1

Load Balancing User Association inMillimeter Wave MIMO Networks

Alireza Alizadeh, Student Member, IEEE, Mai Vu, Senior Member, IEEE

Abstract—User association is necessary in densemillimeter wave (mmWave) networks to determinewhich base station a user connects to in order tobalance base station loads and maximize throughput.Given that mmWave connections are highly direc-tional and vulnerable to small channel variations,user association changes these connections and hencesignificantly affects the user’s instantaneous rate aswell as network interference. In this paper, we intro-duce a new load balancing user association schemefor mmWave MIMO networks which considers thisdependency on user association of user’s transmissionrates and network interference. We formulate the userassociation problem as mixed integer nonlinear pro-gramming and design a polynomial-time algorithm,called Worst Connection Swapping (WCS), to find anear-optimal solution. Simulation results confirm thatthe proposed user association scheme improves net-work performance significantly by moving the trafficof congested base stations to lightly-loaded ones andadjusting the interference accordingly. Further, theproposed WCS algorithm outperforms other genericalgorithms for combinatorial programming such asthe genetic algorithm in both accuracy and speed atseveral orders of magnitude faster, and for small net-works where exhaustive search is possible it reachesthe optimal solution.

Index Terms—User association; mmWave MIMO sys-tems; load balancing; MINLP; polynomial-time algo-rithm.

I. Introduction

BANDWIDTH shortage is a major challenge forcurrent wireless networks. Most of the available

spectrum at microwave frequencies is occupied whilethere is a pressing need for higher throughputs andlarger bandwidths [1]. During the past few years, mil-limeter wave (mmWave) frequencies have attracted theinterest of academia and industry due to the capabilityof multi-Gbps data rates and the huge amount ofbandwidth available at frequencies between 30 - 300GHz. MmWave communication for the new 5th gen-eration (5G) cellular systems is a promising solutionto the current spectrum shortage and ever-increasingcapacity demand [2], where its feasibility has beendemonstrated not only for backhaul but also for mobileaccess [3]-[5].

User association plays an important role in resourceallocation for cellular networks. While user associ-ation has been studied in the context of heteroge-neous networks (HetNets) [6], massive multiple-inputmultiple-output (MIMO) systems [7], and mmWave

Wi-Fi networks [8], the problem in mmWave cellu-lar networks is significantly different. Because of thehigher path loss and sensitivity to blockage, mmWavesignals can only propagate a much shorter distancethan the current RF signals, making it necessary forthe mmWave cellular network to be dense. The basestation (BS) density is expected to be higher by anorder of magnitude [9], and each user equipment (UE)is likely to be surrounded by a number of BSs, wherethe channel to each of these BS can vary significantlygiven the erratic nature of mmWave propagation withprobabilistic line-of-sight (LOS), non-LOS (NLoS) andoutage events. In addition, mmWave systems requirethe use of MIMO beamforming at both the transmitterand receiver ends [5], different from massive MIMOwhich has beamforming at only the BS. Hence theformulation of a user association problem for mmWaveis different from other systems and requires a newapproach.

A. Background and Related WorkMax-SINR user association is a traditional method

of using the highest signal to interference plus noiseratio (SINR) or largest received power to associate anUE with a BS [10]. This technique has been workingwell in cellular networks at microwave frequencieswhere all cells are homogeneous macro cells. Theemergence of HetNets with smaller, lower power pico,femto and relay BSs requires a different look at userassociation to ensure load balancing across the net-work, avoiding the case that all users connect to onlya macro BS because of its strongest SINR and overloadit.

Cell breathing is a technique that balances the loadin a single tier network consisting of BSs of the sametype by allowing each BS to adjust its transmit power,based on its load, to change its coverage area [11]-[12].If the BS is underloaded (overloaded), the transmitpower increases (decreases) to expand (contract) thecoverage region. HetNets also requires balancing thenetwork load among different tiers. A conventionalapproach is biasing in which the transmit power of BSsin a low-power lightly loaded tier is artificially biasedin order to seem more attractive than congested BSs[13]. Although biasing can be done both within andamong tiers, it cannot precisely control the load oneach BS [14]-[15].

A load balancing user association scheme for Het-Nets uses an optimization theoretic approach to

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achieve the balance among all BSs across differenttiers in the network [6]. This reference presented anovel user association technique with integer con-straints on association coefficients and showed thatthis unique association problem is a combinatorialprogram, which is NP-hard. Considering a heuristicapproach, the researchers first relaxed the uniqueassociation constraints to solve for a joint associationproblem (allowing a UE to be served by multiple BSs),and then rounded the relaxed solution. This relaxationreduced the complexity of the problem and providedan upper performance bound for the original uniqueassociation.

In a similar work to [6], the problem of user as-sociation in HetNets with massive MIMO BSs andsingle-antenna UEs is studied in [7]. Because of thechannel hardening effect in massive MIMO, the consid-ered association approach is independent of the userinstantaneous rate or instantaneous channel stateinformation (CSI), but only depends on convergingdeterministic rate values as a function of the systemparameters and large-scale CSI. Even though uniqueassociation is considered, the association coefficientsare not necessarily integers, but represent the limit ofthe fraction of time slots in which each UE is servedby one BS. Using this interpretation of fractional userassociation as long-term time average, the researchersthen formulated user association as a convex networkutility maximization problem and solved it using La-grangian duality.

Optimizing user association is also studied in a 60-GHz mmWave Wi-Fi network [8]. The optimizationproblem in [8] is similar to the one appeared in [6], butinstead of maximizing network throughput it is formu-lated to minimize the maximum per-BS load subject todemanded user data rates. At this high atmospheric-absorption frequency, the user instantaneous ratesconverge to deterministic values and the interferenceis assumed to be negligible due to directional steerableantenna arrays. This assumption simplifies the userassociation problem compared to the case when wemust take into account the effect of interference, asin a dense mmWave cellular network at the proposedfrequencies (28, 38, and 73 GHz), for which interfer-ence has been noted for outdoor environments [3]-[4],[16].

A number of other existing works also consideredthe joint problem of user association, beamformingdesign, and power allocation [17]-[19]. This joint prob-lem is shown to be NP-hard, and researchers usuallyproposed iterative algorithms to achieve near-optimalsolutions. Algorithms solving such an NP-hard jointoptimization problem may be too complex to be imple-mented in practice. As such, we take the alternativeapproach of fixing the beamforming designs and opti-mize for user association alone.

