# Toward a Unified Framework for Analysis of Multi-RAT Heterogeneous Wireless Networks

Authors: Murk Marvi; Adnan Aijaz; Muhammad Khurram
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

The increased penetration of different radio access technologies (RATs) and the growing trend towards their convergence necessitates the investigation of wireless heterogeneous networks (HetNets) from coverage and capacity perspective. This paper develops aunified framework for signal-to-interference-plus-noise ratio and rate coverage analysis of multi-RAT HetNets, with each RAT employing either a contention-free or a contention-based channel access strategy. The proposed framework adopts tools from stochastic geometry, with the location of APs and mobile users modeled through independent Poisson point processes (PPPs). We specifically focus on a two-RAT scenario (i.e., cellular and Wi-Fi), where for multi-tier Wi-Fi RAT, with contention-based channel access like CSMA/CA, the location dependent distribution of interfering APs has been approximated through a homogeneous PPP. Moreover, by using some simple yet realistic set of assumptions, the distance to nearest active AP has been defined which results in simplified expressions. The medium access probability for a random and a tagged AP under a multi-tier Wi-Fi RAT has also been derived and discussed. By keeping in view the tremendous effect of temporal domain on overall network performance, the stable queue probability has been derived by assuming a non-saturated traffic model. The results have been validated through extensive simulations and compared with existing approaches. Some useful insights have also been presented that shed light on design and analysis of multi-RAT HetNets and provide motivation for further research in this direction.


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## Body

## 1. Introduction

According to Cisco’s forecasts [1], 8.3 billion hand-held devices and 3.3 billion machine-to-machine (M2M) devices will be connected by 2021. The number of connected devices would clearly exceed the expected global population of 7.8 billion by that time. The monthly global mobile data traffic is expected to reach 49 exabytes and the annual traffic will exceed half a zettabyte by 2021. Extreme densification and offloading in wireless heterogeneous networks (HetNets) is the key technique for meeting the ever increasing capacity requirements which can potentially bring the 1000× improvement in aggregate data rate as envisioned for 5G networks [2]. With networks becoming denser, more traffic can be offloaded to small cells, especially because a significant proportion of traffic is generated by users in indoor locations (home, offices, etc.). Offloading to Wi-Fi networks is a natural and most effective technique for reducing congestion on cellular networks [3]. The integration of cellular and Wi-Fi networks has been a topic of interest, in both industrial and academic communities, since early 2000s. Wi-Fi is experiencing a trend towards ubiquity with operator-deployed Wi-Fi networks becoming a norm. According to recent projections [1], 66% of mobile data traffic would be offloaded from 4G networks to Wi-Fi by 2021.Witnessing these technological trends, it can be easily inferred that understanding the fundamental limits of HetNets comprising different radio access technologies (RATs) and employing different channel access techniques becomes particularly important. Analytical tools for HetNets can provide valuable insights to system designers in meeting the future coverage and capacity demands through integration of different RATs, with multiple tiers, and by exploiting licensed and unlicensed resources in an effective manner.


### 1.1. Related Work and Motivation

Over the last few years, tools from stochastic geometry have been used for analysis of single-tier and multi-tier cellular networks [4, 5], where mostly Poisson point process (PPP) is used for modeling the spatial location of access points (APs). The PPP assumption provides a lower bound, whereas the traditional grid model provides an upper bound on coverage. Other point processes like Gibbs [6], Strauss [7], Ginibre [8], and Detriminantal [9] have also been used for coverage analysis of cellular networks. Such processes provide better accuracy as compared to PPP but at the cost of limited tractability. Therefore, PPP has been a popular choice for analysis of cellular networks. On the other hand, when it comes to Wi-Fi RAT, PPP cannot be used for spatial modeling of active APs due to contention-based nature of carrier sense multiple access with collision avoidance (CSMA/CA) scheme. However, approximated solution for SINR coverage has been presented in [10] by exploiting Modified Matern Hard Core Process (MMHCP) for estimating the set of active APs, where the APs are originally distributed using a PPP. The work is further extended from spatial averages to spatial distributions in [11], and throughput analysis has also been conducted against various parameters of interest. Due to approximation of the set of interfering APs through a non-homogeneous PPP, the resulting expressions in [10, 11] are extremely complicated. The asymptotic expression, for outage probability of general ad-hoc networks, has been obtained in [12]; however, the results are limited to high signal-to-interference ratio (SIR) cases.Although closed-form expressions for analysis of cellular RAT, under special cases, are available in literature [4, 13], no such results are reported for Wi-Fi RAT due to difficulty in characterizing the interference effect of active APs [10, 11]. This limits the in-depth analysis of multi-RAT HetNets which could be an important tool for coverage and capacity planning of future wireless networks. A general model for multi-RAT HetNets has been presented in [14] by assuming independent PPPs for distribution of APs and users in a given region. However, the effect of channel access schemes, associated with different RATs, has not been taken into account which directly affects the interference and, hence, the coverage analysis. Thus, such a framework cannot provide accurate insights into the different characteristics exhibited by multi-RAT HetNets.Recently, the coexistence problem of multi-RAT networks has received considerable attention. In [15, 16] the coverage and capacity analysis has been presented by assuming the coexistence of cellular and Wi-Fi RAT in unlicensed spectrum. After investigating various transmission mechanism, the authors in [15] reported that the LTE can coexist with Wi-Fi but under certain conditions. On the other hand, in [16] a similar issue has been addressed by assuming the operation of LTE users in both licensed and unlicensed band. Two different user association schemes, i.e., crossing and non-crossing-RAT, have been considered, where the users of licensed RAT can access unlicensed band by exploiting opportunistic CSMA/CA scheme. The crossing-RAT user association is shown to provide better performance as compared to non-crossing-RAT user association. However, the coexistence of LTE and Wi-Fi users in unlicensed band cannot bring significant gain in performance because increase in the capacity of LTE RAT is achieved at the cost of decrease in the capacity of Wi-Fi RAT [16]. Therefore, in this research, the coexistence of different RATs, in unlicensed band, has not been covered.Some recent studies have investigated the fundamental limits on densification of cellular network by assuming different path loss models and fading distributions [17–20]. According to [17], the SINR coverage decreases after a certain threshold, due to increased interference with smaller path loss exponents, which is in contrast to widely used results as reported in [4]. A detailed investigation has been presented in [18] by exploring four different performance regimes while transitioning from sparse to dense networks. Results similar to [17] are reported, where, after a certain threshold, the SIR coverage no longer remains constant; however, the area spectral efficiency increases linearly. According to [19], the SIR coverage can be increased by exploiting the idle mode capability of APs, under dense scenarios. Further, in [20] energy efficiency analysis has been presented by defining the optimal transmission power, for APs, as a function of RAT density. However, in all of the mentioned investigations, a single-tier dense cellular network, with small cells, has been considered. Moreover, in few of the existing studies [21, 22], multi-RAT networks have been investigated by assuming operation of APs in different bands. In [21] the sub-6GHz macrocells are overlaid with mm-wave small cells, and in [22] the small cells can operate on both bands. Various cell association schemes, considering both uplink and downlink channels, have been investigated in [21]. Following a similar thought, in [22] a biasing based strategy has been proposed for load balancing across a multi-RAT network. In contrast to traditional approaches, in [22] two biasing thresholds are exploited; one for offloading users from macrocells to small cells and other for offloading users from sub-GHz band to mm-wave band. In a nutshell, the integration of mm-wave communication into existing infrastructure is one of the potential contributors for increasing capacity of future wireless networks. However, the communication in mm-wave exhibits the characteristics which are different from sub-GHz band, and, hence, its realization requires a lot of enhancements both at component and architecture level [23].Dense multi-RAT HetNets would be a key aspect of future wireless networks [2]. According to some studies [17, 18], the degradation in coverage, provided by dense small cell networks, after a certain limit is expected. Therefore, in order to meet the demands in coming future, the focus must be shifted from single-RAT to multi-RAT HetNets. Although in few of the recent studies [21, 22] the multi-RAT networks have been analyzed by exploiting sub-GHz and mm-wave band, this work is focused around the analysis of multi-RAT HetNets, where each RAT operates on a different pool of resources and can use either contention-based or contention-free channel access schemes. As cellular, contention-free, and Wi-Fi, contention-based, RATs are already deployed at a wide scale, and, hence, their integration can be considered as a potential contributor for improving the capacity of future wireless networks, without many modifications into existing infrastructure. Thus, the key motivating factor behind this work is the growing convergence of the two RATs, and, as opposed to standalone RATs, their integration can lead to better network performance.


