On Uplink User Capacity for Massive MIMO Cellular Networks
aa r X i v : . [ c s . I T ] J un On Uplink User Capacity for Massive MIMOCellular Networks
Anand Sivamalai and Jamie S. Evans
Abstract —Under the conditions where performance in a mas-sive MIMO network is limited by pilot contamination, thereverse link signal-to-interference ratio (SIR) exhibits differentdistributions when using different pilot allocation schemes. Byutilising different sets of orthogonal pilot sequences, as opposedto reused sequences amongst adjacent cells, the resulting SIRdistribution is more favourable with respect to maximising thenumber of users on the network while maintaining a givenquality of service (QoS) for all users. This paper provides asimple expression for uplink user capacity on such networks andpresents uplink user capacity figures for both pilot allocationschemes for a selection of quality of service targets.
I. I
NTRODUCTION
Massive MIMO is a technology which is to play a sig-nificant role in tomorrow’s 5G networks. From the currentlyavailable research, it is evident that multi-user massive MIMOtechnology when deployed on future cellular networks willbring significant gains to network sum rate data capacity,whereas in previous cellular generation advancements, theattention was additionally on increasing the number of usersthe network could service, i.e. user capacity. As we move intothe “Internet of Things” paradigm, the number of connecteddevices is estimated to be more than 25 billion by 2020[1], it is clear we will need to turn our attention again toensuring user capacity. While a large amount of research inmassive MIMO systems focuses on cell sum rate capacityand spectral efficiency [2] [3] [4] [5], there has been littlewith a focus on user capacity [6] [7]. The recent work in[7] extends the single-cell user capacity expressions in [6]by presenting expressions for multi-cell user capacity andlike [6] examines the downlink user capacity while derivingoptimal pilot sequences. In contrast to [7], this paper presentsa comparison of two simple pilot allocation schemes, witha focus on uplink user capacity and the gains that can beachieved through statistical multiplexing when admitting userson a network with the constraint of a minimum SIR and outageprobability.The term “massive” in massive MIMO, refers to the oper-ating scenario where the BS has several orders of magnitudemore antenna elements than users. It has been shown in [2],that as the number of base station antennas M becomeslarge, the effects of noise and uncorrelated intra and inter-cellinterference disappear, and only inter-cell interference due topilot contamination remains - the interference resulting fromchannel estimates which are contaminated from pilots of usersin neighbouring cells. And hence a common tighter definitionof “massive” exists, where the system operates under such a regime where pilot contamination is the dominant impairment[4].This paper starts by looking at the asymptotic result whenthe number of BS antennas is large [2], and derives an analyt-ical approximation for the distribution of the SIR, primarilybased on a distance based path loss model and random userlocations. We then introduce the idea of statistical multiplexingwith regards to user admission, and use our analytical SIRdistribution to evaluate the number of users the network cansupport.Under the scenario where each BS uses the same set oforthogonal pilots, we expect one non-orthogonal interfererper adjacent cell. A way to combat such interference is toemploy frequency reuse factors which are less than one, whereadjacent cells utilise a different frequency resource, and conse-quently connected users of this cell are now orthogonal and nolonger interferers. Such cellular frequency reuse patterns havea fixed granularity (Figure 3) due to arranging the dividedfrequency resources in a regular fashion. Under a differentpilot allocation scheme, where each cell uses an independentorthogonal set of pilots, every user in an adjacent cell is aninterferer, albeit a fraction of the interference from a userunder the former scheme. This provides a finer granularityto control the inter-cell interference before having to resort toreducing the frequency reuse - limiting the number of userswill now effectively reduce the inter-cell interference, allowingit to come closer to the required QoS under user admission.On top of this, the reduced variability of the interferers