Most of recent works on user association ignore theeffect of small-scale fading (instantaneous CSI) on the

user instantaneous rate and consider a scalar channelmodel with only large-scale fading [20]-[25], hencethe SINR used for user association is only a functionof large-scale parameters (e.g. distance) regardless ofsmall-scale channel variations. Specifically, the SINRterm in the capacity formula is not a matrix but ascalar value indicating that the mmWave MIMO small-scale variations were ignored and the system is usuallysimplified by only considering directional gains formmWave antenna. This consideration reduces the com-plexity of the analysis, but this approach is not suitablefor mmWave systems where the channel variation canbe fast and the instantaneous CSI may change rapidly[9]. Our simulation results confirm that consideringboth large-scale and small-scale CSI significantly im-proves the average network spectral efficiently.

B. Our contributionsIn this paper we introduce a new user association

problem formulation suitable for mmWave networks.Specifically, unlike the typical approach in user associ-ation for HetNets [6], massive MIMO [7], and mmWaveWi-Fi [8] which accounts only for large-scale CSI, weconsider both the large-scale and instantaneous CSIin our formulations. This consideration is suitable formmWave networks where the channels can vary signif-icantly even during two consecutive time slots. In thecase that instantaneous CSI is unavailable, however,we can still use the proposed user association schemebut only with large-scale CSI which will provide amuch longer time span. Adaptation to instantaneousCSI will further improve the network performance. Asour user association scheme is independent of channelmodel, the ideal assumption of having both large-scaleand instantaneous CSI can be considered as a perfor-mance upper bound. In practical scenarios we may usethe proposed user association scheme with only large-scale CSI or with both types of CSI depending on theavailability.

Furthermore, previous works ignore the effect ofuser association on network interference; consequentlyin the downlink for example, each user suffers fullinterference from all other BSs, regardless of theirassociation. While this assumption makes sense in anomnidirectional transmission, it no longer holds in adirectional transmission as is the case in mmWave,where the interference will change substantially de-pending on which UE beamforms with which BS.The network interference structure in mmWave ishighly dependent on user association and we need toconsider this dependency while computing the userrates. Not adapting the interference to user associationunfortunately degrades the network spectral efficiencyseverely.

The main contributions of this paper are summa-rized as follows:• We propose a new load balancing user association

scheme which takes into account the dependency

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of network interference on user association. Inparticular, we formulate the user instantaneousrate as a function of user association, as is the casein mmWave. Consequently, the total interferencecoming from other BSs (while serving other UEs)also depends on association coefficients.

• We introduce an Activation Matrix which spec-ifies the BS each UE connects to during eachtime slot, from which association coefficients arederived as its time average. The activation ma-trix allows per-time-slot user association whichis necessary because the instantaneous CSI of ammWave channel may change significantly evenduring two consecutive time slots.

• For the proposed user association scheme, weformulate an optimization problem as a mixedinteger nonlinear programming (MINLP), whichis known to be NP-hard due to its non-convexnonlinear structure and presence of integer vari-ables. This optimization problem is similar in formto those derived in [6]-[7], but instead of beinga linear function of the user association coeffi-cients, its objective function is highly dependenton the network interference structure and userassociation in a non-linear fashion. This makes theoptimization problem highly non-convex and chal-lenging, which in turn makes our proposed userassociation scheme suitable for mmWave cellularnetworks and reveals its novelty.

• We design a polynomial-time specialized efficientalgorithm, called Worst Connection Swapping(WCS), to solve the formulated MINLP and finda near-optimal solution for the user association.This algorithm is based on the intuition that swap-ping the worst UE-BS connection is likely to pro-vide the UE a stronger link to another BS and/orreduce the interference, which consequently im-proves the user’s rate. Complexity analysis revealsthat our proposed algorithm’s runtime grows asK2 log(K)M2 with K as the number of UEs andM as the number of antennas at each BS.

• Our simulation results show that considering in-stantaneous CSI significantly improves the net-work spectral efficiency. Further, the proposedload balancing scheme outperforms max-SINRand other load balancing schemes by includingthe dependency of interference on user associa-tion. Numerical results also confirm the analysisof algorithm complexity and show that the WCSalgorithm has fast convergence with the numberof iterations growing linearly with the numberof UEs, and the runtime growing quadratically,which is several orders of magnitude faster thangenetic algorithm (GA) and exhaustive search.

• Our proposed user association scheme can be inte-grated with any mobility model since it is designedto be performed per-time slot. As an illustration,we provide simulation results of our scheme ap-

plied to a simple user mobility model where theUEs have fixed locations during each time slot,and in the consecutive time slot each UE movesto a new random location within a given rangefrom its location in the previous time slot. Oursimulations show the effect of mobility on theassociation results, for example, a UE with strongsignal from a particular BS tends to be associatedwith that BS even as it moves around within avicinity, whereas a UE with comparable signalsfrom several BSs can switch association frequentlyas it mobilizes.

C. Paper OrganizationThe rest of the paper is organized as follows. We

start with the channel and system model in Section II.In Section III, we introduce our load balancing userassociation scheme and define the activation and as-sociation matrices. The user association optimizationproblem is formulated in Section IV. In Section V, weintroduce the WCS algorithm and analyze its complex-ity. We present the numerical results in Section VI,and Section VII contains our conclusion.

D. NotationThroughout this paper, scalars are represented by

lowercase letters, vectors are denoted by lowercaseboldface letters and matrices by uppercase boldface let-ters. Superscript (.)∗ denotes the conjugate transpose,log(.) stands for base-2 logarithm, mod(.) representsthe modulo operation, and big-O notation O(.) ex-presses the complexity. IN is theN×N identity matrix,and we use tr(A) and |A| to denote the trace anddeterminant of matrix A, respectively. The distributionof a complex Gaussian vector x with mean µ andcovariance matrix Q is denoted by x ∼ CN (µ,Q).

II. Channel and System ModelA. mmWave Channel Model

The mmWave channel has completely different char-acteristics compared to the typical microwave modelof a rich scattering independent and identically dis-tributed (i.i.d.) channel. The channel model consideredin this paper is based on the clustered channel modelintroduced in [26] and the 3GPP-style 3D channelmodel proposed for the urban micro (UMi) environ-ments in [27], which is based on the measurementsobtained in New York City. This channel model has Cclusters with L rays per cluster, and it can be expressedas

H =1√CL

C∑c=1

L∑l=1

√γc a(φUE

c,l , θUEc,l ) a∗(φBS

c,l , θBSc,l ) (1)

where γc is the power gain of the cth cluster. Theparameters φUE, θUE, φBS, θBS represent azimuth an-gle of arrival (AoA), elevation angle of arrival (EoA),

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azimuth angle of departure (AoD), and elevation angleof departure (EoD), respectively. These parameters aregenerated randomly based on different distributionsand cross correlations as given in [27, Tables 1-3].The vector a(φ, θ) is the antenna array response vec-tor, which depends on antenna geometry, includinguniform linear array (ULA) or uniform planar array(UPA). In order to enable beamforming in the elevationdirection (3D beamforming), we use the uniform U×Vplanar array given by [26]

a(φ, θ) =[1, ..., ejkda(u sin(φ) sin(θ)+v cos(θ)), ...,

ejkda((U−1) sin(φ) sin(θ)+(V−1) cos(θ))]T (2)

where da is the distance between antenna elements,and u ∈ {1, ..., U} and v ∈ {1, ..., V } are the indices ofantenna elements.

