On Stochastic Geometry Modeling of Cellular Uplink Transmission with Truncated Channel Inversion Power Control
Abstract
Using stochastic geometry, we develop a tractable uplink modeling paradigm for outage probability and spectral efficiency in both single and multitier cellular wireless networks. The analysis accounts for per user equipment (UE) power control as well as the maximum power limitations for UEs. More specifically, for interference mitigation and robust uplink communication, each UE is required to control its transmit power such that the average received signal power at its serving base station (BS) is equal to a certain threshold . Due to the limited transmit power, the UEs employ a truncated channel inversion power control policy with a cutoff threshold of . We show that there exists a transfer point in the uplink system performance that depends on the tuple: BS intensity (), maximum transmit power of UEs (), and . That is, when is a tight operational constraint with respect to [w.r.t.] and , the uplink outage probability and spectral efficiency highly depend on the values of and . In this case, there exists an optimal cutoff threshold , which depends on the system parameters, that minimizes the outage probability. On the other hand, when is not a binding operational constraint w.r.t. and , the uplink outage probability and spectral efficiency become independent of and . We obtain approximate yet accurate simple expressions for outage probability and spectral efficiency which reduce to closedforms in some special cases.
Keywords: Multitier cellular networks, uplink communication, power control, truncated channel inversion, stochastic geometry.
I Introduction
Due to the variations of the capacity demand across the service areas (e.g., downtowns, residential and suburbans areas, etc.) as well as the deployment of multitier cellular networks (i.e., cellular networks consisting of different types of base stations [BSs] such as macro, micro, pico, and femto BSs), the practical cellular infrastructure deviates from the wellknown regular grid topology. Instead, it is better modeled by a randomized topology with uncertainties in the locations of the BSs [1, 2]. Therefore, modeling and analysis of cellular networks based on stochastic geometry has recently received much attention due to its tractability and accuracy [3]. Stochastic geometry is a very powerful tool to deal with networks with random topologies. Stochastic geometry abstracts the cellular network topology to a convenient point process to facilitate its modeling and analysis. The Poisson point process (PPP) is an appealing abstraction for the cellular network topology due to its simplicity and tractability [3, 4]. Furthermore, the PPP has been proven to provide pessimistic bounds on the performance metrics (e.g., the outage probability and the mean rate) that are as tight as the optimistic bounds provided by the idealized gridbased model for actual cellular networks [1, 2]. Further validations of stochastic geometrybased modeling of cellular networks via PPP can be found in [5, 6, 7].
There have been significant developments in the stochastic geometry modeling of cellular wireless networks. However, most of the available literature focus on modeling and analysis of downlink transmissions due to its relative simplicity. In a multitier cellular network, if the users associate to the BSs in the downlink based on their average received signal strength, the average useful signal power received at each user equipment (UE) from its corresponding BS will be strictly greater than the average interference power from any individual interfering BS. Therefore, power control is not crucial^{1}^{1}1Indeed power control is important in downlink to enhance the network performance (see [8]) but not crucial for the basic network operation due to the inherent interference protection introduced by the association criterion. for the network operation, and hence, power control is ignored in most of the stochastic geometry models on downlink cellular networks and it is generally assumed that all the BSs in the same tier transmit with equal power [1, 2, 3, 9, 10, 11, 12, 13, 14]. In contrast, uplink analysis is quite more involved due to the following reasons:

Per user power control: in the uplink, due to the random cell sizes, an interfering UE in a neighboring cell can be much closer to a BS than its tagged UE (cf. Fig. 1). Therefore, power control per UE is crucial for basic uplink operation in order to mitigate the intercell interference. As will be shown later, per UE power control introduces a new source of randomness to the uplink system model which makes the uplink analysis more involved.

Correlation among interferers: orthogonal channel assignment per BS ensures no channel reuse in the same Voronoi^{2}^{2}2A Voronoi tessellation is the planar graph constructed by perpendicular lines bisecting the distances between the points of a point process. The Voronoi cells constructed from a point process representing the BS locations correspond to the coverage regions of the BSs. cell. That is, given that a UE is transmitting in the uplink on a certain channel, this channel cannot be reused within the coverage of its serving BS, and hence, the locations of the UEs using the same uplink channel are correlated.
Compared to the downlink, only few efforts were invested to understand and model the uplink transmissions in cellular networks. Uplink modeling has been done in an ad hoc manner where different works in the literature made different assumptions based on the problem in hand. For instance, [15] derived the uplink network capacity region for a twotier cellular network consisting of macro BSs (MBSs) modeled via the hexagonal grid, femto access points (FAPs) modeled via PPP, and UEs modeled via an independent PPP. Due to the small coverage radius of the FAPs, in [15], the interference seen from all uplink UEs associated with the same FAP was approximated by an isotropic point source of interference with the worstcase sum transmit powers of the FAP UEs (i.e., power control was ignored). [16] investigated the spectrum sharing between a cellular network uplink and a mobile ad hoc network. In [16], the service areas of all BSs were approximated by circles and power control was ignored. Note that the assumption of circular coverage areas for the BSs eliminates the aforementioned interference problem in the uplink where an interfering UE can be much closer to a BS than the tagged UE which communicates to that BS. Therefore, under the assumption of circular coverage area, power control can be ignored.
In [17], the authors modeled uplink transmission with fractional channel inversion power control in a singletier cellular network. However, for analytical tractability and to avoid the singularity at the origin imposed by the unbounded pathloss model[3], the authors in [17] approximated the entire network model and assumed that the Voronoi cells are divided with respect to (w.r.t.) the users (rather than w.r.t. the BSs) and each user has her own serving BS randomly located in her Voronoi cell. In [18], the authors modeled uplink UEs with channel inversion power control. However, the authors assumed that the tagged UE is uniformly distributed in the tagged BS’s coverage which is approximated by a circle having the radius , where is the BS intensity^{3}^{3}3More discussions on the work presented in [17, 18] will be provided in Section V.. Each of the available stochastic geometry models [15, 16, 17, 18] approximates the system model in a different way to simplify the analysis and none of them accounts for the maximum transmit power of the UEs.
In this paper, we propose a framework for modeling and analysis of uplink transmission in single and multitier cellular networks. We resort to some approximations in order to maintain the tractability of analysis. In particular, the proposed framework, as will be discussed later in details, partially ignores the correlations among the interfering UEs. That is, it captures the correlation between the interfering sources and the test network elements (i.e., the tagged UE and the corresponding BS) and ignores the mutual correlations among the interfering sources. The accuracy of the proposed analysis is validated via simulations. To the best of the authors’ knowledge, when compared to the existing literature on stochastic geometry analysis of uplink cellular systems, this paper uses the least amount of approximations and obtains the simplest expressions for outage probability and spectral efficiency of uplink transmission. The obtained simple expressions characterize the uplink system performance in both single and multitier cellular networks and reveal important design insights. We believe that the presented framework, results, and design insights fill in the gap between the wellunderstood downlink performance and the lagging uplink analysis. The contributions of the paper can be summarized in the following points:

The paper presents a tractable framework for uplink modeling and analysis in a Poisson cellular network. The model is general and extends to multitier cellular networks. The model accounts for limited transmit power of the UEs, per UE power control, and cutoff threshold for the power control.

Approximate yet accurate simple closedform equations are derived for the outage probability and simple forms with only one numerical integral are derived for the spectral efficiency.

The paper discusses the tradeoffs introduced by the cutoff threshold and the maximum transmit power and shows that there exists an optimal cutoff threshold for power control that minimizes the outage probability and transmit power of the UEs.

The paper characterizes the uplink performance and shows the commonalities and differences between the downlink and uplink performances in cellular networks. In particular, we show the existence of a transfer point in the uplink system performance which depends on the tuple (, , ), where is the BS intensity, is the maximum transmit power of a UE, and is the average received power required at the serving BS. That is, when the relative values of , , and lead to a binding maximum transmit power constraint for the uplink operation, the uplink operation is quite different from the downlink operation and depends on both the cutoff threshold and the BS intensity. In contrast, when the relative values of , , and lead to a nonbinding maximum transmit power constraint for the uplink operation (i.e., transmissions are not constrained by the maximum transmit power), the uplink operation becomes analogous to the downlink operation (i.e., becomes independent of the BS intensity and cutoff threshold).
The rest of the paper is organized as follows. The system model, assumptions, and methodology of analysis are presented in Section II. In Section III, the baseline uplink modeling paradigm for singletier cellular networks is presented. Section IV generalizes the developed uplink paradigm for multitier cellular networks. Numerical and simulation results are presented in Section V and the paper is concluded in Section VI.
Ii System Model and Assumptions
Iia Network Model
We consider an independent tier Poisson cellular network. That is, the BSs of each tier are spatially distributed in according to an independent homogenous PPP , with intensity , where is the location of the BS in the tier. The UEs are spatially distributed in according to an independent PPP with intensity .^{4}^{4}4With a slight abuse of notation we will use to denote both the location of the BS in the tier and the BS itself, and the same for . It is assumed that the intensity of the UEs is high enough such that each BS will have at least one user served per channel and that UEs have data to transmit in the uplink (i.e., saturation conditions are assumed). The BSs in each tier have equal receiver sensitivity , however, two BSs from different tiers do not necessarily have the same receiver sensitivity. For successful uplink communication, it is required that the received signal power at the BS is greater than the receiver sensitivity. Therefore, each of the UEs associated to tier controls its transmit power such that the average signal received at the corresponding serving BS is equal to the threshold , where .
It is assumed that all the UEs have the same maximum transmit power of . Due to the maximum transmit power constraint for uplink communications, the UEs use truncated channel inversion power control where the transmitters compensate for the pathloss to keep the average received signal power equal to the threshold [19, chapter 4]. That is, an uplink connection is established between a UE and its serving BS if and only if the transmit power required for the pathloss inversion is less than . Otherwise, the UE does not transmit and goes into an outage (denoted hereafter as truncation outage) due to the insufficient transmit power.
It is worth mentioning that the truncated channel inversion power control mechanism is a realistic power control scheme for codedivision multiple access (CDMA) networks to eliminate the nearfar effect. Moreover, for orthogonal frequencydivision multiple access (OFDMA) networks, it has been shown in [20] that if the edge users (i.e., users who do not have sufficient power for their channel inversions) are allowed to transmit with their maximum power, the interference in the system increases significantly. Consequently, the network performance deteriorates without much improvement in the cell edge user performance. Hence, in this paper we consider the truncated channel inversion power control with the cutoff threshold , where the cutoff threshold is a network design parameter that highly impacts the system behavior. As will be shown later, the relative values of the BS intensity, , and control the tradeoff among transmit power, signaltointerferenceplusnoiseratio (SINR), and truncation outage (i.e., outage due to insufficient transmit power).
Fig. 2 shows the network model for different values of for a singletier cellular network. As shown in Fig. 2, if the value of is relatively high (i.e., relative to and ), not all the UEs can compensate for the pathloss and the cell edge UEs suffer from truncation outage. In contrast, Fig. 2 shows that if the value of is relatively low (or equivalently, the BSs are dense enough and/or the maximum transmit power is high enough), all of the UEs can compensate for the pathloss and none of the UEs suffers from truncation outage. Hereafter, we will denote the scenario in Fig. 2 as the uplink operation under binding maximum transmit power constraint and the scenario in Fig. 2 as the uplink operation under nonbinding maximum transmit power constraint. Universal frequency reuse is used within each tier and across different tiers. Within a network tier, there is no intracell interference. That is, in a network tier, each BS assigns a unique channel to each of its associated UEs.
IiB Radio Channel Model
A general powerlaw pathloss model is considered in which the signal power decays at the rate with the propagation distance , where is the pathloss exponent. For simplicity, it is assumed that all BSs in all tiers share the same pathloss exponent . The case where each network tier has its own pathloss exponent will be discussed in Section IV. The channel (power) gain between two generic locations is denoted by . All the channel gains are assumed to be independent of each other, independent of the spatial locations, symmetric and are identically distributed (i.i.d.). Therefore, for the brevity of exposition, hereafter, the spatial indices are dropped. For analysis, only Rayleigh fading is assumed^{5}^{5}5Techniques to relax the Rayleigh fading assumption to general fading channels can be found in [3]. and the channel power gain is assumed to be exponentially distributed with unit mean. A signaltointerferenceplusnoise ratio (SINR) model is considered where a message can be successfully decoded at the tagged BS in tier if and only if the SINR of the useful signal is greater than a certain threshold . If the SINR at the tagged BS does not exceed the threshold , the link experiences an outage (hereafter referred to as SINR outage).
IiC Criterion for Uplink Association
Without loss of generality, all BSs in all tiers are assumed to have an open access policy^{6}^{6}6Closed access policy can be easily captured by thinning the PPP representing the complete set of BSs in the association analysis and simple modifications in the interference analysis [12]., and hence, all UEs can associate with all BSs. The UEs are assumed to associate to the BSs according to their average link quality^{7}^{7}7Note that user association is not based on the instantaneous link quality to avoid the pingpong effect in handovers that may arise due to the rapid fluctuations of the channel gains [21].. That is, a generic UE associates with its nearest BS from all tiers (i.e., if , , where denotes the Euclidean norm).
IiD Modeling Methodology
The SINR is a very important parameter that affects many performance metrics such as outage, rate, delay, and energy efficiency. We characterize the SINR by driving its cumulative distribution function (cdf) [3]. Due to the randomized network topology, the distances between the UEs and their serving BSs are random. Therefore, the transmit powers of the UEs are random (due to the truncated channel inversion power control).
We will first characterize the transmit power of a generic active UE (i.e., a generic user not in truncation outage) by deriving its probability density function (pdf) and moments. Then, we derive the cdf of SINR. We first develop the modeling paradigm for singletier cellular networks. Then, we show that the developed modeling paradigm for singletier cellular networks can be naturally extended to multitier networks.
Iii Uplink Modeling in a Singletier Cellular Network
In this section, we develop the baseline uplink modeling framework for a singletier cellular network. For the sake of an organized presentation, we further divide this section into two subsections, namely, transmit power analysis and SINR analysis.
Iiia Transmit Power Analysis
Due to the random network topology and the use of truncated channel inversion power control, each UE will transmit with a different power to invert the pathloss towards its serving BS. In this section, we derive the distribution and the moments of the transmit power of a generic UE. Fig. 2 shows the uplink association for a singletier cellular network for different values of the cutoff threshold . As shown in Fig. 2, due to the truncated channel inversion power control, not all of the UEs can communicate in the uplink when the cutoff threshold is relatively high (i.e., relative to and ). That is, the UEs located at a distance greater than from their nearest BS are unable to communicate in the uplink due the insufficient transmit power. Therefore, the complete set of UEs is divided into two nonoverlapping subsets, namely, the subset of active UEs and the subset of inactive UEs. The inactive UEs do not transmit and experience outage due to insufficient transmit power. The distribution for the transmit power of a generic active UE is obtained from the following lemma.
Lemma 1
In a singletier Poisson cellular network with truncated channel inversion power control with the cutoff threshold , the pdf of the transmit power of a generic active UE in the uplink is given by
The moments of the transmit power can be obtained as
(1) 
where is the lower incomplete gamma function.

