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Stochastic geometry modeling and analysis of cognitive heterogeneous cellular networks
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 141 (2015)
Abstract
In this paper, we present a cognitive radio (CR)based statistical framework for a twotier heterogeneous cellular network (femtomacro network) to model the outage probability at any arbitrary secondary (femto) and primary (macro) user. A system model based on stochastic geometry (utilizing the spatial Poisson point process (PPP) theory) is applied to model the random locations and network topology of both secondary and primary users. A considerable performance improvement can be generally achieved by mitigating interference in result of applying the CR idea over the above model. Novel closedform expressions are derived for the downlink outage probability of any typical femto and macro user considering the Rayleigh fading for the desired and interfering links. We also study some important design factors which their role and importance in the determination of outage and interference cannot be ignored. We conduct simulations to validate our analytical results and evaluate the proposed schemes in terms of outage probability for different values of signaltointerferenceplusnoise ratio (SINR) target.
1 Introduction
The best solution to the spectrum saturation and bandwidth availability problems in multitier cellular networks is to adopt technologies that make the most efficient use of existing spectrum through frequency reuse schemes [1,2]. In universal frequency reuse scheme, the existing spectrum can be aggressively and effectively reused by all of the coexisting network tiers. This will lead to higher spatial spectrum utilization and network usage capacity at the expense of an increased possibility of interference among network tiers and of a reduced quality of service (QoS). In multitier cellular networks, interference is increasingly becoming a major performancelimiting factor, and hence, interference modeling, coordination, and avoidance are the primary focus of interest for both the industry and academic communities. Applying the cognitive radio (CR) technology in multitier cellular networks to be aware of and adapt to communication environments, some of the above challenges can be tackled. In fact, CR is the key enabling technology for interference management and avoidance in multitier cellular networks [2,3]. On the other hand, the aggregate interference environment is more complicated to model, and evaluating the performance of communication techniques in the presence of heterogeneous interference is challenging. For interference characterization, if the base stations (BSs) of the cellular network follow a regular grid (e.g., the traditional hexagonal grid model), then the SINR characterization will be either intractable [3,4] or inaccurate due to unrealistic assumptions [5]. Moreover, as urban areas are built out, the BS infrastructure is becoming less like points on a hexagonal lattice and more random. Hence, the use of a hexagonal grid to model the BS locations is violated and is considered too idealized [6]. Furthermore, according to [3,4,6] for snapshots of a cellular network at different locations, the positions of the BSs with respect to each other follow random patterns due to the size and unpredictability of the BSs in these kind of networks. Therefore, the need for a powerful mathematical and statistical tool for modeling, analysis, and design of wireless networks with random topologies is quite obvious.
A new modeling approach called ‘stochastic geometry’ has been recently applied to the analysis of multitier cellular networks due to its ability to capture the topological randomness in the network and its aim at deriving accurate and tractable expressions for outage probability [3,6]. Stochastic geometry stems from applied probability and has a wide range of applications in the analysis and design of wireless networks in particular for modeling and analyzing systems with random channel access (e.g., ALOHA [7,8] and carrier sensing multiple access (CSMA) [9]), single and multitier cellular networks [6], and networks with cognitive abilities [7,10]. Multitier cellular networks have been investigated from different perspectives such as power control [11,12], spectrum allocation [13,14], and exploiting CR techniques [15,16], and recently, many works have been done based on the similar concepts to adopt and extend the stochastic geometric approach to different network models and scenarios (see [1720]). This paper discusses this new theoretical model to provide a better understanding of the heterogeneous cellular networks of tomorrow and their challenges (interference modeling, coordination, and avoidance) that must be tackled in order for these networks to reach their potential. We focus on a twotier femtomacro network where lowpower and smallcoverage local nodes (femto nodes) are distributed in the coverage of macro nodes. We provide an insight into the role of CR in interference mitigation in twotier heterogeneous networks. We derive closedform expressions for the outage probability of any typical femto and macro user in the network. We also study the effect of several important design factors which play vital roles in the determination of outage and interference.
Our main contributions in this work which is an extension of [21] are therefore the following: (1) We analyze the Laplace transforms of all four types of aggregate interference between macro and CR femto networks (including the interference between macro nodes among themselves and femto nodes among themselves, the crossinterference from femto to macro network and vice versa) in perfect and imperfect spectrum sensing CRbased femto networks, considering simultaneously the Poisson point process (PPP) model, and some important design factors (such as spectrum access probability) which can play a major role in determining interference and outage. (2) This article provides an insight into the role of CR in interference mitigation in orthogonal frequencydivision multipleaccess (OFDMA) twotier heterogeneous networks. (3) Closedform expressions are derived for the outage probability of any typical femto and macro user considering the Rayleigh fading assumption for the desired and interfering links with the possibility of using the CR ability for the femto network. It should be noted that in most of the available studies in this area, none of the network tiers is equipped with the CR capability; they are mostly based on the existence of only one macroBS (along with the macro users and the femto network); and the effect of considering multiple macroBSs is ignored in the analysis of outage probability. Authors in [2225] have considered the twotier heterogeneous networks imposing the CR ability to the femto tier. Different from [22], in our work, we consider both the perfect and imperfect sensing scenarios for the CR femtoBSs, however authors in [22] ignore the effect of sensing errors on the opportunistic channel access probability and consequently the outage probability of each tier. On the other hand, in our work, the mathematical demonstration of the obtained expressions (channel access probability and outage probability expressions) is quite different from the mentioned works.
2 Downlink system model
2.1 Model description
We consider infinite spatially collocated macroBS and femtoBS node heterogeneous networks (see Figure 1). It is assumed that the spatial distribution of the nodes is captured using two collocated and independent homogenous Poisson point processes (HPPPs) [6,7] i.e., Φ_{ M } and Φ_{ F } with intensities λ_{ M } and λ_{ F }, respectively. In other words, the locations of the macroBS nodes constitute an HPPP Φ_{ M }, where λ_{ M } is the average number of the macroBS nodes per unit area. Similar statement can be made for the HPPP formed by the femtoBS nodes Φ_{ F } with intensity λ_{ F } . According to Superposition Theorem [26], the overall node process over the network formed by both the macroBS and femtoBS nodes also is an HPPP with intensity λ (λ = λ _{ F } + λ _{ M }). Furthermore, the macro and femto users are scattered about the plane according to some independent PPPs with different densities compared to λ _{ M } and λ _{ F }, respectively. However, our interference analysis is fundamentally concerned with the distribution of the transmitters (BSs).
