 Research
 Open Access
On muting mobile terminals for uplink interference mitigation in HetNets—systemlevel analysis via stochastic geometry
 Francisco Javier Martín^{3},
 Xiaojun Xi^{2},
 Marco Di Renzo^{2}Email authorView ORCID ID profile,
 Mari Carmen AguayoTorres^{1} and
 Gerardo Gomez^{1}
https://doi.org/10.1186/s136380191400x
© The Author(s) 2019
 Received: 21 December 2018
 Accepted: 13 March 2019
 Published: 25 April 2019
Abstract
We investigate the performance of a scheduling algorithm where the mobile terminals (MTs) may be turned off if they cause a level of interference greater than a given threshold. This approach, which is referred to as interference aware muting (IAM), may be regarded as an interferenceaware scheme that is aimed to reduce the level of interference. We analyze its performance with the aid of stochastic geometry and compare it against other interferenceunaware and interferenceaware schemes, where the level of interference is kept under control in the power control scheme itself rather than in the scheduling process. IAM is studied in terms of average transmit power, mean and variance of the interference, coverage probability, spectral efficiency (SE), and binary rate (BR), which accounts for the amount of resources allocated to the typical MT. Simplified expressions of SE and BR for adaptive modulation and coding schemes are proposed, which better characterize practical communication systems. Our systemlevel analysis unveils that IAM increases the BR and reduces the mean and variance of the interference. It is proved that an operating regime exists, where the performance of IAM is independent of the cell association criterion, which simplifies the joint design of uplink and downlink transmissions.
Keywords
 Cellular networks
 Scheduling
 Stochastic geometry
1 Methods/experimental
The methods used in the paper are based on the mathematical tools of point processes and stochastic geometry. A new analytical framework for performance analysis is introduced. The theoretical frameworks are validated against Monte Carlo simulations.
2 Introduction
Interference awareness can be exploited at both the physical and medium access control (MAC) layers to boost the performance of mobile networks. It is especially useful in the uplink (UL) of heterogeneous cellular networks (HCNs) for interference mitigation and performance enhancement.
In current HCNs, the mobile terminals (MTs) are associated with the same base station (BS) in the UL and downlink (DL) [1]. The cell association is performed based on DL pilot signals and the serving BS is chosen based on a given criterion, e.g., the highest average received power in the DL. In the UL, the same BS is used [1] which leads to a situation where MTs are associated with distant BSs. In this context, the use of fractional power control (FPC) accentuates the detrimental effect of the MTs that cause strong interference to neighboring BSs.
2.1 UL analysis: stateoftheart
The aforementioned complex interactions between power control and association in the UL require accurate mathematical frameworks to gain insights about the performance trends and limitations of existing and future networks. Unfortunately, the mathematical analysis of the UL of HCNs is more involved than the analysis of the DL for two main reasons: (i) due to the use of power control, the transmit power of the MTs depends on the distance to their serving BSs and (ii) even though the locations of BSs and MTs are drawn from two independent Poisson point processes (PPPs), the locations of the interfering MTs scheduled in a given orthogonal resource block (RB) do not follow a PPP. These two peculiarities, as compared to the DL, make the mathematical analysis of the UL intractable without resorting to approximations [2]. In [3], the case of homogeneous cellular networks with FPC is studied. To avoid such a mathematical intractability, it is assumed that the MTs that are scheduled on a given RB form a Voronoi tessellation, and a single BS is available in each Voronoi cell. However, such an approach does not consider HCNs.
The case of the UL of HCNs is accurately modeled in recent works like [2, 4–6], where the spatial correlation between the location of the probe BS and those of the interfering MTs is considered.
In [4], a framework to model HCNs with a truncated channel inversion power control under the smallest path loss association is introduced. In this work, a homogeneous PPP is used as a generative process for the locations of the interfering MTs, but the spatial correlation is accounted for by means of an indicator function that discards interfering MTs’ locations based on their received powers.
The case of UL and DL with decoupled access is considered in [2]. The association is based on maximum weighted received powers and FPC is considered in the UL. To account for the spatial correlations, a nonhomogeneous PPP is considered to model the locations of the interfering MTs.
A framework for the UL of HCNs with multiantenna BSs is studied in [5]. In this work, FPC is considered under a generalized association criteria and two extreme detection techniques in terms of complexity and performance: maximumratio combining (MRC) and optimum combining (OC). It is demonstrated that OC, which can be regarded as an interferenceaware detection technique for multiantenna receivers, greatly outperforms MRC when MTs use aggressive power control, i.e., when the interference is high. The spatial correlation is imposed by means of a conditional thinning that takes into account the generalized cell association procedure.
Interferenceawareness is also studied in [6], which considers HCNs with singleantenna BSs. In this work, a power control mechanism [7] is studied, which is referred to as interference aware fractional power control (IAFPC). This approach consists of introducing a maximum interference level, i_{0}, that the transmission of each MT is allowed to cause to its most interfered BS.
In simple terms, the MTs adjust their transmit power in order to cause a maximum interference level of i_{0} to their most interfered BS.
In the present paper, we investigate another option for interference mitigation in the UL and compare it with previously reported schemes. The approach consists of exploiting interferenceawareness when scheduling the transmission of the MTs, rather than in the power control scheme itself (IAFPC) or in the detection process of the receiver (OC). As a result, interference management is conducted at the MAC layer rather than at the physical layer. The considered approach is referred to as interference aware muting (IAM) and consists of turning off, i.e., muting, the MTs whose interference towards the most interfered BS is above a given threshold.
The main difference between IAFPC and IAM can be summarized as follows. In IAFPC, all the MTs are active and adjust their transmit power for interference mitigation. In IAM, on the other hand, the transmit power of the MTs does not account for any interference constraints but some MTs may not be allowed to transmit if they produce too much interference. As a result, IAM has the potential of reducing the aggregate interference in the UL and of enabling the active MTs to better use the available resources, i.e., the transmission bandwidth.
On the other hand, it reduces the fairness of allocating the resources among the MTs, since some of them may be turned off. Nevertheless, thanks to mobility and shadowing, the muted MTs are only inactive for a given period of time. Hence, from the perspective of MTs, the question to answer is whether this muting increases its achievable binary rate (BR), taking into account both the active and inactive periods. In the present paper, this issue is analyzed as well, and some conclusions are drawn.
2.2 Technical contribution

We study the IAM scheme in terms of average transmit power of the MTs, mean, and variance of the interference. The mathematical analysis reveals that IAM is capable of reducing the three latter performance metrics compared with IAFPC, which results in several advantages for practical implementations.
Reducing the variance of the interference, e.g., is beneficial for better estimating the SINR and, thus, for reducing the error probability of practical decoding schemes, e.g., turbo decoding, [8], and for making easier the selection of the most appropriate modulation and coding scheme (MCS) to use in LTE systems [7].

To make our study and conclusions directly applicable to current communication systems that are based on adaptive modulation and coding (AMC) transmission, we provide tractable expressions of SE and BR based on practical MCSs that are compliant with the LTE standard and whose parameters are obtained from a linklevel simulator [9, 10].