See missingAppendix A.
Lemma 1 shows that the smaller the cutoff threshold , the lower is the transmit power of the UEs. That is, when the maximum transmit power is unlimited (i.e., , where is the gamma function), the expected transmit power linearly increases with . This is because as increases the UEs are required to transmit at a higher power. Lemma 1 also shows that the average transmit power decreases with . That is, as the BS intensity increases, the distance between a generic UE and the corresponding serving BS decreases, and hence, a lower transmit power is required to invert the pathloss. The truncation outage probability (i.e., the probability that a UE experiences outage due to the insufficient power) is given by
(2) 
which is increasing in . From Lemma 1 and equation (2) it appears that the lower the cutoff threshold, the better is the network performance in terms of truncation outage probability and transmit power consumption. However, as will be discussed later, a low may highly deteriorate the SINR outage and spectral efficiency, and hence, introduces a tradeoff for the network performance. Equation (2) shows that the truncation outage probability exponentially decreases with increasing BS intensity.
IiiB Sinr Analysis
In this section we derive the SINR outage probability for active UEs (i.e., users not in truncation outage). Note that the inactive UEs do not transmit and are in truncation outage due to the insufficient transmit power. Without any loss in generality, the SINR analysis is conducted on a tagged BS located at the origin. According to Slivnyak’s theorem [4], conditioning on having a BS at the origin does not change the statistical properties of the coexisting PPPs. Hence, the analysis holds for a generic BS located at a generic location. For the tagged active UE operating on a tagged channel, the SINR experienced at the BS located at the origin can be written as
(3) 
where the useful signal power is equal to due to the truncated channel inversion power control, is the noise power, and the random variable denotes the aggregate interference at the tagged BS from the uplink transmissions by other active UEs on the tagged channel. Note that is not identified with the channel index because all channels have i.i.d. interference. The SINR outage probability can be calculated as
(4) 
where the expectation in the second line of (4) is w.r.t. , and denotes the Laplace transform (LT) of the pdf of the random variable ^{8}^{8}8With a slight abuse of terminology, we will denote the LT of the pdf of a random variable by the LT of .
As discussed in Section I, in the uplink, the interfering UEs do not constitute a PPP due to the correlations among them. The correlation among the UEs is due to the unique channel assignment per user in each BS. Hence, the interfering UEs are better modeled using a softcore process (e.g., Strauss point process [5]) to capture the pairwise correlations among the locations of the active UEs per channel. Generally, softcore processes are not analytically tractable [3, 4], and hence, an exact expression for the LT of the aggregate interference cannot be obtained. For this reason, we will approximate the locations of the interfering UEs with a PPP of the same intensity. Note that this approximation only partially ignores the correlations introduced by the system model because the correlation with the tagged BS and the tagged UE is captured by our model.
It is worth mentioning that the PPP assumption for the interference sources has been widely exploited in the literature (even with the hard core point processes which introduce stronger pairwise correlation between points) and has been proved to be accurate if the correlation among the interfering nodes and the tagged receiver is captured [17, 18, 22, 23, 24]. It is also important to note that, due to the correlations among the shapes of the neighboring Voronoi cells and the fact that the sytem allows only one UE per Voronoi cell, the distances between the scheduled (i.e., interfering) UEs and their serving BSs are identically distributed but are not independent [17]. Therefore, as a direct consequence of the employed channel inversion power control, the transmit powers of the scheduled UEs are identically distributed (i.e., follow the distribution derived in Lemma 1) but are not independent. However, the authors in [17] showed that the dependence among the distances between the scheduled UEs and their serving BSs is weak and can be ignored. Therefore, for analytical tractability, we will assume that the active UEs constitute a PPP and their transmit powers are independent. The accuracy of this assumption will be validated via simulations. Exploiting the PPP approximation and the i.i.d. transmit powers for the set of interfering UEs in the uplink, the outage probability for a generic active UE can be given by the following theorem.
Theorem 1
In a singletier Poisson cellular network with truncated channel inversion power control with the cutoff threshold , in the uplink, assuming that the interfering UEs constitute a PPP and that their transmit powers are independent, the SINR outage probability for a generic active UE is given by
(5) 

See missingAppendix B.
Generally, it can be shown that the SINR outage is nonincreasing in (c.f. 3). Recalling (from Section IIIA) that both the truncation outage and the average transmit power are nondecreasing in , it is concluded that introduces a tradeoff in the system performance as will be shown in the numerical results.
The SINR statistics also controls the average uplink spectral efficiency obtained via Shannon’s formula. The average uplink spectral efficiency can be obtained from the following theorem.
Theorem 2
In a singletier Poisson cellular network with truncated channel inversion power control with the cutoff threshold , in the uplink, assuming that the interfering UEs constitute a PPP and that their transmit powers are independent, the average spectral efficiency of transmission by an active UE is given by
(6) 