We consider two link states for each channel, LoSand NLoS, and use the following probability functionsobtained based on the New York City measurementsin [28]

pLoS(d) =[min

(dBPd, 1).(1− e−

dη

)+ e−

dη

]2(3)

pNLoS(d) = 1− pLoS(d) (4)

where d is the 3D distance between UE and BS inmeters, dBP is the breakpoint distance at which the LoSprobability is no longer equal to 1, and η is a decay pa-rameter. The obtained values based on measurementsfor these parameters are dBP = 27 m and η = 71 m.

Moreover, we use the following path loss model forLoS and NLoS links [27]

PL[dB] = 20 log10

(4πd0λ

)+ 10n log10

( dd0

)+XσSF (5)

where λ is the wavelength, d0 is the reference distance,n is the path loss exponent, and XσSF is the lognor-mal random variable with standard deviation σSF (dB)which describes the shadow fading. These parametersvary depending on LoS of NLoS propagation at 73GHz, the path loss exponents and the shadowing fac-tors are nLoS = 2, nNLoS = 3.4, σSF, LoS = 4.8 dB, andσSF, NLoS = 7.9 dB.

In 4G cellular networks, pilot signals are used toestimate CSI at the receiver. Once the CSI is availableat the receiver, it can be shared with the transmittervia limited feedback or channel reciprocity. However,in 5G dense HetNets these conventional approachesare inapplicable due to network densification and thelimited amount of pilot resources [29]. To address thenew challenges emerging from network densification,a promising radio access technology, called cloud ra-dio access network (C-RAN), has been proposed [30].In this radio network, CSI at both the transmitterand the receiver can be estimated through new CSIacquisition schemes and shared via C-RAN for cen-tralized signal processing, coordinated beamforming,and resource allocation in 5G new radio [31], [32]. Inthis paper, we assume that the CSI is estimated for

the aforementioned applications and we can utilize itfor the purpose of user association. We note that C-RAN is an architecture that enables this shared CSIand centralized user association, but it may not be theonly one, and the CSI required for beamforming in 5Gcan be used for user association. Further, distributedalgorithms can also be developed for the same problemformulation which we will consider here, but do notrequire C-RAN because of the distributed nature.

Different user association schemes make use of dif-ferent types of channel models: when an algorithmuses the detailed clustered channel model in (1),we call this instantaneous CSI, and when an algo-rithm uses only path loss, shadowing and LOS/N-LOS channel models as in (3)-(5), we call this large-scale CSI. Our proposed user association scheme asdiscussed in Section III utilizes both large-scale andinstantaneous CSI as given in (1) . The scheme, how-ever, is also applicable if the channel model insteadcontains only large-scale fading as considered in [8,eq. (1)]. In the simulation section, we show examples ofboth cases and provide comparison with other schemeswhich mostly rely on large-scale CSI.

B. System and Signal ModelWe consider a downlink scenario in a cellular

mmWave MIMO network with J BSs and K UEs.Mj and Nk are the number of antennas at BS j andUE k, respectively, where we assume Mj ≥ Nk, ashandsets usually have fewer antennas than BSs. LetJ = {1, ..., J} denotes the set of BSs and K = {1, ...,K}represents the set of UEs. Each UE k aims to re-ceive nk data streams from its serving BS such that1 ≤ nk ≤ Nk, where the upper inequality comes fromthe fact that the number of data streams for each UEcannot exceed the number of its antennas.

Thus, we can define the total number of downlinkdata streams sent by BS j as

Dj =∑

k∈Qj(t)

nk (6)

We define Qj(t) as the Activation Set which representsthe set of active UEs in BS j within time slot t, suchthat Qj(t) ⊆ K and |Qj(t)| = Qj(t) ≤ K. Note that thetotal number of downlink data streams sent by each BSshould be less than or equal to its number of antennas,i.e., Dj ≤ Mj . For notational simplicity, we drop thetime index t in definition of Dj , and only keep the timeindex for Qj(t) due to its importance.

The Mj × 1 transmitted signal from BS j is given by

xj = Fjdj =∑

k∈Qj(t)

Fk,jsk (7)

where sk ∈ Cnk is the data stream vector for UE kconsists of mutually uncorrelated zero-mean symbols,with E[sks∗k] = Ink . The column vector dj ∈ CDjrepresents the vector of data symbols of BS j, which

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is the vertical concatenation of the data stream vec-tors sk, k ∈ Qj(t), such that E[djd∗j ] = IDj . MatrixFk,j ∈ CMj×nk is the linear precoder for each UE kassociated with BS j, and Fj ∈ CMj×Dj is the completelinear precoder matrix of BS j which is the horizontalconcatenation of all Fk,j , k ∈ Qj(t).

The power constraint at BS j can be described as

E[x∗jxj ] =∑

k∈Qj(t)

Tr(Fk,jF∗k,j) ≤ Pj (8)

where Pj is the transmit power of BS j. Now we canexpress the Nk × 1 received signal at UE k antennasas

yk =∑j∈J

Hk,jxj + zk, (9)

and the post-processed signal received by each UE is

yk =∑j∈J

W∗kHk,jxj + W∗

kzk (10)

where Wk ∈ CNk×nk is the linear combiner matrixof UE k, Hk,j ∈ CNk×Mj represents the channelmatrix between BS j and UE k, and zk ∈ CNkis the white Gaussian noise vector at UE k, withzk ∼ CN (0, N0INk). The presented signal models (7)-(10) are applicable for all types of transmit beam-forming and receive combining. Later for user associ-ation optimization, we will specify the transmit andreceive beamforming schemes. Further, based on C-RAN architecture as discussed earlier, each BS knowsits channels to all UEs and can share that CSI withall other BSs for both beamforming design and userassociation purposes.

III. MmWave Load Balancing User AssociationIn the literature, when computing the instantaneous

rate for a specific UE (connected to a BS), the in-terference coming from other UE-BS connections isassumed to be both independent of the user associationand present all the time (full interference) [6]-[8].This assumption is not realistic in mmWave systemsdue to the use of beamforming and results in lowerinstantaneous user rates. In mmWave systems, thenetwork interference highly depends on user associa-tion and we need to consider this while computing userrates. Moreover, user association depends on channelrealizations which is highly directional and can varyfast in mmWave frequencies. Thus, we cannot use thefull interference structure in mmWave systems.

