Case 3, pa = 0.5 nPC, pa = 0.5 Case 1, pa = 1 Case 2, pa = 1 Case 3, pa = 1 nPC, pa = 1
0.9 0.8
Coverage Probability
Corollary 1. From a mathematical point of view, when tm = t s , κ = 1, we can get λ1 = λ2 = λ s + λm = λ, B = 1. So, the b-moment of the conditional success probability for the whole network is Z ∞ Mb = exp (−z (1 + fb (z)))dz, (30)
7
where
0.7 0.6 0.5 0.4 0.3 0.2
Z Z pa λ s + λm ∞ ∞ −xz 1 dxdy ze 1 − fb (z) = b α/2 −α/2 λ s + λm 1 0 1 + θx y Z ∞Z ∞ −xz (b) 1 dxdy. ≈ pa ze 1 − b α/2 −α/2 1 0 1 + θx y
(31)
0.1 0 -20
-15
-10
-5
0
5
10
15
20
25
30
Required SIR (dB)
Fig. 6: The coverage probabilities in three cases vary with pa , where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
where (b) follows λ s λm . 0.16
Proof of Corollary 1. The proof is given in Appendix 6.3.
0.12 0.1
Var
Eq. (30) is consistent with Eq. (25) in [11] when pa = 1. It is the moment of the conditional success probability for a one-tier cellular network with FPC.
Case 1, pa = 0.5 Case 2, pa = 0.5 Case 3, pa = 0.5 nPC, pa = 0.5 Case 1, pa = 1 Case 2, pa = 1 Case 3, pa = 1 nPC, pa = 1
0.14
0.08 0.06 0.04
Coverage Probability
0.8
0.18
0.14
0.6
0.12
0.5
0.1
0.4
0.08
0.3
0.06
0.2
0.04
0.1
0.02
-15
-10
-5
0
5
10
15
20
25
0 -20
-15
-10
-5
0.16
0.7
0 -20
0.02
0.2 P - Num P - Sim P - nPCm V - Num V - nPC
0
5
10
15
20
25
30
Required SIR (dB)
Variance
1 0.9
0 30
Required SIR (dB)
Fig. 5: The coverage probability and variance of the special case, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4, and pa = 1.
Fig. 7: The variance of coverage probabilities in three cases vary with pa , where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
lower when the network is partially loaded. The performance of cases 3 is the best among the three cases, but lower than the case of nPC in the high-θ regime. This is because power control benefits the edge-UEs by sacrificing interests of the inner-UEs. 1 Case 1, pa = 0.5 Case 2, pa = 0.5 Case 3, pa = 0.5 nPC, pa = 0.5 Case 1, pa = 1 Case 2, pa = 1 Case 3, pa = 1 nPC, pa = 1
0.8 0.7 0.6
1-F(x)
Fig. 5 illustrates the coverage probability and its variance of the special case when the network is fully loaded. The black solid line is the coverage probability when all SBSs transmit with a fixed maximum power 2000 mW (nPC) but other parameters remain unchanged in a one-tier UDN. As shown in Fig. 5, the coverage probability with FPC (red solid line) is greater than that of nPC in the low-θ regime but lower than that of nPC in the high-θ regime. This is because FPC can balance the performance of the innerUEs and edge-UEs. This can be proved from the variance where the variance of nPC (black dotted line with plus) is greater than that with FPC (red dotted line with plus). Fig. 6 and Fig. 7 illustrate the coverage probabilities and their variances of the SBS-tier in three cases when the network is fully loaded or partially loaded. We only consider the SBS-tier due to the fact that λ s λm . We add the nPC as the baseline. Obviously, the coverage probabilities are higher and variances are
0.9
0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Fig. 8: The meta distribution by beta distribution approximation varies with pa , where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , α = 4.