See missingAppendix C.
Equation (6) incorporates the tradeoff between and spectral efficiency. On one hand, increasing improves the spectral efficiency and decreases the SINR outage. On the other hand, increasing deteriorates the truncation outage probability and increases the transmit power of the UEs. In the following, we show some interesting special cases that help understanding the uplink system performance.
IiiB1 Special case 1 (infinite )
The case of infinite maximum transmit power is of particular interest because it captures the scenario where the transmit power is not a binding constraint for the uplink communication (cf. Fig.2). In other words, the BSs are dense enough (with respect to the required cutoff threshold) such that the distance between a generic UE and its serving BS is relatively small, and hence, the transmit power will be less than the maximum value almost surely. In the analysis, we capture this case by setting . In this case, the SINR is given by
(7) 
and the spectral efficiency reduces to
(8) 
Equations (7) and (8) show that when the transmit power is not a binding constraint, the SINR outage and the average spectral efficiency in a singletier cellular network are independent of the BS intensity. This result is in compliance with the results in [1] and it can be concluded that, in terms of SINR outage and average spectral efficiency, both the uplink performance (in case of nonbinding maximum transmit power constraint) and the downlink performance are independent of the BS intensity.
It is quite insightful to see that the SINR outage probability and the average spectral efficiency are independent of the intensity of the BSs. That is, when the maximum transmit power is not a binding operational constraint for the UEs, increasing the intensity (number) of the BSs neither degrades nor improves the SINR outage probability and the average spectral efficiency. For the uplink, this behavior can be explained as follows: since each BS will be serving one user per channel, as the intensity of the BSs increases, the intensity of the interfering UEs increases^{9}^{9}9This is due to the saturation condition assumption which occurs in a congested network scenario.; however, the average distance between a UE and its serving BS decreases which decreases the transmit power required to maintain the received signal power at . Hence, the increased intensity of interfering UEs is compensated by the decreased transmit power per interfering UE, and the SINR statistics does not change if is kept constant, and vice versa. Therefore, the coverage probability and the average spectral efficiency can only be improved through interference management techniques such as frequency reuse [25], interference cancellation [26], multipleinputmultipleoutput (MIMO) antennas [27], interference avoidance via cognition [28], or multicell cooperation [29]. Although these results are only valid for the PPP network model, they are insightful because they reflect the worstcase network performance. More specifically, deploying more BSs, in the worst case, will not degrade the SINR statistics.
IiiB2 Special case 2 ()
The integral in (5) can be expressed in closedform for integer values of . For instance, if , the SINR outage probability reduces to the following closedform:
(9) 
and the spectral efficiency reduces to the following:
(10) 
IiiB3 Special case 3 (, infinite , interferencelimited scenario)
In the interferencelimited case, is assumed to be large enough such that and hence noise can be ignored. In this case, the SINR outage reduces to the following simple closedform:
(11) 
and the spectral efficiency reduces to
(12) 
Equation (11) gives a closedform expression for the outage probability in its simplest form and (12) shows that the average rate reduces to a constant value^{10}^{10}10The correspondence between this result and the result for the downlink case will be discussed in Section. VB. in this special case. These equations clearly show that when is not a binding constraint and the interference is much larger than the noise, both the SINR outage and the spectral efficiency are independent of the cutoff threshold as well as the BS intensity.
Iv Uplink Modeling in Multitier Cellular Networks
In this section, we show that the developed baseline model for uplink transmission in singletier cellular networks can be extended for uplink transmission in multitier cellular networks. First, we present the analysis for multitier cellular networks with common pathloss exponent. Then, we will show and comment on the case with different pathloss exponents in the different tiers.
Iva Common Pathloss Exponent
The developed model naturally captures multitier cellular networks with different intensities and different cutoff thresholds for the different tiers but with a common pathloss exponent . In this case, the distribution of the transmit power for the active UEs is obtained from the following lemma.
Lemma 2
In a tier Poisson cellular network with a common pathloss exponent and a truncated channel inversion power control where each tier has the BS intensity of and the cutoff threshold , the distribution of the transmit power of the active UEs in the uplink in the tier is given by
where . The moments of the transmit power of a UE in the tier can be obtained as
(13) 

Since the UEs associate to the BSs based on the average uplink link quality, following the superposition theorem of the PPP [4], the association regions for the BSs form a Voronoi tessellation for a PPP with intensity . Note that, for the uplink in a multitier network, the association regions form a Voronoi tessellation rather than a weighted Voronoi tessellation due to the homogenous transmit powers of the UEs. The rest of the proof follows the same steps as in missingAppendix A.
The truncation outage probability at the tier is given by
(14) 
Without loss of generality, let SINR be the SINR experienced by a tagged BS in the tier and the BS is located at the origin. By Slivnyak’s theorem, the analysis holds for any BS in the tier. Let each tier has its own SINR threshold , then, following (4), the SINR outage in the tier can be expressed as
(15) 
where is the aggregate interference from the tier, and the second line of equation (15) follows from the independence of and . Note that represents the cotier interference and for represents the crosstier interference. As discussed in the singletier case, the interfering UEs do not constitute a PPP and their transmit powers are correlated. However, for analytical tractability, similar to the singletier case, we approximate the interfering UEs in each tier with a PPP and ignore the transmit power correlations. The SINR outage for a generic active UE in the tier is then given by the following theorem.
Theorem 3
In a tier Poisson cellular network with a common pathloss exponent and truncated channel inversion power control where each tier has BS intensity and cutoff threshold , in the uplink, assuming that the interfering UEs in each tier constitute an independent PPP and that their transmit powers are independent, the SINR outage probability for a generic active UE in the tier is given by
(16) 

See missingAppendix D.
The SINR outage in (16) reduces to the singletier case given in (5), but with intensity , when all tiers have the same cutoff threshold despite of their different intensities. The spectral efficiency in the multitier case can be obtained from the following theorem.
Theorem 4
In a tier Poisson cellular network with a common pathloss exponent and truncated channel inversion power control where each tier has BS intensity and cutoff threshold , in the uplink, assuming that the interfering UEs in each tier constitute a independent PPP and that their transmit powers are independent, the average spectral efficiency of transmission by an active UE in the tier is given by
(17) 