In this section, we introduce a new user associationmodel for mmWave MIMO networks. First, we needto introduce our definition of a time slot throughoutthis paper. Each time slot t is a duration of timecomparable to channel coherence time such that theinstantaneous CSI remains relatively the same withinit and only changes from one time slot to another.During time slot t, each UE is connected to exactly one

BS. Because of beamforming, the interference struc-ture in each time slot depends on the user associationon that specific slot. This interference structure isappropriate for mmWave channels where the channelis probabilistic and can vary fast. Moreover, we assumeit is possible to split the data streams of each UEand transmit them in different time slots. Consideringabove definitions, we study two association approachesin this paper: (i) instantaneous user association, whichis performed within each time slot and results inunique association (each UE can be associated withonly one BS during each time slot), and (ii) fractional(joint) user association, which is obtained by averagingthe instantaneous association over T time slots (for anarbitrarily large T ). At each time slot, the mmWavechannels are generated independently based on thechannel model presented in Section II-A. We considerboth approaches to evaluate the performance of ourproposed user association model and compare it withexisting user association schemes.

It is worth mentioning that our proposed per-time-slot user association can be carried out under thesynchronizations requirement of modern cellular sys-tems. In LTE TDD networks, the basic synchronizationrequirement for any pair of cells with overlappingcoverage areas which operate on the same frequencyis 3 µs [33], and it would be in the order of 1 µs orbelow for 5G systems [34]. Moreover, based on thework in [35], a time slot of duration Tslot = 100 µsis considered as a sufficiently short time duration toensure channel coherence at mmWave systems. Sincethe synchronization time is relatively small comparedto the time slot duration, synchronization will not bean issue for performing per-time-slot user association.

We differentiate between user association and userscheduling. Note that user association determines theUE-BS connections among multiple BSs, while userscheduling allocates the available resources (in thiscase, data streams) at each BS to its associated UEs.Thus, user scheduling can be considered as a post-association stage. In the literature, it is shown that thejoint problem of user association and user schedulinghas high computational complexity and researchersusually decouple the joint optimization problem intosub-problems to find the optimal solutions [6], [36]. Inthis paper, we assume that the BSs’ loads are deter-mined based on their available resources. Specifically,we consider the multi-user MIMO scenario where ateach time slot each BS can simultaneously transmit tomultiple UEs by constraining the available number ofdata streams at each BS to be smaller than the numberof its antennas. Therefore, at each time slot, the BScan serve all its associated users simultaneously, bybeamforming simultaneously to multiple users usingmultiple beams, and there are enough spatial degreesof freedom for all the associated users as long as theaforementioned constraint is satisfied. As such, in this

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case, there is no need to perform scheduling amongthe associated users. In the case that the number ofrequired data streams at each time slot is larger thanthe number of BS antennas, then user scheduling isinevitable. In such a case, a simple scheduling policy(e.g. round robin scheduling [36]) or more efficientscheduling schemes (e.g. proportional fairness schedul-ing [6] or CDF-based scheduling [37], [38]) can beimplemented to allocate resources of each BS amongits associated users.

We start by defining the Activation Matrix as

B ,[β(1) · · · β(T )

]=

β1(1) · · · β1(T )... . . . ...

βK(1) · · · βK(T )

(11)

where β(t) is called the Activation Vector at time slott, and each element of B is called an activation factorand is the index of BS to whom user k is associatedwith during time slot t, i.e., βk(t) ∈ J with k ∈ K andt ∈ T = {1, ..., T}. Considering above definition, therelationship between the activation set of BS j and theactivation factors can be described as

Qj(t) = {k : βk(t) = j}. (12)As stated earlier, we assume each UE can be associatedwith only one BS at any time slot t, i.e.,

Qj(t) ∩Qi(t) = ∅, j 6= i (13)J⋃j=1

Qj(t) = K (14)

where (14) indicates that during each time slot all UEsare served by the BSs.

The elements of activation matrix should satisfy thefollowing conditions∑

j∈J1βk(t)(j) ≤ 1, ∀k ∈ K (15)∑

k∈K

1βk(t)(j).nk ≤ Dj , ∀j ∈ J (16)

where the indicator function is defined as

1βk(t)(j) ,

{1, βk(t) = j

0, βk(t) 6= j(17)

The activation constraints in (15) reflect the fact thateach UE cannot be associated with more than oneBS in each time slot, and the resource allocation con-straints in (16) denote that the sum of data streamsof UEs served by each BS cannot exceed the totalnumber of available data streams on that BS. Notethat 1βk(t)(j) is equal to one only if βk(t) = j orequivalently, k ∈ Qj(t). Thus, the summation in (16)is actually over the set of active UEs in BS j.

Next, we define the Association Matrix A as follows

A ,

α1,1 · · · α1,J

... . . . ...αK,1 · · · αK,J

(18)

where αk,j ∈ [0, 1] is the (fractional) association coeffi-cient representing the average connectivity of UE k toBS j. If αk,j = 0, we say UE k is not associated withBS j over time. In this model, association coefficientsare considered as a fraction of time. Specifically, αk,j isthe average fraction of time during which UE k is con-nected to BS j. The relationship between associationcoefficients and activation factors is given by

αk,j = limT→∞

1

T

T∑t=1

1βk(t)(j). (19)

According to (19), given the activation matrix B, onecan easily obtain the association matrix A. For thisreason, we formulate our user association problemin terms of the activation factors, which are integervariables representing the serving BS indices.

Since the proposed user association scheme is car-ried out per-time-slot, it can result in frequent han-dover, especially in the case that the UE is mobileand has comparable signal strengths from severalsurrounding BSs. While we do not consider handovercost explicitly in this work, we note that the proposeduser association can be applied using only large-scaleCSI which will result in a longer time span for eachassociation and lessen handover. In Section VI, weconsider a simple mobility model where the locationof each UE is fixed during each time slot, and changesin the subsequent time slot to a new random locationwithin a fixed range from its previous location. Theuser association is then performed at each time slotto assess the effect of mobility on association andhandover.

IV. User Association Optimization ProblemIn this section, we first evaluate the instantaneous

and average per-user throughputs by processing thereceived signal at each user. Then, we formulate anoptimization problem to find the optimal activationfactors that maximize the throughputs while satisfyingload balancing constraints on the BSs.