### 1.2. Contributions and Outline

With the aforementioned background and motivation, the key contributions of this work can be summarized as follows.(i)
A unified framework for multi-RAT HetNets: using tools from stochastic geometry, we develop a unified framework for SINR and rate coverage analysis of multi-RAT HetNets, where RATs can operate on either contention-free or contention-based (CSMA/CA) channel access schemes. It differs from existing framework [14] due to incorporation of contention-based channel access scheme. More specifically, we focus on a two-RAT HetNet scenario which includes a cellular and a Wi-Fi RAT. The Laplace transform of interference for cellular RAT can be derived easily and it is available in existing literature [4, 13]. The main difficulty arises while modeling the cumulative interference effect under Wi-Fi RAT [10, 15], which operates on a contention-based channel access scheme. Thus, by exploiting a few approximations, we derive the Laplace transform of interference for Wi-Fi RAT which provides accuracy comparable to existing studies [15].(ii)
Analysis of heterogeneous Wi-Fi RAT: we present a tractable solution for SINR and rate coverage analysis of a multi-tier Wi-Fi RAT, by exploiting a few approximations. To the best of the authors knowledge, the notion of multi-tier Wi-Fi RAT has not been studied in existing literature. However, as new techniques like dynamic carrier sensing and extreme densification are emerging, it is important to analyze the effect of heterogeneity in Wi-Fi RAT. We have derived the medium access probability (MAP) for a random and a tagged AP under multi-tier Wi-Fi RAT, and results show that, under dense network conditions, the MAP for a typical AP approaches that of a tagged AP.(iii)
Stable queue probability: by assuming a non-saturated traffic model, we derive the stable queue probability for a user under an AP of a RAT. In order to avoid the problem of interacting queues, similar to [24, 25], we assume a dominant and a modified system, where results for each case have been reported and analyzed. It has been found that, for low packet arrival rate, the stable queue probability of a user under Wi-Fi RAT is slightly higher compared to cellular RAT. However, for higher packet arrival rate, the stable queue probability of a user under cellular RAT is better.(iv)
Various insights: we provide various insights by analyzing different HetNet scenarios with the aid of proposed framework. It has been shown that the integration of femto-tier with Wi-Fi tier provides reasonable SIR coverage as compared to multi-tier cellular or Wi-Fi RAT; however, the rate coverage starts declining as the user association with Wi-Fi RAT exceeds the cellular RAT. Further, the SIR coverage increases and gradually approaches unity, as a function of Wi-Fi RAT density; this insight is in contrast to existing results reported in [14]. Although the SIR coverage provided by Wi-Fi RAT is better than cellular RAT, the stable queue probability of a user under cellular RAT is overall better than Wi-Fi RAT. We also explore the trade-off between user and AP density, and the results show that the rate coverage decreases by increasing the AP density of Wi-Fi RAT while maintaining a constant average load per AP. Under such circumstances, it has been suggested to increase the number of non-overlapping channels for Wi-Fi RAT, as it can greatly improve the rate coverage.The rest of the paper has been organized as follows. Section2 introduces the underlying system model in detail along with the considered channel access schemes and performance metrics. The MAP metric has been covered under Section 3. The main results of the paper have been covered in Section 4, where a unified framework has been presented for SINR and rate coverage analysis of multi-RAT HetNets, and stable queue probability for a user under a RAT has been derived. Various results have been reported and discussed in Section 5. Finally, Section 6 concludes the paper.The notation used in the paper and associated details have been provided in Table1. The general parameters considered for generating various results, under Sections 3, 4, and 5, have been provided in Table 2.Table 1
Notation summary.
Notation Description M Total number of RATs N Total number of tiers under a RAT Φ i j, λij PPP for APs belonging tojth tier of ith RAT and its density Φu,λu PPP for users, and its density Pij Transmit power of APs belonging to pair(i,j) αj Path loss exponent forjth tier σi2 Thermal Noise power associated withith RAT θij SINR threshold for pair(i,j) ρij Rate threshold for pair(i,j) Γwn Carrier sensing for an AP pf Wi-Fi RAT Rwn Carrier sensing range for an AP of Wi-Fi RAT e x t, e^xot Medium access indicator for a random and tagged AP under Wi-Fi RAT P w t, P^wt Medium access probability for a random and tagged AP under Wi-Fi RAT S i j, S SINR coverage of pair(i,j) and overall for Φa R i j, R Rate coverage of pair(i,j) and overall for Φa Bij Effective bandwidth of serving AP Nij Load under serving AP P ^ m n, α^n Normalized Power and path loss exponent Φa PPP for all APs deployed in the region Φ c, Φw PPP for cellular and Wi-Fi RAT, respectively B ( x , r ), Bc(x,r) A ball of radiusx with center at origin, and its compliment ϰc,ϰw Stable queue probability for a user under cellular and Wi-Fi RAT, respectively ξ Packet arrival rate for a user during a time-slot q Active probability of an AP during a time-slotTable 2
General parameters and settings.
P m n   ∀ ( m ) and n=1,2, and 3 46, 33, and 23 dBm  λ m n   ∀ ( m ) and n=1,2, and 3 1, 100, and 1000 AP/km2  ∀ ( n )   α n 4  ∀ ( i )   σ i 2 0 dB  ∀ ( i , j )   θ i j 0 dB  ∀ ( i , j )   ρ i j 2 Mbps  ∀ ( m , n )   Γ m n −82 dBm  ∀ ( i , j )   B i j 10 MHz  f c , f w 1.9 GHz, 5 GHz


## 1.1. Related Work and Motivation

Over the last few years, tools from stochastic geometry have been used for analysis of single-tier and multi-tier cellular networks [4, 5], where mostly Poisson point process (PPP) is used for modeling the spatial location of access points (APs). The PPP assumption provides a lower bound, whereas the traditional grid model provides an upper bound on coverage. Other point processes like Gibbs [6], Strauss [7], Ginibre [8], and Detriminantal [9] have also been used for coverage analysis of cellular networks. Such processes provide better accuracy as compared to PPP but at the cost of limited tractability. Therefore, PPP has been a popular choice for analysis of cellular networks. On the other hand, when it comes to Wi-Fi RAT, PPP cannot be used for spatial modeling of active APs due to contention-based nature of carrier sense multiple access with collision avoidance (CSMA/CA) scheme. However, approximated solution for SINR coverage has been presented in [10] by exploiting Modified Matern Hard Core Process (MMHCP) for estimating the set of active APs, where the APs are originally distributed using a PPP. The work is further extended from spatial averages to spatial distributions in [11], and throughput analysis has also been conducted against various parameters of interest. Due to approximation of the set of interfering APs through a non-homogeneous PPP, the resulting expressions in [10, 11] are extremely complicated. The asymptotic expression, for outage probability of general ad-hoc networks, has been obtained in [12]; however, the results are limited to high signal-to-interference ratio (SIR) cases.Although closed-form expressions for analysis of cellular RAT, under special cases, are available in literature [4, 13], no such results are reported for Wi-Fi RAT due to difficulty in characterizing the interference effect of active APs [10, 11]. This limits the in-depth analysis of multi-RAT HetNets which could be an important tool for coverage and capacity planning of future wireless networks. A general model for multi-RAT HetNets has been presented in [14] by assuming independent PPPs for distribution of APs and users in a given region. However, the effect of channel access schemes, associated with different RATs, has not been taken into account which directly affects the interference and, hence, the coverage analysis. Thus, such a framework cannot provide accurate insights into the different characteristics exhibited by multi-RAT HetNets.Recently, the coexistence problem of multi-RAT networks has received considerable attention. In [15, 16] the coverage and capacity analysis has been presented by assuming the coexistence of cellular and Wi-Fi RAT in unlicensed spectrum. After investigating various transmission mechanism, the authors in [15] reported that the LTE can coexist with Wi-Fi but under certain conditions. On the other hand, in [16] a similar issue has been addressed by assuming the operation of LTE users in both licensed and unlicensed band. Two different user association schemes, i.e., crossing and non-crossing-RAT, have been considered, where the users of licensed RAT can access unlicensed band by exploiting opportunistic CSMA/CA scheme. The crossing-RAT user association is shown to provide better performance as compared to non-crossing-RAT user association. However, the coexistence of LTE and Wi-Fi users in unlicensed band cannot bring significant gain in performance because increase in the capacity of LTE RAT is achieved at the cost of decrease in the capacity of Wi-Fi RAT [16]. Therefore, in this research, the coexistence of different RATs, in unlicensed band, has not been covered.Some recent studies have investigated the fundamental limits on densification of cellular network by assuming different path loss models and fading distributions [17–20]. According to [17], the SINR coverage decreases after a certain threshold, due to increased interference with smaller path loss exponents, which is in contrast to widely used results as reported in [4]. A detailed investigation has been presented in [18] by exploring four different performance regimes while transitioning from sparse to dense networks. Results similar to [17] are reported, where, after a certain threshold, the SIR coverage no longer remains constant; however, the area spectral efficiency increases linearly. According to [19], the SIR coverage can be increased by exploiting the idle mode capability of APs, under dense scenarios. Further, in [20] energy efficiency analysis has been presented by defining the optimal transmission power, for APs, as a function of RAT density. However, in all of the mentioned investigations, a single-tier dense cellular network, with small cells, has been considered. Moreover, in few of the existing studies [21, 22], multi-RAT networks have been investigated by assuming operation of APs in different bands. In [21] the sub-6GHz macrocells are overlaid with mm-wave small cells, and in [22] the small cells can operate on both bands. Various cell association schemes, considering both uplink and downlink channels, have been investigated in [21]. Following a similar thought, in [22] a biasing based strategy has been proposed for load balancing across a multi-RAT network. In contrast to traditional approaches, in [22] two biasing thresholds are exploited; one for offloading users from macrocells to small cells and other for offloading users from sub-GHz band to mm-wave band. In a nutshell, the integration of mm-wave communication into existing infrastructure is one of the potential contributors for increasing capacity of future wireless networks. However, the communication in mm-wave exhibits the characteristics which are different from sub-GHz band, and, hence, its realization requires a lot of enhancements both at component and architecture level [23].Dense multi-RAT HetNets would be a key aspect of future wireless networks [2]. According to some studies [17, 18], the degradation in coverage, provided by dense small cell networks, after a certain limit is expected. Therefore, in order to meet the demands in coming future, the focus must be shifted from single-RAT to multi-RAT HetNets. Although in few of the recent studies [21, 22] the multi-RAT networks have been analyzed by exploiting sub-GHz and mm-wave band, this work is focused around the analysis of multi-RAT HetNets, where each RAT operates on a different pool of resources and can use either contention-based or contention-free channel access schemes. As cellular, contention-free, and Wi-Fi, contention-based, RATs are already deployed at a wide scale, and, hence, their integration can be considered as a potential contributor for improving the capacity of future wireless networks, without many modifications into existing infrastructure. Thus, the key motivating factor behind this work is the growing convergence of the two RATs, and, as opposed to standalone RATs, their integration can lead to better network performance.