The proof is similar to the one in missingAppendix C.
For brevity, we consider only one special case. For infinite , interferencelimited network scenario, and , the SINR outage and average spectral efficiency in the uplink in the tier reduce to the following:
(18) 
(19) 
Equations (18) and (19) show that, in general, the SINR outage probability and spectral efficiency in a certain tier depend on the relative cutoff thresholds and the relative BS intensities. The SINR outage probability decreases and the spectral efficiency improves as the cutoff threshold in the target tier increases and the cutoff thresholds in the other tiers and/or the BS intensities in the other tiers decrease. This is because, a higher cutoff threshold in the target tier increases the useful signal power and lower cutoff thresholds and/or lower BS intensities in the other tiers reduce the crosstier interference.
Theorems 3 and 4 show an important difference between multitier and singletier cellular networks. That is, regardless of the maximum transmit power value (i.e., binding or nonbinding maximum transmit power constraint), the SINR outage probability and spectral efficiency in multitier cellular networks depend on the relative values of the cutoff thresholds and the relative BS intensities. However, note that if all the tiers have the same cutoff threshold, regardless of the relative BSs intensities, for an interferencelimited scenario with and infinite , (18) and (19) reduce to (11) and (12), respectively. Hence, the multitier cellular network can be reduced to a singletier network with intensity , and both the SINR outage probability and the average spectral efficiency become independent of the BS intensities in the different tiers if is nonbinding. It is worth mentioning that similar performance was observed in [2] for the downlink multitier case. In particular, it was shown that if each UE associates to the BS offering the highest average received power, the outage probability and spectral efficiency become independent of the relative intensities and transmit powers of BSs in different network tiers.
IvB Different Pathloss Exponents
In the previous section, it has been shown that when all network tiers share the same pathloss exponent, the association of the UEs does not change from that in the singletier cellular networks. That is, the association regions of the BSs will form a Voronoi tessellation. On the other hand, if different tiers have different pathloss exponents, the association of UEs in the multitier case deviates from that in the singletier case where the association regions of the BSs form a weighted Voronoi tessellation. That is, the BSs in tiers with lower pathloss exponents will have larger service areas than BSs in tiers with higher pathloss exponents. The transmit power in a multitier cellular network with different pathloss exponents can be characterized by the following lemma.
Lemma 3
In a tier Poisson cellular network with truncated channel inversion power control where each tier has BS intensity , cutoff threshold , and pathloss exponent , the pdf of the transmit power of the active UEs in the uplink in the tier is given by
(20) 
The moments of the transmit power of a UE in the tier can be obtained as
(21) 

See missingAppendix E.
The truncation outage probability in the uplink for a UE at the tier is given by
(22) 
From Lemma 3 we can see that the moments of the transmit power of the UEs cannot be obtained in closedform due to the complications introduced by the different pathloss exponents. Note that, since the transmit power appears in the LT of the interference as , the calculations of outage probability and spectral efficiency do not require obtaining the moments of the transmit power in closedforms. It can be shown that, for a common pathloss exponent, Lemma 3 reduces to Lemma 2. Similar to the previous cases, the interfering UEs do not constitute a PPP and their transmit powers are correlated. However, for analytical tractability, we will approximate the locations of the interfering UEs by a PPP and ignore the correlations in transmit power. The SINR outage in multitier cellular networks with different pathloss exponents can be characterized by the following theorem.
Theorem 5
In a tier Poisson cellular network with truncated channel inversion power control where each tier has BS intensity , cutoff threshold , and pathloss exponent , assuming that the interfering UEs in each tier constitute an independent PPP and that their transmit powers are independent, the SINR outage probability in the uplink for a generic active UE in the tier is given by
(23) 

See missingAppendix F.
Note that is just a number that can be obtained numerically from (21). Hence, (23), as shown earlier, can be obtained in closedform for integer values of . The spectral efficiency can be characterized via the following theorem.
Theorem 6
In a tier Poisson cellular network with truncated channel inversion power control where each tier has the BS intensity , cutoff threshold , and pathloss exponent , assuming that the interfering UEs in each tier constitute an independent PPP and that their transmit powers are independent, the average spectral efficiency of transmission in the uplink by an active UE in the tier is given by
(24) 

