IEEE ICC 2014 - Communication QoS, Reliability and Modeling Symposium
Aggregate Interference Statistical Modeling and User Outage Analysis of Heterogeneous Cellular Networks Tiankui Zhang,Lu An
Yue Chen, Kok Keong Chai
School of Information and Communications Engineering Beijing University of Posts and Telecommunications Beijing, China Email: zhangtiankui,
[email protected]
School of Electronic Engineering and Computer Science Queen Mary University of London London, UK Email: yue.chen,
[email protected]
Abstract—The heterogeneous cellular networks (HCNs) will be the typical layout of the next generation mobile networks. Understanding the aggregate interference from multi-tier heterogeneous base stations (BSs) of HCNs is the key for research on network deployment and interference management. In this paper, we propose a statistical model for quantifying the aggregate interference in HCNs and evaluating its impact on system performance. We first model the distribution of multitier heterogeneous BSs as spatial Poisson point process and derive the characteristic function (CF) of the downlink aggregate interference for a specific target user. We review the CF of single-tier network interference and proof that the aggregate interference of HCNs follows the stable distribution, based on which, we derive statistical characterization of aggregate interference amplitude and power, respectively. Then, we propose an aggregate interference statistical model based on truncated-stable distribution. Finally, the users outage probability of the HCNs is analysed via the proposed model. The proposed model is validated with simulation. This work provides essential understanding of interference of HCNs and gives insights which can facilitate system performance analysis and interference management.
I. I NTRODUCTION In the future, the cellular networks will be a heterogeneous cellular network (HCN), in which, different types of base stations (BSs) of small cell networks are distributed throughout traditional macrocell networks [1]. The HCNs consist of different tiers of small cell networks. Fig. 1 shows an example of a three-tier heterogeneous cellular network which contains picocell and femtocell networks share the same frequency spectrum as the macrocell network. In cellular networks, the spatial location of the BSs can be modeled either deterministically or stochastically. The most popular deterministic network model is the two-dimensional hexagonal grid model, which is used as the basis of system-level simulations, but tractable theoretical performance analysis is not possible [2]. In addition, the accuracy of such model in the case of HCNs is questionable, since the HCNs have random location of multitype BSs and different coverage of small cells. 1 This research is financially supported by the National Science and Technology Major Projects (2012ZX03001031-004).
978-1-4799-2003-7/14/$31.00 ©2014 IEEE
Fig. 1. Example of a three-tier heterogeneous cellular network with a mix of macro, pico and femtocell BSs.
Recently, a tractable analytical modeling method for the traditional one-tier macrocell networks and HCNs has been researched in[3-5], in which, the locations of BSs is modeled as Poisson point process (PPP). Importantly, such a model allows useful mathematical tools from stochastic geometry to trace the system capacity and cell coverage. Besides, interference modeling has gained a lot of attention from both academy and industry in the context of ultra wideband (UWB) [6] and cognitive network [7-9], using the PPP [6-7], lognormal distribution [8] and the stable distributions [9-10]. The rationale for modeling interference as a stable distributed statistical model are: 1) the ability to capture the spatial distribution of the interfering nodes; and 2) the ability to accommodate heavy tail behavior with the dominant contribution of a few interferers in the vicinity of the target user [10]. However, only when the interferers are assumed to be distributed in an infinite region, the aggregate interference converges to a stable distribution [9]. This is obvious unrealistic in practical scenarios in cellular networks. While the aforementioned literatures laid a solid foundation in modeling the traditional cellular networks and HCNs but there is still missing a practical and reliable tractable interfer-
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IEEE ICC 2014 - Communication QoS, Reliability and Modeling Symposium
ence model that able to capture the essential physical parameters of the interferers. These parameters include the spatial distribution of the interferers and also the transmission and propagation characteristic of the interferers, such as transmit power and channel fading. In this paper, we extend the statistical model of the singletier case of [10] to the multi-tier case. Based on this, we use the truncated-stable distribution to model the aggregate interference of multi-tier networks in a finite region. We propose a statistical model of the aggregate interference and quantify the impact of the aggregate interference on the users outage performance in HCNs. The remainder of the paper is organized as follows. In Section II, we describe the system model of multi-tier networks. In Section III, the statistical characterization of the interference amplitude and power are derived, respectively. Section IV discuses the system performance in terms of outage probability. The simulation results are given in Section V. Finally, Section VI concludes the paper. II. S YSTEM M ODEL We model a K-tier heterogeneous cellular network as one tier marcocell and K-1 tiers small cells. The BSs across tiers may differ in terms of the transmit power, the network capacity and their spatial density. We model the spatial distribution of the BSs in the kth tier network according to a homogeneous PPP of density λk in the two dimensional plane. The probability of n BSs being inside a region (not necessarily connected) depends only on the total area A of the region and is given by
Set the interference power from the ith BS of the kth tier network to target user as Ik,i , which is expressed as 2
−2b Ik,i = Pk |Hk,i | Rk,i .