With the aid of the proposed mathematical frameworks, we compare IAFPC and IAM schemes in terms of SE and BR, which provide information on their strengths and weaknesses. The SE provides information on how well the MTs exploit the available resources (e.g., bandwidth) that are shared among the MTs served by the same BS, whereas the BR accounts for the specific fraction of resources that is allocated to each MT served by a given BS. While the IAFPC scheme is superior in terms of SE, the IAM scheme is superior in terms of RB. This implies that IAM provides service to fewer users, which get better performance compared with IAFPC. To characterize this tradeoff, we investigate the fairness of both schemes, which is defined as the probability that a randomly chosen MT gets access to the resources, and provide a tractable frameworks for its analysis.

In light of the emerging ULDL decoupling principle, we develop the mathematical frameworks for a general cell association (GCA) criterion, whose association weights may be appropriately optimized for performance enhancement. By direct inspection of the mathematical frameworks, we prove that three operating regimes can be identified as a function of the interference threshold i_{0}: (i) the first, where the performance is independent of i_{0}, (ii) the second, where the performance depends on i_{0} but it does not depend on the cell association, and iii) the third, where the performance depends on i_{0} and the cell association. Of particular interest in this paper is the second regime, which highlights that ULDL decoupling may not be an issue for some system setups, which in turn simplifies the design of HCNs.

As for the relevant case study for the UL where the serving BS of the typical MT is identified based on the smallest path loss association (SPLA) criterion with channel inversion power control [1], we provide simple and closedform frameworks for relevant performance indicators and prove that two operating regimes exist: (i) the first, where the performance depends on i_{0} (interferenceaware) and (ii) the second, where the performance is independent of i_{0} (interferenceunaware). We prove, in addition, that (i) the scaling law of the average transmit power of the MTs, the mean interference, and the probability that a MT gets access to the resources is a polynomial function of i_{0} whose exponent depends on the path loss exponent, (ii) the distance towards the serving BS gets smaller as i_{0} increases, and (iii) the CCDF of the SINR is independent of the density of BSs.
To the best of authors’ knowledge, all these contributions are new in the literature and are not included in previous works. For instance, the muting mechanism introduces further correlations that do not exist in [6] and need to be taken into account. This muting makes the whole analysis different. New metrics like the BR, which accounts for the amount of resources allocated by the scheduler, are obtained and a new framework to compute the SE and BR with AMC, which is closer to real systems than Shannon formula, is introduced as well. Finally, closedform expressions and remarks are obtained which provide important insights about system performance, fairness, and cell association.
The remainder of this paper is organized as follows. Section 3 introduces the system model and the approach for systemlevel analysis. In Sections 4 and 5, the analysis of IAM is presented for GCA and SPLA criteria, respectively. The BR of AMC schemes is analyzed and discussed in Section 6. In Section 7, we introduce and study a hybrid scheme that allows us to overcome some fairness issues introduced by the IAM scheme. In Section 8, IAM and IAFPC schemes are compared against each other via numerical simulations and the main findings and performance trends derived in the paper are substantiated with the aid of Monte Carlo simulations. Finally, Section 9 concludes this paper.
Summary of main symbols and functions used throughout the paper
Symbol/function  Definition 

_{2}F_{1}(·,·,·,·)  Gauss hypergeometric function 
\(\mathcal {K} = \left \{ {1,2} \right \}\)  Tier set: tier 1 is related to macro BSs, and tier 2 is related to small cell BSs 
\(\tilde j = \left \{ {k \in \mathcal {K}:k \ne j} \right \}\)  Complementary tier, i.e., \(\tilde 1~=~2\) and \(\tilde 2~=~1\) 
Φ^{(j)},λ^{(j)}  PPP and its density related to the locations of macro (j = 1) and small cell BSs (j = 2) 
λ _{MT}  Density of the PPP of MTs’ positions 
Φ,λ  PPP and its density related to the locations of all BSs 
t ^{( j)}  Association weight for tier j 
i_{0},p_{0},ε,p_{max}  Interference threshold, target receive power, partial compensation factor, and maximum transmit power 
τ,α  Path loss slope and path loss exponent 
MT_{0},MT_{i}  Position of the probe MT and position of a generic MT, e.g., an interfering MT 
Ψ ^{( k)}  PPP of interfering MTs’s locations 
\(R_{x,(q)}^{(j)}\)  Distance (including shadowing) between location x and the qth nearest BS from tier j 
\(\phantom {\dot {i}\!}R_{{\text {MT}}_{\mathrm {i}}},U_{{\text {MT}}_{\mathrm {i}}},D_{{\text {MT}}_{\mathrm {i}}}\)  Distances (including shadowing) between MTi and its serving BS, its most interfered BS and the probe BS 
\(\phantom {\dot {i}\!}H_{{\text {MT}}_{\mathrm {i}}}\)  Power gain of the multipath fading which is exponentially distributed 
p_{MT}(r)=p_{0}(τr)^{αε}  Transmit power for a given distance towards the serving BS for active MTs. Muted MTs has 0 transmit power 
\(\sigma _{n}^{2},I\)  Noise power and aggregate interference according to Assumption 1 
\(\mathcal {X}_{{\text {MT}}_{i}}^{(j)}\)  Event defined as MT_{i} is associated with tier j 
\(\mathcal {Q}_{{\text {MT}}_{i}}^{(m)}\)  Event defined as the most interfered BS of MT_{i} belongs to tier m 
\(\mathcal {X}_{{\text {MT}}_{i}}^{(j,m)}\)  Event defined as MT_{i} is associated with tier j and the most interfered BS of MT_{i} belongs to tier m 
\(\mathcal {A}_{{\text {MT}}_{i}}\)  Event defined as MT_{i} is active, i.e., nonmuted 
\(\overline {\mathcal {A}_{{\text {MT}}_{i}}}\)  Event defined as MT_{i} is muted 
\(\mathcal {O}_{{\text {MT}}_{i}}^{(j,k)}\)  Event defined as the interfering MT_{i} of tierk receives higher weighted average power 
from its serving BS than from the probe BS that belong to tier j  
\(\mathcal {Z}_{{\text {MT}}_{i}}\)  Event defined as the interfering MT_{i} causes a level of interference less than i_{0} to the probe BS 
f_{X}(·)  PDF (Probability Density Function) of random variable X 
\(\bar F_{X} (\cdot)\)  CCDF (Complementary Cumulative Distribution Function) of random variable X 
\({\mathcal {L}}_{X} (\cdot)\)  Laplace transform of random variable X 
f^{′}(x_{0}),f^{′′}(x_{0})  First and second derivatives of function f(x) evaluated at x=x_{0} 
\(\mathbb {E}\left [\cdot \right ],\Pr \left (\cdot \right),{\bf {1}}\left (\cdot \right)\)  Expectation operator, probability measure and indicator function 
Γ(z), Γ(a,z)  Euler gamma function and incomplete gamma function 
3 System model
We consider the UL of a HCN composed of two tiers, \(j\in \mathcal {K}=\{1,2\}\), e.g., macroand smallcell BSs, which are spatially distributed according to two independent PPPs, Φ^{(j)}, of intensities λ^{(j)}. Each transmitted signal goes through an independent multipath fading channel with Rayleigh fading and lognormal shadowing. The path loss is modeled by using a path loss slope τ and a path loss exponent α>2^{1}. The cell association among MTs and BSs is based on the weighted average received power criterion, similar to [2], where the association weights are denoted by t^{(j)} for tier \(j\in \mathcal {K}\). Hence, the ith MT is associated with the nth BS of tier j if the MT is in the weighted Voronoi cell of \(\text {BS}^{(j)}_{n}\) with respect to \(\Phi = \bigcup _{j \in \mathcal {K}} {{\Phi ^{(j)}}}\). With these assumptions, shadowing can be modeled as a random displacement [11] of Φ^{(j)} [6, 12].