Since femtoBSs are installed and maintained by the paying home users for better indoor performance, they are only accessible by their own mobile subscribers (femto users) (known as closedaccess policy). On the other hand, macroBSs can be accessed only by unauthorized users (macro users). In practice, macro network is deployed usually without awareness of the distributed femto network. To this end, wireless operators can consider giving priority to the macro users, and the femto network has to be selfoptimized to mitigate its interference to the macro users. Motivated by this insight, the macroBSs (along with the macro users) and the CRenabled femtoBSs (along with the femto users) are analogous to primary and secondary systems in the CR model, respectively.
2.2 System structure
In OFDMA, the spectrum is orthogonally divided into timefrequency resource blocks (RBs), which increases flexibility in resource allocation, thereby allowing high spectral efficiency. As shown in Figure 2, we consider a spectrum of N RBs, out of which M (M ≤ N) random RBs are idle or unoccupied by the macro users (primary system). With the CR capability, a femtoBS could actively acquire knowledge about its environment and access to the RBs without the aid of a macrocell in a decentralized fashion (clearly, no synchronization between the macro and femto network is needed any more) and automatically prevent disturbing the macro users [2].

1)
As shown in Figure 2, each femtoBS’s transmission strategy is divided into consecutive slots, each having a duration of T. Each slot is divided into two consecutive stages, i.e., sensing and data transmission, with durations of T _{ S } and T _{ D }, respectively. Each femtoBS periodically senses the spectrum to identify which RBs are occupied by the macro network. Indeed, each femtoBS accomplishes sensing one RB in one unit slot T _{ SRB } within T _{ S }. Each femtoBS senses N _{ s } RBs in sequence which is randomly selected from the N available RBs, and detects its idle RB set. Clearly, the time required for sensing the N _{ s } RBs is T _{ S } = T _{ SRB } N _{ s }. Note that the femtoBSs cannot perform data transmission within the sensing time T _{ S }. We assume that all femtoBSs are perfectly synchronized and have the same time as the sensing time. Methods for implementing a perfect synchronization among the femtoBSs are outside the scope of this paper; however, a set of possible candidates exist, including GPS synchronization, the wired backhaul (IEEE 1588), and leveraging synchronization signals broadcasted by the femtoBSs [27].

2)
Each femtoBS senses the received interference power on each RB within the sensing duration:

If the received interference power on an RB at a typical femtoBS exceeds a certain threshold, the RB is identified as being occupied by one or more macro nodes but not by the femto network since all the femtoBSs have the same sensing time (It should be noted that if an RB is identified as being occupied at a typical femtoBS, it does not necessarily mean that it is also seen as an occupied RB at the other femtoBSs, as this status determination process depends only on the received interference power level on the RB at each individual femtoBS).

Otherwise, the RB is unoccupied by the macro network.


3)
In the data transmission time (T _{ D }), each femtoBS only allocates an unoccupied RB sensed in the sensing time to its user (by only utilizing these unoccupied RBs, crosstier interference can be consequently avoided). Since the determination of each individual RB status as busy/idle is subject to (occasional) error, determined by the probability of (correct) detection of the presence of PUs’ signals P _{ d } and probability of false alarm P _{ f } (probability of falsely declaring an idle RB as busy), we study the effect of both the ideal detection, i.e., P _{ d } = 1 and P _{ f } = 0, and the cases involving imperfect sensing (see [28,29]), i.e., P _{ d } ≠ 1 and P _{ f } ≠ 0 on the outage probabilities of femto and macro users.
In each realization of the point process, each macro and femto user communicates only with its nearest macroBS and femtoBS, respectively. As shown in Figure 3, the macro users’ exclusion regions with radius D are used to guarantee that the femtoBSs will, on average, not generate an aggregate interference leading to the outage of macro (primary) users. We assume that the macro users can be localized, e.g., based on pilot signals or transmitted acknowledgements. Therefore, the femtoBSs inside the macro users’ exclusion regions may be able to detect the macro signals and cease their transmissions. As shown in the figure, for example, those femtoBSs located in the tagged macro user’s exclusion region are not allowed to transmit data whether they pick the same RB as the tagged macro user or a different one. It should be noted that the tagged macro user is not disturbed by the femtoBSs transmitting on different RBs (from that of the tagged macro) even if they are inside its exclusion region. However, we deactivate them to protect the tagged macro user from any harmful interference as a result of possible errors in their sensing and location detection processes (we consider this law throughout the paper even in the case of the perfect sensing scenario). For instance, if a femtoBS, located at a very close distance from the tagged macro user, wrongly (the cases involving imperfect sensing) picks the occupied RB by the tagged macro user, it will cause a severe interference to the tagged macro user if it is not able to perfectly detect the location of the tagged macro user (e.g., because of the hidden node problem in CR systems [30]) to cease its transmission.
3 Stochastic geometrybased network configuration
3.1 Femto outage probability formulation
We derive the probability of outage for a typical femto user (p _{OF}) in a downlink heterogeneous cellular network defined as the probability that a randomly chosen femto user cannot achieve a target SINR θ (or equivalently as the average fraction of femto users who do not achieve a target SINR θ, i.e., the averaged outage probability of all femto users) considering a collocated spectrum sensing CRbased femto network and macroBSs as follows [6]:
In fact, the outage probability evaluates the CDF of SINR over the entire network. The experienced SINR by a typical femto user is calculated as
where P _{ F } is the transmission power from the nearest femtoBS (tagged femtoBS) located in the random distance r from its tagged femto user (we assume that the tagged femto user under consideration is located at the origin), and α is the pathloss exponent. I _{FB} and I _{MB} are the aggregate interference power at the origin from the other femtoBSs and macroBSs, respectively, and σ ^{2} is the noise power. It should be noted that the transmission power values of all the femtoBSs in the network are kept constant, i.e., P _{ F }.