Fig. 8 illustrates the beta distribution approximation of the meta distribution through matching the mean and variance with the required SIR threshold θ = 5 dB for the three cases and the nPC case considering that the network is either fully loaded or partially loaded. It can reveal the percentage (y-axis) of UEs whose suc-
8
Qiaoshou Liu, Zhongpei Zhang.
cess probability is greater than some values (x-axis) with a given SIR threshold θ. As shown in Fig. 8, a larger proportion of UEs have a larger probability to achieve the SIR requirement in case 3 among the three cases. For example, there are about 10 percent of UEs whose success probability is greater than 0.8 (x-axis) in case 3 when the the network is fully loaded. Meanwhile, the UEs in case 1 and case 2 are less than 10 percent. The proportions of case 1, case 2, and case 3 increase to 21%, 23%, and 25% when the network is partially loaded with pa = 0.5 for the same x. The UEs’ proportion of nPC is highest in the high-x regime (means high-θ regime), but lower than those of case 2 and 3 in low-x regime (means low-θ regime). This further proves that power control can benefit the edgeUEs by sacrificing interests of the inner-UEs.
SBSs is much more than that of MBSs. Moreover, case 3 achieves the highest ASE among the three FPC cases. The ASE of nPC is greater than the ASE of the three FPC cases in the high-θ regime by guaranteeing the inner-UEs’ link quality at the expense of more energy consumption. Obviously, when the network is partially loaded, the spatial reuse efficiency of spectrum resources reduces due to BSs’ sleeping although the coverage probability increases. It leads to the descending of all ASE (line curves). The ASE of the network being fully loaded is the maximum ASE that the network can reach and the ASE of the network being partially loaded can be regarded as the real-time ASE of the network.
4
10-5 Case 1 ,p a = 0.5
3.5
4. ASE and EE
Case 2 ,p a = 0.5 Case 3 ,p a = 0.5
In our HetUND model, MBSs and SBSs are randomly deployed in a given area and all base stations totally reuse the same spectrum resources. There are not only intra-tier but also inter-tier interference in the HetUDN. In the previous section, we have evaluated the network performance in three FPC cases by coverage probability, its variance, and the meta distribution of SIR to reveal the distribution of UEs whose success probability is greater than some values with a given SIR threhold. In fact, power control not only mitigates the interference, balances the performances between inner-UEs and edge-UEs in the cell, but also reduces the energy consumption of BSs. In this section, we further measure the network performance by the ASE and EE. Definition 1. In this section, the ASE is the spectral efficiency in a unit area, and the ASE of the SBS-Tier is defined as ASE s =pa λ s Pcs (θ) log2 1 + SIRreq , (32) the ASE of the MBS-tier is defined as ASEm =λm Pm c (θ) log2 1 + SIRreq ,
(33)
and the ASE of the whole network is ASE = ASE s +ASEm ,
(34)
m where Pcs (θ) = M1s and Pm c (θ) = M1 , i.e., the coverage probability of SBS and MBS, respectively; SIRreq = θ is the required SIR threshold for demodulation.
Fig. 9 illustrates the ASE of the three FPC cases in the previous section compared with the case of nPC. As shown in Fig. 9, the three colour solid lines of FPC with pa = 1 basically coincide and the three colour dotted lines of FPC with pa = 0.5 basically coincide too. This is because the SBS-tier is set with same parameters in the three FPC cases and the number of
ASE (bps/Hz/m 2 )
3
no PC,p = 0.5 a
Case ,p = 1 1
2.5
a
Case ,p = 1 2
a
Case 3 ,p a = 1
2
no PC,p = 1 a
1.5 1 0.5 0 -20
-15
-10
-5
0
5
10
15
20
25
30
Required SIR (dB)
Fig. 9: The area spectral efficiency of three cases, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
Let BW denote the total bandwidth which is reused by all SBSs and MBSs, S denote the area of the network, and τ denote the running time. The EE is defined as the number of bits sent at unit bandwidth by consuming one Joule energy.