The proof is similar to that in missingAppendix C.
Due to the complicated expressions for the moments of the transmit power as given in Lemma 3, Theorems 5 and 6 do not give simple formulas for the SINR outage and spectral efficiency. However, Theorems 5 and 6 do give general formulas that reduce to all of the previously presented special cases. Note that without simple expressions, significant insights may not be extracted from the results obtained in Theorems 5 and 6 and Lemma 3 for multitier networks with different pathloss exponents. In contrast, with a common pathloss exponent, Theorems 5 and 6 and Lemma 3 simplify to Theorems 3 and 4 and Lemma 2, respectively. The main conclusion from this section is that the developed paradigm is general and flexible to capture different practical system parameters.
V Results and Discussions
Va Results
In this section, we validate our model against simulations and present some numerical results for a singletier cellular network (or equivalently, a multitier cellular network with common cutoff threshold and pathloss exponent ). Unless otherwise stated, we set the BS intensity to BSs/km, the maximum transmit power W, the receiver sensitivity dBm, the cutoff threshold dBm, the SINR threshold , dBm, and the number of channels . In the simulation setup, we realize a PPP cellular network with intensity in a 400 km area. Then we drop the UEs randomly and uniformly over the simulation area until all BSs are active (i.e., each BS has a scheduled UE for which is sufficient to invert its pathloss and communicate in the uplink) to ensure that the saturation condition is satisfied. All UEs employ the channel inversion power control. The simulation is repeated 10000 times.
First, we validate our model against simulation results obtained for a Poisson cellular network and compare it with the circular approximation of the target BS coverage as used in [16, 18]. The reason for the model validation is that the derived cdf of the SINR assumes that the set of active UEs constitutes a PPP and that their transmit powers are independent. Note that our proposed analysis captures the correlations between the location of the tagged BS and the locations of the interfering UEs as well as the correlations between the transmit power of the tagged UE and the transmit powers of the interfering UEs (cf. Facts – in Appendix B). Hence, our model partially ignores the correlations imposed by the uplink system model.
Fig. 4 shows that the derived model accurately captures the SINR outage. The figure also shows that approximating the coverage area of the tagged BS with a circle with radius underestimates the outage at dBm and overestimates the outage at dBm. The reason is that at a low cutoff threshold , the interference exclusion region around each BS is large (cf. Fact in Appendix B), and hence, the radius estimates a more aggressive interference. On the other hand, for a high cutoff threshold , the interference exclusion region around the tagged BS is small, and hence, the radius estimates a more conservative interference.
Fig. 5 shows the tradeoff introduced by on the total outage probability . As shown in the figure, tunes the tradeoff between the two outage probabilities and there exists an optimal cutoff threshold that minimizes the total outage probability. That is, at lower values of , the SINR outage dominates the outage probability. On the other hand, at high values of , the truncation outage dominates the outage probability. The figure shows the two regions of operation for the uplink, namely, when is a binding constraint and when is a nonbinding constraint. Note that becomes binding when the truncation outage probability is nonzero. For small values of , induces a nonbinding constraint, and hence, for relatively high BS intensity (e.g., and ) the SINR outage is independent of the BS intensity (i.e., the two curves for and coincides). This reinforces case in Section IIIB. Note that when is comparable to the noise power , the SINR outage depends on . However, when is much greater than the noise power and is nonbinding, the SINR outage is independent of both the cutoff threshold and BS intensity (case in Section IIIB). For high values of , becomes a binding constraint and the SINR outage depends on both the BS intensity and cutoff threshold.
To show the two regions of operation for the uplink more clearly, we plot Fig. 6 with a negligible noise power value of dBm. This figure shows that when is nonbinding (i.e., the truncation outage probability is almost zero), the SINR outage is completely independent of both the cutoff threshold and BS intensity (case in Section IIIB). Note that the independence w.r.t. the BS intensity can be seen from the coincidence of the curves for and as long as is nonbinding for both intensities (i.e., the truncation outage probability is equal to zero). In contrast, when becomes binding (i.e., the truncation outage probability is nonzero), the SINR outage is highly affected by both the BS intensity and cutoff threshold (case in Section IIIB).
Fig. 7 shows the tradeoff introduced by on the effective spectral efficiency defined as . The effective spectral efficiency captures the average spectral efficiency for active users (i.e., users with no truncation outage). As shown in this figure, when is a nonbinding constraint (i.e., for with and with [cf. Fig. 5]), the effective spectral efficiency is independent of the BS intensity. Note that the effective spectral efficiency depends on when induces a nonbinding constraint due to the relatively (i.e., relative to ) high noise power (case in Section IIIB). However, for the effective spectral efficiency is independent of the cutoff threshold in the range of dBm because induces a nonbinding constraint and the noise power is negligible w.r.t. the value of (case in Section IIIB). On the other hand, when becomes a binding constraint, the effective spectral efficiency depends on both the BS intensity and cutoff threshold. This figure also shows the existence of an optimal cutoff threshold which maximizes the effective spectral efficiency.
To show the tradeoff introduced by on the effective spectral efficiency more clearly, we plot Fig. 8 for negligible noise power dBm. This figure shows that when induces a nonbinding constraint, the effective spectral efficiency is completely independent of the BS intensity and cutoff threshold. On the other hand, when induces a binding constraint, the effective spectral efficiency depends on both the BS intensity and the cutoff threshold.
Fig. 9 shows the average transmit power of the UEs vs. the cutoff threshold . As the cutoff threshold increases, the UEs are required to transmit at higher powers to invert the pathloss and maintain a high threshold at their serving BSs. Therefore, the average transmit power is nondecreasing in . Note that the average transmit power saturates at .
VB Discussions
In the light of the proposed uplink framework and the results provided in [1, 2], we highlight the commonalities and differences between the uplink and downlink transmission performances. The criticality of the power control is the first main difference between the uplink and downlink transmission performances. While power control enhances the downlink transmission performance, it is not crucial for the basic network operation due to the inherent interference protection provided by the user association policy. On the other hand, power control is essential for the case of uplink operation to mitigate potential severe interference caused by the arbitrary close interfering UEs (cf. Fig. 1).