## 1.2. Contributions and Outline

With the aforementioned background and motivation, the key contributions of this work can be summarized as follows.(i)
A unified framework for multi-RAT HetNets: using tools from stochastic geometry, we develop a unified framework for SINR and rate coverage analysis of multi-RAT HetNets, where RATs can operate on either contention-free or contention-based (CSMA/CA) channel access schemes. It differs from existing framework [14] due to incorporation of contention-based channel access scheme. More specifically, we focus on a two-RAT HetNet scenario which includes a cellular and a Wi-Fi RAT. The Laplace transform of interference for cellular RAT can be derived easily and it is available in existing literature [4, 13]. The main difficulty arises while modeling the cumulative interference effect under Wi-Fi RAT [10, 15], which operates on a contention-based channel access scheme. Thus, by exploiting a few approximations, we derive the Laplace transform of interference for Wi-Fi RAT which provides accuracy comparable to existing studies [15].(ii)
Analysis of heterogeneous Wi-Fi RAT: we present a tractable solution for SINR and rate coverage analysis of a multi-tier Wi-Fi RAT, by exploiting a few approximations. To the best of the authors knowledge, the notion of multi-tier Wi-Fi RAT has not been studied in existing literature. However, as new techniques like dynamic carrier sensing and extreme densification are emerging, it is important to analyze the effect of heterogeneity in Wi-Fi RAT. We have derived the medium access probability (MAP) for a random and a tagged AP under multi-tier Wi-Fi RAT, and results show that, under dense network conditions, the MAP for a typical AP approaches that of a tagged AP.(iii)
Stable queue probability: by assuming a non-saturated traffic model, we derive the stable queue probability for a user under an AP of a RAT. In order to avoid the problem of interacting queues, similar to [24, 25], we assume a dominant and a modified system, where results for each case have been reported and analyzed. It has been found that, for low packet arrival rate, the stable queue probability of a user under Wi-Fi RAT is slightly higher compared to cellular RAT. However, for higher packet arrival rate, the stable queue probability of a user under cellular RAT is better.(iv)
Various insights: we provide various insights by analyzing different HetNet scenarios with the aid of proposed framework. It has been shown that the integration of femto-tier with Wi-Fi tier provides reasonable SIR coverage as compared to multi-tier cellular or Wi-Fi RAT; however, the rate coverage starts declining as the user association with Wi-Fi RAT exceeds the cellular RAT. Further, the SIR coverage increases and gradually approaches unity, as a function of Wi-Fi RAT density; this insight is in contrast to existing results reported in [14]. Although the SIR coverage provided by Wi-Fi RAT is better than cellular RAT, the stable queue probability of a user under cellular RAT is overall better than Wi-Fi RAT. We also explore the trade-off between user and AP density, and the results show that the rate coverage decreases by increasing the AP density of Wi-Fi RAT while maintaining a constant average load per AP. Under such circumstances, it has been suggested to increase the number of non-overlapping channels for Wi-Fi RAT, as it can greatly improve the rate coverage.The rest of the paper has been organized as follows. Section2 introduces the underlying system model in detail along with the considered channel access schemes and performance metrics. The MAP metric has been covered under Section 3. The main results of the paper have been covered in Section 4, where a unified framework has been presented for SINR and rate coverage analysis of multi-RAT HetNets, and stable queue probability for a user under a RAT has been derived. Various results have been reported and discussed in Section 5. Finally, Section 6 concludes the paper.The notation used in the paper and associated details have been provided in Table1. The general parameters considered for generating various results, under Sections 3, 4, and 5, have been provided in Table 2.Table 1
Notation summary.
Notation Description M Total number of RATs N Total number of tiers under a RAT Φ i j, λij PPP for APs belonging tojth tier of ith RAT and its density Φu,λu PPP for users, and its density Pij Transmit power of APs belonging to pair(i,j) αj Path loss exponent forjth tier σi2 Thermal Noise power associated withith RAT θij SINR threshold for pair(i,j) ρij Rate threshold for pair(i,j) Γwn Carrier sensing for an AP pf Wi-Fi RAT Rwn Carrier sensing range for an AP of Wi-Fi RAT e x t, e^xot Medium access indicator for a random and tagged AP under Wi-Fi RAT P w t, P^wt Medium access probability for a random and tagged AP under Wi-Fi RAT S i j, S SINR coverage of pair(i,j) and overall for Φa R i j, R Rate coverage of pair(i,j) and overall for Φa Bij Effective bandwidth of serving AP Nij Load under serving AP P ^ m n, α^n Normalized Power and path loss exponent Φa PPP for all APs deployed in the region Φ c, Φw PPP for cellular and Wi-Fi RAT, respectively B ( x , r ), Bc(x,r) A ball of radiusx with center at origin, and its compliment ϰc,ϰw Stable queue probability for a user under cellular and Wi-Fi RAT, respectively ξ Packet arrival rate for a user during a time-slot q Active probability of an AP during a time-slotTable 2
General parameters and settings.
P m n   ∀ ( m ) and n=1,2, and 3 46, 33, and 23 dBm  λ m n   ∀ ( m ) and n=1,2, and 3 1, 100, and 1000 AP/km2  ∀ ( n )   α n 4  ∀ ( i )   σ i 2 0 dB  ∀ ( i , j )   θ i j 0 dB  ∀ ( i , j )   ρ i j 2 Mbps  ∀ ( m , n )   Γ m n −82 dBm  ∀ ( i , j )   B i j 10 MHz  f c , f w 1.9 GHz, 5 GHz


## 2. System Model

We consider aM-RAT, N-tier HetNet scenario wherein RATs can employ either contention-free (OFDMA, TDMA, CDMA, etc.) or contention-based (CSMA/CA) channel access scheme. We specifically consider a two-RAT scenario (M=2) which includes a cellular and a Wi-Fi RAT each with N≥1 number of tiers. More RATs (M>2) can be considered provided that each RAT operates on a different pool of resources and the user equipment supports connection to all considered RATs. We adopt a homogeneous PPP Φmn, with density λmn, for drawing the locations of APs belonging to the nth tier of the mth RAT whereas m∈c,w and n∈1,2,…,N. Another independent PPP Φu, with density λu, has been considered for the distribution of users in a given region. We assume that Φc=∪j=1NΦcj and Φw=∪j=1NΦwj denote the set of all APs under cellular and Wi-Fi RAT, respectively. Moreover, all APs provide open access; i.e., there is no closed subscriber group, and Φa=∪i∈c,wΦi denotes the set of all APs deployed in the given region.We consider a downlink channel wherein single resource block (i.e., time, frequency, and code) is utilized in every cell of cellular network. For Wi-Fi RAT, we assume single downlink channel. A saturated traffic model has been considered, where APs transmit continuously even without any packet in queue for transmission. Further, APs of one RAT cannot interfere with those of the other RATs as they operate in different pools of wireless resources. However, APs of different tiers under the same RAT interfere with each other due to shared resources. All APs of(m,n) transmit at the same power Pmn over the bandwidth Bmn. We consider both large-scale path loss and small-scale fading. Free space path loss (FSPL) model with reference distance of 1 meter, as given by l[dB](d)=20log10⁡4π/λc+10αnlog10⁡(d), has been assumed for all links; here, λc and αn>2 denote the operating wavelength and path loss exponent, respectively. The fading channels are Rayleigh distributed with average power of unity i.e., h~exp⁡(1). The noise is assumed additive with power σm2 corresponding to the mth RAT. We assume that user association is based on the maximum average received signal strength. However, it can be easily extended to a generic user association scheme, as given in [14], by just introducing a weight or bias variable. For simplification, normalized parameters for a pair (m,n) with respect to serving pair (i,j) have been defined as P^mn≜Pmn/Pij, α^n≜αn/αj, and f^mn≜fmn2/fij2. Similar to [13, 14], the probability density function (PDF) fYij(y) of the distance Yij between a typical user and the tagged AP is given by(1)fYijy=2πλijAijexp⁡-π∑m,n∈ΦaλmnP^mnyαjf^mn2/αny,where Aij is the probability that a typical user associates with an AP of pair (i,j), and it can be given as(2)Aij=2πλij∫0∞exp⁡-π∑m,n∈ΦaλmnP^mnyαjf^mn2/αnydy.Due to assumption of FSPL model, the association of a user to an AP of pair (i,j) is dependent on the operating frequencies of RATs, as clear from (1) and (2). The normalized component (f^mn) for standalone RATs becomes unity as we have assumed that all tiers under a single-RAT share the same resources; hence, for such cases, we get simplified expressions for user association, which are similar to those in [4, 13].


### 2.1. Channel Access

Contention-free channel access schemes are employed by cellular RAT, where some of the operators deploy frequency reuse factor of unity and others go for fractional frequency reuse. Under contention-based channel access schemes, like CSMA/CA used by Wi-Fi RAT, only the APs with different contention domains are allowed to transmit simultaneously, and, therefore, the set of active APs can be less than the deployed one. Under such a scheme for channel contention, each AP maintains a random back-off timer and waits for its expiry, when the channel is sensed as free. Meanwhile the transmission starts if no other AP accesses the channel. Otherwise, it freezes the timer and repeats the procedure. Due to various reasons, a collision may occur when two APs, in the same contention domain, transmit simultaneously. However, there are defined procedures in Wi-Fi for handling such situations.Under cellular RAT, with contention-free channel access, all deployed APs are active; therefore, the original PPP(Φc) can be used for capturing the cumulative interference effect. However, under Wi-Fi RAT, with CSMA/CA channel access, APs sharing the same contention domains are not allowed to transmit simultaneously. Therefore, the original homogeneous PPP (Φw), used for drawing the location of Wi-Fi APs across a given region, cannot be used for interference modeling. In literature, Modified Matern Hard Core Process (MMHCP), also known as MHCP-2, is widely used for estimating the set of active APs [10, 15]. MMHCP is basically obtained by mark tx dependent thinning of original PPP (Φw), where tx represents the back-off timer of an AP located at x. Thus, any point (xo) of the original PPP (Φw) is retained, only if it has a mark (txo) smaller than all marks associated with the APs in its contention domain; i.e., Φwt=xo∈Φw:txo<tx,∀z:Pxox>Γwn. MMHCP does not take into account the effect of variable back-off timer window size or collisions. However, in [10, 11] it has been proved that the model provides a reasonable conservative representation of active APs, by comparing it against an actual CSMA/CA networks.


### 2.2. Performance Metrics

We consider four performance metrics described as follows.


#### 2.2.1. Medium Access Probability

For cellular RAT, the MAP, denoted byPct, is unity as all APs are allowed to transmit simultaneously. On the other hand, due to contention-based channel access, the MAP for Wi-Fi RAT, denoted by Pwt, can be less than unity. According to MMHCP defined in [10, 15], a random AP under Wi-Fi RAT can access medium only if it has the smallest mark among all the APs in its contention domain (3). Hence, the medium access indicator ejt for an AP xj is given by(3)ejt=∏xn∈Φw∖xj1tn≥tj+1tn<tj1hnj/lxn-xj≤Γwn/Pwn.For further details, please refer to Section 3.


#### 2.2.2. SINR Coverage

A typical user is said to be under coverage if the received SINR from a tagged AP of pair(i,j), located at xo=(y,0), is greater than some defined threshold θij, and it is given by(4)Sijθij=EyPSINRijy>θij∣xo=y,0,exot=1,where(5)SINRijy=Pijhy/ly∑n∈ΦiIin+σi2,hy denotes the channel gain from a tagged AP located at y distance from the user, and Iin=Pin∑x∈Φn∖xoexthx/l(x) is the cumulative interference from all APs of serving RAT-tier pair (i,j), outside the disk of radius y with center at origin. By using total probability theorem, the overall SINR coverage provided to a randomly located user can be given as(6)S=∑i,j∈ΦaAijSijθij.Further details are covered under Section 4.1.


#### 2.2.3. Rate Coverage

The probability that a user, which is associated with an AP of pair(i,j), receives a rate greater than a certain threshold (ρij) is given by(7)Rijρij=PCij>ρij,where(8)Cij=BijNijlog⁡1+SINRijP^ijtrepresents the rate of a user, Nij denotes the number of users served by an AP of pair (i,j), and P^ijt represents the MAP for a tagged AP. By exploiting total probability theorem, the overall rate coverage provided to a randomly located user can be given as(9)R=∑i,j∈ΦaAijRijρij.Further details are covered under Section 4.2.


