I Introduction
The tedious and timeconsuming nature of systemlevel performance evaluations in wireless networks is exacerbated with massive MIMO [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], as the number of active users per cell becomes hefty and the dimensionality of the channels grows very large. This reinforces the interest in analytical solutions, and such is the subject of this paper. To embark upon the analysis of massive MIMO settings, we invoke tools from stochastic geometry that have been successfully applied already in nonMIMO [12, 13, 14, 15, 16, 17, 18] and in MIMO contexts [19, 20, 21, 22].
We present expressions for the forwardlink signaltointerference ratio (SIR) and the spectral efficiency in macrocellular networks with massive MIMO conjugate beamforming, both with a uniform and a channeldependent power allocation, and for different alternatives concerning the number of users per cell. The derived expressions apply to very general network geometries in the face of shadowing. These expressions allow:

Testing and calibrating systemlevel simulators, e.g., determining how many cells need to be featured for desired levels of accuracy in terms of interference.

Gauging the impact of parameters such as the path loss exponent.

Optimizing the number of active users as a function of the number of antennas and the path loss exponent.

Assessing the benefits of a channeldependent power allocation.
At the same time, the analysis is not without limitations, chiefly that pilot contamination is not accounted for. We explore this aspect by contrasting our analysis with simulationbased results that include the contamination, thereby delineating the scope of our results.
Ii Network Modeling
We consider a macrocellular network where each base station (BS) is equipped with antennas while users feature a single antenna.
Iia Largescale Modeling
The BS positions form a stationary and ergodic point process of density , or a realization thereof, say a lattice network. As a result, the density of BSs within any region converges to as this region’s area grows [23]. In turn, the user positions conform to an independent point process of density , also stationary and ergodic. Altogether, the models encompass virtually every macrocellular scenario of relevance.
Each user is served by the BS from which it has the strongest largescale channel gain, and we denote by the number of users served by the th BS.
The largescale channel gain includes path loss with exponent and shadowing that is IID across the links. Specifically, the largescale gain between the th BS and the th user served by the th BS is
(1) 
with the path loss intercept at a unit distance, the link distance, and
the shadowing coefficient having standard deviation
and satisfying , where we have introduced .Without loss of generality, we declare the th BS as the focus of our interest and, for notational compactness, drop its index from the scripting. For the largechannel gains, for instance, this means that:

relates the BS of interest with the th user served by the th BS.

relates the th BS with the th user served by the BS of interest.

relates the BS of interest with its own th user.

