 Research
 Open Access
Stochastic geometry modeling and analysis of cognitive heterogeneous cellular networks
 Fereidoun H Panahi^{1}Email author and
 Tomoaki Ohtsuki^{1}
https://doi.org/10.1186/s1363801503639
© Panahi and Ohtsuki; licensee Springer. 2015
 Received: 1 September 2014
 Accepted: 16 April 2015
 Published: 22 May 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.
Keywords
 Heterogeneous cellular network
 Cognitive radio
 Outage probability
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
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
 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.
3 Stochastic geometrybased network configuration
3.1 Femto outage probability formulation
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 }.
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).
 (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} .
3.1.2 Scenario II
Imperfect detection (P _{ d } ≠ 1 and P _{ f } ≠ 0):

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)
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.
3.2 Macro outage probability formulation
3.2.1 Scenario I
Ideal detection (P _{ d } = 1 and P _{ f } = 0):
3.2.2 Scenario II
Imperfect detection (P _{ d } ≠ 1 and P _{ f } ≠ 0):
 (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.
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.
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].
\( \mathcal{Q}(x)=\frac{1}{\sqrt{2\pi }}{\displaystyle \underset{x}{\overset{\infty }{\int }}} \exp \left(\frac{{t}^2}{2}\right)\mathrm{d}t \) 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}.
Symbols used in Section 4
Symbols  Descriptions 

N  Number of RBs 
M  Number of idle RBs 
N _{ s }  Number of sensed RBs 
M _{ s }  Number of idle RBs within the N _{ s } sensed RBs 
M _{ D }  Number of RBs detected as idle within the N _{ s } sensed RBs 
m _{ID}  Number of idle RBs (out of the M _{ D } detected idle RBs) detected correctly. (m _{ID} ∈ [max{1, M _{ s } – (N _{ s } – M _{ D })}, min{M _{ s }, M _{ D }}]) 
m _{OD}  Number of busy RBs (out of the M _{ D } detected idle RBs) detected as idle. (m _{OD} = M _{ D } − m _{ID}) 
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.
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].
 (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):
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)
Declarations
Authors’ Affiliations
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