#### 2.2.4. Stable Queue Probability

The stable queue probability has been defined as the probability that a user queue under an AP of a RAT is stable. A queue is stable only if the provided service rate(μ) is greater than the arrival rate of packets (ξ) during a time-slot.(10)μ>ξHowever, the service rate provided by the network is dependent on the queues status, and vice versa is also true. This creates the problem of interacting queues and it becomes difficult to analyze the combined effect of spatial and temporal domain on overall performance of the network. Thus, in order to avoid this issue, the concept of dominant and modified systems has been exploited in existing literature [24, 25]. Where the dominant system provides a lower bound on performance, by assuming full buffer model for interfering APs, and modified system provides an upper bound by assuming that the active probability of APs is equal to the packet arrival rate of users, hence, the packets not transferred successfully are dropped. Further details are included under Section 4.3.


## 2.1. Channel Access

Contention-free channel access schemes are employed by cellular RAT, where some of the operators deploy frequency reuse factor of unity and others go for fractional frequency reuse. Under contention-based channel access schemes, like CSMA/CA used by Wi-Fi RAT, only the APs with different contention domains are allowed to transmit simultaneously, and, therefore, the set of active APs can be less than the deployed one. Under such a scheme for channel contention, each AP maintains a random back-off timer and waits for its expiry, when the channel is sensed as free. Meanwhile the transmission starts if no other AP accesses the channel. Otherwise, it freezes the timer and repeats the procedure. Due to various reasons, a collision may occur when two APs, in the same contention domain, transmit simultaneously. However, there are defined procedures in Wi-Fi for handling such situations.Under cellular RAT, with contention-free channel access, all deployed APs are active; therefore, the original PPP(Φc) can be used for capturing the cumulative interference effect. However, under Wi-Fi RAT, with CSMA/CA channel access, APs sharing the same contention domains are not allowed to transmit simultaneously. Therefore, the original homogeneous PPP (Φw), used for drawing the location of Wi-Fi APs across a given region, cannot be used for interference modeling. In literature, Modified Matern Hard Core Process (MMHCP), also known as MHCP-2, is widely used for estimating the set of active APs [10, 15]. MMHCP is basically obtained by mark tx dependent thinning of original PPP (Φw), where tx represents the back-off timer of an AP located at x. Thus, any point (xo) of the original PPP (Φw) is retained, only if it has a mark (txo) smaller than all marks associated with the APs in its contention domain; i.e., Φwt=xo∈Φw:txo<tx,∀z:Pxox>Γwn. MMHCP does not take into account the effect of variable back-off timer window size or collisions. However, in [10, 11] it has been proved that the model provides a reasonable conservative representation of active APs, by comparing it against an actual CSMA/CA networks.


## 2.2. Performance Metrics

We consider four performance metrics described as follows.


### 2.2.1. Medium Access Probability

For cellular RAT, the MAP, denoted byPct, is unity as all APs are allowed to transmit simultaneously. On the other hand, due to contention-based channel access, the MAP for Wi-Fi RAT, denoted by Pwt, can be less than unity. According to MMHCP defined in [10, 15], a random AP under Wi-Fi RAT can access medium only if it has the smallest mark among all the APs in its contention domain (3). Hence, the medium access indicator ejt for an AP xj is given by(3)ejt=∏xn∈Φw∖xj1tn≥tj+1tn<tj1hnj/lxn-xj≤Γwn/Pwn.For further details, please refer to Section 3.


### 2.2.2. SINR Coverage

A typical user is said to be under coverage if the received SINR from a tagged AP of pair(i,j), located at xo=(y,0), is greater than some defined threshold θij, and it is given by(4)Sijθij=EyPSINRijy>θij∣xo=y,0,exot=1,where(5)SINRijy=Pijhy/ly∑n∈ΦiIin+σi2,hy denotes the channel gain from a tagged AP located at y distance from the user, and Iin=Pin∑x∈Φn∖xoexthx/l(x) is the cumulative interference from all APs of serving RAT-tier pair (i,j), outside the disk of radius y with center at origin. By using total probability theorem, the overall SINR coverage provided to a randomly located user can be given as(6)S=∑i,j∈ΦaAijSijθij.Further details are covered under Section 4.1.


### 2.2.3. Rate Coverage

The probability that a user, which is associated with an AP of pair(i,j), receives a rate greater than a certain threshold (ρij) is given by(7)Rijρij=PCij>ρij,where(8)Cij=BijNijlog⁡1+SINRijP^ijtrepresents the rate of a user, Nij denotes the number of users served by an AP of pair (i,j), and P^ijt represents the MAP for a tagged AP. By exploiting total probability theorem, the overall rate coverage provided to a randomly located user can be given as(9)R=∑i,j∈ΦaAijRijρij.Further details are covered under Section 4.2.


### 2.2.4. Stable Queue Probability

The stable queue probability has been defined as the probability that a user queue under an AP of a RAT is stable. A queue is stable only if the provided service rate(μ) is greater than the arrival rate of packets (ξ) during a time-slot.(10)μ>ξHowever, the service rate provided by the network is dependent on the queues status, and vice versa is also true. This creates the problem of interacting queues and it becomes difficult to analyze the combined effect of spatial and temporal domain on overall performance of the network. Thus, in order to avoid this issue, the concept of dominant and modified systems has been exploited in existing literature [24, 25]. Where the dominant system provides a lower bound on performance, by assuming full buffer model for interfering APs, and modified system provides an upper bound by assuming that the active probability of APs is equal to the packet arrival rate of users, hence, the packets not transferred successfully are dropped. Further details are included under Section 4.3.


## 2.2.1. Medium Access Probability

For cellular RAT, the MAP, denoted byPct, is unity as all APs are allowed to transmit simultaneously. On the other hand, due to contention-based channel access, the MAP for Wi-Fi RAT, denoted by Pwt, can be less than unity. According to MMHCP defined in [10, 15], a random AP under Wi-Fi RAT can access medium only if it has the smallest mark among all the APs in its contention domain (3). Hence, the medium access indicator ejt for an AP xj is given by(3)ejt=∏xn∈Φw∖xj1tn≥tj+1tn<tj1hnj/lxn-xj≤Γwn/Pwn.For further details, please refer to Section 3.


## 2.2.2. SINR Coverage

A typical user is said to be under coverage if the received SINR from a tagged AP of pair(i,j), located at xo=(y,0), is greater than some defined threshold θij, and it is given by(4)Sijθij=EyPSINRijy>θij∣xo=y,0,exot=1,where(5)SINRijy=Pijhy/ly∑n∈ΦiIin+σi2,hy denotes the channel gain from a tagged AP located at y distance from the user, and Iin=Pin∑x∈Φn∖xoexthx/l(x) is the cumulative interference from all APs of serving RAT-tier pair (i,j), outside the disk of radius y with center at origin. By using total probability theorem, the overall SINR coverage provided to a randomly located user can be given as(6)S=∑i,j∈ΦaAijSijθij.Further details are covered under Section 4.1.


## 2.2.3. Rate Coverage

The probability that a user, which is associated with an AP of pair(i,j), receives a rate greater than a certain threshold (ρij) is given by(7)Rijρij=PCij>ρij,where(8)Cij=BijNijlog⁡1+SINRijP^ijtrepresents the rate of a user, Nij denotes the number of users served by an AP of pair (i,j), and P^ijt represents the MAP for a tagged AP. By exploiting total probability theorem, the overall rate coverage provided to a randomly located user can be given as(9)R=∑i,j∈ΦaAijRijρij.Further details are covered under Section 4.2.


## 2.2.4. Stable Queue Probability

The stable queue probability has been defined as the probability that a user queue under an AP of a RAT is stable. A queue is stable only if the provided service rate(μ) is greater than the arrival rate of packets (ξ) during a time-slot.(10)μ>ξHowever, the service rate provided by the network is dependent on the queues status, and vice versa is also true. This creates the problem of interacting queues and it becomes difficult to analyze the combined effect of spatial and temporal domain on overall performance of the network. Thus, in order to avoid this issue, the concept of dominant and modified systems has been exploited in existing literature [24, 25]. Where the dominant system provides a lower bound on performance, by assuming full buffer model for interfering APs, and modified system provides an upper bound by assuming that the active probability of APs is equal to the packet arrival rate of users, hence, the packets not transferred successfully are dropped. Further details are included under Section 4.3.