Let h ~ exp (μ) (with mean 1/μ) be a random variable accounting for the random channel gain of the link between the tagged femto user and its corresponding femtoBS, and then we have
where (⋅⋅) is derived by rearranging the terms in SINR; (⋯) is derived by noting that h is an exponential random variable; and \( {\mathrm{E}}_{I_{\mathrm{FB},}{I}_{\mathrm{MB}}}\left[\cdot \right] \) is the expectation operator with respect to the joint distribution of the random variables I _{FB} and I _{MB}.
Note that due to the assumption of independent PPPs for the femto and macro networks, the aggregate interference received from the femtoBSs is independent of the aggregate interference received from the macroBSs [10], therefore, we can write
where \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{FB}}}(s) \) and \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{MB}}}(s) \) are the Laplace transform of random variables I _{FB} and I _{MB} evaluated at \( s\left(s=\mu \frac{\theta {r}^{\alpha }}{P_F}\right) \), respectively. Thus, the probability of outage averaged over the plane is derived as
where \( {f}_r(r)={e}^{{\uplambda}_F\pi {r}^2}2\pi {\lambda}_Fr \) (as mentioned before, r is the random distance between the tagged femto user and its corresponding femtoBS) is the probability density function (pdf) of r [6]. Then, we have
3.1.1 Scenario I
Ideal detection (P _{ d } = 1 and P _{ f } = 0):
Each secondary node (femtoBS) has perfect knowledge of each primary (macroBS) signaling. In other words, sensing at each femtoBS is done perfectly. Therefore, an RB occupied by a macroBS is not chosen for data transmission by the nearby femtoBSs. Under this condition, the tagged femto user, during the data transmission time, does not experience any interference from the macroBSs since it always communicates with its corresponding femtoBS on an idle RB. In fact, we assume that the received interference power from the macro network under this scenario can be neglected if it is measured to be less than a specified threshold (if we do not neglect the received interference power under the explained condition, then the outage probability formulations will be the same as in Scenario II except for the RB selection probability expressions, p _{RB}, as we explain later. Similar arguments can be made for the outage probability of the macro tier as discussed in the next subsections).
Therefore, under this assumption, \( {\mathrm{E}}_{I_{\mathrm{MB}}}\left[{e}^{\mu \frac{\theta {r}^{\alpha }}{P_F}\kern0.24em {I}_{\mathrm{MB}}}\right]=1 \) and \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{MB}}}\left(\mu \frac{\theta {r}^{\alpha }}{P_F}\right)=1 \), consequently. Then (3) is rewritten as follows
The Laplace transform of the aggregate interference from all the active femtoBSs except the tagged femtoBS denoted by fbs _{ 0 } is given as follows (Note: some of the femtoBSs located in the macro users’ exclusion regions with radius D are deactivated, therefore, λ'_{ F } ≤ λ _{ F } (see Figure 3))
Where R _{ i } is the distance of the ith interferer from the tagged femto receiver captured by the point process Φ_{ F } The interference channel gains g _{ i } are assumed to be mutually independent and have identical pdfs. Each of the active interfering femtoBSs transmits with the same power P _{ F }. Using the definition of the Generating functional [6] for the Poisson point process (PPP), which states for some function f(x) that \( \mathrm{E}\left[{\displaystyle {\prod}_{x\in \Phi}}f(x)\right]= \exp \left({\displaystyle \underset{{\mathrm{\mathbb{R}}}^d}{\int }}\left(1f(x)\right)\uplambda \mathrm{d}\mathrm{x}\right) \), (5) can be rewritten as
where we flipped the order of integration and expectation. Since the closest interfering femtoBS is at least at distance r from the tagged user, the integration limits are from r to ∞. In other words, interference is encountered from all the active femtoBSs located in the area ℝ^{d}\b(0, r) (where b(x, y) is ball of radius y centered at point x). However, not all the femtoBSs will contribute towards the aggregate interference, i.e., only those active femtoBSs located outside the mentioned ball which at minimum satisfy all of the following conditions qualify as potential contributors.
Before explaining these conditions ((a) and (b)), it is useful to translate the point process into polar coordinates. Therefore, according to [7,26], the intensity of the HPPP Φ_{ F } is shown as
where R is the distance between an arbitrary femtoBS and the tagged femto receiver. b _{ d } is the volume of a unit sphere in ℝ^{d} (\( {b}_d=\frac{\sqrt{\pi^d}}{\Gamma \left(1+\raisebox{1ex}{$d$}\!\left/ \!\raisebox{1ex}{$2$}\right.\right)},\Gamma (x)={\displaystyle {\int}_0^{\infty }}{t}^{x1}{e}^{t}\mathrm{d}t \) denotes the standard Gamma function).

(a)
Satisfying the aforementioned condition, any active femtoBS contributes towards the interference at the tagged femto receiver, if it picks the same RB as the tagged femtoBS to communicate with its user. We show the probability of picking a same RB from a pool of all RBs as p _{RB} (the calculation of p _{RB} for this case is derived in Section 4, Scenario I, Case 1).

(b)
We assume that the CR femtoBSs employ a slotted ALOHA MAC (medium access control) protocol to schedule their transmission. Therefore, they only transmit with probability p _{tx} in the current time slot and defer the transmission with probability 1 − p _{tx} .