is the number of users served by the BS of interest.
The same scripting and compacting is applied to other quantities.
Let us consider an arbitrary user served by the BS of interest. It is shown in [23, 24, 25] that, as , irrespective of the actual BS positions, the propagation process from that user’s vantage, specified by and seen as a point process on , converges to what the typical user—e.g., at the origin—would observe if the BS locations conformed to a homogeneous PPP on with density . Moreover, by virtue of independent shadowing in each link, as the propagation process from the vantage of each user becomes independent from those of the other users. Relying on these results, we embark on our analysis by regarding the propagation processes
(2) 
as IID and Poisson, anticipating that the results be applicable under relevant network geometries and realistic values of ; this is validated in Section IX.
IiB Number of Users per BS
The modeling of is a nontrivial issue. Even in a lattice network with equalsize cells, and let alone in irregular networks, disparities may arise across BSs because of the shadowing and the stochastic nature of the user locations.
In the absence of shadowing, the users served by a BS are those within its Voronoi cell, and their number—conditioned on the cell area—is a random variable with mean
times the cell area; for instance, it is a Poisson random variable if is a PPP [14]. Depending on how the cell area is distributed, then, the distribution of the number of users per cell can be computed. For instance, ifis PPP, such number in a lattice network is Poissondistributed with mean
while, for an irregular network with PPPdistributed BSs, the corresponding distribution is computed approximately in [26, 27].With shadowing, a user need not be served by the BS in whose Voronoi cell it is located. Remarkably though, with strong and independent shadowing per link, a Poisson distribution with mean turns out to be a rather precise model—regardless of the BS locations—for the number of users per BS. This is because, as the shadowing strengthens, it comes to dominate over the path loss and, in the limit, all BSs become equally likely to be the serving one. Consider a region of area having BSs and users, all placed arbitrarily. As
, the number of users associated served by each BS becomes binomially distributed,
, because each user has equal probability
of being served by any of the BSs. Now, letting while keeping constant, the binomial distribution converges to the Poisson distribution [14].In accordance with the foregoing reasoning, which is formalized in [28], we model as IID Poisson random variables with mean . Noting that the unbounded tail of the Poisson distribution needs to be truncated at , because such is the maximum number of users that can be linearly served by a BS with antennas, our analysis is conducted with Poisson and subsequently we verify the accuracy against simulations where the truncation is effected (see Examples 10 and 12). For the BS of interest specifically, we apply the Poisson PMF (probability mass function)
(3) 
whose corresponding CDF (cumulative distribution function) is
(4) 
where is the upper incomplete gamma function.
IiC Smallscale Modeling
Let us focus on the local neighborhood of the th user served by the BS of interest, wherein the largescale gains apply. Without loss of generality, a systemlevel analysis can be conducted from the perspective of this user, which becomes the typical user in the network once we uncondition from .
Upon data transmission from the BSs, such th user observes
(5) 
where is the (normalized) reverselink
channel vector,
is its forwardlink reciprocal, and is AWGN. The signal vector emitted by the th BS, intended for its users, satisfies where is the perbase transmit power.Iii Conjugate Beamforming Transmission
The signal transmitted by the th BS is
(6) 
where is the power allocated to the data symbol , which is precoded by and intended for its th user. The power allocation satisfies
(7) 
and, with conjugate beamforming and an average power constraint,
(8) 
where
are the channel estimates gathered by the
th BS from the reverselink pilots transmitted by its own users.Iv Sir
We consider receivers reliant on channel hardening [1], whereby user served by the BS of interest regards as its precoded channel, with the expectation taken over the smallscale fading. The fluctuations of the actual precoded channel around this expectation constitute selfinterference, such that (9) can be elaborated into
(10) 
and the SINR is
(11) 
with
(12) 
As indicated, the analysis in this paper ignores pilot contamination. It follows that, in interferencelimited conditions (), the conjugate beamforming precoders at BS are
(13) 
from which
(14)  
(15) 
and
(16)  
(17)  
(18) 
while, likewise, for both and ,
(19) 
Altogether, in interferencelimited conditions,
(20) 
which, invoking (7), further reduces to
(21) 
The foregoing ratio of channel gains can be seen to equal
(22)  
(23)  
(24) 
with
(25) 
being the localaverage SIR in singleuser transmission [19]. Hence, (21) can be rewritten as
(26) 
Two different power allocations, meaning two different formulations for , are analyzed in this paper; the corresponding SIRs are presented next.
Iva Uniform Power Allocation
With a uniform power allocation, and
(27) 
IvB EqualSIR Power Allocation
Alternatively, the SIRs can be equalized by setting [29]
(28)  
(29) 
The ensuing SIR, common to all the users served by the BS of interest, equals
(30) 
Introducing the harmonic mean of
, namely(31) 
we can rewrite (30) as
(32) 
Having formulated the SIRs for given largescale link gains, let us next characterize the systemlevel performance by releasing the conditioning on those gains.
3.5  0.571  0.672 
3.6  0.556  0.71 
3.7  0.540  0.747 
3.8  0.526  0.783 
3.9  0.513  0.819 
4  0.5  0.854 
4.1  0.488  0.888 
4.2  0.476  0.922 
V Spatial Distribution of
