User Association and Interference Management in Massive MIMO HetNets

Smart densification & heterogeneity in base station (BS) deployments and massive MIMO are considered as two of the most important technologies in 5G cellular systems [1, 2, 3].¹¹1As higher-frequency spectrum being available, large arrays become practical even for small cells. For example, at 3.5GHz band, a 36-antenna (arranged on a square grid at half-wavelength separation) can be implemented on a 26cm $\times$ 26cm surface. The required implementation area would be smaller as the carrier frequency becomes higher. The massive MIMO regime is the setting when the number of antennas at a BS is significantly larger than the number of users that are simultaneously served by the BS [4, 5, 6]. In this paper, various aspects such as user association, load balancing, scheduling and interference management are considered for future networks with massive MIMO deployments where these technologies are adapted together.

In this paper, we focus on best-effort traffic and consider downlink (DL) transmission in a HetNet with $J$ BSs and $K$ single-antenna users. We let $j\!\in\!\mathcal{B}\!=\!\{1,2,\dots,J\}$ and $k\!\in\!\mathcal{U}\!=\!\{1,2,\dots,K\}$ denoted the indices of BSs and users, respectively. Without loss of generality, we focus on two-tier HetNets comprising a macro layer and a small-cell layer, such as the one considered in Fig. 1. Letting $\mathcal{B}_{\rm m}$ and $\mathcal{B}_{\rm s}$ be the set of macro and small cell BSs, respectively, the BSs belonging to $\mathcal{B}_{\rm m}$ and $\mathcal{B}_{\rm s}$ can differ in terms of transmit power, size, density, and number of antennas [1]. The number of antennas at BS $j$ is denoted by $M_{j}$ with $M_{j}\gg 1$. We assume time division duplex (TDD) operation with reciprocity-based CSI acquisition [4, 42]. Hence, each user sends a single uplink (UL) pilot to train multiple nearby BSs. In contrast to feedback-based CSI acquisition, this enables the training of large antenna arrays with overhead proportional to the number of simultaneously served users. Moreover, it enables CoMP with practical CSI acquisition overheads. We also assume a block-fading channel model where the channel coefficients remain constant within each RB [43, 4, 42, 5].

With massive MIMO, a subset of users are scheduled (or, in other words, are active) on each RB. Various transmission schemes in terms of BS-cluster options are possible with this setup as shown in Fig. 1. At one extreme is cellular transmission shown in Fig. 1(a), where the coded data for any given active user is transmitted from a single BS and each BS manages interference only to the users it serves. This corresponds to having $|\mathcal{B}|$ many disjoint clusters of size $1$. Fig. 1(c) shows the other extreme, Global Joint Transmission (ideal/full CoMP) where all BSs serve and coordinate interference to all active users in the network. In this case there is a single cluster serving all of the users. Other transmission schemes with cluster options in between these two extremes are also possible. DCC [39] can be used to describe any of these transmission schemes. Fig. 1(b) shows the scheme considered in this work where BS-clusters are user-specific. According to this scheme, different active users can be served simultaneously by different potentially overlapping clusters.

CoMP has some well-known advantageous over cellular transmission:
(1) Performance gain at the cell edge: The beamforming (BF) gain becomes intra-cluster BF gain in CoMP, as the same coded data is transmitted from all BSs serving the user. Similarly, the intra-cell interference mitigation is extended across the cluster of BSs by which the user is served. As a result, the performance gain can be realized at the cell edge.

A major disadvantage of the general CoMP [44, 45, 18] is that it incurs additional overhead compared to cellular transmission for CSI exchanges among cooperating BSs. To overcome this challenge, we focus on schemes that perform local precoding at each BS: The user beams (i.e., precoding vectors) at any RB for any BS $j$ are designed as if BS $j$ was in cellular multi-user (MU)-MIMO transmission over all the users it serves. All BSs serving a user transmit the same coded data stream to the user [35]. Each BS transmits the stream on a beam that is (independently) designed for the users at that BS. For instance, the beam with Linear Zero forcing Beamforming (LZFBF) for each user served by BS $j$ is chosen within the null space of the channels of all the other users served by BS $j$, no matter whether there are additional BSs serving the user on the same RB or not. Due to local precoding, BS $j$ only needs CSI between the users it serves and its own antennas to generate the user beams. Hence, the challenge of costly CSI exchanges between BSs is eliminated.