Applying both conditions (a) and (b) (which will result in reducing the number of interferers and hence outage probability at the tagged femto user), Φ_{ F } is thinned using two independent thinning. Indeed, first, Φ_{ F } is reconstructed by p _{RB} thinning, where each point of Φ_{ F } is retained with probability p _{RB} (to accommodate the condition (a)), and then it is thinned again by applying the other independent thinning (to accommodate the condition (b)). Thus, the intensity of the process (the number of the interfering CR femtoBSs) becomes
Now, (6) is rewritten as
By using the change of variables R ^{d} → x and then \( {x}^{\frac{\alpha }{d}}\to y \), and doing some simple calculations, the above expression is simplified as:
where
Note that \( \Gamma \left(a,x\right)={\displaystyle {\int}_x^{\infty }}{t}^{a1}{e}^{t}\mathrm{d}t \) indicates the incomplete gamma function. From (4) and (9), taking d = 2, then replacing r ^{2} with z, we get the final formula for the outage probability of the tagged femto user as follows
in which \( M\left(\theta, \alpha \right)=\mathrm{E}\left[{(g)}^{\frac{2}{\alpha }}\left(\Gamma \left(\frac{2}{\alpha },\mu \theta g\right)\Gamma \left(\frac{2}{\alpha}\right)\right)\right] \). According to [13], the downlink femtocell networks are assumed to be interferencelimited, i.e., the noise can be neglected as the interference dominates the whole performances of the system σ ^{2} → 0.
Thus, (11) reduces to (12) as follows (α = 4)
Similar with the desired link, we consider the Rayleigh fading model for the femto interfering links as well (Rayleigh fading links with equal parameter μ). Then following the derivation of M(θ, α) in Appendix I, p _{OF} is reexpressed as follows
3.1.2 Scenario II
Imperfect detection (P _{ d } ≠ 1 and P _{ f } ≠ 0):
In this scenario, each secondary node (femtoBS) has imperfect knowledge of each primary (macroBS) signaling. In other words, sensing at each femtoBS is done imperfectly. Therefore, an occupied RB by a macroBS may also be wrongly considered idle by the femtoBSs, causing collision between the two networks. Two cases can take place under the imperfect sensing scenario:

Case 1. The tagged femtoBS transmits data on an idle RB (for this case, the outage probability formulations can be considered the same as in the perfect sensing scenario except for the calculation of p _{RB} (see Section 4, Scenario II, Case 1))

Case 2. The tagged femtoBS transmits on an occupied RB (outage probability formulation in this case is explained as follows and the calculation of p _{RB} is presented in Section 4, Scenario II, Case 2)
In Case 2, the tagged femto user can experience interference from both the active femtoBSs (which pick the same occupied RB as the tagged femtoBS) and macroBSs. Indeed, the \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{MB}}}\left(\mu \frac{\theta {r}^{\alpha }}{P_F}\right) \) (the Laplace transform of the aggregate interference from the macroBSs) in (3) is not ignored. The Laplace transform of the aggregate interference power generated by the macroBSs at the tagged femto user is given by
where L _{ i } is the distance of the ith interfering macroBS from the tagged femto receiver captured by the point process Φ_{ M }. The interference channel gains G _{ i } are assumed to be mutually independent and have identical pdfs. Each of the active interfering macroBSs transmits with the same power P _{ P }.
Again, using the definition of the generating functional for the PPP, we can write
The interference is encountered from all the macroBSs located in the area ℝ^{d}\b(0, 0). It should be noted that not all the macroBSs in ℝ^{d} will contribute towards the aggregate interference, i.e., only those macroBSs which are transmitting on the same RB as the tagged femto user qualify as potential interferers. The intensity of the HPPP Φ_{ M } process can be therefore written as follows
where L is the distance between an arbitrary macroBS and the tagged femto receiver and \( {\lambda}_M^{\hbox{'}} \) is the intensity of those macroBSs transmitting on the same RB as the tagged femto user at a time.
Now, (15) is rewritten as
By using the change of variables and doing some simple calculations, \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{MB}}}(s) \) is obtained as follows [26]
From (3), (9), and (17), taking d = 2, then replacing r ^{2} with z, and assuming σ ^{2} → 0, we get the final formula as follows
in which \( M\left(\theta, \alpha \right)=\mathrm{E}\left[{(g)}^{\frac{2}{\alpha }}\left(\Gamma \left(\frac{2}{\alpha },\mu \theta g\right)\Gamma \left(\frac{2}{\alpha}\right)\right)\right] \) and \( N\left(\alpha \right)=\Gamma \left(1\frac{d}{\alpha}\right)\mathrm{E}\left[{(G)}^{\frac{d}{\alpha }}\right] \).
Finally, the closedform expression for the outage probability of the tagged femto user, under the imperfect sensing scenario for all CR femtoBSs, is obtained as follows (α = 4)
Similar with the desired link, we consider the Rayleigh fading model for the femto interfering links as well as the macro interfering links (Rayleigh fading links with equal parameter μ for the femto interfering links and μ _{ p } for the macro interfering links). Following the derivation of M(θ, α) in Appendix I, and after the simplification of N(α) using the definition of expectation and the standard gamma function, p _{OF} is reexpressed as follows α = 4
3.2 Macro outage probability formulation
We derive the outage probability for a typical macro user (p _{OM}) in a downlink heterogeneous cellular network defined as the probability that a randomly chosen macro user cannot achieve a target SINR γ (or equivalently as the average fraction of macro users who do not achieve a target SINR γ, i.e., the averaged outage probability of all macro users) considering a collocated spectrum sensing CRbased femto network and macroBSs as follows:
The experienced SINR by a typical macro user is calculated as
where P _{ P } is the transmission power from the nearest macroBS located in the random distance r _{ p } from its tagged macro user located at the origin and α is the pathloss exponent. I _{FB} and I _{MB} are the aggregate interference power to the tagged macro user (located at the origin) from the surrounding femtoBSs and macroBSs, respectively, and σ ^{2} is the noise power. It should be noted that the transmission power values of all the macroBSs in the network are kept constant, i.e., P _{ P }.
Let h _{ p } ~ exp(μ _{ p }) (with mean 1/μ _{ p }) be a random variable accounting for the random channel gain of the link between the tagged macro user and its corresponding macroBS, then the similar approach to obtain the outage probability of the tagged femto user is used for the outage probability calculation of the tagged macro user. Therefore, from (21) and (22), we have
where \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{FB}}}(s) \) and \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{MB}}}(s) \) are the Laplace transform of random variables I _{FB} and I _{MB} evaluated at \( s\left(s={\mu}_p\frac{\gamma {r_p}^{\alpha }}{P_P}\right) \), respectively.