in thecontext of statistical multiplexing under the second schememakes it superior to the former scheme when maximising thenumber of users that can be admitted onto a network. The ap-plication of statistical multiplexing has been used extensivelyin the dimensioning of effective bandwidths for users of CodeDivison Multiple Access (CDMA) cellular networks. In [8]several effective bandwidth models are presented, which arethen used to develop capacity specifications and consequentlycall admission procedures for multi-cell CDMA networks withmultiple classes. A very similar approach is employed in thispaper to dimension an effective interference for each user,which then enables the calculation of a user capacity.II. S YSTEM M ODEL AND LIMITING
SIR
EXPRESSIONS
The problem of pilot contamination occurs when the basestation receives indistinguishable uplink training pilot signalsfrom both the intended user and other interfering users. Hencethe estimated channel for the intended user is compromised,and subsequent data decoding suffers. Figure 1 depicts such a ell l Cell j β jkj β lkl β jkl Fig. 1: Uplink Pilot Contamination - Pilot reception from userof interest in cell j is contaminated with pilot transmission frominterfering user in neighbouring cell l. scenario where the BS of cell j receives uplink pilot transmis-sions from both its user and an interfering pilot transmissionfrom a user in the neighbouring cell l .We start with the result from [2], which shows that in acellular network with L cells with K users in each cell, asthe number of antennas M approaches infinity at BS j , allthe effects of uncorrelated receiver noise and fast fading areeliminated, and we are left with pilot contamination as thedominant impairment. Consequently, as M goes to infinity,the reverse link SIR at the BS for the k -th terminal, in the j -th cell, reduces to: SIR = β jkj P l = j β jkl , (1)where β represents the slow fading gain incorporating the ef-fects of distance based path loss and shadow fading. Therefore,(1) is the ratio of the slow fading gains of the k -th user in the j -th cell of interest, to the sum of slow fading gains to thetarget base station from all users in the network that employthe same pilot sequence. Interestingly, as highlighted in [2] theSIR is proportional to a ratio of the squares of the slow gains,which is a result of the MRC processing and the interferencebeing the dominant impairment.Note, due to the elimination of noise terms in (1), SIN R iswritten throughout as
SIR for clarity. Given the fact that thenoise and fast fading terms are no longer present, the simplicityof (1) allows us to analyse its distribution further, which canthen be used as a basis for evaluating the admissibility of auser on a network.We acknowledge that the convergence to the asymptoticbehaviour of (1) is slow, typically requiring an impracticalnumber of BS antenans even in the absence of noise, atleast M = 10 were required in our simulations to comewithin 15% of the mean of the large M SIR, where similarobservations are noted in [3]. However the large M resultserves as a means to compare the two pilot allocation schemesanalytically, where later, the finite M Monte Carlo simulationresults confirm the differences of the two schemes. Some veryrecent work by [9] presents some accurate approximationsto the SIR distribution for the reverse link under a similar system model for finite M , but with the main difference thatthe users and BSs are distributed as Poisson point processes.Our approach follows the more traditional fixed hexagonalgrid of BSs and uniformly distributed users, and allows fora simple explicit expression for user capacity from the keysystem parameters. A. Uplink Power Control
In order to improve inter-cell interference, uplink powercontrol (ULPC) can be utilised on the reverse link. We in-corporate a modest open loop scheme which does not attemptto compensate for fast fading. We assume perfect path lossestimation at the terminal, with each terminal transmitting witha power which is the inverse of the path loss to its own basestation. The terminal transmits the pilots and the correspondingdata at the same power, and (1) becomes:
SIR = ( β jkj β jkj ) P l = j ( β jkl β lkl ) = 1 P l = j ( β jkl β lkl ) . (2) B. Pilot allocation