Definition 2. The EE of SBS-tier is defined as total capacity o f S BS − tier (total energy o f S BS − tier) · BW ASE s · BW · τ · S = s E Pt · BW · τ · S · λa ASE s = s , E Pt · λa
EE s =
(35)
the EE of MBS-tier is defined as total capacity o f MBS − tier (total energy o f MBS − tier) · BW ASEm · BW · τ · S = m E Pt · BW · τ · S · λm ASEm , = m E Pt · λm
EEm =
(36)
The analysis of coverage probability, ASE and EE in heterogeneous ultra-dense networks with power control the EE of the whole network is defined as total capacity (total energy) · BW ASE · BW · τ · S = s E Pt · τ · S · λa + E Pm t · τ · S · λm · BW ASE = s . E Pt · λa + E Pm t · λm (37)
EE =
Based on Eq. (5) and (6), we can get α ts α E Pts = · ·Γ , α/2 2 2 (λ1 π)
(38)
and tm α α α/2 · 2 · Γ m 2 , with FPC (λ π) 2 E Pt = T m , without FPC.
(39)
Fig. 10 illustrates the EE in the three cases with FPC compared with the case of nPC. As shown in Fig. 10, power control greatly improves the EE, especially in case 3. Power control reduces the energy consumption of BSs but almost does not affect the ASE. In Fig. 9, the ASE of nPC is greater than that of the three cases with FPC in the high-θ regime, but the EE of nPC is the lowest among all cases in Fig. 10. This is because power control sacrifices the performance of the inner-UEs to guarantee the performance of the edge-UEs and mitigate the interference to improve the network’s EE. The EE of all cases increases when the network is partially loaded because its coverage probability is greater than that of the full loaded network. This means the BSs can send more bits with the same energy consumption.
Case ,p = 0.5 a
Case ,p = 0.5
6
2
a
Case 3 ,p a = 0.5 no PC,p a = 0.5
5
EE, will increase if we reasonably offload UEs from the MBS-tier to the SBS-tier, especially in case 3 by changing the HetUDN into a one-tier UDN.
5. Conclusion In this paper, we model a heterogeneous ultra-dense cellular network based on spatial PPP. In our model, we consider three cases of FPC strategies, and derive the probability and variance of the UEs’ successfully being associated with BSs utilizing the meta distribution. The numerical results and Monte Carlo simulation results show that the network performance is mainly influenced by the SBS-tier in the HetUDN because the number of SBSs is much larger than that of the MBSs and most edge-UEs of MBSs are offload to the SBS-tier. Power control can mitigate the interference by sacrificing the performance of inner-UEs to guarantee the performance of edge-UEs. Furthermore, power control can balance the performances of two tiers if both tiers transmit with power control. Especially, in case 3, we change the two-tier HetUDN into a one-tier homogeneous network by setting appropriate parameters. And the numerical results show it can greatly improve the EE with little loss of the network performance. Finally, in the HetUDN, some BSs which no UE needs to be served can be turned off to save energy and to increase the coverage probability and EE. But BSs’ sleeping will reduce the ASE due to the decrease of active BSs.
Acknowledgement This work was supported in part by the Major Program of the National Nature Science Foundation of China (Grant No. 61831004).
7 1
9
EE (bits/Hz/J)
Case 1 ,p a = 1 Case 2 ,p a = 1
4
Case 3 ,p a = 1 no PC,p a = 1
6. Appendix
3
2
6.1. Proof of theorem 1
1
0 -20
-15
-10
-5
0
5
10
15
20
25
30
Required SIR (dB)
Fig. 10: The energy efficiency of three cases, where λs = 1 · 10−6 pieces/m2 , λs = 5 · 10−5 pieces/m2 , and α = 4.