## 3. Medium Access Probability

According to the given definition (3) for MAP, a Wi-Fi AP cannot transmit whenever any of its contender AP has a smaller back-off timer, which is similar to one in [10, 15]. As we have assumed a multi-tier Wi-Fi RAT, the APs operate at different power levels (Pwn) based on the tier to which they belong to; hence, it is possible that the APs operating at higher power levels do not sense the presence of low power APs in their vicinity. This effect needs to be captured carefully in order to derive the MAP for a multi-tier Wi-Fi RAT. For better illustration, a two-tier Wi-Fi RAT scenario has been shown in Figure 1, where “Rn<N” and “RN” represent the sensing radius for APs operating at high (n<N) and low (n=N) power levels, respectively. The sensing radius has been obtained by using (14) which does not include small-scale fading; however, this is just an illustration of possible effects on contention domains while considering multi-tier Wi-Fi RAT. The contention domain of each AP, for scenario shown in Figure 1, is AP-0:[1], AP-1:[0], AP-2:[0], AP-3:[3], AP-4:[3]. It must be clear that AP-2 is not part of the contention domain of AP-0, as the received signal strength at AP-0 is below the required threshold (Γwn). On the other hand, AP-0 is in the contention domain of AP-2. AP-1 is sufficiently close to AP-0, and the required threshold is maintained; hence, it belongs to the contention domain of AP-0. AP-3 and AP-4 are at a far distance from AP-0 such that the received signal strength is less than the required threshold. If AP-0 get a chance to access medium, AP-1 and AP-2 remain in silent mode. On the other hand, if AP-2 access the medium, then AP-0 can also transmit given it has a smaller back-off timer than AP-1, as it cannot detect the presence of AP-2. Thus, the MAP, under multi-tier case, can easily be obtained by exploiting the given definition (3).Figure 1
Contention domains of APs under a two-tier Wi-Fi RAT.Lemma 1.
Given a Wi-Fi RAT withN-tiers, each with transmission power (Pwn) and sensing threshold (Γwn), then the MAP for a typical AP is given by(11)Pwt=1-exp⁡-∑n∈ΦwλwnA¯wn∑n∈ΦwλwnA¯wn,where(12)A¯wn=∫R2exp⁡-ΓwnPwnlxdx.Proof.
See AppendixA.Remark 2.
If either∑n∈Φwλwn or A¯wn→∞, Pwt→1/∑n∈ΦwλwnA¯wn. Furthermore, Pwt decays at a faster rate with respect to A¯wn as compared to λwn.Remark 3.
The MAP for any random AP is the same irrespective of the tier to which it belongs. As clear from Figure1, the contention domain of an AP operating at either high or low power level includes both low and high power APs within the sensing range “RN” and “Rn<N”, respectively.The obtained expression (11) can be approximated by following expression (13)Pwt~≈1-exp⁡-∑n∈ΦwλwnAwn∑n∈ΦwλwnAwn,which provides a lower boun on MAP, where Awn=πRwn2 and(14)Rwn=λc4πPwnΓwn2/αnis the sensing radius of APs belonging to the nth tier. Based on the parameters listed in Table 2, the MAP for a single-tier and a two-tier Wi-Fi RAT has been plotted in Figure 2, against density parameter. The numerical and analytical results are obtained by using (11) and (13), respectively, whereas the simulation results are generated by using given definition (3). It must be noted that the simulation results are closely following the numerical ones. The results of analytical expression (13) are fairly close and providing a lower bound. As tier-3 operates at a lower power as compared to tier-2, under single-tier scenario, the MAP for tier-3 is higher as compared to tier-2. In accordance to Remark 2, it must be clear from the reported results that, with gradual increase in λw or Rwn, the MAP approaches 1/∑n∈ΦwλwnAwn.Figure 2
Comparison of numerical, analytical, and simulation results for the MAP of a random AP against Wi-Fi RAT density.Remark 4.
The approximated expression (13) provides a lower bound on Pwt; therefore it is reasonable to say that A¯wn<Awn; here Awn takes into account only large-scale path loss, whereas A¯wn also considers the effect of small-scale fading. This implies that the expected sensing area, or equivalently sensing radius, for an AP is small when fading effects are taken into account; hence, the expected number of contenders are less which results in improved MAP; i.e., Pwt>Pwt~.According toCorollary 1 of [15], the MAP of a tagged AP is the biased version of the MAP for a typical AP. However, we argue that, as the density or power of tier increases, the MAP for a tagged AP approaches the MAP for a typical AP. For better illustration please refer to Figure 3, where three different cases are considered i.e., low, moderate, and high density, by assuming single-tier scenario. Part (a) shows moderate density case because the distance between a user and its tagged AP is ro≤Rwn. As the user associates with the nearest AP, the shaded region does not include any AP other than the tagged one. That is why, in [15] it has been suggested that the MAP for a tagged AP is the biased version of MAP for a random AP. Now let us consider the sparse case in part (c) of Figure 3 where ro>>Rwn. Although the MAP is high in this case, the link between user and its tagged AP is of no use because the received signal strength is less than the required threshold (Γwn), assuming that the received signal strength required for user is the same as that for the tagged AP. Thus, under sparse condition the MAP for a tagged AP and even for a random AP approaches unity but at the cost of decrease in received signal strength. Finally moving to dense case, part (c) of Figure 3 where ro<<Rwn, it must be clear that as density of the RAT increases, ro decreases; hence, the shaded region starts shrinking and the MAP for a tagged AP approaches that of a typical AP.Figure 3
The relationship between approximated sensing range of a tagged AP and its distance to user.
(a) (b) (c)Lemma 5.
The MAP for a tagged AP belonging to thejth tier of Wi-Fi RAT, with transmission power (Pwj) and sensing threshold (Γwj), is given by (15)P^wjt=∫0∞1-exp⁡-∑n∈ΦwλwnA¯wn-A~wny∑n∈ΦwλwnA¯wn-A~wnyfYwjy, where (16)A~wny=2∫02y′arccos⁡x2y′exp⁡-ΓwnPwnlxxdx.A¯wn is defined in (12), and fYwj(y) is given by (1).Proof.
See Appendix B.Remark 6.
By using total probability theorem, the overall MAP for a tagged AP inΦw can be given as P^wt=∑j∈ΦwAwjP^wjt.As we have assumed a multi-tier Wi-Fi RAT scenario, Lemma5 provides the MAP for a tagged AP which belongs to the jth tier of Wi-Fi RAT. It is an extension of Lemma 2 from [26] in which the retention probability for an associated AP has been defined, when LTE APs coexist with single-tier Wi-Fi RAT in unlicensed band. In Figure 4, the numerical results have been plotted for a tagged and a random AP, under single-tier and multi-tier scenarios, against density parameter. It must be clear that, under low density with smaller power of transmission (n=3), the MAP for a tagged AP is slightly higher than random AP. However, as the density or power of transmission increases (n=2), the MAP for a tagged AP approaches that of a random AP. Further, in Figure 5 the void probability, given in [4], for no AP within a region of radius R has been plotted, and the approximated sensing radii for tier-2 and tier-3 are also denoted with markers. It must be clear that, under sparse case when λw = 100 AP/km2, the probability that the distance between a user and the tagged AP is greater than the corresponding sensing radius is around 80% for tier-3 and 40% for tier-2. As already mentioned while discussing Figure 3, such an event does not provide a successful connection to a user because of low received signal strength. As density increases to 1500 AP/km2, the probability of such an event approaches zero, and the MAP for a tagged AP approaches that of a random AP which is evident from Figure 4.Figure 4
Comparison of MAP for a typical and a tagged AP against Wi-Fi RAT density for single-tier and multi-tier scenarios.Figure 5
Probability that the distance from a typical user to a tagged AP is greater than approximated sensing radius of an AP.Remark 7.
Under dense network scenario, it is reasonable to approximateP^wt by Pwt, whereas by dense here we mean that the probability of no AP within the approximated sensing region approaches zero; hence, the required received signal strength for a successful connection is fulfilled across the region. This can be achieved by either increasing the transmission power of APs or density of the RAT.


## 4. Coverage

Under this section we cover the rest of the three performance metrics, namely, SINR coverage, rate coverage, and stable queue probability. The key factor which plays an important role for derivation of each of the mentioned metric is the Laplace transform of cumulative interference. We have assumed a multi-RAT HetNet scenario, where APs can access channel by using either contention-free or contention-based schemes; therefore the interference distribution vary under each RAT, and hence, the corresponding Laplace transform. Moreover, it is also important to consider if the user equipment can support multi-RAT connection. Thus, in this work we specifically focus on a two-RAT scenario, by assuming a cellular and a Wi-Fi RAT, each withN-tiers such that the APs of tier-1 have maximum and tier-N have minimum power of transmission. Please note that the framework is generalized and can be extended to more RATs.


### 4.1. SINR Coverage

Cellular RAT is deeply investigated in existing literature by using tools from stochastic geometry; therefore, we refer to [4, 13] for the Laplace transform of cumulative interference under cellular RAT. Due to contention-based nature of channel access in Wi-Fi RAT, it is hard to characterize the cumulative interference effect. As the distribution of interfering APs is non-independent thinning of Φw, the Laplace transform of interference is not known in closed-form [10, 15]. Therefore, in [15] the set of interfering APs, under Wi-Fi RAT, is approximated by non-homogeneous PPP with certain density which has been defined by exploiting the conditional MAP and Bayes’ rule. On the other hand, in [26] the set of interfering APs has been approximated by a homogeneous PPP (Φwt) with density λwt=Pwtλw, and it has been assumed that the repulsion among APs is captured by Pwt, which is reasonable as per discussions in [10, 27]. Two main factors for capturing the cumulative interference effect are (1) the density of active APs and (2) the distance to those APs. In this work, similar to [26], we approximate the conditional MAP for an interfering AP (P[ext=1∣exo=1]) by the conditional MAP of a tagged AP (P[exot=1∣xo=(y,0)]). As per an alternative definition, given in [10], the MAP represents the probability of successful simultaneous transmissions. This implies that if a tagged AP transmits, then on average the number of simultaneous transmissions, and hence, the number of active APs in a given region remain constant. Thus, we can approximate the set of interfering APs by a PPP (Φwnt) with density λwnt=P^wjtλwn. The other important factor, in modeling the interference effect, is the distance to nearest active AP. As heavy portion in interference is mainly contributed by the closest active APs, the distance to nearest interfering AP has been approximated by using some simple yet effective set of assumptions. The following lemma provides an approximated Laplace transform of cumulative interference for Wi-Fi RAT. Although our framework is based on a few approximations, it provides reasonable accuracy when compared with simulated and existing results.Lemma 8.
The Laplace transform of cumulative interference for Wi-Fi RAT withN-tiers is approximated by (17)LIwns^wj≈exp⁡-πP^wjtλwnZswjPwn,αn,zwnaαn∣y<R¯wN4+ZswjPwn,αn,zwnbαn∣y>R¯wN4,where R¯wN represents the mean sensing radius for a tier with lowest power of transmission (N) and zwna and zwnb are defined in (C.7) and (C.8), respectively.Proof.
See Appendix C.Following Lemma8 and existing studies [4, 13], for Laplace transform of cumulative interference under cellular RAT, the SINR coverage for a typical user has been defined in the following theorem.Theorem 9.
The SINR coverage of a randomly located user under a multi-RAT HetNet, as defined in Section2, is approximated by (18)S≈∑j∈Φc2πλcj∫0∞exp-s^cjσc2-π∑n∈ΦcλcnZscjPcn,αn,zcnbαn+∑m,n∈ΦaλmnP^mnyαjf^mn2/αnydy+∑j∈Φw2πλwj∫0∞exp⁡-s^wjσw2-πP^wjt∑n∈ΦwλwnZswjPwn,αn,zwnaαn∣y<R¯wN4+λwnZswjPwn,αn,zwnbαn∣y>R¯wN4+∑m,n∈ΦaλmnP^mnyαjf^mn2/αnydy,where s^ij=θijl(y)Pij-1, θij is the SINR threshold for the jth tier of the ith RAT, and(19)Za,b,c=a2/b∫c/a2/b∞du1+ub/2.Proof.
By following given definition (4) for SINR coverage, we get(20)Sijθij=a∫y>0Eexp⁡-θijlyPij∑n∈ΦiPin∑x∈Φn∩Bc0,y′exthxlx+σi2∣exot=1fYijydy(21)≈b∫y>0Eexp⁡-s^ij∑n∈ΦitIin+σi2fYijydy(22)=c∫y>0e-s^ijσi2∏n∈ΦitLIin-s^ijfYijydy,where (a) is the result of deconditioning with respect to xo=(y,0) and assumption that hy~exp⁡(1), (b) follows from an approximation P[ext=1∣exot=1]≈P[exot=1∣xo=(y,0)] for i∈w, and an assumption that Pext=1=1 for i∈c, (c) follows from independent random variable Iin, and LIin(s^ij) is the Laplace transform of interference. We refer to existing results from [4, 13] for LIcn(s^cj). By using Lemma 8, we get an approximated LIwn(s^wj) for Wi-Fi RAT, and the final expression (18) is obtained by using total probability theorem (6) which completes the proof.Corollary 10.
By assuming an interference-limited scenario, i.e.,σw2=0, with αj=4 and P^wjt≈Pwjt, the SIR coverage of a randomly located user under a single-tier (jth) Wi-Fi RAT is given by (23)Swj≈1-exp⁡-πλwjR¯wj2/16γwj3γwj3+exp⁡-πλwjR¯wj2/16γwj1γwj1,where(24)γwjv=1+Pwjtθwjarctan⁡θwjv2.Proof.
Substituting given parameters in (18), performing some mathematical operations, and re-arranging variables proof the given corollary.In Figure6, the numerical results obtained through (18) are compared against the simulated ones for two single-tier (λw=λw2, λw=λw3) and two multi-tier cases under Wi-Fi RAT. The simulation environment was created by randomly deploying APs of given density in a region of size 1 km × 1 km. The results were averaged over number of iterations, and under each iteration the SIR was evaluated for 2000 randomly chosen points. It must be clear that the approximated expression (18) is closely following the simulated results and provides a lower bound on coverage, which is according to discussions under Lemma 8 and Theorem 9. Although the interfering APs are very close to the tagged one, under high density regime, the distance between a user and tagged AP is also very less as compared to the sensing radius of APs; that is why in Figure 6, the numerical results provide an upper bound on SIR coverage for tier-2 as density of APs increases. Further in Figure 7, the numerical results are plotted for various network configurations including both standalone and multi-RAT HetNets. Standalone cellular (Φc) and Wi-Fi (Φw) RAT, each with two tiers, have been considered, where Φc is providing a lower bound, and it is according to reported results [4, 13]. On the other hand, Φw is providing better coverage as some of the APs are prohibited to transmit, because of the contention domains. The results for two multi-RAT HetNets are also reported, where in Φam a macro-tier (n=1) has been overlaid with a Wi-Fi tier (n=2) and in Φaf a femto-tier (n=2) is overlaid with a Wi-Fi tier (n=3). Although the power of tier-2 >> tier-3, the considered density for tier-3 >> tier-2 which reduces the MAP and, hence, improves the SIR coverage; that is why, all configurations which include tier-3 of Wi-Fi RAT are providing better coverage as compared to those with tier-2.Figure 6
Comparison of numerical results with simulated ones for single-tier and two-tier Wi-Fi RAT only.Figure 7
Numerical results for SIR coverage, under various network configurations, obtained through (18).