For ease of writing, we introduce the event \(\mathcal {X}^{(j)}_{\text {MT}_{i}}\) as follows:
Definition 1
The event \(\mathcal {X}^{(j)}_{\text {MT}_{i}}\) is defined as “ MT_{i}is associated with tier j.”
where (τR_{MT})^{−α} is the path loss at a distance ^{2} R_{MT} from the transmitter, \(\tilde {j} = \left \{k \in \mathcal {K}: k \neq j \right \}\) is the complementary tier of j, i.e., \(\tilde {1}~=~2\), and \(\tilde {2}~=~1\), \({R}^{(\tilde {j})}_{x,(q)}\) is the distance from x to the qth nearest BS of tier \(\tilde {j}\), i.e., \({R}^{(\tilde {j})}_{x,(1)}\) is the distance to the nearest BS. The association weights t^{(1)} and t^{(2)} allow us to model the GCA criterion, which encompasses the SPLA criterion for t^{(1)}=t^{(2)}.
Throughout this paper, the analysis is performed for the probe or typical MT, i.e., for a randomly chosen MT, which is denoted by MT_{0}. Its serving BS is referred to as the probe BS.
3.1 Scheduling
We consider fullfrequency reuse, where all the BSs share the same bandwidth. Each BS has available a bandwidth of b_{w} Hertz that is shared among the MTs that are in its Voronoi cell. In practice, b_{w} is divided in orthogonal RBs and each scheduled MT in each cell transmits in one (or several) of these RBs. Thus, no intracell interference is available. This implies that a single MT per BS can interfere with the probe MT. The set of active interfering MTs of tier k that are scheduled for transmission in a given RB is denoted by Ψ^{(k)}. For tractability, we assume that the number of RBs is large enough to be regarded as a continuous resource by the scheduler.
 1
To determine the set of active MTs. The active transmitters are the MTs that, simultaneously, cause less interference than i_{0} to any BSs and that transmit with less power than p_{max}. The MTs that do not fulfill these two constraints are turned off (muted).
 2
Resource allocation. Once the active MTs in each cell are identified, the bandwidth of each BS is equally divided among the active MTs associated with it. Let \(N^{\mathcal {A}}_{\text {BS}^{(j)}_{n}}\) be the number of MTs associated with BS \(\text {BS}^{(j)}_{n}\). Each of them is allocated a bandwidth \(b_{w}/N^{\mathcal {A}}_{\text {BS}^{(j)}_{n}}\)^{3} Hz.
This scheduling process characterizes the IAM scheme and makes it different from the IAFPC scheme in [6]. In [6], all the MTs are active and power control is responsible for controlling the level of interference, by making sure that the interference level at any BS is less than i_{0}.
To better understand the implications of interference awareness on turning off (muting) some MTs, we analyze the case study i_{0}→∞ as well, which is referred to as interferenceunaware muting (IUM)^{4}.
For ease of writing, we introduce some definitions that are useful for mathematical analysis.
Definition 2
The event \(\mathcal {Q}^{(m)}_{\text {MT}_{i}}\) is defined as “the most interfered BS of MT _{i}belongs to tier m.”
Definition 3
The event \(\mathcal {X}^{(j,m)}_{\text {MT}_{i}} = \mathcal {X}^{(j)}_{\text {MT}_{i}} \cap \mathcal {Q}^{(m)}_{\text {MT}_{i}}\) is defined as “MT _{i}is associated with tier j and the most interfered BS of MT_{i} belongs to tier m.”
According to IAM, the MTs that either cause higher interference than i_{0} or transmit with higher power than p_{max} are kept silent. The set of active MTs is defined as follows:
Definition 4
The event \(\mathcal {A}_{\text {MT}_{i}}\) is defined as “MT _{i}is active.”
where p_{MT}(r), p_{0}, and ε are related to power control, and they are described in Table 1, \(\phantom {\dot {i}\!}R_{\text {MT}_{i}}\) is the distance between MT_{i} and its serving BS, and \(\phantom {\dot {i}\!}U_{\text {MT}_{i}}\) is the distance between MT_{i} and its most interfered BS. If the probe MT is associated with tier j, i.e., the event \(\mathcal {X}^{(j)}_{\text {MT}_{i}}\) is true, then \(R_{\text {MT}_{i}} = R^{(j)}_{\text {MT}_{i},(1)}\). The distance \(\phantom {\dot {i}\!}U_{\text {MT}_{i}}\) depends, on the other hand, on the event \(\mathcal {X}^{(j,m)}_{\text {MT}_{i}}\). Accordingly, \(U_{\text {MT}_{i}} = R^{(m)}_{\text {MT}_{i},(1)}\) if j≠m and \(U_{\text {MT}_{i}} = R^{(j)}_{\text {MT}_{i},(2)}\) if j=m. The aim of event \(\mathcal {A}_{\mathrm {MT}_{i}}\) is to capture the spatial correlation between the position of a given MT, its serving BS, and its most interfered BS, which follows from the muting process.
As far as IAM is concerned, fractional power control is applied at the physical layer and is interferenceunaware, i.e., the transmit power of the MTs that are not turned off depends only on path loss and shadowing and it can be expressed as \(p_{\text {MT}} \left (R_{\text {MT}_{0}} \right)\). If the MTs are muted, on the other hand, their transmit power is equal to zero. This implies that their associated SINR, BR, etc. are, by definition, equal to zero as well.
3.2 SINR
where \(\phantom {\dot {i}\!}H_{\text {MT}_{0}}\) is the channel gain, \(\phantom {\dot {i}\!}R_{\text {MT}_{0}}\) is the distance from the serving BS, \(p_{\text {MT}} \left (R_{\text {MT}_{0}} \right)\) is the transmit power, I is the othercell interference, and \(\sigma _{n}^{2}\) is the noise power.
In the UL, as discussed in Section 2, the set of interfering MTs does not constitute a PPP, even though the MTs and BSs are distributed according to a PPP. Further details can be found in [5] and [6]. This makes the mathematical analysis intractable. In the present paper, the distinctive scheduling process of IAM negatively affects the mathematical tractability of the problem at hand even further. To make the analysis tractable, some approximations for modeling the set of active MTs are needed. In [5] and [6], it is shown that a tractable approximation consists of assuming that the set of active MTs can still be modeled as a PPP, provided that appropriate spatial constraints on the locations of the MTs are introduced. Stated differently, the set of active MTs is modeled as a spatiallythinned PPP or equivalently as a nonhomogeneous PPP.
Before introducing the approach to model interfering MTs’ locations, the following events need to be defined:
Definition 5
The event \(\mathcal {O}^{(j,k)}_{\text {MT}_{i}}\) is defined as “the interfering MT _{i}of tier k receives higher weighted average power from its serving BS than from the probe BS that belongs to tier j.”