A. Formulation of instantaneous user rateConsidering (7), the received signal by UE k in (10)

can be decomposed as

yk(t) = W∗kHk,jFk,jsk︸ ︷︷ ︸

Desired signal

+W∗kHk,j

∑l∈Qj(t)l 6=k

Fl,jsl

︸ ︷︷ ︸Intra-cell interference

+ W∗k

∑i∈Ji 6=j

∑l∈Qi(t)

Hk,iFl,isl

︸ ︷︷ ︸Inter-cell interference

+W∗kzk︸ ︷︷ ︸

Noise

(20)

where the first term is the desired received signalfrom the serving BS j, the second term represents theinterference coming from the same BS j by signals

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intended for its other active UEs, the third term isthe interference coming from other BSs i 6= j bysignals sent to their active UEs, and the last term isthe thermal noise at UE k. The activation sets Qj(t)and Qi(t) appeared in the interference terms indicatethat the interference is highly dependent on the userassociation, which highlights the novelty of this work.Again, we note that all vectors and matrices in (20)are time-dependent, and the time index t is droppedfor the sake of notational simplicity.

When UE k is connected to BS j in time slot t, itsinstantaneous rate can be obtained as [39]

Rk,j(t) = log2

∣∣∣Ink+(Vk,j(t))−1W∗

kHk,jFk,jF∗k,jH

∗k,jWk

∣∣∣(21)

where Vk,j is the interference-plus-noise covariancematrix given as

Vk,j(t) = W∗kHk,j

( ∑l∈Qj(t)l 6=k

Fl,jF∗l,j

)H∗k,jWk

+ W∗k

(∑i∈Ji 6=j

∑l∈Qi(t)

Hk,iFl,iF∗l,iH

∗k,i

)Wk +N0W

∗kWk.

(22)

The instantaneous rate given in (21) is a function ofactivation sets Qj(t). Thus, the instantaneous per-userthroughput at time slot t can be expressed as

rk(t) =∑j∈J

1βk(t)(j)×Rk,j(t), (23)

and the average per-user throughput is given by

rk = limT→∞

1

T

T∑t=1

rk(t). (24)

B. Optimization ProblemAs stated before, the channel variation can be fast

and unpredictable in mmWave frequencies and theclustered channel as modeled in (1) may change signifi-cantly even during two consecutive time slots. Thus, weneed to perform the user association in each time slot,which is comparable with the channel coherence time.Defining the instantaneous user throughput vectorr(t) , (r1(t), ..., rK(t)), we wish to find the optimal acti-vation vector β(t) which maximizes an overall networkutility function. This utility function should be concaveand monotonically increasing. Our proposed user asso-ciation scheme can implicitly incorporate user fairnessvia the choice of network utility function including thefamily of utility functions defined in [7].

In this paper, we consider two well-known andwidely used utility functions. The sum-rate utilityfunction defined as

Us(r(t)) ,∑k∈K

rk(t) (25)

and the min-rate utility function given by

Um(r(t)) , mink∈K

rk(t). (26)

Maximizing the first utility function corresponds toachieving the highest network throughput, while max-imizing the second utility function guarantees a fairuser association and improves the throughput for cell-edge UEs who usually suffer from low data rates.

Thus, the user association optimization problem attime slot t can be written as

maximizeβ(t)

U(r(t)) (27a)

subject to∑j∈J

1βk(t)(j) ≤ 1, ∀k ∈ K (27b)∑k∈K

1βk(t)(j).nk ≤ Dj , ∀j ∈ J (27c)

This is an optimization problem with integer variablesβk(t) ∈ J . Here, we use equal power allocation schemeto split the BS power among its active users. Thus,the power constraint in (8) is automatically satisfiedand can be ignored. Note that the set of constraints in(27c) allows our user association scheme to limit eachBSâĂŹs load separately, and assuming the number ofdata streams (Dj) is chosen based on the availableresources at each BS, this set of constraints guaranteesthat each BS can serve all its associated UEs simulta-neously and thus scheduling is not required.

As discussed earlier, we adopt the approach of fixingthe beamforming scheme and only optimize for userassociation. For beamforming in the ideal conditionof instantaneous CSI and absence of interference, theoptimal beamforming vectors of a channel from UE kto a BS j are the eigenvectors corresponding to the firstnk largest singular values, where nk is the number ofdata streams intended for UE k [40]. This ideal caseprovides an upper bound on beamforming performancein a realistic network. Considering (21), it is clearthat our formulation of the user instantaneous rateincludes the precoder and combiner matrices of all UEsand BSs which are applicable using any beamformingmethod. Thus, it is plausible that a user associationscheme based on singular value decomposition (SVD)beamforming is compatible with user association basedon other beamforming approaches and can achievenear-optimal results.

We use SVD beamforming to obtain the precoderand combiner matrices. To this end, we first need todecompose the channel matrix Hk,j ∈ CNk×Mj as

Hk,j = ΦΣΓ∗ (28)

where Φ ∈ CNk×r is the unitary matrix of left singularvectors, Σ ∈ Cr×r is the diagonal matrix of singularvalues (in decreasing order), Γ ∈ CMj×r is the unitary

8

matrix of right singular vectors, and r = rank(Hk,j).Then, we partition the channel matrix as

Hk,j =[Φ1 Φ2

] [Σ1 00 Σ2

] [Γ∗1Γ∗2

]= Φ1Σ1Γ

∗1 + Φ2Σ2Γ

∗2

(29)

where Φ1 ∈ CNk×nk , Σ1 ∈ Cnk×nk , and Γ1 ∈ CMj×nk .The above partitioning is done to extract the precoderand combiner of appropriate sizes to support the num-ber of data streams of each user k. Specifically, eachprecoder Fk,j and combiner Wk must be of size Mj×nkand Nk×nk, respectively. Then, the SVD precoder andcombiner can be obtained as

Fk,j = Γ1 (30)Wk = Φ1. (31)

The optimization problem in (27) is a highly non-convex MINLP, which is known to be NP-hard due toits non-convex and nonlinear structure and presence ofinteger variables [41]. The non-convexity and nonlin-earity come from the fact that interference structurecontains association variables, specifically the indica-tor function appeared in (23), and constraints (27b-c).This feature differentiates our formulated optimizationproblem from user association formulations consideredin [6]-[7]. These MINLP problems are typically difficultto solve due to their combinatorial structure and po-tential presence of multiple local maxima in the searchspace. Exhaustive search is the only known techniqueto guarantee the optimal solution but quickly becomesinfeasible with problem size due to an exponentialcompletion time.