The result in (2) represents a pilot allocation where eachcell in the network re-uses the same set of orthogonal pilots.As a result, when M is large, there is no intra-cell interferencein the j -th cell, and there is one interfering user in every othercell in the network, resulting in a denominator that is a sumof L − terms.An alternative pilot allocation, which will be used tocompare against the aforementioned re-used pilot scheme inthis paper, is to allocate different orthogonal pilot sets in allcells of the network. Such an approach still leaves us withno intra-cell interference but now every inter-cell user in thenetwork is an interferer, since all these users have used a pilotsequence which is no longer orthogonal to the pilot of theuser of interest. This results in an interference which is thesum of K × ( L − terms. In our SIR expression, this non-orthogonality is represented by φ kl which multiplies each ofthe interference terms, and is equal to the square of the innerproduct of the pilot sequence of the j -th user, with the pilotsequence of the interfering user. In contrast, under the reusedpilot set scheme, φ kl = 1 for only one user in the l -th cell and φ kl = 0 for all remaining users of the l -th cell. (This approachmodels φ kl as a random quantity as opposed to (26) in [2],where only the expected value of φ kl is used instead in thelimit expression). As a result, (2) becomes : SIR = 1 P l = j P Kk =1 φ kl ( β jkl β lkl ) . (3)III. A NALYSIS OF THE
SIR
DISTRIBUTION
In our analysis of (3), we model the slow fading gain bydistance based path loss alone, where β jkl = r − γjkl . The usersare uniformly distributed over the cell area, where the randomuantity r jkl , is the distance between the k -th user in the l -th cell and BS j . The constant γ is the path loss exponent.Including the effects of log-normal shadowing in the slowfading gain of our analysis of the SIR expression is non-trivial,and is further discussed in Section VI-B.Therefore, we begin by defining, y kl = φ kl (cid:18) r lkl r jkl (cid:19) γ , (4)so that the SIR in (3) can be expressed as: SIR = 1 P l = j P Kk =1 y kl . (5)Since we are interested in determining the distribution ofthe SIR, it is clear that we need to determine the distributionof the sum of the random variables in the denominator. Therandom quantity y kl is a function of three different randomvariables r lkl , r jkl and inner product of the pilot sequences φ kl . Therefore the nature of the SIR expression is still quitecomplex. Rather than attempting to find the exact distributionwe will approximate the sum of y kl random variables by aGaussian random variable, based on the central limit theorem.In order to make this approximation, we need the mean andvariance of the quantity y kl .In the interests of readability for the remaining part of thissection, we can drop the kl subscript by just considering thetwo BS scenario shown in Figure 2, with our BS of interest(BS j ) and an arbitrary interferer in an adjacent interferingcell (cell l ). Note that we have approximated the hexagonalcell by a circle as will be explained in more detail shortly.Due to the statistical independence between the pilot se-quences and the remaining random quantities, the mean canbe written as E [ y ] = E [ φ ] E [ x ] , where the random variable x = ( r l /r j ) γ . The E [ x ] can be determined analytically asfollows.Assuming users which are uniformly distributed over eachcell of our cellular network, the probability distribution func-tions (PDFs) of the two random variables r l (the distance ofthe interferer to their own BS) and r j (the distance of theinterferer to the base station of interest) are dependent onthe geometry of the individual cells and the layout of thecells within the network. These two random quantities areclearly not independent and therefore, computing a joint PDFis difficult.An alternative approach, which is also used in [10], is towrite x = (cid:18) r l r j (cid:19) γ = v ( r l , θ ) = ( r l r l + a − ar l cos θ ) γ . If we now approximate the hexagon shaped cell with a circularcell, we achieve two important points. The circle cell approx-imation results in very simple PDFs, where f Rl ( r l ) = 2 r l /b for ≤ r l < b and f Θ ( θ ) = 1 /π for ≤ θ < π . Secondly, we r j θ r l Cell l BS j ab BS l Fig. 2: Circular cell approximation with user in cell l interfering withreverse link reception at BS j have statistical independence between r l and θ , and can write E [ x ] = Z π Z b v ( r l , θ ) f Rl ( r l ) f Θ ( θ ) dr l dθ (6) σ x = Z π Z b ( v ( r l , θ ) − E [ x ]) f Rl ( r l ) f Θ ( θ ) dr l dθ (7)Given (6) and (7) we are able to make the Gaussian ap-proximation in (5) under the pilot allocation scheme where thesame orthogonal set is reused in every cell, (since effectively φ is a constant). If we now consider the scheme where eachcell uses a different orthogonal set, we will have to derivethe mean µ y and variance σ y where the random nature of φ is treated. The pilots sequences from the orthogonal set areassigned randomly to the K users of the cell, therefore φ is statistically independent from x , (the user locations). Thepilot sequences ψ are the coloumns from a K × K unitarymatrix, distributed uniformly according to the Haar measure.Consequently, the random quantity φ = | h ψ j , ψ l i | , has anexpectation of /K and a variance of /K , and the mean isgiven by: µ y = E [ φx ] = E [ φ ] µ x = µ x /K. (8)In order to derive the variance σ y , a series of simplesubstitutions can be made, resulting in: σ y = 1 K (2 σ x + µ x ) . (9)IV. U SER CAPACITY AND EFFECTIVE I NTERFERENCE
One of the objectives of any user admission policy is toensure that admitting a new user to the network will still guar-antee a certain QoS for the existing users of a network. In thecontext of our massive MIMO network, whose performance isinterference limited, such an admission policy would attemptto predict if admitting a new user would add an acceptablelevel of interference to all users of the network.As seen in the previous section, the SIR expression in (3)is random in nature and hence not easy to predict accurately.Just as [8] describes an effective bandwidth , we introduce thenotion of an effective interference , y E for each user. We assignan effective interference for each user which is somewhere inbetween the mean interference and the maximum interference.Assigning an effective interference for each user which isqual to the maximum interference results in a very conser-vative admission policy which does not benefit at all fromstatistical multiplexing. Assigning an effective interferenceequal to the mean interference implies the unrealistic scenarioof an infinite number of users on the network, where thesample mean is infact the true mean.A user admission policy would ensure that every userof the network experiences a quality of service, governedby a minimum SIR, S and an outage probability α . Usingthe Gaussian approximation for the denominator of (3), thiscondition can be written as: P ( 1 N ( µ, σ ) > S ) ≥ − α, which is equivalent to the condition, /S − µσ ≥ Q − ( α ) , (10)where Q ( x ) = 1 − Φ( x ) , and Φ( x ) is the CDF function for N (0 , .The central limit theorem allows us to approximate thesum of interference from the n users, which are i.i.d , bya Gaussian random variable with µ = nµ y and σ = nσ y .However, depending on the network topology, the surroundingcells containing the interferers could be at different distances,resulting in interference with different means and variances.The Gaussian approximation can be extended to approximatethe sum of these T different types of interferers, where thetotal interference is now approximated by a sum of T normaldistributions, which is itself also a normal distribution, with µ = P Tt =1 n t µ y t and σ = P Tt =1 n t σ y t , resulting in: /S − P Tt =1 n t µ y t qP Tt =1 n t σ y t ≥ Q − ( α ) . (11)If we consider the traditional cellular layouts as shownin Figure 3, we initially only need to consider the Tier 1interferers ( t ≤ T = 1 ) - the set of interferers (marked in darkblue for different frequency reuse factors) which are closestto our user/cell of interest (marked in red). This is becauseunder our current model, the interference contribution fromusers in the next tiers of interfering cells ( < t ≤ T ), isnegligible. For instance, given a frequency reuse factor of 1(with x t = n denoting the random interference at tier n ), wehave E [ x t =1 ] ≃ E [ x t =2 ] , and var [ x t =1 ] ≃ var [ x t =2 ] .Under some extensions to our model, the outer tier interferersdo become relevant as discussed in Section VI-B2.Using (11) with T = 1 , and for a given α and S , we cansolve for the maximum number of interferers n max , where n max = ⌊ n ⌋ . As a result, each of these interferers contributesan effective interference y E , given in [3] as : y E = µ / (1 + 2 z (1 − √ z )) , (12) In this approximation, we consider the weak correlation in the interferenceterms induced by the unitary property of the pilot sequence matrix negligible. where z = 4 µ ( Q − ( α )) σ S .