Definition 6
The event \(\mathcal {Z}_{\text {MT}_{i}}\) is defined as “the interfering MT _{i}causes a level of interference less than i_{0}to the probe BS.”
Hence, inspired by [5] and [6], our mathematical framework is based on the following approximation.
Assumption 1
where Ψ^{(k)} is a PPP of intensity λ^{(k)} whose points constitute the locations of the interfering MTs that are scheduled for transmission in the same RB as that of the typical MT, the events \(\phantom {\dot {i}\!}\mathcal {O}^{(j,k)}_{\text {MT}_{i}}\) and \(\phantom {\dot {i}\!}\mathcal {Z}_{\text {MT}_{i}}\) take into account the necessary spatial constraints imposed by the cell association criterion and the maximum interference and power constraints, respectively, \(\phantom {\dot {i}\!}R_{\text {MT}_{i}}\) and \(\phantom {\dot {i}\!}D_{\text {MT}_{i}}\) are the distances from MT_{i} to its own serving BS and to the probe BS, respectively.
More specifically, (i) the event \(\mathcal {O}^{(j,k)}_{\text {MT}_{i}}\) is necessary to account for the spatial correlation that exists between the locations of the probe BS, the interfering MTs and their serving BSs, since the interfering MTs must lie outside the Voronoi cell of the probe BS by definition of cell association, and (ii) the event \(\mathcal {Z}_{\text {MT}_{i}}\) is necessary to account for the fact that the interfering MTs need to cause less interference than i_{0} according to the IAM scheduling process.
The next two sections provide mathematical expressions of the CCDF of the SINR and of the mean and variance of the othercell interference for GCA and SPLA cell association criteria respectively.
4 General cell association criterion
We start introducing some enabling results for proving the main theorems of this section.
Proposition 1
Proposition 1 is useful for understanding and quantifying the fairness of the IAM scheme. The higher \(\Pr \left ({\mathcal {A}_{{\text {MT}}_{0}}} \right)\) is, in fact, the higher the probability that a randomly chosen MT is served in a given RB and, thus, the higher the fairness that it gets access to the available resources is^{5}.
Proof
See Appendix A. □
Lemma 1
where ν^{(j)}(v) and η^{(j)}(v) are defined in (9) and (10), respectively.
Proof
The Cumulative Distribution Function (CDF) of the distance between the typical MT and its serving BS by conditioning on the MT being active and on \(\mathcal {X}_{\text {MT}_{\text {0}}}^{(j,m)}\) is obtained by using steps similar to Appendix A. The PDF is obtained from the CDF by computing the derivative. □
In the UL, an important performance metric to study is the average transmit power of the typical MT, which is related to its power consumption. Since some MTs may be turned off in the IAM scheme, this implies that some MTs may transmit zero power, which results in reducing their power consumption. The following proposition provides the average transmit power of the typical MT, by taking into account that the typical MT may be a MT that is turned off as it does not fulfill either the maximum power constraint or the maximum interference constraint.
Proposition 2
where \(f_{R_{\text {MT}_{0}}}\left (v \mathcal {X}_{\text {MT}_{0}}^{(j,m)},\mathcal {A}_{\text {MT}_{0}}\right)\) is in (11) and \({\Pr \left (\mathcal {X}_{\text {MT}_{0}}^{(j,m)},\mathcal {A}_{\text {MT}_{0}} \right)}\) is defined in Appendix A.
Proof
It follows by computing the average transmit power by conditioning on the events \(\mathcal {A}_{\text {MT}_{0}}\) and \(\mathcal {X}_{\text {MT}_{\text {0}}}^{(j,m)}\). The final result is obtained from the total probability theorem. □
Remark 1
(Exact analysis) The previous propositions and lemmas are exact, since they do not depend on the set of active interfering MTs but depend only on the locations of the BSs, which constitute a PPP, and on the typical MT. In other words, Assumption 1 is not applied.
The next lemma provides the Laplace transform of the othercell interference based on its mathematical formulation in (7), which exploits Assumption 1.
Lemma 2
Proof
See Appendix B. □
From the Laplace transform in (47), the moments of the interference can be obtained as shown in the next proposition. Of particular interest is the variance of the interference, since its affects the performance of AMC schemes [7]: the smaller the variance is, the more robust and accurate the estimation of the SINR is, which makes easier the choice of the best MCS to use.
Proposition 3
Proof
It directly follows from the first and second derivative of 47 evaluated at s=0. □
Remark 2
(Impact of i_{0}) By inspection of Propositions 1 and 3, we evince that the average transmit power, the mean, and variance of the interference decrease by decreasing i_{0}. Since the interferenceunaware setup is obtained by setting i_{0}→∞, this implies that IAM is beneficial in terms of reducing the power consumption of the MTs and of implementing AMC schemes. The system fairness may, however, be negatively affected if i_{0} decreases, as more MTs are muted.
The next theorem provides a tractable expression of the coverage probability of HCNs.
Theorem 1
Proof
The proof follows by computing the two remaining expectations. □
Corollary 1
Proof
It follows from (21) by setting ε=1 and some algebra. □
Remark 3
(Operating regimes as a function of i_{0}) By direct inspection of Corollary 1, three operating regimes as a function of i_{0} can be identified: (i) interferenceunaware, where the CCDF of the SINR is independent of i_{0}. This occurs if i_{0}>p_{0} and p_{0}/i_{0}< min(t^{1}/t^{(2)},t^{(2)}/t^{(1)}), (ii) interferenceaware and cell association independent, where the CCDF of the SINR depends on i_{0} but does not depend on the cell association weights t^{(1)} and t^{(2)}. This occurs if i_{0}<p_{0} and p_{0}/i_{0}> max(t^{(1)}/t^{(2)},t^{(2)}/t^{(1)}), (iii) interferenceaware and cell association dependent, where the CCDF of the SINR depends on i_{0} and \(t^{(\tilde j)}/t^{(j)}, \, \forall j\in \mathcal {K}\). This occurs if the conditions above are not satisfied. The same operating regimes can be identified from Propositions 1 and 2.
Proof
It follows by inspection of \(\Pr \left ({{\mathcal {X}}_{{\text {MT}}_{\text {0}} }^{(j,m)},{\mathcal {A}}_{{\text {MT}}_{\text {0}} } } \right)\), ν^{(j)}(v) and η^{(j)}(v). □
The second operating regime, i.e., the performance is independent of the cell association weights, is of particular interest for making the design of HCNs easier: it implies that, for some system parameters, optimizing the DL results in optimizing the UL as well.
It is worth mentioning, in addition, that the conditions that identify the three operating regimes in Remark 3 can be conveniently formulated in decibels as well, which provides further information for system design. More precisely, regime (i) emerges if i_{0}>p_{0} dB and t^{(1)}/t^{(2)}∈[−i_{0}/p_{0},i_{0}/p_{0}] dB and regime (ii) emerges if i_{0}<p_{0} dB and t^{(1)}/t^{(2)}∈[−p_{0}/i_{0},p_{0}/i_{0}] dB.