Suboptimal and often heuristic methods exist forsolving such an MINLP. For example, genetic algo-rithm (GA) is a such tool which can be effective forsolving combinatorial optimization problems. GA is amethod based on natural selection which simulates bi-ological evolution. It iteratively generates and modifiesa population of individual solutions. After successivegenerations, the population eventually evolves towarda near-optimal solution [42]. In [43], we used theGA solver provided in Global Optimization Toolbox ofMATLAB to solve the optimization problem in (27).We showed that the GA works effectively for smallnetworks. However, we found that GA does not workwell when the network size increases, taking an ex-ceedingly long time to run and still not reaching a good(close to optimal) solution. In this paper, we propose anefficient algorithm to find a near-optimal solution forour MINLP.

V. Worst Connection Swapping (WCS) AlgorithmIn the previous section, we formulated the optimiza-

tion problem in (27) which is an NP-hard MINLP.In this section, we introduce an efficient iterativealgorithm which converges to a near-optimal solutionin a polynomial time. The algorithm is based on the

Algorithm 1: WCS User Association Algorithm1 initialization:2 - Generate an arbitrary feasible activation

vector β(1);3 - Find the initial worst connection

(k(1), j(1)) = argmink,j Rk,j(β(1));

4 repeat5 Swap the k(i)th element (activation factor)

with all other elements of β(i) to obtain(K − 1) new vectors β(i)

n with n ∈ K, n 6= k(i);6 Find the best activation vector

β(i+1) = arg max{β(i)n |n∈K, n6=k(i)}, β(i) U(r(β));

7 Find the worst connection(k(i+1), j(i+1)) = argmink,j Rk,j(β

(i+1));8 if β(i+1) = β(i) then9 l(m) ← mod(m− 1,K) + 1;

10 Switch the k(i+1)th and l(m)th elements ofβ(i+1) to obtain β(i+1)

sw ;11 β(i+1) ← β

(i+1)sw ;

12 m← m+ 1;13 end14 Stopping criterion: Break if the best β does

not change after K consecutive swappingiterations;

15 i← i+ 1;16 until stopping criterion achieved;

rational that a user association throughput may besuboptimal because of the weakest UE suffering froma weak direct link or high interference from otherBSs in the network. Thus, swapping the worst UE-BS connection is likely to provide the UE a strongerlink (better connection to another BS) and/or reducethe interference, which consequently improves userinstantaneous rates and hence the throughput. Thisreasoning leads us to design an algorithm to swap theBS in the worst UE-BS connection with all other BSsand then take the best configuration (activation vector)which results in the highest network utility function.Next, we describe the algorithm in detail.

A. Algorithm DescriptionThe proposed algorithm is summarized in Algorithm

1 and works as follows. It starts with a random fea-sible initial activation vector β(1) which satisfies theconstraints in (27b-c), and an initial worst UE-BSconnection k(1) obtained from β(1) as described later in(33). The algorithm has two main steps: i) Swappingstep, ii) Switching step. In what follows, we describethese steps in detail.

1) Swapping step: At each iteration i, we swap theworst connection β(i)(k(i)), which is the index of theBS with worst UE-BS connection in current activationvector β(i), with all other connections (i.e. other BSentries in the same activation vector β(i)). Since the

9

size of an activation vector is K, this swapping resultsinK−1 new activation vectors β(i)

n with n ∈ K, n 6= k(i).Then, the network utility function, defined in (25)-(26),is computed for the current and new activation vectors,and the one corresponding to the largest network util-ity function is chosen as the best activation vector forthe next iteration, i.e.,

β(i+1) = arg max{β(i)n |n∈K, n6=k(i)}, β(i)

U(r(β)) (32)

Next, we find the worst UE-BS connection in β(i+1) bycomparing the user instantaneous rates as

(k(i+1), j(i+1)) = arg mink,j

Rk,j(β(i+1)). (33)

and then repeat the process until no improvement inthe utility function is achieved. Note that the algorithmis applied for any given time slot t to obtain the optimalactivation vector β(t). Here, we dropped the time indext for the ease of notation.

Proposition 1: The sequence of network utility func-tion values produced by Algorithm 1 is non-decreasing.

Proof : Since the user instantaneous rate Rk,j andthe network utility function U(r) are both functions ofthe activation vector, at each iteration based on (32),the network utility function either remains the same(when β(i+1) = β(i)) or improves (when β(i+1) 6= β(i)).

The above procedure works effectively if the objectivefunction is always increasing in all iterations, i.e.β(i+1) 6= β(i), ∀i. However, at some point, the currentand best activation vectors may be equal. In thiscase, the algorithm repeatedly produces the same bestactivation vector (and consequently the same worstconnection) in the following iterations. When that hap-pens, swapping does not lead to any more changes inthe connections of other users or the network utilityfunction. Thus, we need to design another step, theswitching step, in order to overcome this problem. Notethat if β(i+1) = β(i), then the worst connections areequal, i.e. k(i+1) = k(i), but the reverse is not true ingeneral.

2) Switching step: In order to resolve the aboveissue and find new possibility to improve the objec-tive function, we introduce a switching step. At eachiteration, we compare β(i+1) and β(i). If they are notequal, no further action is required and the algorithmproceeds to the next iteration. If they are equal, thenwe switch the worst connection k(i+1) (which is equalto k(i)) with another user to find a potentially betteractivation vector. To this end, at each switching stepm, we compute a UE index as l(m) = mod(m−1,K)+1and switch the k(i+1)th and l(m)th elements of thecurrent β(i+1) to obtain the switched activation vectorβ(i+1)sw . Then, we use this vector as the initial activation

vector for the next swapping iteration. Note that theswitching process does not affect the non-decreasingnature of the algorithm, since in the swapping stepof next iteration, the pre-switching activation vector

β(i+1) appears in β(i+2)n and therefore is included in

the utility function comparisons (see (32)). Moreover,the switching process guarantees that the connectionof all users is considered for improvement, since l(m)

sweeps the set of all users in order.The algorithm stops when there is no further im-

provement in the network utility function (no changein optimal activation vector) after K consecutive swap-ping iterations. This criterion is reasonable becausel(m) = l(m+K) means after K iterations the algorithmstarts to switch the same worst connection with thesame user. We note that the feasibility of the solutionis also guaranteed since the algorithm starts with afeasible activation vector and at each iteration justswaps the elements of this vector to find the bestactivation vector.