### 4.2. Rate Coverage

Under this section, in the following theorem, we derive the rate coverage probability of a randomly located user.Theorem 11.
The probability that a randomly located user, in a network setting as defined in Section2, receives a rate greater than some defined threshold (ρij) is approximated by(25)R≈∑i,j∈ΦaAijSijθij=τijN-ij,where N¯ij denotes expected load under the serving AP and τij(N¯ij)=2ρijN¯ij/P^ijtBij-1Proof.
The proof simply follows from [14]; however, for readability the details are included in Appendix D.Remark 12.
The rate coverage is function of four parameters including rate threshold(ρij), average load under serving AP (N¯ij), MAP (P^ijt), and bandwidth (Bij). Under cellular RAT, the relation of rate coverage with the mentioned parameters can be explained with the help of the following expression(26)ρcjN¯cjP^cjtBcj=ρcjBcj+1.28ρcjλuAcjλcjBcj,whereP^cjt=1. It must be clear that the rate coverage of a user under cellular RAT is directly proportional to Bcj and λcj of the tier, whereas it is inversely proportional to ρcj and λu. In case of Wi-Fi RAT, by using an approximation P^wjt≈Pwjt, we get(27)ρwjN¯wjP^wjtBwj≈ρwjλwjA¯wj+1.28ρwjλuAwj1-exp⁡-λwjA¯wjBwj.Similar to cellular RAT, the rate coverage under Wi-Fi RAT is inversely proportional to ρwj and λu, and it is directly proportional to Bwj. Moreover, the rate coverage is indirectly proportional to the product λwjA¯wj and at the same time directly proportional to the negative exponent of it. For lower values of λwj, the negative exponential effect dominates, and, therefore, the rate coverage increases. On the other hand, as λwj→∞ the term 1-exp⁡-λwjA¯wj approaches unity, and, hence, the rate coverage starts declining.Remark 13.
The rate coverage under Wi-Fi RAT is inversely proportional toA¯wj and directly proportional to the negative exponent of it; please see (27). Therefore, for lower values of A¯wj the term in denominator of (27) dominates, and, hence, the rate coverage improves. As A¯wj→∞, the term 1-exp⁡-λwjA¯wj approaches unity, and, hence, the rate coverage starts declining. Thus, in either case the tiers operating at low power levels provide better rate coverage as compared to high power tiers. Equivalently, we can also conclude that the rate coverage increases as a function of sensing threshold (Γwj).In Figure8 the numerical results obtained through (25) have been plotted by considering network configurations similar to those of Figure 7. It must be noted that, in Figure 7, the SIR coverage was slightly affected by the changes in configuration as compared to the rate coverage, in Figure 8, which is significantly varying for various network configurations. The reason behind such a result is the dependence of rate coverage over four different parameters, as clear from Theorem 9 and Remark 12. Moreover, for all those configurations the rate coverage is high which include tier-3 of Wi-Fi RAT, because of its high density and lower power of transmission; please see Remarks 12 and 13 for further details. In Figure 9 the rate coverage for different network configurations has been plotted and the results are in accordance with Remarks 12 and 13. The rate coverage increases for standalone cellular RAT; however, for Wi-Fi RAT it initially increases and then it starts declining. Similarly, under multi-RAT case as the user association with Wi-Fi RAT exceeds the cellular RAT, the rate coverage starts declining. Moreover, the rate of low power Wi-Fi tier is better than high power tier which is in accordance with Remark 13.Figure 8
Numerical results for rate coverage, under various network configurations, obtained through (25).Figure 9
Rate coverage as a function of Wi-Fi RAT density or, in case of standalone cellular RAT, it is function of cellular RAT density.


### 4.3. Stable Queue

Most of the existing studies assume a saturation model for traffic which do not capture the randomness introduced by the temporal domain. In few of the recent works [24, 25, 28], both the temporal and spatial domains have been analyzed by exploiting tools from queuing theory and stochastic geometry. In [25, 29], the conditions for a network to be stable have been derived by assuming a dominant and a modified system. In [28], the probability for a user queue to be unstable has been derived by assuming a Poisson and a uniform distribution for arrival rate of packets, where PPP and Poisson cluster process (PCP) have been used for the distribution of APs across a given region. In all of the aforementioned works, single-tier cellular RAT and a downlink channel have been assumed. As the PPP realization is random and irregular, there are some APs with good and others with poor transmission environment, resulting in some users near APs with good experience and others at the edge under outage [25]. In [30], the outage probability has been derived as a function of distance from a user to the tagged AP, and it has been shown that the outage increases, as the distance increases. By exploiting the given concepts mainly from [25, 29, 30], we derive the stable queue probability for a user under an AP of a given RAT.In this section, for simplified analysis, we follow a different set of assumptions [24, 28]. We assume standalone single-tier cellular and Wi-Fi RAT, and an interference-limited scenario; i.e., σi2=0, and αn=4. A non-saturated traffic model has been considered, where packets arrive at a user with probability ξ∈[0,1] during a time-slot. Further, we assume that q represents the probability that an AP is active during a time-slot. For avoiding interacting queues problem, similar to [24, 29], we assume a dominant and a modified system. Under a dominant system, the interfering APs have full buffers and transmit continuously, i.e., q=1, whereas, under modified system, the interfering APs are active with probability q=ξ; the packets not delivered successfully are, hence, assumed to be dropped. With the aforementioned assumptions, the following theorem provides the probability that a user queue is stable.Theorem 14.
The stable queue probability of a user under a single-tier cellular(Φc) or a Wi-Fi (Φw) RAT, with a packet arrival rate of ξ, is given by (28)ϰcθ,ξ,q=1-exp⁡log⁡ξqθarctan⁡θ,(29)ϰwθ,ξ,q=1-explog⁡ξ/P^wtqP^wtθarctan⁡θ/91ro3<R¯w/4+1-exp⁡-πλwR¯w2161ro3≥R¯w/4+exp⁡-πλwR¯w216-exp⁡log⁡ξ/P^wtqP^wtθarctan⁡θ1ro1>R¯w/4,respectively, where(30)rov2=-log⁡ξ/P^wtπqP^wtλwθarctan⁡θ/v2.Proof.
S ee Appendix E.Remark 15.
From the given condition (E.5), for a stable queue of a user under an AP of Wi-Fi RAT, it is clear that the MAP for an AP must be greater than the arrival rate of packets during a time-slot. Hence, (30) is valid only when P^wt>ξ.By assuming a dominant(q=1) and a modified (q=ξ) system, the numerical results for stable queue probability have been reported in Figures 10, 11, and 12 against different parameters of interest. The dominant system in each result is providing a lower bound, whereas the modified system is providing an upper bound [24, 25]. It must be clear from Figure 10 that the stable queue probability for a user under Wi-Fi RAT (ϰw) is slightly better than cellular RAT (ϰc) when the packet arrival rate is low. As ξ increases, ϰw decreases and eventually approaches zero when ξ→P^wt, which is in accordance with Remark 15; please see Figures 10 and 12 for clarification. Moreover, it must also be noted that the decay in ϰ as a function of ξ is faster as compared to θ in Figure 11 and λw in Figure 12, which is in agreement with the results reported in [29]. ξ as a function of AP density is constant for cellular RAT, because, under interference-limited scenario with αc=4, the SIR coverage becomes independent of density of the RAT [4]. On the other hand, under Wi-Fi RAT ϰw first decreases because P^wt decreases as a function of λw. After that it increases slightly as the probability of distance ro>R¯w/4 between a user and its tagged AP approaches zero; hence, the second indicator function in (29) becomes active; as all other factors are constant, thus increase in λw results in an increase in ϰw. Finally, when the distance between a user and its tagged AP is ro≤R¯w/4, the very first indicator function in (29) becomes active, and, hence, ϰw starts declining and finally approaches zeros as P^wt→ξ.Figure 10
Stable queue probability(ϰ) as a function of packet arrival rate (ξ), by assuming a dominant and a modified systems.Figure 11
Stable queue probability(ϰ) as a function of SIR threshold (θ), by assuming a dominant and a modified system.Figure 12
Stable queue probability(ϰ) as a function of AP density, by assuming a dominant and a modified system.