The aggregate interference power of all the BSs in the kth tier networks is 2 −2b |Hk,i | Rk,i . (6) I k = Pk i∈Sk
The aggregate interference power of all the BSs in systems is I=
where, b > 1 is the amplitude path loss exponent, and the corresponding power path loss exponent is 2b. The aggregate interference amplitude of all the BSs in the kth tier networks is −b Y k = Pk Hk,i Rk,i , (3)
Y =
K k=1
Yk .
(4)
Ik .
(7)
III. S TATISTICAL M ODEL OF AGGREGATE I NTERFERENCE A. Stable Distribution The stable distribution of a real random variable (RV) X can be denoted as X ∼ S (α, β, γ) by the characteristic exponent α ∈ (0, 2], skewness β ∈ [−1, 1], and dispersion γ ∈ [0, ∞). The corresponding characteristic function (CF) is
α , α = 1 exp −γ |ω| 1 − jβsign (ω) tan πα 2
ϕ (ω) = exp −γ |ω| 1 + j π2 βsign (ω) ln |ω| , α = 1 (8) and when β = 0, X follows symmetric stable distribution. There is no close form of probability density function (PDF) of stable distribution, we can give the express of its PDF by Fourier transform, 1 f (x; α, β, γ) = 2π
+∞ ϕ (ω) exp (−jωx) dω.
(9)
−∞
Stable distribution has the two important properties, the stability property and the generalized central limit theorem, for aggregate interference modeling. B. Statistical Characterization of Interference Amplitude For the kth tier network, let Rk,i denote the distance of the
∞ user and the ith BS, and the sequence of distances target is a two-dimensional Poisson process with spaRk,i i=0
∞ √ tial density λk . Let Qk,i = Pk Hk,i i=0 be a sequence ∞
. Let of real RV, independent of the sequence Rk,i i=0 ∞ −b Rk,i Qk,i denote the aggregate interference at the Yk = i=1
origin (received by the target user) generated by the kth tier BSs scattered in the infinite plane. However, considering the limitation of the practical system, the aggregate interference amplitude should be
i∈Sk
where Sk is the BSs active set in the kth tier network which are the interferer of the target user. The aggregate interference amplitude of all the BSs in systems is
K k=1
n
(λk A ) exp (−λk A ) , n ≥ 0. (1) P {n} = n! Without loss of generality, we conduct analysis on a typical mobile user located at the origin, this user is named as target user. We set the frequency reuse factor is one, so except the serving BS of the target user, all the other BSs in the system are potential interferers. The distance between the ith BS of kth tier network and the target user is Rk,i , and the corresponding channel fading (including shadowing, and fast fading) is Hk,i . The received interference signal of target user from the ith BS of the kth tier network is −b Yk,i = Pk Hk,i Rk,i , (2)
(5)
Yk =
∞ i=1
−b Rk,i Qk,i ISk (h, r) ,
(10)
in which, ISk (h, r) is the indicator function, defined as, 1, (h, r) ∈ Sk . (11) ISk (h, r) 0, otherwise
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IEEE ICC 2014 - Communication QoS, Reliability and Modeling Symposium
It means that only the BSs in the active set Sk can interfere the target user, then the corresponding values of Qk,i , Rk,i make sense. The BSs set Sk can be defined according to different scenario of the systems. For example, 1) if Sk = {(q,r) : u < r ≤ v}, (10) represents the aggregate interference resulting from all the BSs inside a region described by u < r ≤ v.
2 −2b < Ith , (10) represents the 2) if Sk = (q,r) : Pk |h| |r| aggregate interference resulting only from the BSs for which 2 −2b at the target use is below the received power Pk |Hk,i | Rk,i the threshold Ith , which can be seen as a effect of power control. The CF of Yk is ϕYk (ω) = E [exp (jωYk )], using the Campbells theorem, ϕYk (ω) can be expressed as,
⎧ ⎪ ⎨
1−α , α = 1 Γ (2 − α) cos (πα/2) , (18) Cα ⎪ ⎩ 2, α = 1 π where Γ (·) denotes the gamma function. So in the case of infinite plane, the CF of Yk is α
ϕYk (ω) = exp (−γk |ω| ) ,
(19)
α
in which, α = 2b , γk = λk πCα−1 E {|Qk,i | }. The above analysis gives the CF of aggregate interference in a single tier network, based on which, we deduce the multi-tier network interference statistical characterization. According to the property of CF, when Y1 , Y2 , · · · , YK are K independent RVs, and Y = Yk , the CF of Y is k=1
ϕYk (ω) = ϕY (ω) = ϕY1 (ω) ϕY2 (ω) · · · ϕYK (ω) K , jωq . (20) α 1Sk (q, r) fQ (q) rdqdr exp −2πλk 1 − exp = exp −|ω| γk rb k=1 (12) in which, fQ (q) is the PDF of Q. From (8) we know that, the RV whose CF is given by (20) K r ≤v}, according For the BS set Sk = {(q,r) :u