B. Complexity AnalysisIn this subsection we consider the sum-rate util-

ity function and analyze the algorithm complexity bycomputing the number of floating multiplications. Thecost of multiplication of two matrices, A ∈ Cn×m andB ∈ Cm×p, is O(nmp). SVD decomposition of matrixA ∈ Cn×m has a cost of O(nm2) (m ≥ n), matrix inver-sion by Jordan-Gauss elimination scales as O(n3), andthe cost of computing the matrix determinant is O(n3).Finding arg max or arg min over a set of n variables,also known as sorting algorithms, requires n log(n)comparisons [45]. When computing the multiplicationof three matrices A ∈ Cn×m, B ∈ Cm×p, and C ∈ Cp×q,the order in which the product is parenthesized isimportant, and we take this into account to minimizethe cost [44]. Based on these complexity orders, wecan analyze the complexity of our WCS algorithm. Wefirst look at the complexity order of a single swappingiteration, then examine the number of iterations. Ateach swapping iteration of the algorithm, Steps 6 and7 have the highest computational cost because of thesearch/sort procedures, and according to above com-plexity bases, they exhibit similar computation order.Thus, we focus on computing the cost of Step 6 whichinvolves two major operations: computing the networksum-rate Us(r(t)), and sorting the resulted sum-ratevector. Computing the network sum-rate involves com-puting the interference plus noise term Vk,j in (22)which is of order O(M2

j ), and this leads to the costof computing user instantaneous rate Rk,j in (21) alsoas O(M2

j ). Thus, the cost of computing the networksum-rate in (25) is of the order O(M2

j ). Next, the costof sorting and finding arg max over the set of K − 1new activation vectors β(i)

n and the previous activationvector β(i) is O(K log(K)). Therefore, the total cost ofcomputing Step 6 is O(K log(K)M2

j ), and since Step6 is computationally dominant in each iteration, thisis also the order of the cost for a single swappingiteration. Based on the algorithm’s stopping criterion,the number of swapping iterations is between K and

10

cK iterations with c as a constant factor, 1 ≤ c � K,and we can conclude that the number of iterationsincreases linearly with K. Thus, the final complexityof the algorithm is O(K2 log(K)M2

j ).

VI. Numerical ResultsIn this section, we evaluate the performance of the

proposed user association scheme in the downlink ofa mmWave MIMO network with J BSs and K UEsoperating at 73 GHz. In our simulations, we considerthe network sum-rate utility function defined in (25),except when study user fairness and cell-edge users’rate in Section VI.A.5. The mmWave links are gener-ated as described in Section II-A, each link is composedof 5 clusters with 10 rays per cluster. Each BS isequipped with a 8 × 8 UPA antenna, and each UE isequipped with a 2 × 2 mmWave UPA antenna for 5Gconnections. The noise power spectral density is −174dBm/Hz. Unless otherwise stated, all BSs transmit atthe same power level Pj . Moreover, we assume thatthe network nodes are deployed in a region of size300 m×300 m. The BSs are placed at specific locationsand the UEs are distributed randomly within the givenarea. There are nk data streams for each UE and Qj(t)is the maximum number of allowed active users ateach BS. Thus, the total number of data streams sentby each BS is Dj = nkQj(t). A BS is considered tobe overloaded (congested) if more than Qj(t) UEs areassociated with that BS.

We present our numerical results in two mainthrusts. First is the performance of the proposed userassociation scheme for mmWave with respect to itsnovel features: (i) using both instantaneous and large-scale CSI, (ii) considering the dependency of inter-ference on user association, and (iii) using MIMObeamforming at both ends. Second is the numericalanalysis of our proposed WCS algorithm in termsof convergence and complexity to verify the derivedtheoretical complexity order.

A. Performance of the proposed user associationscheme

1) Comparison of max-SINR and proposed load bal-ancing user association schemes: First, we comparethe conventional max-SINR user association schemewith our proposed load balancing user associationscheme. In order to better reflect the capabilities of theproposed user association scheme, in this simulationstudy, we consider a two-tier HetNet with 1 macro BS(centered at the middle of the network), 3 pico BSs and20 UEs. The transmit power of macro BS is 30 dBm,while pico BSs transmit at 20 dBm. The microwavelinks are i.i.d. Rayleigh fading MIMO channels, and weassume that each UE is also equipped with a single-antenna module designed for LTE/4G connections atmicrowave band in addition to their mmWave anten-nas. The macro BS can accommodate 8 UEs, and each

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(b)Fig. 1. Comparison of user assocaition schemes in a HetNet with 1Macro BS and 4 Pico BSs. (a) Max-SINR user association with fullinterference, (b) Proposed load balancing user association.

pico BS can only serve 3 UEs. Fig. 1.a shows the resultof max-SINR association in the network. As we cansee from the figure, the macro BS is overloaded by 4extra UEs, and BS #3 is overloaded with one extra UE.User association using our proposed scheme is shownin Fig. 1.b. It can be seen from the figure that theproposed method perfectly balances the BSs’ load bypushing the overloading UEs from the congested BSsinto the lightly-loaded BSs.

2) Effect of network interference dependency on userassociation: Next, we compare throughput perfor-mance of the proposed user association method by com-paring it with three other user association schemes:(i) load balancing user association [7], (ii) max-SINRuser association with user drop, and (iii) max-SINRuser association with resource (data stream) sharingand user drop. The first scheme is a load balancinguser association which results in fractional associationcoefficients, because the integer associaiton constraintis relaxed to solve the optimization poblem [7]. In thesecond method, those UEs who overloaded the con-

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only large-scale CSI

both large-scale and instantaneous CSI

Fig. 2. Average network spectral efficiency for user associationschemes in a mmWave network with 3 BSs, 12 UEs, 8 × 8 UPAat each BS, and 2 × 2 UPA at each UE. The UEs are distributedrandomly in an area of size 300 m × 300 m. All BSs transmit atthe same power level, and the noise power spectral density is −174dBm/Hz.

gested BS are dropped. For the third scheme, the datastreams of the congested BS are shared among themaximum number of UEs it can serve, and subsequentUE connections are dropped. For example, consider anetwork with 4 BSs and 8 UEs where nk = 2, ∀k, andDj = 4, ∀j. If a BS is overloaded with 5 UEs, it sharesits 4 data streams among 4 UEs (reducing the numberof data stream for each UE to 1) and drops the re-maining UE. In all these three reference schemes, theinterference is assumed to be both independent of userassociation and present all the time (full interference),while our proposed scheme considers the dependencybetween the network interference structure and userassociation.

Fig. 2 compares the network spectral efficiency(given in (25)) for the user association schemes withtwo different CSI assumptions: (i) both large-scale andinstantaneous CSI, and (ii) only large-scale CSI. As wecan see from the figure, our proposed scheme showsa better performance in both cases since we adaptthe network interference to the association. Thus, Itcan be inferred from the figure that user associationis highly dependent on network interference, sinceour proposed method outperforms the other user as-sociation schemes which all ignore the effect of userassociation on the network interference. Further, theavailability of instantaneous CSI in mmWave channelscan dramatically boost performance, and this effect isobserved across all schemes, though most pronouncedin our proposed scheme.