Therefore, from (12), the maximum number of allowedinterferers n max across all Tier 1 interfering cells, in our powercontrolled system is given by: n max ≤ y E S . (13)V. N
UMERICAL R ESULTS
A. Scenario
Our system is modelled around the LTE reverse-link in[2]. Consequently we consider the possible frequency reusefactors w ∈ { , , } , assume a coherence bandwidth of 14subcarriers, and our coherence time is divided into 7 symbols,3 of which are used for uplink training. As a consequence,our pilot sequence is of length ×
14 = 42 , and can thereforesupport a maximum of K = 42 terminals. Each cell has aradius a = 1600 metres and with cell-hole radius a h = 100 metres. A path loss exponent of 4 is used.We wish to model a simple non-cooperative user admissionpolicy, and therefore impose the requirement that each BS candecide to admit a user autonomously, (i.e. BSs do not requireinformation from other BSs for user admission) which in turnimplies that the maximum number of users connected to a BSis the same for all BSs of the network.In order to meet higher QoS requirements, we employfrequency reuse factors w using traditional cellular frequencyreuse patterns to reduce the interference experienced by theuser of interest. By utilising different frequency resources inadjacent cells, all users in adjacent cells no longer interferewith the user of interest (located in the cell marked in red ofFigure 3), and non-orthogonal interferers are essentially movedfurther away (who are located in the cells marked in darkerblue). Of course this reduction of interference to the user ofinterest comes at the cost of a reduction (by the frequencyreuse factor) to the maximum number of users that can besupported by each cell, since the available frequency resourceswithin the cell to train its users has been reduced.Figure 3 shows the different frequency reuse factors w , withthe cell of interest in red, and the cells which have interferingusers in darker blue. The frequency resource utilised by thecell is indicated by the number in the cell. B. M in the Limit
As highlighted earlier we consider only Tier 1 interferingcells, and from (12), the effective interference y E is computedfor a given minimum SIR and outage probability. Using (13),the total maximum number of interferers across all Tier 1interfering cells n max is given, and hence the unconstrained maximum number of admissible users per cell is given by, k u = n max / . However, given that the number of users a cellcan support is constrained by the length of the pilot sequenceand our simple non-cooperative admission policy, the number ig. 3: Non-orthogonal Tier 1 interferers (dark blue) under differentfrequency reuse factors w . Di (cid:1) erent Pilot SetsReused Pilot Sets Fig. 4: Maximum number of users per cell, k max vs. QoS of users per cell which we can support in practise is given by k max .Figure 4 shows clearly that for the region of relevantoutage and SIR requirements, a pilot allocation scheme whichemploys different pilot sets, as opposed to reused sets, issuperior with regards to k max .In order to examine the differences further, Figure 5 showsa cross-section of Figure 4, with both k u and k max plotted fora fixed outage probability of 0.05. Firstly, for both w = 1 and w = 3 , it is noteworthy that k u is significantly higher underthe relevant SIR range when different pilot sets are used.Under low SIR requirements, for both schemes, the numberof users in the cell is restricted by the resources allocated touplink training - as k u shows we would be able to support many more users at this QoS based on their interferencecontribution, but can only support the 42 based on our pilotlength. Therefore the maximum number of users that can besupported by the BS, k max , is the same, regardless of thepilot allocation scheme. However, to meet an SIR requirementwhich is greater than 1dB when pilot sets are reused, we areforced to employ w = 3 , indicated by the “switching point”circled on Figure 5, while when using different pilot sets weare able to continue using w = 1 . Moving to w = 3 reduces theeffective interference y E by a factor of 500 since the interferersare now positioned in cells further away, and as can be seenfrom the steep slope of the k u curves for both schemes under w = 3 , we would be able to support many users (for allSIRs up to 26dB). However, frequency resources for uplinktraining have consequently been reduced by a factor of 3, andin practice we can only train a maximum of / users.Under the different pilot set scheme, k max only starts todecline for SIRs greater than 5dB, where we are able tosupport the increasing SIR requirements by simply admittingfewer users in the cell. For SIRs greater than 9dB we wouldhave to admit less than 14 users to meet this requirement,and therefore it makes more sense that we employ w = 3 (asindicated by the switching point, where the red k u curve fallsunder the k max line). It is the region between these switchingpoints, indicated by the shaded blue in Figure 5, which isthe region of gain when using the different pilot sets scheme.Both schemes now support the same number of users up untilan