The second difference between the uplink and downlink operation is the maximum transmit power constraint for the UEs. For a singletier cellular network, the relative values of , , and may lead to a binding maximum transmit power constraint for the uplink operation. In this case, the uplink performance is highly affected by the intensity of the BSs as well as the cutoff threshold for power control. On the other hand, when the relative values of , , and lead to an uplink operation under nonbinding maximum transmit power constraint, the uplink transmission performance is analogous to the downlink transmission performance. More specifically, the uplink network can be transformed into an equivalent network where all UEs have a constant transmit power of (see (26))^{11}^{11}11The expression for LT of the interference in (26) shows the equivalence of the interference distributions in the uplink and downlink transmission scenarios where the randomness in the transmit power is captured via its fractional moments in the expression for LT as discussed in [21, 30].. Hence, the uplink performance with nonbinding is similar to the downlink performance with BS transmit power of and a constant downlink distance of . It is important to note that the similarity between the uplink performance (with nonbinding ) and the downlink performance is due to the power control in the uplink which enforces an interference protection for the BSs. This is analogous to the interference protection imposed by the association in the downlink for the UEs. However, the power control maintains a fixed power level of at the BSs which increases the SINR outage probability (e.g., as seen by comparing (11) to the downlink outage given by [1]), and reduces the spectral efficiency (e.g., nat/sec/Hz obtained in (12) for the uplink when compared to the nat/sec/Hz obtained for the downlink in [1]). A better performance is achieved in the downlink because the UEs which are close to their serving BSs experience high received power (w.r.t. the fixed in the uplink) from their serving BSs. To this end, it can be concluded that, in a singletier cellular network with a loose maximum transmit power constraint, the SINR outage and spectral efficiency performances for uplink communication are independent of the BS intensity. Therefore, in the worst case (due to the PPP assumption), deploying more BSs will not affect the SINR statistics in either the uplink or the downlink.
For multitier cellular networks, regardless of whether is binding or not, the uplink performance depends on the relative cutoff thresholds and relative BS intensities in the network tiers. However, in a multitier cellular network with a common cutoff threshold, a common pathloss exponent, and a loose maximum transmit power constraint, the SINR outage and the spectral efficiency performances for uplink communications are independent of the BS intensity. Therefore, in the worst case (because of the PPP assumption), deploying more BSs or more tiers will not affect the SINR statistics in either the uplink or the downlink.
In addition to its practicality, the channel inversion power control with the cutoff threshold highly facilitates the analysis in this paper and leads to simple expressions for the outage probability and rate. However, the presented analysis can be considered as a special case of the fractional channel inversion power control (i.e., full inversion) and cannot be used to capture the fractional channel inversion power control. Nevertheless, the presented analysis provides design insights and helps understanding the uplink system behavior. In addition, in contrast to fractional channel inversion power control, the channel inversion power control with the cutoff threshold results in a location independent outage probability and average rate for active users (i.e., homogenous performance for all active users). The fractional channel inversion power control imposes a location dependent performance for the active users. Furthermore, the fractional channel inversion power control highly complicates the system model, and hence, necessitates more approximations as in [17] in order to maintain the tractability. Although [17] provided a mathematically elegant technique to deal with uplink cellular networks with fractional channel inversion power control, our results reveal that the model provided in [17] will not be accurate when considering the maximum transmit power constraint for the UEs and the UEs operating in a binding maximum transmit power scenario. This is because, in [17] the system model was approximated and it was assumed that the Voronoi cells are realized w.r.t. the users not the BSs. In other words, the uplink system model was converted to an equivalent downlink (i.e., the users are treated as BSs) which, as we have shown here, is inaccurate if is a binding operational constraint. Moreover, the model presented in [17] resulted in a relatively more complex expression for the outage probability^{12}^{12}12Note that, the expressions presented in [17] for the outage probability and rate do not reduce to the expression presented in this paper due to the approximation in system model used in [17]..
Vi Conclusion
We have introduced a novel modeling approach for uplink transmission in Poisson cellular networks with truncated channel inversion power control. The model assumes a practical system model and accounts for power control, maximum transmit power of the UEs, and cutoff threshold for the power control. The developed modeling approach is general and captures the uplink performance of multitier cellular networks. The results show that the cutoff threshold introduces tradeoffs in the system performance and there exists an optimal cutoff threshold that minimizes the outage probability. When multiple outage optimal cutoff thresholds exist, the minimum cutoff threshold minimizes the average transmit power of the users. Approximate yet accurate simple expressions for the outage probability and spectral efficiency, which reduce to closed forms is some special cases, have been obtained.
We have characterized the uplink performance and showed the existence of a transfer point for the uplink operation which depends on the relative values of the BS intensity, the maximum transmit power of the UEs, and the cutoff threshold for power control. When the BSs are dense enough such that the maximum transmit power is not a binding constraint to establish an uplink connection with the nearest BS, the SINR outage and the spectral efficiency are independent of the BS intensity. Since the maximum transmit power is, in general, not a binding constraint in dense cellular networks, deploying more tiers and more BSs may decrease the transmit power consumption for UEs, improve the spatial frequency reuse and system capacity, but will neither improve nor deteriorate the SINR outage probability and spectral efficiency. Hence, the SINR outage probability and the spectral efficiency could only be improved via interference management/avoidance techniques.
Appendix A Proof of Lemma 1
For a generic UE , let . The uplink distance has the Rayleigh distribution , [1, 4]. The transmit power for a generic UE is given by such that . Therefore, the pdf of is given by
Note that the pdf of is normalized due to the truncated channel inversion power control. The moment of is given by and the lemma is obtained.
Appendix B Proof of Theorem 1
For the interference experienced by a cellular uplink, we find the Laplace Transform (LT) of the aggregate interference at a tagged BS located at the origin. Note that orthogonal channel assignment per BS brings correlations among the locations of the interfering UEs as well as with the location of the tagged BS, which highly complicates the analysis. The derivation here is based on the three facts and a key assumption listed below:

Fact : the average useful signal received at any BS is equal to the cutoff threshold .

Fact : the average interference received from any single interfering UE is strictly less than .

Fact : at any time instant each BS will have a single user served per channel, and hence, the intensity of interfering UEs on each channel is .

Key assumption: the interfering UEs constitute a PPP and their transmit powers are independent.
Note that Fact and Fact are a direct consequence of the association policy and power control, while Fact holds because each BS assigns a unique channel for each of its associated users. Hence, the aggregate interference received at the tagged BS can be written as
(25) 
where is a PPP with intensity representing the interfering UEs, and is the indicator function which takes the value if the statement is true and zero otherwise. Note that the indicator function in (25) captures the correlation between the interfering UEs and the tagged BS. The LT of the aggregate interference from the interfering UEs received at the tagged BS is obtained as
(26) 
where is the expectation with respect to the random variable , follows from independence between , , and , follows from the probability generation functional (PGFL) of the PPP [4], follows from the LT of , and obtained by changing the variables . The theorem is obtained be substituting (26) in (4) for and substituting the value of from Lemma 1. It is worth mentioning that follows from the assumption of i.i.d. transmit power, and follows from the PPP assumption for the UEs’ locations (i.e., using the PGFL of the PPP).
Appendix C Proof of Theorem 2
Since the SINR is a strictly positive random variable, the average spectral efficiency can be obtained as
Appendix D Proof of Theorem 3
For the interference experienced by a cellular uplink from UEs in tier