3) Effect of antenna array size on network spectralefficiency: Fig. 3 depicts the network spectral efficiencyversus the number of antennas at each BS and each

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UE. There are 12 UEs and 4 BSs in the network andthe number of data stream at each UE is nk = 2.The average is taken over 500 mmWave channel re-alizations, and the error bars represent the standarddeviation. In the upper subfigure, we fix the numberof antennas at each UE (Nk = 4), and increase theantenna array size at each BS. The result shows thatincreasing the antenna array size from a 3×3 UPA to a10×10 UPA improves the network spectral efficiency by164%. Similarly, in the lower subfigure we increase theantenna array size at each UE while fixing the numberof antennas at each BS at Mj = 64. The simulated setof UPA at each UE is: {1×2, 2×2, 2×3, 2×4, 2×5, 3×4, 4×4, 5×4}. The results show a 170% improvementin the network spectral efficiency when we increase theUE antenna array size from 1×2 to 5×4. Both resultsshow that increasing the number of antennas at eitherthe UE or BS has a positive impact on the networkspectral efficiency, although the improvement is moresignificant at small numbers of antennas.

4) Effect of mobility on user association: We considera simple mobility model where the location of UEs arefixed during each time slot. In the subsequent slot,each UE moves to a new random location within a5m x 5m square box centered at its location in theprevious time slot. The user association is performedat each time slot and the fractional user associationis obtained by taking average over T = 1000 timeslots. Fig. 4 depicts the network topology with randomlocations of UEs shown in small circles. Note that thenumber of random locations for each UE is 1000, butwe only show 10 random locations (t = 1 : 100 : 1000)for each UE to avoid a messy figure. Fig. 5 depictsthe average per-user throughputs and the fractionalassociations. As we can see from these simulations,

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Fig. 4. A mmWave network with 3 BSs and 12 UEs. All UEs moverandomly from one time slot to another. The trajectory is shown onlyfor couple of UEs.

UE 4 is always moving around BS 1 and thus it isassociated with BS 1 for about 87 percent of the time.The same result can be inferred for UE 1 which isconnected to BS 2 for most of the time. UE 2, however,gradually moves away from BS 3 and gets closer to BS2, and that is the reason this UE is associated withboth BSs for about half of the time. UE 2 thereforecould experience frequent handover in this case. Thesesimulations confirm that our proposed user associationis well-designed to be performed per-time slot, and itcan be applied with any mobility model.

5) Effect of fairness on cell-edge users’ throughputs:Cell-edge users usually suffer the worst throughputin the network. In order to study the the effect offairness on cell-edge throughput, we compare the twoutility functions defined in (25)-(26). Fig. 6 showsthe probability density function (PDF) and cumulativedistribution function (CDF) of users’ data rates in ammWave network with 3 BSs and 15 UEs. We considerthe sum-rate utility function in (25) for the “Max-throughput” case, and use the min-rate utility functionin (26) to implement âĂIJMax-min FairnessâĂİ. Byfocusing on the low rate region (bottom left corner) ofeach figures, it can be inferred that compared to max-throughput, max-min fairness shows a lower probabil-ity of users having an extremely low (close to zero)rate. For example, in Fig. 6.b the probability of havingusers with data rate smaller than 1 bps/Hz is 5.1%for sum-rate utility function, while this probability isabout 2.8% when using the max-min fairness. Sincemost cell-edge users would be represented in the low-rate region, this result implies that max-min fairnessresults in higher data rates for the cell-edge users inthe network, especially ensuring that these users’ rates

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start at a reasonable non-zero value. This is in contrastto max-throughput in which the CDF figure shows asmall portion of users actually having zero rates. Thisresult confirms that max-min fairness improves thethroughput for cell-edge users who usually suffer fromlow data rates.

B. Efficiency of the proposed WCS algorithm1) Convergence performance: Fig. 7 compares the

performance of the proposed algorithm with GA in anetwork with 12 UEs and 4 BSs. As a benchmark,exhaustive search result is included to provide theoptimal solution. The “improvement” curve shows theincrement of network sum-rate at each swapping iter-ation. It can be seen from the figure that the proposedWCS algorithm rapidly converges to the optimal so-lution in 32 iterations which execute in 1.08 seconds.However, the GA (with 50 stall generations) stops after215 iterations and still does not reach the optimal solu-tion. GA can, in theory, reach to the optimal solution byincreasing the number of stall generations, but it takesdrastically more time to reach this. Our simulationsshow that it takes 417 iterations which is about 86seconds for GA (with 200 stall generations) to reach

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the optimal solution, compared to the proposed WCSwhich takes only 1.08 seconds. Note that the GA stopsif there is no improvement in the objective functionfor a sequence of consecutive generations of the lengthequal to the number of stall generations.

2) Complexity results: Comparing the convergencetime of the proposed algorithm and GA in Fig. 7, onecan easily confirm that the WCS algorithm has muchlower complexity than GA. Fig. 8.a shows the num-ber of swapping iterations of the proposed algorithmversus the number of UEs while the number of BSs isfixed (J = 3). The average is taken over 500 channel re-alizations and polynomial data fitting is implemented.As expected, there is a linear relationship betweenthe number of iterations and the number of UEs. Fig.8.b depicts the actual elapsed time of the proposedalgorithm. Based on our complexity analysis in SectionV-B, we fit the elapsed time results with a function ofthe form f(K) = aK2 log(K) + bK2 + cK + d, whereK is the number of UEs and a, b, c, and d are some

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Run Time: 2951 seconds = 49 minutes

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constants. The result confirms the complexity analysisand shows that the fitting model matches well with thesimulation results.

VII. ConclusionIn this paper we investigated the problem of optimal

user association in a mmWave MIMO network. We firstintroduced the activation matrix and showed that theuser instantaneous rate is a function of the activa-tion factors. Then, we formulated a new associationscheme in which network interference depends on userassociation. We designed an efficient polynomial-timealgorithm to solve the formulated MINLP optimizationproblem. Simulation results showed that the proposeduser association scheme outperforms existing associa-tion methods, which ignore the effect of interferenceon user association, confirming the fact that the net-work interference structure is highly dependent onuser association. Moreover, we designed an efficientalgorithm to solve the formulated MINLP optimizationproblem. Numerical results confirmed the polynomialcomplexity of the proposed algorithm and showed thatit has significantly lower complexity than the geneticalgorithm, and reaches a near-optimal solution in areasonable time and number of iterations.

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14

0 10 20 30 40 50 60Number of UEs (K)

-50

0

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100

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ber

of it

erat

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