The tedious and time-consuming nature of system-level 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 non-MIMO [12, 13, 14, 15, 16, 17, 18] and in MIMO contexts [19, 20, 21, 22].
We present expressions for the forward-link signal-to-interference ratio (SIR) and the spectral efficiency in macrocellular networks with massive MIMO conjugate beamforming, both with a uniform and a channel-dependent 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 system-level 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 channel-dependent 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 simulation-based 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.
Ii-a Large-scale 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 . 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 large-scale channel gain, and we denote by the number of users served by the th BS.
The large-scale channel gain includes path loss with exponent and shadowing that is IID across the links. Specifically, the large-scale gain between the th BS and the th user served by the th BS is
with the path loss intercept at a unit distance, the link distance, and
the shadowing coefficient having standard deviationand 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 large-channel 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
as IID and Poisson, anticipating that the results be applicable under relevant network geometries and realistic values of ; this is validated in Section IX.
Ii-B Number of Users per BS
The modeling of is a nontrivial issue. Even in a lattice network with equal-size 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 meantimes the cell area; for instance, it is a Poisson random variable if is a PPP . Depending on how the cell area is distributed, then, the distribution of the number of users per cell can be computed. For instance, if
is PPP, such number in a lattice network is Poisson-distributed with meanwhile, for an irregular network with PPP-distributed 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 probabilityof being served by any of the BSs. Now, letting while keeping constant, the binomial distribution converges to the Poisson distribution .
In accordance with the foregoing reasoning, which is formalized in , 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)
whose corresponding CDF (cumulative distribution function) is
where is the upper incomplete gamma function.
Ii-C Small-scale Modeling
Let us focus on the local neighborhood of the th user served by the BS of interest, wherein the large-scale gains apply. Without loss of generality, a system-level 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
where is the (normalized) reverse-link
channel vector,is its forward-link reciprocal, and is AWGN. The signal vector emitted by the th BS, intended for its users, satisfies where is the per-base transmit power.
Iii Conjugate Beamforming Transmission
The signal transmitted by the th BS is
where is the power allocated to the data symbol , which is precoded by and intended for its th user. The power allocation satisfies
and, with conjugate beamforming and an average power constraint,
are the channel estimates gathered by theth BS from the reverse-link pilots transmitted by its own users.
We consider receivers reliant on channel hardening , whereby user served by the BS of interest regards as its precoded channel, with the expectation taken over the small-scale fading. The fluctuations of the actual precoded channel around this expectation constitute self-interference, such that (9) can be elaborated into
and the SINR is
As indicated, the analysis in this paper ignores pilot contamination. It follows that, in interference-limited conditions (), the conjugate beamforming precoders at BS are
while, likewise, for both and ,
Altogether, in interference-limited conditions,
which, invoking (7), further reduces to
The foregoing ratio of channel gains can be seen to equal
Two different power allocations, meaning two different formulations for , are analyzed in this paper; the corresponding SIRs are presented next.
Iv-a Uniform Power Allocation
With a uniform power allocation, and
Iv-B Equal-SIR Power Allocation
Alternatively, the SIRs can be equalized by setting 
The ensuing SIR, common to all the users served by the BS of interest, equals
Introducing the harmonic mean of, namely
we can rewrite (30) as
Having formulated the SIRs for given large-scale link gains, let us next characterize the system-level performance by releasing the conditioning on those gains.
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 PPP-distributed network of BSs [19, 18], in accordance with the PPP convergence exposed in Section II-A. Specifically, are regarded as IID with CDF , where is the local-average 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.
where is the lower incomplete gamma function. The CDF of satisfies
where ”” indicates asymptotic () equality while
with the Gauss hypergeometric function. Setting
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 .
where denotes imaginary part and is the confluent hypergeometric or Kummer function.
See Appendix A.
While the results in the sequel are derived using either Lemma 1 or Lemma 2, there is yet another useful alternative, namely computing via an approximate numerical inversion of the Laplace transform , presentation of which is relegated to Appendix B.
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.
Vi-a Uniform Power Allocation
Starting from (27), we can determine the CDF of as
which depends on and only through their ratio, .
The CDF of with a fixed satisfies
with constant value within . Alternatively, the CDF can be computed exactly as
With either of these expressions, one can readily compute the percentage of users achieving a certain local-average performance for given and . Furthermore, one can establish minima for given and given some target performance at a desired user percentile.
Let . In order to ensure that no more than of users experience an SIR below dB, it is required that .
Vi-B Equal-SIR Power Allocation
From (32), the CDF of can be expressed as
where, recall, is the harmonic mean of .
The CDF of with a fixed is
See Appendix C.
What can be established explicitly is that, as with ratio ,
which follows from
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 SIR-equalizing power allocation for massive MIMO, this allows establishing rather accurately by directly equating to the desired SIR.
Vii Spatial SIR Distribution with Poisson-distributed
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.
Vii-a Uniform Power Allocation
The CDF of with a Poisson-distributed satisfies
where is as per (35). Exactly,
See Appendix D.
For with fixed , it can be verified that with convergence in the mean-square 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 .
Vii-B Equal-SIR Power Allocation
The CDF of with a Poisson-distributed equals
See Appendix E.
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
and the spatial distribution thereof can be readily obtained as
for each setting considered in the foregoing sections.
Let , and . The -ile user spectral efficiency, computed by numerically solving for in and , respectively under uniform and equal-SIR power allocations, are b/s/Hz and b/s/Hz.
Reconsidering Example 4, but with being Poisson with mean , the -ile user spectral efficiencies, respectively under uniform and equal-SIR 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
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.
Viii-a Average Spectral Efficiency: Uniform Power Allocation
With a uniform power allocation and a fixed , the spatially averaged user spectral efficiency equals
and the spatially averaged sum spectral efficiency is .
See Appendix F.
For the special case of , i.e., for , Prop. 5 reduces to
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).
The corresponding spatial average of the sum spectral efficiency, recalling (56), is
Viii-B Average Spectral Efficiency: Equal-SIR Power Allocation
With an equal-SIR power allocation and a fixed , the spatially averaged user spectral efficiency equals
and the spatially averaged sum spectral efficiency is .
See Appendix F.
For , recalling how the SIR hardens to , we have that hardens to