In contrast to full CoMP⁵⁵5With general CoMP schemes (e.g., cluster ZFBF transmission), the instantaneous rate that is provided to a user by a cluster of BSs is also a function of the identity (and in fact the large-scale channel coefficients) of the set of the other users scheduled for cluster transmission with the user, and can thus vary from slot to slot depending on the scheduling set [45]. As a result, in contrast to the LJT schemes we consider (whereby a user’s instantaneous rate is independent of the identity of the active users in the slot), general CoMP schemes are not amenable to the load balancing and scheduling techniques presented in this work., with local precoding of the form depicted in Fig. 1(b), the instantaneous rate is independent of the other active users’ channels and only depends on the power allocation for user streams served by the BS. By allocating BS power equally across the set of active users⁶⁶6Power allocation is a thoroughly studied topic in the context of MIMO in general (see, e.g.  [46]). With large antenna arrays, massive MIMO systems are able to get substantially better SINRs even without considering any power allocation optimization. For example, [45] considers equal power allocation while Marzetta in his pioneering work [4] allocates power to users proportional to their channel gains, which is in the reverse direction of typical power allocation in the context of a fairness criterion (whereby more power is typically allocated at the cell-edge). Following this trend in massive MIMO and considering the high complexity of power allocation optimization, we consider equal active user-stream power allocation at each BS. This approach simplifies the parametrization of peak-rate calculations in Section III and yields the convex NUM formulation in Section IV. and fixing the scheduling set sizes for BSs belong to common BS clusters, the instantaneous rate can be predicted a priori and independently of the other active-user set, thereby substantially reducing the complexity of the resource allocation problem.

In this work, we restrict our attention to a small set of predefined possible scheduling set sizes, while how to select these sizes depend on many factors (e.g., number of antennas at the BSs, network deployment, etc.) and we leave the study to future work. Due to the fact that an uplink user-pilot trains all antennas at nearby BSs, we assume BSs that serving users at larger-size clusters (during a given scheduling slot) can schedule a larger number of users in the same slot than BSs serving users at smaller-size clusters or with cellular transmission. We thus let the size of scheduling set of any BS $j$ depend on the size of the cluster including BS $j$. To avoid ambiguity on the scheduling set size of different BSs in the same BS cluster, we enforce the following condition: All users served by a given BS $j$ are served in clusters of the same size on a given RB. This constraint ensures that overlapping clusters on any RB are of the same size. Let $S_{j}$ denote the number of users that can be simultaneously served by BS $j$ in the cellular transmission. We summarized the properties of the specific form of LJT considered in this work in the following definition:

As explained in Sec. III, the properties of ALJTS enable us to decouple the problems scheduling and load balancing, since the instantaneous active-user rates depend only on the serving BS cluster and are predictable a priori based on the given $S_{j}(L)$ value.

To illustrate these principles, we give LJT examples that obey these rules in Table II, involving clusters of sizes 1 (i.e., cellular transmission) and 2. Four BSs are considered with $P_{j}\!=\!1$, $S_{j}(1)\!=\!2$ and $S_{j}(2)\!=\!3$. As the table reveals, each BS on RB #1 engages in cellular transmission. On RB #2, pairs of BSs perform LJT with each BS pair serving a triplet of users. RBs #3 and #4 provide additional, more interesting modes. On RB #3, no user is served by the same cluster. On RB #4, BSs 1 and 2 serve users in clusters of size 2, while BSs 3 and 4 serve users in cellular transmission. Note that if orthogonal pilots are used, (at least) 8, 6, 6 and 7 uplink pilot dimensions (one dimension per user) are needed to enable RBs #1, #2, #3 and #4, respectively. Evidently, the choice of scheduled user sizes $S_{j}(L)$ signifies how aggressively pilot dimensions are reused across the network (e.g., $S_{j}$ for fully reused pilots and $LS_{j}$ for orthogonal pilots). Inspection of Table II reveals that the first ALJTS property can be satisfied either if all clusters at an RB are of the same size (RBs #1,2,3) or if different size clusters are disjoint across BSs (RB #4).