## 4.1. SINR Coverage

Cellular RAT is deeply investigated in existing literature by using tools from stochastic geometry; therefore, we refer to [4, 13] for the Laplace transform of cumulative interference under cellular RAT. Due to contention-based nature of channel access in Wi-Fi RAT, it is hard to characterize the cumulative interference effect. As the distribution of interfering APs is non-independent thinning of Φw, the Laplace transform of interference is not known in closed-form [10, 15]. Therefore, in [15] the set of interfering APs, under Wi-Fi RAT, is approximated by non-homogeneous PPP with certain density which has been defined by exploiting the conditional MAP and Bayes’ rule. On the other hand, in [26] the set of interfering APs has been approximated by a homogeneous PPP (Φwt) with density λwt=Pwtλw, and it has been assumed that the repulsion among APs is captured by Pwt, which is reasonable as per discussions in [10, 27]. Two main factors for capturing the cumulative interference effect are (1) the density of active APs and (2) the distance to those APs. In this work, similar to [26], we approximate the conditional MAP for an interfering AP (P[ext=1∣exo=1]) by the conditional MAP of a tagged AP (P[exot=1∣xo=(y,0)]). As per an alternative definition, given in [10], the MAP represents the probability of successful simultaneous transmissions. This implies that if a tagged AP transmits, then on average the number of simultaneous transmissions, and hence, the number of active APs in a given region remain constant. Thus, we can approximate the set of interfering APs by a PPP (Φwnt) with density λwnt=P^wjtλwn. The other important factor, in modeling the interference effect, is the distance to nearest active AP. As heavy portion in interference is mainly contributed by the closest active APs, the distance to nearest interfering AP has been approximated by using some simple yet effective set of assumptions. The following lemma provides an approximated Laplace transform of cumulative interference for Wi-Fi RAT. Although our framework is based on a few approximations, it provides reasonable accuracy when compared with simulated and existing results.Lemma 8.
The Laplace transform of cumulative interference for Wi-Fi RAT withN-tiers is approximated by (17)LIwns^wj≈exp⁡-πP^wjtλwnZswjPwn,αn,zwnaαn∣y<R¯wN4+ZswjPwn,αn,zwnbαn∣y>R¯wN4,where R¯wN represents the mean sensing radius for a tier with lowest power of transmission (N) and zwna and zwnb are defined in (C.7) and (C.8), respectively.Proof.
See Appendix C.Following Lemma8 and existing studies [4, 13], for Laplace transform of cumulative interference under cellular RAT, the SINR coverage for a typical user has been defined in the following theorem.Theorem 9.
The SINR coverage of a randomly located user under a multi-RAT HetNet, as defined in Section2, is approximated by (18)S≈∑j∈Φc2πλcj∫0∞exp-s^cjσc2-π∑n∈ΦcλcnZscjPcn,αn,zcnbαn+∑m,n∈ΦaλmnP^mnyαjf^mn2/αnydy+∑j∈Φw2πλwj∫0∞exp⁡-s^wjσw2-πP^wjt∑n∈ΦwλwnZswjPwn,αn,zwnaαn∣y<R¯wN4+λwnZswjPwn,αn,zwnbαn∣y>R¯wN4+∑m,n∈ΦaλmnP^mnyαjf^mn2/αnydy,where s^ij=θijl(y)Pij-1, θij is the SINR threshold for the jth tier of the ith RAT, and(19)Za,b,c=a2/b∫c/a2/b∞du1+ub/2.Proof.
By following given definition (4) for SINR coverage, we get(20)Sijθij=a∫y>0Eexp⁡-θijlyPij∑n∈ΦiPin∑x∈Φn∩Bc0,y′exthxlx+σi2∣exot=1fYijydy(21)≈b∫y>0Eexp⁡-s^ij∑n∈ΦitIin+σi2fYijydy(22)=c∫y>0e-s^ijσi2∏n∈ΦitLIin-s^ijfYijydy,where (a) is the result of deconditioning with respect to xo=(y,0) and assumption that hy~exp⁡(1), (b) follows from an approximation P[ext=1∣exot=1]≈P[exot=1∣xo=(y,0)] for i∈w, and an assumption that Pext=1=1 for i∈c, (c) follows from independent random variable Iin, and LIin(s^ij) is the Laplace transform of interference. We refer to existing results from [4, 13] for LIcn(s^cj). By using Lemma 8, we get an approximated LIwn(s^wj) for Wi-Fi RAT, and the final expression (18) is obtained by using total probability theorem (6) which completes the proof.Corollary 10.
By assuming an interference-limited scenario, i.e.,σw2=0, with αj=4 and P^wjt≈Pwjt, the SIR coverage of a randomly located user under a single-tier (jth) Wi-Fi RAT is given by (23)Swj≈1-exp⁡-πλwjR¯wj2/16γwj3γwj3+exp⁡-πλwjR¯wj2/16γwj1γwj1,where(24)γwjv=1+Pwjtθwjarctan⁡θwjv2.Proof.
Substituting given parameters in (18), performing some mathematical operations, and re-arranging variables proof the given corollary.In Figure6, the numerical results obtained through (18) are compared against the simulated ones for two single-tier (λw=λw2, λw=λw3) and two multi-tier cases under Wi-Fi RAT. The simulation environment was created by randomly deploying APs of given density in a region of size 1 km × 1 km. The results were averaged over number of iterations, and under each iteration the SIR was evaluated for 2000 randomly chosen points. It must be clear that the approximated expression (18) is closely following the simulated results and provides a lower bound on coverage, which is according to discussions under Lemma 8 and Theorem 9. Although the interfering APs are very close to the tagged one, under high density regime, the distance between a user and tagged AP is also very less as compared to the sensing radius of APs; that is why in Figure 6, the numerical results provide an upper bound on SIR coverage for tier-2 as density of APs increases. Further in Figure 7, the numerical results are plotted for various network configurations including both standalone and multi-RAT HetNets. Standalone cellular (Φc) and Wi-Fi (Φw) RAT, each with two tiers, have been considered, where Φc is providing a lower bound, and it is according to reported results [4, 13]. On the other hand, Φw is providing better coverage as some of the APs are prohibited to transmit, because of the contention domains. The results for two multi-RAT HetNets are also reported, where in Φam a macro-tier (n=1) has been overlaid with a Wi-Fi tier (n=2) and in Φaf a femto-tier (n=2) is overlaid with a Wi-Fi tier (n=3). Although the power of tier-2 >> tier-3, the considered density for tier-3 >> tier-2 which reduces the MAP and, hence, improves the SIR coverage; that is why, all configurations which include tier-3 of Wi-Fi RAT are providing better coverage as compared to those with tier-2.Figure 6
Comparison of numerical results with simulated ones for single-tier and two-tier Wi-Fi RAT only.Figure 7
Numerical results for SIR coverage, under various network configurations, obtained through (18).


## 4.2. Rate Coverage

Under this section, in the following theorem, we derive the rate coverage probability of a randomly located user.Theorem 11.
The probability that a randomly located user, in a network setting as defined in Section2, receives a rate greater than some defined threshold (ρij) is approximated by(25)R≈∑i,j∈ΦaAijSijθij=τijN-ij,where N¯ij denotes expected load under the serving AP and τij(N¯ij)=2ρijN¯ij/P^ijtBij-1Proof.
The proof simply follows from [14]; however, for readability the details are included in Appendix D.Remark 12.
The rate coverage is function of four parameters including rate threshold(ρij), average load under serving AP (N¯ij), MAP (P^ijt), and bandwidth (Bij). Under cellular RAT, the relation of rate coverage with the mentioned parameters can be explained with the help of the following expression(26)ρcjN¯cjP^cjtBcj=ρcjBcj+1.28ρcjλuAcjλcjBcj,whereP^cjt=1. It must be clear that the rate coverage of a user under cellular RAT is directly proportional to Bcj and λcj of the tier, whereas it is inversely proportional to ρcj and λu. In case of Wi-Fi RAT, by using an approximation P^wjt≈Pwjt, we get(27)ρwjN¯wjP^wjtBwj≈ρwjλwjA¯wj+1.28ρwjλuAwj1-exp⁡-λwjA¯wjBwj.Similar to cellular RAT, the rate coverage under Wi-Fi RAT is inversely proportional to ρwj and λu, and it is directly proportional to Bwj. Moreover, the rate coverage is indirectly proportional to the product λwjA¯wj and at the same time directly proportional to the negative exponent of it. For lower values of λwj, the negative exponential effect dominates, and, therefore, the rate coverage increases. On the other hand, as λwj→∞ the term 1-exp⁡-λwjA¯wj approaches unity, and, hence, the rate coverage starts declining.Remark 13.
The rate coverage under Wi-Fi RAT is inversely proportional toA¯wj and directly proportional to the negative exponent of it; please see (27). Therefore, for lower values of A¯wj the term in denominator of (27) dominates, and, hence, the rate coverage improves. As A¯wj→∞, the term 1-exp⁡-λwjA¯wj approaches unity, and, hence, the rate coverage starts declining. Thus, in either case the tiers operating at low power levels provide better rate coverage as compared to high power tiers. Equivalently, we can also conclude that the rate coverage increases as a function of sensing threshold (Γwj).In Figure8 the numerical results obtained through (25) have been plotted by considering network configurations similar to those of Figure 7. It must be noted that, in Figure 7, the SIR coverage was slightly affected by the changes in configuration as compared to the rate coverage, in Figure 8, which is significantly varying for various network configurations. The reason behind such a result is the dependence of rate coverage over four different parameters, as clear from Theorem 9 and Remark 12. Moreover, for all those configurations the rate coverage is high which include tier-3 of Wi-Fi RAT, because of its high density and lower power of transmission; please see Remarks 12 and 13 for further details. In Figure 9 the rate coverage for different network configurations has been plotted and the results are in accordance with Remarks 12 and 13. The rate coverage increases for standalone cellular RAT; however, for Wi-Fi RAT it initially increases and then it starts declining. Similarly, under multi-RAT case as the user association with Wi-Fi RAT exceeds the cellular RAT, the rate coverage starts declining. Moreover, the rate of low power Wi-Fi tier is better than high power tier which is in accordance with Remark 13.Figure 8
Numerical results for rate coverage, under various network configurations, obtained through (25).Figure 9
Rate coverage as a function of Wi-Fi RAT density or, in case of standalone cellular RAT, it is function of cellular RAT density.