SIR around 30dB, where the reused pilot set scheme mustagain switch to w = 7 , while the different pilot set schemeonly switches at an SIR of 35dB, and hence can support moreusers.As can be seen clearly in Figure 4, the outage probabilitycomponent of the QoS has less of an effect on k max , whencompared to changes in the SIR. The shaded regions inFigure 5 clearly show for the given QoS, the scale and regionwhere significant gains in user capacity can be realised byutilising different orthogonal pilot sets amongst cells.We now briefly look at the worst case accuracy of the Gaus-sian approximation under large M used to generate the resultsshown in Figure 5. As posed by the central limit theorem,we expect that as the numbers of interferers increases, thedistribution of interference closer approaches that of a normaldistribution, and the accuracy of our estimation improves.Given our restriction of a simple user admission policy andstandard cellular frequency reuse patterns, our worst caseapproximation would be under a frequency reuse factor of 7.Under this level of frequency reuse we could support a maxi-mum of 6 users in each of the interfering Tier 1 cells. Figure6 shows a comparison between the Gaussian approximation(in combination with our circular cell approximation) and aMonte Carlo simulation, for reverse link SIR distributions forboth pilot allocation schemes under frequency reuse factor7. When specifying meaningful QoS, we typically deal withoutages of 0.1 or less and therefore the main area of interestis in the lower tail of the distributions. As expected, theapproximation of the different pilot set scheme follows more u (Reused sets) kmax (Reused sets) ku (Different sets) kmax (Different sets)Freq. reuse switching points w = 1 w = 3 Fig. 5: Maximum number of users per cell, k max and Unconstrainedmaximum number of users k u vs. SIR under fixed outage probabilityof 0.05
44 45 46 47 48 49 50 5100.010.020.030.040.050.060.070.080.090.1 SIR (dB) F ( x ) Limit M−>inf, 36 interferers,Different Pilot SetsLimit M−>inf, 6 interferers,Reused Pilot setsGaussian approx. Reused Pilot setsGaussian approx. Different Pilot Sets
Fig. 6: SIR CDF under Freq. Reuse factor = 7 - (Worst-case)Gaussian Approximation vs. Simulation, under large M closely the results of the simulation, a consequence of beingthe sum of 36 interferers from these 6 interfering cells, asopposed to 6 interferers in the reused pilots case. Importantly,our approximated user capacity gain from the different pilotset allocation is conservative, since around this region, theapproximation returns results significantly more optimistic inthe reused pilot sets case compared to when different pilot setsare used.
C. Finite M
As discussed previously, the asymptotic behaviour of thereverse link SIR as M grows large results in the noise and fastfading becoming negligible due to the sum of the un-correlatedcross terms approaching zero. Consequently, we are able toderive the approximation for the reverse link SIR analytically. TABLE I: Maximum Number of Admissible Users per cell for Differ-ent QoS, M = 500, MRC detector
QoS Low0db/0.01 Medium10db/0.05 High25db/0.05 Very High30db/0.005ReusedPilot Sets 14 14 6 6DifferentPilot Sets 42 14 14 6
In the case of finite M , all such cross terms are present, andconsequently the expression for the reverse link SIR at the BScontains many more random variables.Given the complexity of the SIR expression under finite M ,we present results from a Monte Carlo simulation in Table Ifor the maximum number of users that can be admitted percell for a given arbitrary set of QoSs. We present some resultshere to show that significant real world gains are also expectedin terms of user capacity, where the values presented here areonly to serve as an indication. Due to the presence of all intraand inter-cell interference terms, we expect estimates of usercapacity which are less than our previous analysis with M inthe limit. Given recent measurements from real world massiveMIMO experiments, antenna arrays of 128 elements [11] [12]have been used and we select an M of similar order, where M = 500 . From Table I, it can be seen that as soon as weincrease the frequency reuse factor w to meet the given QoS,the interference levels fall well below the requirement, andwe are then simply limited by the reduced available uplinktraining resources. VI. E XTENSIONS
A. Generalised Cooperative Admission Policy
Since all Tier 1 interferers in our system are equivalent, a cooperative user admission policy could manage user admis-sion across the set of cells T ( j )1 which are considered Tier 1interfering cells to cell j , by ensuring only the sum of all usersin these cells is less than the combined amount n max . Undersuch a policy, this constraint would have to be satisfied forevery cell in the network. i.e.: X l ∈ T ( j )1 k l ≤ n max , ∀ j ∈ L. (14)Furthermore, the upper limit of users within the cell is stillrestricted by the length of the pilot sequence, i.e. k l ≤ K/w .Such a policy would increase the probability of a randomlyplaced new user being admitted at the expense of cooperationbetween the BSs of the network.