Depending on the availability of bands and the preferences of the operator, in 5G HetNets, different groups of BSs might be allowed to jointly transmit in clusters across groups of RBs (e.g., across different frequency bands). Each such combination of BS clusters is considered separately as a distinct entity in the resource allocation problem. Given the set of BSs $\mathcal{B}$ in the network, there are $2^{|\mathcal{B}|}-1$ many possible different entity/operation options. Although in principle it is straightforward to take into account all these different options, for simplicity and considering the general interest in 5G HetNet deployments, we focus only on the following options: 1) macro and small cell layers may operate together; 2) only the macro layer operates; 3) only the small cell layer operates. We call these 3 different operations as shared, macro-only and blanking operations, respectively. Let $A\!\in\!\{1,2,3\}$ denote the operation type, where $A\!=\!1$, $A\!=\!2$ and $A\!=\!3$ denote shared, macro-only and blanking operations, respectively. Let RBs allocated to operation $A$ form Band-$A$, and the set of BSs that can transmit in Band-$A$ be $\mathcal{B}^{(\!A)}$. We have the following cases:
1) $A\!=\!1$: shared operation is considered for this band, where macro and small cell BSs can both transmit, i.e., $\mathcal{B}^{(1)}\!=\!\mathcal{B}$. In this band, clusters can be formed by BSs from different layers.
2) $A\!=\!2$: macro-only operation is considered for this band, and only macros can transmit, i.e., $\mathcal{B}^{(2)}\!=\!\mathcal{B}_{m}$.
3) $A\!=\!3$: blanking is applied to this band, where all of the macro BSs are muted, i.e., $\mathcal{B}^{(3)}\!=\!\mathcal{B}_{s}.$⁸⁸8In general, blanking can be applied to only a subset of of macro BSs. In fact, [29] has shown gains compared to blanking all macro BSs. Our formulation can include partial blanking by considering new operation options. To simplify the exposition, in this work we confine ourselves to blanking all of the macro BSs.

Different resource allocations among operations refer to different scenarios. For example, scenarios with $A\!\in\!\{2,3\}$ correspond to cases where orthogonal RBs are allocated to macro and small cells, while scenarios with $A\!\in\!\{1,3\}$ can be applied to cases with eICIC. In some scenarios, resource allocation among operations can be fixed a priori. In more flexible scenarios, resource allocation among operations can be a part of the optimization problem. Our formulation in Sec. IV can be applied to both cases.

To show how these different transmission operations of practical interest can be considered within our specific LJT, we give a small example in Table III. As different bands may prefer different cluster sizes, we consider clusters up to size $L_{\max}^{(A)}$ in Band-$A$. Then the potential BS clusters that can be active in this band is given by the subsets of $\mathcal{B}^{(A)}$ with size less than or equal to $L_{\max}^{(A)}$.⁹⁹9Many of these clusters are not necessary. For example, clusters between BSs that are geographically distant are not necessary to consider, as in a practical system no user would be assigned to these clusters. This type of practical observations can eliminate many cluster options for all RBs. In the following, to avoid cumbersome notation, while listing potential clusters, we only use the cluster size and the active BS set of the corresponding band to describe potential BS clusters. Table III is extended from Table II by adding the macro BS (BS #5) and considering other BSs as small cell BSs, i.e., $\mathcal{B}_{m}\!=\!\{5\}$ and $\mathcal{B}_{s}\!=\!\{1,2,3,4\}$. Assume $L_{\max}^{(1)}\!=\!2,L_{\max}^{(2)}\!=\!1$ and $L_{\max}^{(3)}\!=\!2$ in this example. RBs $b$ and $d$ are in Band-$1$ (shared operation), RB $e$ is in Band-$2$ (macro only operation), and RBs $a$ and $c$ are in Band-$3$ (blanking operation). In RB $b$, each BS is performing cellular transmission, while RB $d$ considers clusters of size 2 including both macro and small cell BSs. In RB $e$, the macro BS (BS #5) serves users via cellular transmission. In RBs $a$ and $c$, only small cells serve users while the macro BS is muted. In fact, clusters in RBs $a$ and $c$ are the same as in RBs #1 and #3 in Table II, respectively.

In this section, we provide proxy expressions for instantaneous rates and long-term rates (throughput) with either LZFBF or Maximum Ratio Transmission (MRT, also known as Conjugate Beamforming). We denote the transpose and conjugate transpose of matrices by $(\cdot)^{T}$ and $(\cdot)^{H}$, respectively.