In the HetUDN, the inter-cell interference exists on both the same tier and different tiers. It is very important to balance loads and jointly optimize resources [22, 23]. The numerical and simulation results show the network’s performance, such as the coverage probability, the proportion of UEs achieving some success probabilities with a given SIR threshold, ASE, and
SBSs transmit with FPC, and MBSs transmit with m fixed power, i.e., Pts = t x Rα s , and Pt = T m , respecmo m tively. Then Pt /Pt = 1. From Eq. (9), the Mbs follows as
Mbs
Y =ER ERs ,Ds
Φa \{so }
Y EDm Φm
1 1+
−α α(1−) b θRa s Ds R
. b −α α(1−) 1 + θT m t−1 D R m s 1
· (40)
10
Qiaoshou Liu, Zhongpei Zhang. Let x = t/r, y = v/r, z = λ1 πr2 . We can get
and "
1 a −α α(1−) b Φa \{so } 1 + θR s D s R Z ∞ Z ∞ 2 (c) 2λ1 πte−λ1 πt 1 − = exp −2λa π dt vdv b r 0 1 + θtα v−a rα(1−) Z Z ∞ λa ∞ e−u 1 − dw = exp − du b α/2 −α/2 α(1−)/2 λ1 z 0 1 + θu w z Z ∞ Z ∞ −u λa e−u du dw = exp − e − b α/2 −α/2 α(1−)/2 λ1 z 0 1 + θu w z Z ∞Z ∞ −xz λa 1 ze 1 − , = exp − z dxdy b α/2 −α/2 λ1 1 0 1 + θx y
Y ERs ,Ds
ERs ,Ds
# 1
Q
−α α b 1+θT m−1 t s Rα s Ds R
) Φa ( R R∞ 2 ∞ 2λ1 πte−λ1 πt = exp −2λa π κr 1 − 0 b dt vdv −1 α −a α (1+θTm ts t v r ) ! ! R∞ R∞ e−u = exp − λλa1 κz 1 − 0 du dw (1+θTm−1 ts (λ1 π)−α/2 uα/2 w−α/2 zα/2 )b ! ! R∞R∞ e−u = exp − λλa1 κz 0 e−u − dudw (1+θTm−1 ts (λ1 π)−α/2 uα/2 w−α/2 zα/2 )b ! ! R R ∞ ∞ ze−zx = exp − λλa2 Bz k 0 ze−zx − dxdy −α/2 α/2 −α/2 α/2 b −1 (1+θTm ts (λ1 π) x y z )
(45) where B = λ2 /λ1 , s = Bz. Combining Eq. (3), (43), (44), and (45) yields Eq. (21).
(41) 6.2. Proof of theorem 2
and
SBSs and MBSs transmit with FPC, Pts = t s Rα s , and mo m α −α = tm Rα , respectively, then P /P = R . m mR t t The rest of the proof is similar to Appendix 6.1. Pm t
Y 1 EDm −1 D−α Rα(1−) 1 + θP P m s m Φm Z ∞ (c) 1 vdv = exp −2λm π 1 − b −1 −a α(1−) r/κ 1 + θT m t s v r λm Z ∞ 1 1 − = exp − b dw λ1 z/κ α/2 −α/2 α(1−)/2 (λ w z 1 + θT m t−1 π) 1 s Z ∞ 1 λm 1 − = exp − z dy , b λ1 1/κ α/2 −α/2 −α/2 −1 z 1 + θT m t s (λ1 π) y
6.3. Proof of corollary 1 mo SBSs and MBSs transmit with FPC, then Pm t /Pt = Moreover, tm = t s , κ = 1, and we can get λ1 = λ2 = λ s + λm = λ, B = 1. The rest of the proof is similar to Appendix 6.1. −α . Rα mR
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(42) where (c) follows the PGFL. Combining Eq. (2), (40), (41), and (42) yields Eq. (18). From Eq. (10), the Mbm follows as
Mbm
Y 1 =ER EDm · −α α b Φm \{mo } (1 + θDm R ) Y ERs ,Ds Φa
(43) b . α −α α m −1 1 + θ Pt ts Rs Ds R 1
Let x = t/r, y = v/r, s = λ2 πr2 = Bz, and we can get Y 1 EDm −α α b (1 ) + θD R m Φm \{mo } ! ! Z ∞ 1 vdv = exp −2λm π 1− (1 + θv−a rα )b r Z ∞ λm 1 1 − = exp − b dw (44) −α/2 α/2 λ2 s 1 + θw s Z ∞ λm 1 1 − dy = exp − s b λ2 1 1 + θy−α/2 Z ∞ λm 1 = exp − Bz 1 − b dy , −α/2 λ2 1 1 + θy
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Conflict of interest statement
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service or company that could be construed as influencing the position presented in the manuscript entitled “The analysis of coverage probability, ASE and EE in heterogeneous ultradense networks with power control”
Yours Sincerely, Qiaoshou Liu, Zhongpei Zhang 07/01/2020