3.2.1 Scenario I
Ideal detection (P _{ d } = 1 and P _{ f } = 0):
Each secondary node (femtoBS) has perfect knowledge of each primary (macroBS) signaling. In other words, sensing at each femtoBS is done perfectly. Therefore, an RB occupied by a macroBS is not chosen for data transmission by any nearby femtoBS. Under this condition, the tagged macro user, during the data transmission time, does not experience any interference (or a negligible interference) from the surrounding femtoBSs; however, those macroBSs operating on the same RB as the tagged macro user and located in the area ℝ^{d}\b(0, r _{ p }) make interference to the tagged macro user. Therefore, (23) becomes
Taking a similar approach to what we had before, \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{MB}}}(s) \) is obtained as follows
and
in which W is the distance between an arbitrary macroBS (captured by the point process Φ_{ M }) and the tagged macro receiver and \( {\lambda}_M^{\hbox{'}} \) is the intensity of those macroBSs transmitting on the same RB as the tagged macro user at a time. Similarly, the interference channel gains g _{ p } (between the interfering macroBSs and the tagged macro user) are assumed to be mutually independent and have identical pdfs. Each of the active interfering macroBSs transmits with the same power P _{ P } as for the tagged macroBS. Same as before, (25) is simplified as
From (24) and (27) and the previous assumptions, the closedform formula is expressed as follows
in which \( V\left(\gamma, \alpha \right)=\mathrm{E}\left[{\left({g}_p\right)}^{\frac{2}{\alpha }}\left(\Gamma \left(\frac{2}{\alpha },{\mu}_p\gamma {g}_p\right)\Gamma \left(\frac{2}{\alpha}\right)\right)\right]. \)
The Rayleigh fading model is also considered for the macro interference links (Rayleigh fading links with equal parameter μ _{ p }). The derivation of V(γ, α) is similar to the derivation of M(θ, α) in Appendix I. Therefore, p _{OM} is reexpressed as follows (α = 4)
3.2.2 Scenario II
Imperfect detection (P _{ d } ≠ 1 and P _{ f } ≠ 0):
In this scenario, each secondary node (femtoBS) has imperfect knowledge of each primary (macroBS) signaling. In other words, sensing at each femtoBS is done imperfectly and subjected to (occasional) error. Therefore, an occupied RB by a macroBS may also be wrongly considered idle by the femtoBSs. In this situation, the tagged macro user can experience interference on its RB, from both the femto and macro BSs. Indeed, the \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{FB}}}\left({\mu}_p\frac{\gamma {r_p}^{\alpha }}{P_P}\right) \) (the Laplace transform of the aggregate interference to the tagged macro user from the surrounding femtoBSs) in (23) is not ignored. Using the same approach as described before, \( {\mathrm{\mathcal{L}}}_{I_{\mathrm{FB}}}(s) \) is also given by
Since the closest interfering femtoBS is at least at distance Kr _{ p } from the tagged macro user, the integration limits are from Kr _{ p } to ∞. In other words, interference is encountered from all the femtoBSs located in the area ℝ^{d}\b(0, Kr _{ p }) (see Figure 3). It should be noted that not all the femtoBSs outside this ball will contribute towards the aggregate interference, i.e., only those femtoBSs which are outside the mentioned ball and at minimum satisfy all of the following conditions are considered as potential contributors

(c)
Satisfying the above condition, any arbitrary femtoBS contributes towards the interference at the tagged macro receiver, if it wrongly picks the same RB as the tagged macroBS to communicate with its user. We show the probability of picking a same RB for data transmission from a pool of all RBs as p _{RB} (the calculation of p _{RB} for this case is seen in Section 4, Scenario II, Case 2).

(d)
Same as the condition (b) in subsection 3.1.1.
Applying these two independent thinning, the intensity of the process (the number of the interfering CR femtoBSs at the tagged macro user) becomes
Taking the similar approach, (30) is simplified as follows
in which \( O\left(\gamma, \alpha \right)=\mathrm{E}\left[{\left({G}_p\right)}^{\frac{d}{\alpha }}\left(\Gamma \left(\frac{d}{\alpha },\frac{\mu_p\gamma {P}_F{G}_p}{K^{\alpha }{P}_P}\right)\Gamma \left(\frac{d}{\alpha}\right)\right)\right]. \)
From (23), (27), and (32) and the previous assumptions, the closedform expression for the outage probability of the tagged macro user is obtained as follows
Similar with the desired link, the Rayleigh fading model for the femto interfering links (with equal parameter μ) as well as the macro interfering links (with equal parameter μ _{ p }) is considered. Considering Appendix I and after replacing V(γ, 4) and O(γ, 4) with their expanded versions, p _{OM} can be reexpressed as follows
where
and
4 Resource block selection probability (p _{RB}) calculations under perfect and imperfect sensing
In this section, we discuss how the optimal values of the RB selection probability (p _{RB}) for a secondary transmitter (femtoBS) can be determined under each femtoBS’s perfect and imperfect sensing scenarios.