As the SIR formulations in (27) and (30) indicate, a key ingredient is the distribution of . To characterize this distribution, we capitalize on results derived for the typical user in a PPPdistributed network of BSs [19, 18], in accordance with the PPP convergence exposed in Section IIA. Specifically, are regarded as IID with CDF , where is the localaverage SIR of the typical user in such a network [19, 18]. For our analysis in the sequel, we employ a slightly simplified version of the result in [19, Eq. 18] as presented next.
Lemma 1.
[19] Define as the solution—common values are listed in Table I—to
(33) 
where is the lower incomplete gamma function. The CDF of satisfies
(34) 
where ”” indicates asymptotic () equality while
(35) 
with the Gauss hypergeometric function. Setting
(36) 
we ensure and the CDF can be taken as constant therewithin.
As an alternative to the foregoing characterization, an exact but integral form can be obtained for .
Lemma 2.
(37) 
where denotes imaginary part and is the confluent hypergeometric or Kummer function.
Proof.
See Appendix A.
Vi Spatial SIR Distributions with Fixed
To begin with, let us characterize the SIR distributions for a fixed number of served users. Besides having their own interest, the ensuing results serve as a stepping stone to their counterparts for conforming to the Poisson or to any other desired distribution.
Via Uniform Power Allocation
Starting from (27), we can determine the CDF of as
(38)  
(39) 
which depends on and only through their ratio, .
Now, invoking Lemmas 1–2, we can express the SIR distribution for the typical user in a massive MIMO network with fixed and a uniform power allocation.
Proposition 1.
The CDF of with a fixed satisfies
(40) 
with constant value within . Alternatively, the CDF can be computed exactly as
(41) 
With either of these expressions, one can readily compute the percentage of users achieving a certain localaverage performance for given and . Furthermore, one can establish minima for given and given some target performance at a desired user percentile.
Example 1.
Let . In order to ensure that no more than of users experience an SIR below dB, it is required that .
ViB EqualSIR Power Allocation
While an explicit expression such as (40) seems difficult to obtain for , an integral form similar to (41) is forthcoming.
What can be established explicitly is that, as with ratio ,
(45) 
which follows from
(46)  
(47) 
Example 2.
Note from the foregoing example that, for and onwards, over 90% of users have an within dB of its spatial average. Besides confirming the effectiveness of the simple SIRequalizing power allocation for massive MIMO, this allows establishing rather accurately by directly equating to the desired SIR.
Vii Spatial SIR Distribution with Poissondistributed
Let us now allow to adopt a Poisson distribution, which, as argued, captures well the variabilities caused by network irregularities and shadowing. Similar derivations could be applied to other distributions if, for instance, one wishes to further incorporate activity factors for the users or channel assignment mechanisms.
Viia Uniform Power Allocation
Proposition 3.
Proof.
See Appendix D.
For with fixed , it can be verified that with convergence in the meansquare sense, and therefore in probability. As a consequence, , which depends on and only through their ratio, progressively behaves as if this ratio were fixed at despite the Poisson nature of . This behavior is clearly in display in Fig. 2, where can be seen to approach its value for fixed .
ViiB EqualSIR Power Allocation
Again, because of the convergence for , hardens to its fixed value with . In this case, this corresponds to as demonstrated in Fig. 2.
Viii Spectral Efficiency
With each user’s SIR stable over its local neighborhood thanks to the channel hardening, the spectral efficiency of the typical user is directly
(53) 
and the spatial distribution thereof can be readily obtained as
(54)  
(55) 
for each setting considered in the foregoing sections.
Example 4.
Let , and . The ile user spectral efficiency, computed by numerically solving for in and , respectively under uniform and equalSIR power allocations, are b/s/Hz and b/s/Hz.
Example 5.
Reconsidering Example 4, but with being Poisson with mean , the ile user spectral efficiencies, respectively under uniform and equalSIR power allocations, are b/s/Hz and b/s/Hz.
From its user spectral efficiencies, the sum spectral efficiency at the BS of interest can be found as
(56) 
which is zero whenever the BS serves no users.
Explicit expressions for the spatial averages of the user and the sum spectral efficiencies, and with expectation over all possible propagation processes, are presented next.
Viiia Average Spectral Efficiency: Uniform Power Allocation
Proposition 5.
With a uniform power allocation and a fixed , the spatially averaged user spectral efficiency equals
(57) 
and the spatially averaged sum spectral efficiency is .
Proof.
See Appendix F.
For the special case of , i.e., for , Prop. 5 reduces to
(58) 
thanks to , an equivalence that is further applicable in the characterizations that follow.
Since the denominator of the integral’s argument in (58), and more generally in (57), is strictly positive for , we can deduce by inspection that shrinks if we increase with a fixed , i.e., if we add more users with a fixed number of antennas. However, this reduction is sublinear in and thus grows as we add more users. Thus, should be set to the largest possible value that ensures an acceptable performance for the individual users—recall Example 1—and the corresponding average performance for both individual users and cells can then be readily computed by means of (57) or (58).
Example 6.
ViiiB Average Spectral Efficiency: EqualSIR Power Allocation
Proposition 6.
With an equalSIR power allocation and a fixed , the spatially averaged user spectral efficiency equals
(63) 
and the spatially averaged sum spectral efficiency is .
Proof.
See Appendix F.
For , recalling how the SIR hardens to , we have that hardens to
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