On a generic RB $t$, we let $g_{kj,i}=\sqrt{\beta_{kj}}h_{kj,i}$ be the channel between the $i^{\rm th}$ transmit antenna of BS $j$ and any user $k$, and it includes both slow fading $\beta_{kj}$ and fast fading $h_{kj,i}$. The slow fading $\beta_{kj}$ characterizes the combined effect of distance-based path loss and location-based shadowing. Let $\mathcal{K}_{j}(t)$ be the set of users served by BS $j$ and $S_{j}(t)=|\mathcal{K}_{j}(t)|$ denote the number of users scheduled by BS $j$ on RB $t$. The channel matrix between BS $j$ and its active users (the users scheduled by this BS at this RB) is denoted by $\mathbf{G}_{j}$, where the dimension of $\mathbf{G}_{j}$ is $M_{j}\!\times\!S_{j}(t)$. The $m^{\rm th}$ column of $\mathbf{G}_{j}$ corresponds to the channel of user $k=k_{j}(m)$ for some $k\in\{1,2,\ldots,\mathcal{K}\}$. That is, for a given $m$ we have $[\mathbf{G}_{j}]_{im}=g_{kj,i}$, with $k=k_{j}(m)$. This expression can also be interpreted in terms of the inverse mapping $m=m_{j}(k)$, that is, for a given $m$ we have $[\mathbf{G}_{j}]_{im}=g_{kj,i}$, with $m=m_{j}(k)$.

We assume each link experiences independent Rayleigh fading, i.e., $\mathbf{h}_{kj}=[h_{kj,1},\cdots,h_{kj,M_{j}}]^{T}$ are complex Gaussian i.i.d. random variables.¹⁰¹⁰10The instantaneous user-rate expressions in this paper assume no spatial correlation in the user channels. In principle, instantaneous user-rate expressions can be developed for spatially correlated user channels with a given spatial correlation structure by using the method of deterministic equivalent [5]. The framework presented in Secs. IV–VI, however, is not directly applicable with spatially correlated user channels, since a user’s instantaneous rate would in general depend on the spatial correlation of the other scheduled user channels. Although beyond the scope of this paper, extensions involving spatially correlated user channels is an area worth further investigation. We let $\mathbf{F}_{j}$ denote the precoding matrix at BS $j$ with dimension $M_{j}\!\times\!S_{j}(t)$, whose $m^{\rm th}$ column $\mathbf{f}_{mj}$ is the beam (i.e., the precoding vector) for the $m^{\rm th}$ user of BS $j$, i.e., the user whose channel to the BS is given by the $m$-th column of $\mathbf{G}_{j}$. The signal symbol of user $k$ is denoted by $s_{k}$, where $s_{k}$ has unit energy. The thermal noise at user $k$ is denoted by $w_{k}$, which is assumed to be additive white Gaussian noise (AWGN) with variance $\sigma^{2}$.

We consider a scheduling policy on RBs $\{1,\,2\,\cdots,T\}$ and assume that all the large-scale coefficients stay fixed within this period. Any such scheduling policy can be described in terms of the scheduling sets $\{\mathcal{U}_{\mathcal{C}}(t);\forall\mathcal{C},\forall t\in\{1,\,2\,\cdots,T\}\}$, where $\mathcal{U}_{\mathcal{C}}(t)$ denotes the set of users served by cluster $\mathcal{C}$ on RB $t$ and $\forall\mathcal{C}$ denotes all of the possible cluster options considered for all bands taking into account active BS sets, cluster sizes and possible cluster selections/eliminations. Without loss of generality, we assume that RB $t$ is in Band-$A$ and $\mathcal{C}\!\subset\!\mathcal{B}^{(A)}$. Thus, the received signal at user $k\in\mathcal{U}_{\mathcal{C}}(t)$ on RB $t$ is

Adopting LZFBF, the precoding matrix at BS $j$ is $\mathbf{F}_{j}\!=\!\mathbf{G}_{j}\left(\mathbf{G}_{j}^{H}\mathbf{G}_{j}\right)^{-1}\mathbf{A}_{j}^{1/2}$, where $\mathbf{A}_{j}$ is the normalizing coefficients matrix. Specifically, $\mathbf{A}_{j}$ is a $S_{j}(L)\times S_{j}(L)$ diagonal matrix with the $k$th diagonal element being $a_{k,k}=\frac{1}{\left[\left(\mathbf{G}_{j}^{H}\mathbf{G}_{j}\right)^{-1}\right]_{k,k}}$, where $[\left(\mathbf{G}_{j}^{H}\mathbf{G}_{j}\right)^{-1}]_{k,k}$ denotes the $k$th row and $k$th column element of the matrix $\left(\mathbf{G}_{j}^{H}\mathbf{G}_{j}\right)^{-1}$. In this case, the intra-cluster interference is $0$. With MRT, the precoding matrix at BS $j$ is $\mathbf{F}_{j}$ with the $m$th column being $\mathbf{f}_{mj}\!=\!\frac{\mathbf{g}_{k_{j}(m)j}}{\|\mathbf{g}_{k_{j}(m)j}\|}$. Note that the set of interfering BSs depends on the operation band.