4.1 Scenario I
Ideal detection for all the CR femtoBSs (P _{ d } = 1 and P _{ f } = 0) [31,32]:
Case 1. The tagged femtoBS assigns the ith idle RB to its femto user.
p _{RB}: The probability that the ith idle RB being selected for data transmission by any of the other active CR femtoBS [32].
where
The first term (p _{idle}(M _{ s })) indicates the probability of M _{ s } idle RBs sensed by a femtoBS (during the sensing time, T _{ S }), the second term is the probability that the ith idle RB is inside the M _{ s } idle RBs, and finally the third term indicates that the probability of selecting the ith idle RB (out of the M _{ s } idle RBs) by that femtoBS is equal to \( \frac{1}{M_s} \) (since each of the idle RBs within the M _{ s } idle RBs has an equal probability of being chosen). It should be noted that each CR femtoBS will fail to access when M _{ s } = 0 (the maximum value of M _{ s } is equal to min{M, N _{ s }}). Therefore, from (35),
If the number of RBs sensed by a CR femtoBS (i.e., N _{ s }) is more than or equal to N − M + 1, then at least we have one idle RB within the N _{ s } detected RBs, i.e., p _{idle}(M _{ s } = 0) = 0. On the other hand, if N _{ s } is smaller than or equal to N − M, the CR femtoBS will fail to access when the RBs within the N _{ s } sensed RBs are all occupied by the macroBSs. Therefore, we can write
From (36) and (37), the probability that a CR femtoBS selects the ith idle RB for data transmission is obtained as follows
4.2 Scenario II
Imperfect detection for all the CR femtoBSs (P _{ d } ≠ 1 and P _{ f } ≠ 0) [31,32]:
Case 1. The tagged femtoBS assigns the ith idle RB to its femto user.
p _{RB}: The probability that the ith idle RB being selected for data transmission by any of the other active CR femtoBS [32].
We show the detection result indicator of the nth RBs by D _{ n } (n ∈ {1, 2, …, N}). If D _{ n } = 1, the nth RB is detected as idle RB, otherwise, D _{ n } = 0. The probability of one idle RB detected with no false alarm is 1 − P _{ f } and the probability for an occupied RB detected as an idle RB is 1 − P _{ d }. In other words,
in which P _{ f } is the false alarm probability and can be obtained as follows [3234]
\( \mathcal{Q}(x)=\frac{1}{\sqrt{2\pi }}{\displaystyle \underset{x}{\overset{\infty }{\int }}} \exp \left(\frac{{t}^2}{2}\right)\mathrm{d}t \) and P _{ d } is the predefined detection probability. τ is the spectrum sensing time, f _{ s } the sampling frequency, and η the received interference power on an RB to each femtoBS.
Indeed, the probability that the ith idle RB is detected with no false alarm by a CR femtoBS is Pr(D _{ i } = 1) = V _{0}.
To obtain the probability that the ith idle RB being selected for data transmission by a CR femtoBS (p _{RB}), first, we calculate the probability that the ith idle RB is sensed and included in the M _{ s } idle RBs out of the N _{ s } sensed RBs in the sensing period (T _{ S }), and it is expressed as follows
where
Conditioning on M _{ D } (see Table 1) and M _{ s }, the probability that the ith idle RB being detected as idle is obtained as follows
in which Φ is the set of the detected RBs by a femtoBS (see Table 1 for the definitions of m _{ID} and m _{OD}). Replacing m _{OD} with M _{ D } − m _{ID}, we have
Having the M _{ D } detected idle RBs (including the ith idle RB), the probability of a CR femtoBS accessing the ith idle RB is equal to \( \frac{1}{M_D} \) . Thus, the probability that the ith idle RB is selected for data transmission by any CR femtoBS (under imperfect sensing scenario) is obtained as follows
Finally,
Case 2. The tagged femtoBS assigns the ith busy RB (occupied by the macro network) to its femto user.
p _{RB}: The probability that the ith busy RB being selected for data transmission by any of the other active CR femtoBS.
To obtain the probability of the ith busy RB selected for data transmission by a femtoBS (p _{RB}), first, we calculate the probability that the ith busy RB is sensed and included in the (N _{ s } − M _{ s }) busy RBs out of the N _{ s } sensed RBs in the sensing period, and it is expressed as follows
Conditioning on M _{ D } and (N _{ s } − M _{ s }), the probability that the ith busy RB being detected as idle is obtained as follows
in which Φ is the set of the detected RBs by a femtoBS. Again, replacing m _{OD} with M _{ D } − m _{ID}, we have
Having the M _{ D } detected idle RBs (including the ith busy RB), the probability of a CR femtoBS accessing the ith busy RB is \( \frac{1}{M_D} \). Thus, the probability that the ith busy RB is selected for data transmission by any CR femtoBS (under imperfect sensing scenario) is obtained as follows
Finally,
5 Simulation results and discussions
Before we present the obtained results, a brief discussion on the spatial distribution of the BSs and the macro exclusion regions is conducted as follows.
Talking about the femto outage probability, each femto user suffers from two sources of interference, i.e., macro and femto networks. For the macro network, the aggregate interference results from all macroBSs that use the same RB as the tagged femto user (i.e., we define a homogenous PPP with intensity \( {\lambda}_M^{\hbox{'}} \)). For the femto network, the aggregate interference results only from the other femtoBSs that (1) pick the same RB as the tagged femto, (2) are allowed to transmit in the current time slot, and (3) are not inside the macro users’ exclusion regions. Hence, the interfering femtoBSs do not constitute a homogeneous point process anymore, and analytical characterization of interference and outage in this case is hard to characterize (the resulting point process is called Poisson hole process). In the analysis, to keep the modeling tractable, we ignored the possible correlation between the locations of the interfering femtoBSs and approximated the spatial distribution of them by a homogeneous PPP of intensity \( {\lambda}_F^{\hbox{'}}{p}_{\mathrm{RB}}{p}_{\mathrm{tx}} \). Authors in [10,35] use the same approximation approach, where its accuracy is also justified by simulation in [10]. Similar arguments and approximations were considered for the macro tier outage probability.
As interferences are experienced at receivers, we centered the macro exclusion regions around the macro users. The femtoBSs inside these areas may be able to detect the macro signals and cease their transmissions. The exclusion regions are usually chosen to be centered at the location of the macroBSs not the macro receivers based on the argument that it is easier to detect the macroBSs than the macro receivers especially if the receivers are passive like TV receivers. However, if the macro receivers (users) can be localized, e.g., based on pilot signals or transmitted acknowledgments, our obtained results directly apply and the exclusion regions around macro users can make sense. If the macro users cannot be localized, the exclusion regions have to be formed around the macroBSs. This scenario can be evaluated with slight changes in the proposed model. It should be noted that the location detection of the macro users is outside the scope of this paper, however, many schemes have been already proposed. Measuring the power leakage of local oscillator is a possible way to detect the presence of the macro passive users (see [10,36]). The hidden node problem in CR systems which makes it difficult to detect the macro users can be also tackled, e.g., by adding a margin to the RB access detection threshold accounting for shadow fading and receiver location uncertainty for worstcase scenarios [30].