## 4.3. Stable Queue

Most of the existing studies assume a saturation model for traffic which do not capture the randomness introduced by the temporal domain. In few of the recent works [24, 25, 28], both the temporal and spatial domains have been analyzed by exploiting tools from queuing theory and stochastic geometry. In [25, 29], the conditions for a network to be stable have been derived by assuming a dominant and a modified system. In [28], the probability for a user queue to be unstable has been derived by assuming a Poisson and a uniform distribution for arrival rate of packets, where PPP and Poisson cluster process (PCP) have been used for the distribution of APs across a given region. In all of the aforementioned works, single-tier cellular RAT and a downlink channel have been assumed. As the PPP realization is random and irregular, there are some APs with good and others with poor transmission environment, resulting in some users near APs with good experience and others at the edge under outage [25]. In [30], the outage probability has been derived as a function of distance from a user to the tagged AP, and it has been shown that the outage increases, as the distance increases. By exploiting the given concepts mainly from [25, 29, 30], we derive the stable queue probability for a user under an AP of a given RAT.In this section, for simplified analysis, we follow a different set of assumptions [24, 28]. We assume standalone single-tier cellular and Wi-Fi RAT, and an interference-limited scenario; i.e., σi2=0, and αn=4. A non-saturated traffic model has been considered, where packets arrive at a user with probability ξ∈[0,1] during a time-slot. Further, we assume that q represents the probability that an AP is active during a time-slot. For avoiding interacting queues problem, similar to [24, 29], we assume a dominant and a modified system. Under a dominant system, the interfering APs have full buffers and transmit continuously, i.e., q=1, whereas, under modified system, the interfering APs are active with probability q=ξ; the packets not delivered successfully are, hence, assumed to be dropped. With the aforementioned assumptions, the following theorem provides the probability that a user queue is stable.Theorem 14.
The stable queue probability of a user under a single-tier cellular(Φc) or a Wi-Fi (Φw) RAT, with a packet arrival rate of ξ, is given by (28)ϰcθ,ξ,q=1-exp⁡log⁡ξqθarctan⁡θ,(29)ϰwθ,ξ,q=1-explog⁡ξ/P^wtqP^wtθarctan⁡θ/91ro3<R¯w/4+1-exp⁡-πλwR¯w2161ro3≥R¯w/4+exp⁡-πλwR¯w216-exp⁡log⁡ξ/P^wtqP^wtθarctan⁡θ1ro1>R¯w/4,respectively, where(30)rov2=-log⁡ξ/P^wtπqP^wtλwθarctan⁡θ/v2.Proof.
S ee Appendix E.Remark 15.
From the given condition (E.5), for a stable queue of a user under an AP of Wi-Fi RAT, it is clear that the MAP for an AP must be greater than the arrival rate of packets during a time-slot. Hence, (30) is valid only when P^wt>ξ.By assuming a dominant(q=1) and a modified (q=ξ) system, the numerical results for stable queue probability have been reported in Figures 10, 11, and 12 against different parameters of interest. The dominant system in each result is providing a lower bound, whereas the modified system is providing an upper bound [24, 25]. It must be clear from Figure 10 that the stable queue probability for a user under Wi-Fi RAT (ϰw) is slightly better than cellular RAT (ϰc) when the packet arrival rate is low. As ξ increases, ϰw decreases and eventually approaches zero when ξ→P^wt, which is in accordance with Remark 15; please see Figures 10 and 12 for clarification. Moreover, it must also be noted that the decay in ϰ as a function of ξ is faster as compared to θ in Figure 11 and λw in Figure 12, which is in agreement with the results reported in [29]. ξ as a function of AP density is constant for cellular RAT, because, under interference-limited scenario with αc=4, the SIR coverage becomes independent of density of the RAT [4]. On the other hand, under Wi-Fi RAT ϰw first decreases because P^wt decreases as a function of λw. After that it increases slightly as the probability of distance ro>R¯w/4 between a user and its tagged AP approaches zero; hence, the second indicator function in (29) becomes active; as all other factors are constant, thus increase in λw results in an increase in ϰw. Finally, when the distance between a user and its tagged AP is ro≤R¯w/4, the very first indicator function in (29) becomes active, and, hence, ϰw starts declining and finally approaches zeros as P^wt→ξ.Figure 10
Stable queue probability(ϰ) as a function of packet arrival rate (ξ), by assuming a dominant and a modified systems.Figure 11
Stable queue probability(ϰ) as a function of SIR threshold (θ), by assuming a dominant and a modified system.Figure 12
Stable queue probability(ϰ) as a function of AP density, by assuming a dominant and a modified system.


## 5. Numerical Results and Discussions

Under this section, various numerical results for different performance metrics have been discussed. An interference-limited scenarioσi=0 with αn=4 has been assumed for all RAT-tier pairs (m,n). The parameters have been carefully chosen by considering dense HetNet scenario [18, 31] and summarized in Table 2. In general, if not specified, the parameters mentioned in Table 2 have been used for all the results reported in this paper.The association probability, as a function of Wi-Fi RAT density, for multi-RAT HetNets has been plotted in Figure13. Initially, most of the users are associated with cellular RAT and as the density of Wi-Fi RAT increases, the user association (Awj) increases. For a two-RAT scenario, each with single-tier as assumed for Figure 13, the AP density at which the association probability of Wi-Fi RAT becomes equal to the cellular RAT can be obtained by the following relation:(31)λwj=λcjfwjfcjPcjPwj.For the case when Φa=Φc2+Φw2, in Figure 13, the power of Wi-Fi and cellular tier is the same; i.e., n=2. However, in order to get equal association, i.e., Aw2=Ac2, the required λw2>λc2 as fw2>fc2, and this is evident from (31). In Figure 14, the SIR coverage of two different HetNets has been analyzed against Wi-Fi tier density (λw3). When λw3<200, most of the users are associated with cellular RAT, as clear from Figure 13, and the overall SIR coverage (S) of multi-RAT HetNet becomes equal to the single-tier cellular RAT (Sc2), which is function of the chosen thresholds only (θc2). According to the results of Theorem 9 as λw3 increases, the association of users with Wi-Fi RAT increases and, hence, the coverage. On the other hand, according to [14], the SIR coverage keeps on decreasing and at last it meets Sc1; as the same thresholds are used (i.e., θc1=θc2=θw3), Sc1=Sc2=Sw3, each denoting the SIR coverage of standalone cellular tiers (macro, femto) and the Wi-Fi tier (n=3). It is because of the fact that the framework given in [14], for multi-RAT HetNets, does not capture the effect of different channel accessing schemes. Thus, addition of a new RAT is simply another cellular RAT which operates on a different pool of resources; hence, it does not cause interference to existing RATs. The proposed framework, in this work, captures the effect of both the contention-free and the contention-based channel accessing schemes; therefore, it provides generalization and ease of analysis for various network configurations.Figure 13
Association probability as a function of Wi-Fi RAT density.Figure 14
SIR coverage as a function of Wi-Fi RAT density when overlaid with macro- or femto-tier.In Figure15, the SIR coverage has been analyzed against sensing threshold (Γw3) and λw3. By increasing Γw3, the SIR coverage decreases, because of the increase in density of active APs (λw3t). It must also be noted that, after a certain sensing threshold, the SIR coverage becomes almost constant as Pw3t→1, λw3t→λw3. Similarly, in Figure 16 the rate coverage has been analyzed against Γw3 and λw3. Initially, the rate coverage improves by increasing Γw3 because the density of active APs increases, and, hence, the average load per AP decreases. After a certain limit, it becomes constant as λw3t→λw3. Please see Remark 13, for an alternative and detailed description of the results reported in Figure 16. The rate coverage has been analyzed against users density and bandwidth of Wi-Fi RAT, in Figure 17, which shows that the increase in Bw3 greatly affects the rate coverage. Apart from that, as λu increases, the rate coverage decreases because the average load per AP increases.Figure 15
SIR coverage as a function of sensing threshold and AP density.Figure 16
Rate coverage as a function of sensing threshold under various user and AP density.Figure 17
Variation in rate coverage as a function of bandwidth of Wi-Fi tier(Bw3) and AP density.In Figure18, an interesting result has been reported by keeping the density ratio of users and APs constant. Although the average load per AP has been kept fixed, the rate coverage declines as the density increases and the sensing threshold decreases. This is due to the fact that the rate coverage depends on four factors which include both the average load and the MAP of a serving AP. By increasing the AP density and reducing the sensing threshold, under a constant load, the MAP decreases; hence, the overall rate coverage declines. Please see Remark 12 for further details. Under such situations, increasing the number of non-overlapping channels can improve the rate coverage.Figure 18
Rate coverage against constant user to AP density ratio when femto-tier is overlaid with Wi-Fi tier.


## 6. Conclusion

In this paper, we have proposed a unified framework for SINR and rate coverage analysis of multi-RAT HetNets by considering different channel access schemes. By assuming a multi-tier Wi-Fi RAT, we have derived the MAP for a random and a tagged AP, where the results show that the MAP for a typical AP approaches that of a tagged AP as density of Wi-Fi RAT approaches∞. It has been shown that, by increasing the density of Wi-Fi RAT, the SIR coverage of multi-RAT HetNet increases and gradually approaches unity. Moreover, multi-RAT HetNets, specifically with small cell tiers, provide better SIR coverage; however, as the user association with Wi-Fi RAT increases, the rate coverage starts declining. We have also derived the stable queue probability of a user under cellular and Wi-Fi RAT by assuming a non-saturated traffic model. The results show that the stable queue probability of a user under cellular RAT is better as compared to Wi-Fi RAT, when packet arrival rate is high. Although Wi-Fi RAT provides better SIR coverage, it is hard to maintain the stability of a queue, as the medium access probability of an AP is less than unity. This result suggests that the un-bounded increase in the density of Wi-Fi RAT cannot bring significant improvement in users experience; hence, care must be taken while planning the deployment of Wi-Fi RAT.Recently, research on ultra-dense small cell networks has received significant attention. Various tools and techniques like multi-slope path loss models, LOS and non-LOS channels, and different shadowing effects have been used to provide new insights. However, such investigations are limited to single-tier, single-RAT scenario. A straightforward extension of the proposed work is to incorporate such tools for the analysis of multi-RAT HetNets. Another potential area for future work is the incorporation of queuing theory evaluating the impact of traffic variations on the performance bounds of multi-RAT HetNets.



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*Source: 6918637-2019-01-03.xml*