B. Log-normal shadowing and BS selection1) BS selection:
The path loss due to shadowing can bemodelled by a log-normal random variable z , where typicallyfor macrocell shadowing we have N (0 ,
8) = 10 log z . Whenthis is to be modelled by our slow fading gain, we have β jkl = z jkl r − γjkl , and (4) becomes : kl = φ kl (cid:18) z jkl z lkl (cid:19) (cid:18) r lkl r jkl (cid:19) γ . (15)Due to the large variance of the random quantity ( z jkl /z lkl ) in (15), it can be expected that a given user maynot necessarily experience the best channel to the closest BS,or even to the immediately surrounding BSs. Of course wewould expect that a realistic user admission policy wouldassign the user to the BS to which it experiences the bestchannel gain. As a consequence, for the k -th user in the l -thcell to be considered as an interferer for a user in the j -th cell,the condition β jkl ≥ β lkl must hold. From this inequality, wethen have the constraint on the interference: (cid:18) z jkl z lkl (cid:19) (cid:18) r lkl r jkl (cid:19) γ < , (16)where the l -th cell may or may not be the closest cellto the user of interest. As a result of (16), a true statisticalindependence between these ratios no longer exists. In orderto provide results for k max as in Section V for this model,(16) first needs to be properly incorporated in the analyticalexpression for SIR.
2) Multi Tier user admission:
When realistically modellinglog-normal shadow fading in our system, results from MonteCarlo simulation show we must consider Tier 2 interfererssince they are no longer negligible.
C. Downlink User Capacity
In order to provide the inputs to a comprehensive user ad-mission policy, downlink capacity also needs to be considered.As pointed out in [2], the interferers in the SIR forward linkexpression are no longer strictly i.i.d , and consequently werequire an alternate approach to analyse downlink capacity.The expressions presented in [7] are based on large M SINRresults, and provide an alternative approach into dimensioningthe downlink user capacity.VII. C
ONCLUSION
We derived a reverse link SIR expression for our systemmodel, which was then used as a basis to derive an explicitexpression for the maximum number of users which can beadmitted to a cell in a multi-cell massive MIMO network,when M is large.For both large and finite M , it has been shown that usingdifferent orthogonal pilot sets in each cell, as compared toreusing pilot sets amongst all cells, allows us to admit thesame or significantly more users while upholding a given QoSfor all users of the network.R Wireless Communications, IEEE Transactionson , vol. 9, no. 11, pp. 3590–3600, November 2010. [3] E. Bjrnson, E. G. Larsson, and M. Debbah, “Massive MIMO formaximal spectral efficiency: How many users and pilots should beallocated?”
IEEE Transactions on Wireless Communications , vol. 15,no. 2, pp. 1293–1308, Feb 2016.[4] J. Hoydis, S. ten Brink, and M. Debbah, “Massive MIMO in the UL/DLof cellular networks: How many antennas do we need?”
IEEE Journal onSelected Areas in Communications , vol. 31, no. 2, pp. 160–171, February2013.[5] H. Huh, G. Caire, H. Papadopoulos, and S. Ramprashad, “Achieving”massive MIMO” spectral efficiency with a not-so-large number ofantennas,”
Wireless Communications, IEEE Transactions on , vol. 11,no. 9, pp. 3226–3239, September 2012.[6] J. Shen, J. Zhang, and K. Letaief, “Downlink user capacity of massiveMIMO under pilot contamination,”
Wireless Communications, IEEETransactions on , vol. 14, no. 6, pp. 3183–3193, June 2015.[7] N. Akbar, N. Yang, P. Sadeghi, and R. A. Kennedy, “User capacityanalysis and pilot design for multi-cell multiuser massive MIMOnetworks,”
CoRR , vol. abs/1511.07568, 2015. [Online]. Available:http://arxiv.org/abs/1511.07568[8] J. Evans and D. Everitt, “Effective bandwidth-based admission controlfor multiservice CDMA cellular networks,”
Vehicular Technology, IEEETransactions on , vol. 48, no. 1, pp. 36–46, Jan 1999.[9] T. Bai and R. W. H. Jr., “Analyzing uplink SIR and rate in massiveMIMO systems using stochastic geometry,”
CoRR , vol. abs/1510.02538,2015. [Online]. Available: http://arxiv.org/abs/1510.02538[10] J. Evans and D. Everitt, “On the teletraffic capacity of CDMA cellularnetworks,”
Vehicular Technology, IEEE Transactions on , vol. 48, no. 1,pp. 153–165, Jan 1999.[11] X. Gao, O. Edfors, F. Rusek, and F. Tufvesson, “Massive MIMOin real propagation environments,”