First, the accuracy of our analytical results for the downlink analysis in the proposed model is validated by simulations, as shown in Figures 4 and 5. The simulations which are built on Matlab platform are carried out to plot the curves of outage probability versus the SINR threshold for the tagged femto and macro user, as shown in Figures 4 and 5, respectively. The considered scenario is a twotier network (exactly following the network model described in Sections 2 and 3) over an approximately 1 × 1 km square with the locations of different classes of BSs as realizations of independent PPPs of given densities and the tagged users located at the center. To have an estimate of the outage probability at the tagged users, the simulation results are averaged over both the spatial PPP (500 different positions) and fading distribution (300 realizations) and are conducted using the parameters mentioned in the figures’ captions. Analytical curves are compared with the simulations under both perfect and imperfect sensing. It is observed that the simulation results closely match our analytical model, and the curves of analytical and simulation results match fairly well, which confirms our analysis. The plots exhibit slight discrepancies between analytical results and the corresponding simulation results which are mainly due to the independence assumption used in Section 3.
In Figure 6, the outage probability of the tagged femto user under perfect and imperfect spectrum sensing abilities for the CR femtoBSs is shown for different values of the target SINR θ on the horizontal axis. Our results show that the outage probability at the tagged femto receiver in the absence of a perfect spectrum sensing ability is considerably increased. In either the Scenario I, when all the femtoBSs employ perfect sensing to sense the RBs, or in Scenario II, Case 1, the tagged femto user does not experience any interference from the macroBSs owing to the correct detection at the tagged femtoBS or choice of idle RBs for data transmission (RBs not occupied by the macro network). In this case, the interference seen by the tagged femto user is only the aggregate interference from the other femtoBSs which are transmitting on the same idle RB as the tagged femto. Clearly, the lowest outage probability is achieved for this case (see the red curve). Now, let us consider the imperfect sensing scenario for the CR femtoBSs. Obviously, the tagged femto user is now subject to sensing error and therefore picking an occupied RB for its data transmission period. Under this condition, it may receive interference not only from the other femtoBSs which pick the same busy RB (due to the imperfect sensing) as the tagged femto but also from those macroBSs communicating with their own users on the same RB as the tagged femto. Therefore, the tagged femto user experiences an interference larger than before and consequently a significant increase in the outage probability. Moreover, the tagged femto user will face an outage with a higher probability whenever it picks an RB (occupied RB) already used by a larger number of macro users (a larger \( {\lambda}_M^{\hbox{'}} \) ) (see Figure 6).
Figure 7 depicts the outage probability of the tagged macro user for different values of the target SINR γ and different situations. Considering the results obtained in Figure 6, here, we also investigate the effect of employing the two different sensing scenarios for the CR femtoBSs on the outage probability of the tagged macro user. In the case of perfect sensing, the tagged macro user does not experience any interference from the femtoBSs because only those RBs sensed to be idle (RBs not occupied by the macro network) are always chosen for data transmission by the femto network. In this case, the interference observed at the tagged macro user is only the aggregate interference received from those macroBSs transmitting on the same RB as the tagged macro. Clearly, the lowest outage probability is obtained for this case (see the red curve). Now, the case of imperfect sensing of the CR femtoBS nodes is considered when the femtoBSs are subject to sensing error and therefore the possibility of transmitting on the RB occupied by the tagged macro. Under this condition, the tagged macro user may receive interference not only from the other macroBSs communicating with their own users over the same RB (due to the lack of RBs) as the tagged macro but also from those femtoBSs which pick the same RB. Therefore, the tagged macro user experiences an interference larger than before and consequently a significant increase in the outage probability. Moreover, the tagged macro user will face an outage with a higher probability whenever its own RB is wrongly selected for data transmission by a larger number of femtoBSs (a larger \( {\lambda}_F^{\hbox{'}} \)).
Figure 8 illustrates the effect of K (in D = Kr _{ p }) on the observed outage probability at the tagged macro user in the presence of both the macro and femto networks. Considering the previous explanations and Figure 3, let \( {\Phi}_F^{\hbox{'}} \) include all the points (representing the femtoBSs) in Φ_{ F } except the points inside the exclusion region D of the tagged macro user. Since \( {\Phi}_F^{\hbox{'}}\subset {\Phi}_F \) , the potential aggregate interference at the tagged macro user, caused by the active (considering slotted ALOHA) CR femtoBSs, is less than that in the case with no D. Furthermore, as the exclusion region D becomes larger (when K = 10, for example), the probability of outage is significantly reduced. Indeed, the bigger the value of D, the closer the outage probability curve becomes to the black curve which represents the outage probability in the case when no overall interference from the femto network is observed at the tagged macro user due to the perfect sensing ability of the CR femtoBSs. However, the reduction in the outage probability can be less when the number of macroBSs transmitting on the same RB as the tagged macro is larger (see Figure 8b).
For a commercial network, designers must find a way to achieve a lower probability of outage for a certain SINR as the minimum quality needed for a typical femto or macro user to experience an acceptable QoS. A common way to decrease the outage probability is to reduce the number of interfering BSs encountered at the users. This can be done through applying both the (a) and (b) conditions in the perfect sensing scenario (or both the (c) and (d) in the imperfect sensing scenario). As shown in Figure 9, under these constraints, the outage probability is significantly reduced at the tagged femto user (note that the number of interfering macroBSs at the tagged femto user is considered equal for all curves in the figure). The goal is to see the effect of both p _{RB} and p _{tx} (these two parameters are employed at each CR femtoBS) on the outage probability of the tagged femto user. As can be seen, for the case when p _{RB} = 1 and p _{tx} = 1, outage occurs with higher probability. In other words, when all the existing femtoBSs (except those who are inside the macro users’ exclusion regions) pick the same RB as the tagged femto (p _{RB} = 1) and when they all have data to transmit in the current time slot (p _{tx} = 1), the tagged femto user will experience the maximum value for the outage probability derived for different SINR targets. Clearly, a significant reduction in the outage probability is occurred for the smaller values of p _{RB} and p _{tx} (see Figure 9) (the smaller the values of p _{RB} and p _{tx}, the closer the outage probability becomes to the outage probability in the case when the received interference at the tagged femto user is only the aggregate interference from the macro network). Indeed, this validates that many studies which do not consider these constraints overestimate the interference encountered by a typical femto user.