While ALJTS allows for varying cluster sizes within an RB as revealed in Table II, in the sequel we specialize to the tractable case of practical interest involving equal-size clusters on each RB. Consequently, the following framework allows for cluster options as seen in RBs #1-3 but not for RB #4 in Table II. We call this new, additionally constrained, scheme the Uniform Cluster-Size scheme (UCS).¹³¹³13 Unlike UCS, when different size clusters are allowed to operate together, great care in the scheduling design must be given to avoid overlapping clusters with different sizes to operate in the same RB. This makes scheduling very complicated. The considered UCS provides a lower bound on the performance compared to more general ALJTS, which can serve a useful benchmark.

Then LJT designs considered in the rest of the paper are all Uniform Cluster-Size Schemes. RBs allocated to serving size-$L$ clusters in Band-$A$ comprise what we call the $L^{\rm th}$ subband of Band-$A$. The NUM problem for the UCS optimizes activity fractions and subband/band allocations is as follows:

where the utility function $U(\cdot)$ is a continuously differentiable, monotonically increasing, and strictly concave function [48]. Constraint (5b) signifies that the total activity fractions allocated by BS $j$ in clusters of size $L$ in Band-$A$ cannot exceed the total available resources $\lambda_{AL}S_{j}(L)$. On the other hand, recalling that each user cannot be served by multiple clusters on the same RBs, (5c) signifies that the fraction of RBs over which user $k$ is served by clusters of size $L$ in Band-$A$ cannot exceed RBs allocated to the clusters of size $L$ in Band-$A$, $\lambda_{AL}$. (5d) ensures that the total resources allocated to the subbands in Band-$A$ are no more than the resources allocated to that band. Finally, (5e) signifies the fact that the summation of resources allocated to different bands is equal to all available resources.

Any concave function (e.g., general $\alpha$-fairness, [49]) can be applied to the problem formulation (5). The formulated optimization is convex as long as the utility function is concave [50]. In this paper, among various concave utility functions available in the literature, we work with the logarithmic utility which is also known as “proportional fairness” [14, 33, 15].¹⁴¹⁴14There are many options for the utility function such as (weighted) arithmetic mean, geometric mean, max-min fairness. Each option can be relevant for certain scenario and interest. Both arithmetic mean and max-min fairness have their shortcomings [39, Chapter 1]. Geometric mean (aka proportional fairness), promotes a trade-off of user-rates between the other two utility functions. The “$\log$” function in the utility ensures diminishing returns for individual users rates as they get higher. This de-motivates the optimization to give high rates to a few users. General numerical solvers (e.g., CVX) can be used. Since CVX is not well-suited for large instances [51], we alternatively propose an efficient algorithm that can be applied to large networks in the next section and the complexity difference between the proposed algorithm and a general numerical solver is investigated in Appendix C.

In this section, we propose an efficient algorithm based on the dual subgradient method [50]. We let $\nu_{jL}^{(A)}$ and $\theta_{kL}^{(A)}$ be the Lagrange multipliers corresponding to (5b) and (5c), respectively. The dual problem of (5) is $\min\limits_{\nu_{jL}^{(A)},\theta_{kL}^{(A)}\geq 0}\ \sum_{k\in\mathcal{U}}f_{k}\left(\nu_{jL}^{(A)},\theta_{kL}^{(A)}\right)+g\left(\nu_{jL}^{(A)},\theta_{kL}^{(A)}\right),$ where

and

The constraints of (5) satisfy the Slater condition [50], and thus strong duality holds (i.e., the dual problem and the original problem (5) have the same optimal value).

In this section, we develop scheduling policies that yield activity fractions closely matching the NUM solution. Scheduling is done independently and in parallel for each band. As seen in Fig. 2, a scheduler at a central controller can collect the needed scheduling information and schedule the users according to the proposed scheduling scheme independently for each band (i.e., for each operation option). Considering a scheduling policy for Band-$A$ and letting $L(t)$ be the cluster size in RB $t$, we define the feasible scheduling policy as follows.