5 Smallest path loss association
In this section, tractable mathematical frameworks under the SPLA scheme are provided. In this case, the condition t^{(1)}=t^{(2)} holds and simplified formulas can be obtained. Under the assumption that the path loss exponents of all the tiers of BSs are the same, in fact, multitier HCNs reduce to an equivalent singletier cellular network of intensity \(\lambda = {\sum }_{j \in \mathcal {K}} {{\lambda ^{(j)}}}\) [2].
Proposition 4
Proof
The proof is similar to that of Proposition 1. The difference is that only the joint PDF of the distance of the nearest and second nearest BSs needs to be used (see Appendix A). □
Corollary 2
Proof
It directly follows from (24) by setting ε=1 and computing the integral. □
Remark 4
(Operating regimes as a function of i_{0}) From (25), two operating regimes can be identified: (i) interferenceunaware, i.e., \(\Pr \left ({{\mathcal {A}_{{\mathrm {M}}{{\mathrm {T}}_{0}}}}} \right)\) is independent of i_{0}, which occurs if i_{0}>p_{0} and (ii) interferenceaware, i.e., \(\Pr \left ({{\mathcal {A}_{{\mathrm {M}}{{\mathrm {T}}_{0}}}}} \right)\) depends on i_{0}, which occurs if i_{0}<p_{0}.
Remark 5
(Unlimited transmit power of the MTs) Assume p_{max}→∞, i.e., the MTs have no maximum transmit power constraint. From (25), the following holds: (i) under the interferenceunaware regime (i_{0}>p_{0}), \(\Pr \left ({{\mathcal { A}_{{\mathrm {M}}{{\mathrm {T}}_{0}}}}} \right) \to 1\), and (ii) under the interferenceaware regime (i_{0}<p_{0}), \(\Pr \left ({{\mathcal {A}_{{\mathrm {M}}{{\mathrm {T}}_{0}}}}} \right) = {\left ({{{{i_{0}}} / {{p_{0}}}}} \right)^{{2 \over \alpha }}}\). In both regimes, \(\Pr \left ({{\mathcal {A}_{{\mathrm {M}}{{\mathrm {T}}_{0}}}}} \right)\) is independent of the density of BSs λ.
Lemma 3
Proof
The proof is similar to that of Lemma 1. The difference is that only the joint PDF of the distance of the nearest and second nearest BSs needs to be used (see Appendix A). □
Remark 6
This implies that IAM’s impact is equivalent to increasing the density of BSs from λ to \( \lambda {\left ({{p_{0}}/{i_{0}}} \right)^{{2 \over \alpha }}}\), since the PDF of the distance from the nearest BS in Poisson cellular networks is \(2\pi \lambda v{{\mathrm {e}}^{ \pi \lambda {v^{2}}}}\). Hence, the distance between probe MT and probe BS is reduced, resulting in better performance.
Proposition 5
Proof
If follows from Proposition 2, by setting ε=1 and computing the integral. □
Remark 7
which implies that the average power consumption of the MTs scales polynomially with exponent 2/α+1, as a function of the maximum interference constraint i_{0}.
Lemma 4
Proof
The proof follows from χ(s,r) in (15), by setting t^{(1)}=t^{(2)} and formulating it as χ(s,r)=r^{2}μ(s). Hence, β(s)=−2πλμ(s)θ, where \(\theta = \mathbb {E}\left [ {R_{{\text {M}}{{\text {T}}_{\text {i}}}}^2{A_{{\text {M}}{{\text {T}}_{\text {0}}}}}} \right ]\). □
Proposition 6
Proof
It follows from Lemma 4 evaluating the derivatives of the Laplace transform at zero. □
Remark 8
which implies that the mean and variance of the interference scale polynomially with exponents α+2/α and 2(α+1)/α as a function of i_{0}, respectively, and they do not depend on the BSs’ density.
Finally, the following theorem provides the coverage probability under the SPLA criterion.
Theorem 2
Proof
The proof follows from Theorem 1 by setting t^{(1)}=t^{(2)} and ε=1, and from Lemma 4 by letting p_{max}→∞ and considering i_{0}<p_{0}. □
Remark 9
(SINR invariance as a function of λ) From (34), we evince that the CCDF of the SINR is independent of λ, but it depends on the ratio i_{0}/p_{0} and the path loss exponent α.
Interestingly, the SINR in such a setup is invariant with the BSs’ density. Intuitively, this means that both the desired received power and the interference does not vary with the BSs’ density. On the one hand, the desired power does not vary thanks to full channel inversion power control (ε = 1, p_{max}→∞). Furthermore, although the distances towards the nearest interfering MTs decrease with λ, their transmit power also decrease with λ, making the received interference invariant with λ, as it can be observed from its moments in (33). This density invariance has been also reported in [2, 13] for the case of the SIR.
6 Spectral efficiency and binary rate
This section is focused on the analysis of SE and BR. Unlike the vast majority of papers on stochastic geometry modeling of HCNs that evaluate these key performance indicators based on the Shannon formula, we provide a mathematical formulation that is more useful for current cellular deployments based on practical AMC schemes and, thus, provides estimates of SE and BR that can be achieved at a finite target value of the block error rate (BLER) rather than their theoretically achievable counterparts under the assumptions of unlimited decoding complexity and arbitrarily small BLER. We show, remarkably, that more tractable expressions of SE and BR can be provided, compared to those that can be obtained based on the Shannon definition. The BR accounts for the amount of bandwidth allocated to the typical MT by the scheduler and, thus, accounts for the BS’s load, i.e., the number of MTs that need to be simultaneously served in the cell to which the typical MT belongs to. Accordingly, SE and BR provide information on the advantages and limitations of transmission schemes and, as such, are both employed for assessing the performance of practical LTE systems [14].
where b_{w} is the available bandwidth per BS and \(N^{\mathcal {A}}_{\mathcal {B}_{\text {MT}_{0}}}\) denotes the number of active MTs associated with the probe BS, which is commonly referred to as the cell load [15].
As extensively discussed in, e.g., [15–17], the distribution of \(N^{\mathcal {A}}_{\mathcal {B}_{\text {MT}_{0}}}\) is not available for cell association criteria that are not based on the shortest distance, and, thus approximations need to be used. For mathematical tractability, but without loosing in accuracy, we exploit the approximation in [16] which, for the convenience of the readers, is reported in what follows.
Assumption 2
where, for notational simplicity, the shorthand \(p=\Pr \left (\mathcal {X}^{(j)}_{\text {MT}_{0}},\mathcal {A}_{\text {MT}_{0}}\right)\) is used.
6.1 Adaptive modulation and coding
In modern cellular systems [14], AMC is aimed to adapt the MCS to be used to the channel conditions. This is needed for maximizing the BR while providing a BLER below a desired threshold BLER _{T}. In practice, AMC is implemented as follows. In the UL, the MTs transmit sounding reference signals that are used by the BSs for estimating the SINR. Based on these estimates, the BSs choose the MCS to use (usually identified by an index), which corresponds to a given Channel Quality Indicator (CQI), i_{CQI}∈[1,n_{CQI}], that maximizes the SE while maintaining the BLER below BLER _{T}. The choice of the best MCS to use is made based on lookup tables that provide the SINR thresholds, \(\gamma _{i_{\text {CQI}}}\), associated to each value of CQI. Finally, the BSs inform each scheduled MT of the MCS index to use for its subsequent transmission. To reduce the reporting overhead associated with the CQIs, the LTE standard assumes that the number of bits used for reporting the CQI is equal to 4, which implies n_{CQI} = 15.
where \(\phantom {\dot {i}\!}\gamma _{1}<\cdots <\gamma _{n_{\text {CQI}}}\), i_{CQI}=0 if no transmission, \(\bigcap _{i_{\text {CQI}}=1}^{n_{\text {CQI}}}{[\gamma _{i_{\text {CQI}}},\gamma _{i_{\text {CQI}+1}})=\emptyset }\), \(\gamma _{n_{\text {CQI}+1}} \rightarrow \infty \).
where (a) is obtained by computing the summation over \(n~=~N_{\mathcal {B}_{\text {MT}_{0}}}^{\mathcal {A}}\) in closedform with the aid of the PMF in (36).