It can be seen that even for high values of p _{tx}, the outage probability is relatively less than that in the case where there is no constraint on the femtoBS’s transmission schedule (p _{tx} = 1). Also, it is obvious that in the presence of multiple RBs where each RB is picked with probability p _{RB}, the outage probability is further decreased. Hence, any practical heterogeneous network designed to satisfy both of the mentioned conditions can reap the benefit of opportunistic exploitation of spectrum, while possibly causing little or no harmful interference. The same story exists when the outage probability of the tagged macro user is investigated. Figure 10 shows the effect of p _{RB} and p _{tx} (parameters which are related to the CR femtoBSs) on the outage probability of the tagged macro user. For instance, when all the active femtoBSs select the same RB as the tagged macro (p _{RB} = 1) and when they all have data to transmit (on this busy RB) in the current time slot (p _{tx} = 1), the tagged macro user will experience the maximum value for the outage probability. It should be noted that the number of interfering macroBSs at the tagged macro is considered equal for all curves in this figure.
In Figures 11 and 12, we analyze the performance of the authorized links (femto links) in terms of throughput (achievable with a simple ARQ scheme with errorfree feedback) under both the perfect and imperfect sensing scenarios. Considering the slotted ALOHA scheme, we define the following terms [37,38],
Probabilistic link throughput (τ) of a femto user:

(i)
in the halfduplex (HD) communication scenario: it is defined to be the success probability of a femto user (i.e., 1 − p _{OF}) multiplied by the probability that the corresponding femtoBS actually transmits over a specific RB (i.e., p _{RB} p _{tx}), and the probability that the femto receiver actually receives over that RB (i.e., 1 − p _{RB} p _{tx})

(ii)
in the fullduplex (FD) communication scenario: it is defined to be the success probability of a femto user (i.e., 1 − p _{OF}) multiplied by the probability that the corresponding femtoBS actually transmits over a specific RB (i.e., p _{RB} p _{tx}).
Femto link throughput (T):
The femto link throughput is defined as the product of the probabilistic link throughput (τ) and the rate of transmission, i.e., T = τ log(1 + θ). Therefore, the femto link throughput for the half and fullduplex cases is written as follows
in which p = p _{RB} p _{tx}.
In Figure 11, the performance of half and fullduplex systems is presented for the femto users. More specifically, the link throughput of any typical femto user (e.g., the link between the tagged femto user and its corresponding femtoBS) under perfect and imperfect spectrum sensing abilities for the CR femtoBSs is shown as a function of the transmission probability over a specific RB (i.e., p = p _{RB} p _{tx}). It can be seen that the throughput achieved by the FD system is significantly higher, particularly when p is high. Regarding the performance of the HD system, for both the perfect and imperfect sensing cases, there is a unique optimal p which achieves the maximum throughput (p = 0.3 for the perfect and p = 0.35 for the imperfect sensing scenario). However, for high p, both throughput curves converge to zero due to over many transmissions and interferences on the RB. Obviously, for both the half and fullduplex communications, a higher perlink throughput is achieved when the CR femtoBSs employ perfect sensing.
In Figure 12, the performance of half and fullduplex systems is presented for femto users. More specifically, the link throughput of any typical femto user (e.g., the link between the tagged femto user and its corresponding femtoBS) under perfect and imperfect spectrum sensing abilities for the CR femtoBSs is shown as a function of the target SINR θ. It can be seen that the perlink throughput achieved by the FD system, for both the perfect and imperfect sensing scenarios, is significantly higher than the HD one. As it is seen, the link throughput curves are concave and there is an optimal point in each curve. With a hightarget SINR, we can transmit the user data with high spectral efficiency; however, the outage probability of this transmission is high, too. In contrast, with a lowtarget SINR, we can send many packets that include little information. In other words, a high reliable transmission can be experienced at lowtarget SINRs, while the minimum requirements for the transmission rate cannot be met.
6 Conclusions
In this paper, utilizing the spatial Poisson point process (PPP) theory, we presented a tractable model to derive the outage probability of a typical femto and macro user in a twotier heterogeneous network which provides insight into system design guidelines. In other words, for the case of the node locations modeled by a PPP and the desired and interfering channels are subject to Rayleigh fading, we demonstrated the use of the cognitive radio (CR)based framework to evaluate the outage probability at any arbitrary user. Exact closedform expressions were obtained as a result. In addition, we observed that in the downlink analysis, the outage probability is a function of the network topology and several important system design parameters such as SINR target, exclusion regions, MAC mechanisms such as ALOHA (p _{tx}), and the resource block (RB) selection constraint (p _{RB}) which is controlled by the spectrum sensing measurements.
7 Appendix I
DERIVATION OF M(θ, α)
Proof: From the expression of M(θ, α) in (11), we have
We know that Г(a, x) + γ(a, x) = Г(a) and \( \gamma \left(a,x\right)={x}^a\Gamma (a){e}^{x}{\displaystyle \sum_{k=0}^{\infty }}\frac{x^k}{\Gamma \left(a+k+1\right)} \), therefore, the above equation is simplified as follows
Hence, we have
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Panahi, F.H., Ohtsuki, T. Stochastic geometry modeling and analysis of cognitive heterogeneous cellular networks. J Wireless Com Network 2015, 141 (2015). https://doi.org/10.1186/s1363801503639
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DOI: https://doi.org/10.1186/s1363801503639