The mathematical expressions of SE and BR of AMC schemes are easier to compute than the corresponding formulas obtained from the Shannon definition of SE, since the latter definition requires an extra integral to be computed [15]. This is remarkable, since the SE and BR in (38) and (39) account for feedback’s overhead and limitedcomplexity receivers.
i _{CQI}  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 

\(\text {SE}_{i_{\text {CQI}}}\) [bps/Hz]  0.15  0.23  0.38  0.60  0.88  1.18  1.48  1.91  2.41  2.73  3.32  3.90  4.52  5.11  5.55 
\(\gamma _{i_{\text {CQI}}}\) [dB]  −3.65  −1.60  0.00  2.25  3.75  4.75  9.00  10.50  12.35  15.40  17.18  18.85  20.70  24.0  25.0 
7 On fairness—a hybrid muting scheme
In this section, we introduce a new hybrid interferenceaware muting scheme in order to overcome some of the limitations of the IAM scheme, especially in terms of fairness among the MTs. Due to the specific operations of the IAM scheme, some MTs may be muted for long time, which is unfair for them compared to the MTs that are scheduled for transmission. To overcome this limitation, we propose a transmission scheme whose operation is split into two time slots. In time slot 1, the IAM is applied as described in the previous sections. This implies that some MTs are muted and do not transmit. In time slot 2, the MTs that are muted during the first time slot are the only ones allowed to transmit signals. The rationale for this scheme is that the first time slot is optimized for transmission based on the IAM scheme, while the second time slot allow us to avoid unfairness among the MTs. The duration of each time slot is an optimization variable, which allows us to identify the best transmission scheme to use. If the duration of the first time slot is 100% of the entire time allocated for transmission, then the hybrid scheme boils down to the IAM scheme. If, on the other hand, the duration of the second time slot is 100% of the entire time allocated for transmission, then the hybrid scheme boils down to the conventional scheme where all the MTs are allowed to transmit, which avoids fairness issues. The optimization of the duration of each time slot, given the total time allocated for transmission, is an important parameter to tradeoff performance and fairness. In this section, we analyze the performance of this hybrid scheme.
where t_{1} denotes the duration of the first time slot, and the entire time allocated for transmission is normalized to 1 for simplicity. This implies that the duration of the second time slot is 1−t_{1}. The symbols with the “overline” denote the performance metrics during the second time slots, where the MTs that are muted in the first time slot are the only ones allowed to transmit.
Since the network operation during the first time slot coincides with the IAM scheme that is studied in the previous sections, in this section, we analyze only the performance during the second time slot. In this case, we need to replace \(\mathcal {A}_{\text {MT}_{i}}\) with \(\overline {\mathcal {A}_{\text {MT}_{i}}}\), based on the notation in Table 1.
7.1 General cell association criterion
We start introducing some enabling results to prove the main theorems.
Proposition 7
Proof
See Appendix C. □
Lemma 5
where ν^{(j)}(v), η^{(j)}(v), ω^{(j)}(v), and ζ^{(j)}(v) are defined in (9), (10), (45), and (46) respectively.
Proof
The Cumulative Distribution Function (CDF) of the distance between the typical MT and its serving BS by conditioning on the MT being muted and on \(\mathcal {X}_{\text {MT}_{\text {0}}}^{(j,m)}\) is obtained by using steps similar to Appendix A. The PDF is obtained from the CDF by computing the derivative. □
Lemma 6
Proof
See Appendix D. □
Theorem 3
7.2 Smallest path loss association
In this section, we study the smallest path loss cell association criterion. In this case, the condition t^{(1)}=t^{(2)} holds and simplified formulas can be obtained. Under the assumption that the path loss exponents of all the tiers of BSs are the same, in fact, multitier HCNs reduce to an equivalent singletier cellular network of intensity \(\lambda = \sum \limits _{j \in \mathcal {K}} {{\lambda ^{(j)}}}\) [2].
Proposition 8
Proof
The proof is similar to that of Proposition 7. □
Corollary 3
Proof
It directly follows from 52 by setting ε=1 and computing the integral. □
Lemma 7
Proof
The proof is similar to that of Lemma 5. The difference is that only the joint PDF of the distance of nearest and second nearest BSs needs to be used (see Appendix C). □
Lemma 8
Proof
The proof follows from χ_{all}(s,r) in (49), by setting t^{(1)}=t^{(2)} and computing the integral. □
Theorem 4
Proof
The proof follows by computing the two remaining expectations. □
8 Numerical results and discussion
Simulation setup
Parameter  Value  Parameter  Value 

f_{c} (MHz)  2×10^{3}  h_{BS} (m)  10 
b_{w} (MHz)  9  t^{(1)}/t^{(2)} (dB)  {9,0} 
λ^{(1)} (points/m^{2})  2×10^{−6}  λ^{(2)} (points/m^{2})  4×10^{−6} 
λ_{MT} (points/m^{2})  80×10^{−6}  n_{thermal} (dBm/Hz)  − 174 
n_{F} (dB)  9  σ_{s} (dB)  4 
p_{0} (dBm)  − 70  p_{max} (dBm)  {∞,5} 
i_{0} (dBm)  [−120,− 60]  ε  [0,1] 
8.1 Average transmit power, probability of being active, mean, and variance of the interference
In this section, we analyze the average transmit power of the MTs, the probability that the typical MT is active, which provides information on the system fairness, and the mean and variance of the interference.
In the figures, IAM and IAFPC are compared as well. We observe that IAM reduces the average transmit power and the mean and variance of the interference.
Consider the SPLA criterion, which is illustrated with dashed lines in the figures. We observe that the findings in Remark 4 are confirmed: the system is interferenceaware and interferenceunaware if i_{0}<p_{0} and i_{0}>p_{0}, respectively. As expected, the crossing point occurs at p_{0} = − 70 dBm based on the simulation parameters used. In addition, the scaling laws of average transmit power and average interference are in agreement with the findings in Remark 7 and Remark 8.
All in all, the numerical illustrations reported in Figs. 1, 2, 3, and 4 confirm all the conclusions and performance trends discussed in the previous sections and highlight the advantages of IAM.
8.2 Complementary cumulative distribution function of the SINR
In both figures, we observe a good agreement between mathematical frameworks and Monte Carlo simulations. In particular, the figures confirm, once again, that the coverage probability of IAM increases as i_{0} decreases. In Fig. 5, for example, almost all the active MTs have a SINR greater than 20 dB if i_{0}=− 120 dBm. This good SINR is obtained because IAM keeps under control the interference by muting the MTs that create more interference. Based on Fig. 2, in fact, we note that only a small fraction of the MTs are allowed to be active for i_{0}=− 120 dBm. The active MTs, however, better exploit the available bandwidth. Similar conclusions can be drawn for ε=0.75 shown in Fig. 6. The main difference is that, in this latter figure, IAM provides almost the same coverage probability for i_{0} = − 60 dBm and i_{0} = − 90 dBm. The reason is that the MTs transmit with less power if ε = 0.75 and, thus, there is almost no difference between the two interference constraints. This brings to our attention that the design of the UL of HCNs requires to jointly optimize i_{0}, p_{0}, p_{max}, and ε, in order to identify the desired operating regime that fulfills the requirements in terms of system fairness and interference mitigation. The proposed mathematical frameworks can be used to this end.
8.3 Spectral efficiency and binary rate
In this section, the average SE and average BR are analyzed, as well as the IAFPC and IAM schemes are compared against each other for several system setups.
The SE, however, does not provide information on the amount of bandwidth that the scheduler allocates to each active MT.
8.4 Impact of the association weights: on ULDL decoupling
As shown in [2] and [5], optimizing the performance of HCNs for DL transmission does not necessarily results in optimizing their performance in the UL. Based on the GCA criterion, this implies that different cell association weights (i.e., a different ratio t^{(1)}/t^{(2)} for twotier HCNs) may be needed in the DL and in the UL. However, this approach, which is referred to as ULDL decoupling, introduces additional implementation challenges, which require the modification of the existing network architecture and control plane.
In Fig. 12, in particular, we compare the average BR of IUFPC and IAM schemes. The figure highlights important differences between these two interference management schemes for improving the performance of the UL of HCNs. First of all, we note that the average BR of the IUFPC scheme decreases as the ratio t^{(1)}/t^{(2)} increases. More specifically, the best average BR is obtained if the SPLA criterion is used, which is in agreement with previously published papers [16]. This originates from the fact that the larger t^{(1)}/t^{(2)} is, the more MTs are associated with more distance BSs, which, due to the use of power control, results in increasing the interference in the UL. The performance trend is, on the other hand, different if the IAM scheme is used. In this case, there are several values of i_{0} that provide a better average BR compared with IUFPC. In addition, the average BR increases as t^{(1)}/t^{(2)} increases, since the excess interference that is generated under the IUFPC scheme is now kept under control by imposing the maximum interference constraint i_{0}. As observed in previous figures, Fig. 11 confirms that this gain is obtained since more MTs are turned off.
Figures 11 and 12 confirm the findings in Remark 3 and, in particular, the existence of an operating regime where the performance of IAM is independent of the association weights. Let us consider, for example, the setup for i_{0} = − 60 dBm. In this case, i_{0}>p_{0} and hence, according to Remark 3, the system is interferenceunaware if t^{(1)}/t^{(2)}∈[− 10,+ 10] dB. Figure 12, more specifically, confirms that IAM is interferenceunaware since it provides the same average BR as IUFPC for t^{(1)}/t^{(2)}∈[− 10,+ 10] dB^{6}. Similar conclusions can be drawn for other values of i_{0}, where different operating regimes can be identified as predicted in Remark 3. If i_{0} = − 90 dBm, in particular, then i_{0}<p_{0} and the system is independent of the cell association criterion for t^{(1)}/t^{(2)}∈[− 20,+ 20], which is confirmed in Figs. 11 and 12. It is worth mentioning that the values of t^{(1)}/t^{(2)} for which the considered system model is cell association independent are usually adopted in practical engineering applications. In particular, the authors of [2, 19] have shown that the optimal cell association ratio that optimizes the DL is usually less than 20 dB. This is in agreement and compatible with the findings in Figs. 11 and 12.
 1
Taking into account the periods where the typical MT is active and those where it is muted, the average BR is increased with IAM compared to IAFPC and IUFPC.
 2
Thanks to mobility and shadowing, MTs are only muted for a given period of time.
 3
Since the muted MTs do not transmit, their average transmitted power is reduced compared to IAFPC and IUFPC. This has been studied with Fig. 1.
 4
With IAM, there is a regime where the UL performance is independent of cell association, which eases the joint design of UL and DL transmissions as it have been discussed above.
 5
The IAM scheme can be further enhanced as discussed in Section 7. Some numerical illustrations are provided in the next section.
8.5 Hybrid scheme—complementary cumulative distribution function of the SINR
In this section, we analyze the coverage probability (CCDF of the SINR) of the hybrid scheme introduced in Section 7.
In these figures, we observe a good agreement between analytical frameworks and Monte Carlo simulations.
In Fig. 13, it can be observed that the coverage probability of the hybrid scheme increases as t_{1} increases. In Fig. 14, it can be observed that the coverage probability of the hybrid scheme decreases first and then increases as t_{1} increases. In Fig. 15, it can be observed that coverage probability of the hybrid scheme decreases as t_{1} decreases. As for high i_{0}, these trends are obtained because the coverage probability in time slot 1 is better than in time slot 2 for high i_{0} since more MTs are active. Then, increasing t_{1} improves the coverage probability. As for low i_{0}, the trend is opposite for similar reasons. The comparison of these figures also shows that the coverage probability of the hybrid scheme decreases as i_{0} increases for small value of t_{1} because time slot 2 dominates the performance. On the contrary, the coverage probability of the hybrid scheme increases as i_{0} increases for large value of t_{1}.
8.6 Hybrid scheme—spectral efficiency and binary rate
In this section, the average SE and average BR are analyzed.
This result is reasonable because the lower i_{0} is the fewer MTs transmit in time slot 1 and more MTs transmit in time slot 2, which on average, contributes to reduce the SE if t_{1} = 1 and to increase the SE if t_{1} = 0. As for the hybrid scheme, the SE is somehow in between.
9 Conclusion
In this paper, we have studied the performance of IAM: an interference management scheme for enhancing the throughput of HCNs. With the aid of stochastic geometry, we have developed a general mathematical approach for analyzing and optimizing its performance as a function of several system parameters. Simplified and insightful expressions of the throughput and other relevant performance indicators have been proposed for simplified but relevant case studies, such as in the presence of channel inversion power control and equal cell association weights. Among the many performance trends that have been identified, we have proved that, while optimizing the DL and the UL of HCNs necessitates, in general, to use different cell association weights, there exist some operating regimes where IAM is cell association independent. This is shown to simplify the design of HCNs, since no changes in their control plane is needed compared with conventional cellular networks. The mathematical frameworks and findings have been substantiated against Monte Carlo simulations, as well as the achievable performance of IAM has been compared against other IAFPC and IUFPC schemes, by highlighting several important tradeoffs in terms of system fairness and system throughput. Finally, we have introduced a hybrid scheme that overcomes some fairness issues associated with the operating principle of the IAM scheme, and have highlighted important performance tradeoffs.
10 Appendix
11 A. Proof of Proposition 1
where (a) is obtained by definition of expectation formulated with the aid of indicator functions.
The computation of the twofold integral leads to the function ν^{(j)}(v) that is provided in (9).
The case j=m can be solved by using an approach similar to the previous case. The final result corresponds to the function η^{(j)}(v) available in (10).
The proof follows by computing the summation over \(m\in \mathcal {K}\) in (58).
12 B. Proof of Lemma 2
The proof follows by computing the inner integral.
13 C. Proof of Proposition 7
The computation of the twofold integral leads to the function ω^{(j)}(v) that is provided in (44).
The case j=m can be solved by using an approach similar to the previous case. The final result corresponds to the function ζ^{(j)}(v) available in (45).
The proof follows by computing the summation over \(m\in \mathcal {K}\) in (66).
14 D. Proof of Lemma 6
The proof follows by computing the inner integral.
The proposed framework can be generalized to account for a bounded path loss model; however, an unbounded path loss model has been used for the sake of mathematical tractability.
Although, in practice, the bandwidth is divided in RBs, we assume that it can be treated as a continuous resource and hence that it can be equally divided among the active MTs. This is assumed in [2] as well.
In the present paper, IUM and interference unaware FPC (IUFPC) schemes are similar but slightly different. IUM is referred to a setup where i_{0}→∞ and p_{max}<∞. IUFPC is referred to a setup where i_{0}→∞ and p_{max}→∞. As for IUM, only the constraint on the maximum transmit power exists. As for IUFPC, there is no constraint on either the maximum transmit power or the maximum interference.
Declarations
Acknowledgements
Not applicable.
Funding
This work was supported in part by the European Commission through the H2020MSCA ETN5Gaura project under Grant Agreement 675806.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. The paper is built upon mathematical analysis.
Authors’ contributions
The authors declare that they have equally contributed to the paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 H. Elshaer, F. Boccardi, M. Dohler, R. Irmer, in 2014 IEEE Global Communications Conference. Downlink and uplink decoupling: a disruptive architectural design for 5G networks, (2014), pp. 1798–1803. https://doi.org/10.1109/GLOCOM.2014.7037069.
 S. Singh, X. Zhang, J. G. Andrews, Joint rate and SINR coverage analysis for decoupled uplinkdownlink biased cell associations in HetNets. IEEE Trans. Wirel. Commun.14(10), 5360–5373 (2015). https://doi.org/10.1109/TWC.2015.2437378.View ArticleGoogle Scholar
 T. D. Novlan, H. S. Dhillon, J. G. Andrews, Analytical modeling of uplink cellular networks. IEEE Trans. Wirel. Commun.12(6), 2669–2679 (2013). https://doi.org/10.1109/TWC.2013.050613.120325.View ArticleGoogle Scholar
 H. ElSawy, E. Hossain, On stochastic geometry modeling of cellular uplink transmission with truncated channel inversion power control. IEEE Trans. Wirel. Commun.13(8), 4454–4469 (2014). https://doi.org/10.1109/TWC.2014.2316519.View ArticleGoogle Scholar
 M. D. Renzo, P. Guan, Stochastic geometry modeling and systemlevel analysis of uplink heterogeneous cellular networks with multiantenna base stations. IEEE Trans. Commun.64(6), 2453–2476 (2016). https://doi.org/10.1109/TCOMM.2016.2552163.View ArticleGoogle Scholar
 F. J. MartinVega, et al., Analytical modeling of interference aware power control for the uplink of heterogeneous cellular networks. IEEE Trans. Wirel. Commun.PP(99), 1–1 (2016). https://doi.org/10.1109/TWC.2016.2588469.Google Scholar
 H. Zhang, N. Prasad, S. Rangarajan, S. Mekhail, S. Said, R. Arnott, in Communications (ICC), 2012 IEEE International Conference On. Standardscompliant LTE and LTEA uplink power control, (2012), pp. 5275–5279. https://doi.org/10.1109/ICC.2012.6364333.
 H. Shin, J. H. Lee, Channel reliability estimation for turbo decoding in rayleigh fading channels with imperfect channel estimates. IEEE Commun. Lett.6(11), 503–505 (2002). https://doi.org/10.1109/LCOMM.2002.804246.View ArticleGoogle Scholar
 M. C. AguayoTorres, et al., WMSIM LTE link simulator. Technical report (University of Malaga, 2014). http://riuma.uma.es/xmlui/handle/10630/7438.
 F. J. MartinVega, I. M. DelgadoLuque, F. BlanquezCasado, G. Gomez, M. C. AguayoTorres, JT Entrambasaguas, in Vehicular Technology Conference (VTC Fall), IEEE 78th. LTE performance over high speed railway channel, (2013). https://doi.org/10.1109/VTCFall.2013.6692274.
 M. Haenggi, Stochastic geometry for wireless networks (Cambridge University Press, 2013).Google Scholar
 H. S. Dhillon, J. G. Andrews, Downlink rate distribution in heterogeneous cellular networks under generalized cell selection. Wirel. Commun. Lett. IEEE. 3(1), 42–45 (2014). https://doi.org/10.1109/WCL.2013.110713.130709.View ArticleGoogle Scholar
 J. G. Andrews, F. Baccelli, R. K. Ganti, A tractable approach to coverage and rate in cellular networks. IEEE Trans. Commun.59(11), 3122–3134 (2011). https://doi.org/10.1109/TCOMM.2011.100411.100541.View ArticleGoogle Scholar
 S. Sesia, et al., LTE, The UMTS long term evolution: from theory to practice (Wiley Publishing, 2009).Google Scholar
 M. D. Renzo, et al., The intensity matching approach: a tractable stochastic geometry approximation to systemlevel analysis of cellular networks. IEEE Trans. Wirel. Commun. 15(9) (2016). https://doi.org/10.1109/TWC.2016.2574852.View ArticleGoogle Scholar
 S. Singh, F. Baccelli, J. G. Andrews, On association cells in random heterogeneous networks. Wirel. Commun. Lett. IEEE. 3(1), 70–73 (2014). https://doi.org/10.1109/WCL.2013.111713.130707.View ArticleGoogle Scholar
 J. S. Ferenc, Z. Néda, On the size distribution of Poisson Voronoi cells. Physica A: Statistical Mechanics and its Applications. 385(2), 518–526 (2007).View ArticleGoogle Scholar
 3GPP, Technical specification group radio access network; Smallcell enhancements for EUTRA and EUTRAN  Physical layer aspects. TR 36.872. 3rd Generation Partnership Project (3GPP) (2013).Google Scholar
 S. Singh, J. G. Andrews, Joint resource partitioning and offloading in heterogeneous cellular networks. IEEE Trans. Wirel. Commun.13(2), 888–901 (2014). https://doi.org/10.1109/TWC.2013.120713.130548.View ArticleGoogle Scholar
 Y. Lin, W. Bao, W. Yu, B. Liang, Optimizing user association and spectrum allocation in HetNets: a utility perspective. IEEE J. Sel. Areas Commun.33(6), 1025–1039 (2015). https://doi.org/10.1109/JSAC.2015.2417011.View ArticleGoogle Scholar
 F. J. MartinVega, F. J. LopezMartinez, G. Gomez, M. C. AguayoTorres, in Global Communications Conference (GLOBECOM), 2014. Multiuser coverage probability of uplink cellular systems: A stochastic geometry approach (IEEE, 2014). https://doi.org/10.1109/GLOCOM.2014.7037431.