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IEEE SYSTEMS JOURNAL, VOL. 11, NO. 1, MARCH 2017 7

User Association in Massive MIMO HetNets Yi Xu, Student Member, IEEE, and Shiwen Mao, Senior Member, IEEE

Abstract—Massive multiple-input–multiple-output (MIMO) and small cell are both recognized as key technologies for the future fifth-generation wireless systems. In this paper, we investigate the problem of user association in a heterogeneous network (HetNet) with massive MIMO and small cells, where the macro base station (BS) is equipped with a massive MIMO, and the picocell BSs are equipped with regular MIMOs. We first develop centralized user association algorithms with proven optimality, considering various objectives such as rate maximization, proportional fair- ness, and joint user association and resource allocation. We then develop a repeated game model, which leads to distributed user association algorithms with proven convergence to the Nash equi- librium. We demonstrate the efficacy of these optimal schemes by comparing with several greedy algorithms through simulations.

Index Terms—Game theory, heterogeneous networks (HetNets), massive multiple input multiple output (MIMO), small cells, uni- modularity, user association.

I. INTRODUCTION

EXISTING and future wireless networks are facing thegrand challenge of a 1000-time increase in mobile data in the near future [1]. To boost wireless capacity, two technologies have gained most attention from both industry and academia. The first one is massive multiple input multiple output (MIMO; a.k.a., large-scale MIMO, full-dimension MIMO, or hyper MIMO) [2], [3]. The idea is to equip a base station (BS) with hundreds, thousands, or even tens of thousands of antennas, hereby providing an unprecedented level of degrees of freedom (DoFs) for mobile users. The massive MIMO concept has been successfully demonstrated in recent studies [4], [5]. The second technology is small cell, which is reshaping the future of cellular networks [7]. A great benefit of deploying small cells is that the distance of the user–BS link can be greatly shortened, leading to reduced transmit power, higher data rate, enhanced coverage, and better spatial reuse of spectrum. Both massive MIMO and small cells are recognized as key technologies of the future fifth-generation wireless systems [6].

In this paper, we consider a heterogeneous network (HetNet) with massive MIMO and small cells. Specifically, we consider a HetNet where the macrocell BS (MBS) is equipped with a massive number of antennas and the picocell BSs (PBSs) are equipped with a regular amount of antennas. To fully harvest

Manuscript received December 20, 2014; revised May 16, 2015 and June 29, 2015; accepted August 30, 2015. Date of publication September 22, 2015; date of current version March 10, 2017. This work was supported in part by the National Science Foundation under Grant CNS-1247955 and Grant CNS- 1320664 and in part by the Wireless Engineering Research and Education Center at Auburn University.

The authors are with the Department of Electrical and Computer Engi- neering, Auburn University, Auburn, AL 36849 USA (e-mail: YZX0010@ tigermail.auburn.edu; smao@ieee.org).

Digital Object Identifier 10.1109/JSYST.2015.2475702

the benefits promised by massive MIMO and small cells in an integrated HetNet system, it is critical to investigate the user association problem, i.e., how to assign active users to the BSs such that the system-wide capacity can be maximized and users’ experience can be enhanced.

There are already several recent studies moving forward in this direction. In [8]–[11], the authors considered the problem of user association in massive MIMO systems operated in the frequency-division duplexing mode. These papers focused on a macrocell without small cells. In [12], user associa- tion in time-division duplexing (TDD) massive MIMO system was addressed, where fractional user association was allowed. Bayat et al. in [13] modeled the problem of user association in a femtocell HetNet as a dynamic matching game and derived the optimal user association. In [14], near-optimal user association schemes were proposed for HetNet with WiFi and conven- tional cellular networks. However, massive MIMO was not considered in these studies. In [15], the authors investigated the problem of user association with conventional MIMO BSs and proposed a simple bias-based selection criterion to approximate more complex selection rules. Björnson et al. [16] considered improving energy efficiency without sacrificing the quality of service of users in a massive MIMO and small cell HetNet.

Motivated by these interesting studies, we consider the user association problem in a TDD massive MIMO HetNet in this paper. We take into consideration the practical constraints, such as the limited load capacity at each BS, without allowing fractional user association. The main goal is to maximize the system capacity while enhancing user experience. More specifically, this paper contains two parts: centralized user as- sociation and distributed user association. For centralized user association, we investigate the problems of rate maximization, rate maximization with proportional fairness, and joint resource allocation and user association. We prove the unimodularity of our formulated problem and develop optimal user association algorithms to the problems of rate maximization and rate maxi- mization with proportional fairness. We then propose a series of primal decomposition and dual decomposition algorithms to solve the problem of joint resource allocation and user association and prove the optimality of the proposed scheme. For distributed user association, we model the behavior and in- teraction between the service provider (who owns the BSs) and users as repeated games. We consider two types of operations: 1) the service provider sets the price and the users decide which BS to connect to and 2) the users bid for connection opportu- nities. We prove that, in both cases, the proposed algorithms converge to the respective Nash equilibrium (NE).

In the remainder of this paper, Section II introduces the system model and preliminaries. Optimal centralized and dis- tributed user association schemes are presented in Sections III

1937-9234 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Authorized licensed use limited to: Auburn University. Downloaded on April 13,2020 at 02:19:45 UTC from IEEE Xplore. Restrictions apply.

8 IEEE SYSTEMS JOURNAL, VOL. 11, NO. 1, MARCH 2017

and IV, respectively. Section V presents the simulation study, and Section VI concludes this paper. Throughout this paper, a boldface upper (lower) case symbol denotes a matrix (vector), a normal symbol denotes a scalar, (·)H denotes the Hermitian of a matrix, and (·)T denotes the transpose of a matrix.

II. SYSTEM MODEL AND PRELIMINARIES

The system considered in this paper consists of K users and J BSs inside a macro cell. There is a single MBS with a massive MIMO and (J − 1) PBSs, each with a regular amount of antennas. Following [19], the channel model we consider is hj,k,n = gj,k,nlj,k, where hj,k,n is the channel of antenna n at BS j to user k, gj,k,n represents the small-scale fading coefficient between antenna n of BS j and user k, and lj,k stands for the large-scale fading coefficient between BS j and user k. To obtain the channel vector hj,k between BS j and user k, we could concatenate all the channel coefficients from all the antennas of BS j as hj,k = [hj,k,1, hj,k,2, . . . , hj,k,n]. For the channel coefficient matrix of signals transmitted from BS j, we could obtain it by concatenating channel vector hj,k as Hj = [hj,1,hj,2, . . . ,hj,k].

Let yj denote the signals received by the users connecting to BS j, Wj the precoding matrix of BS j, and dj the data sent from BS j. We have

yj = HjWjdj + nj (1)

where nj is the zero-mean circulant symmetric complex Gaussian noise vector.

Each active user has the options to connect to either the MBS or a PBS. For a user k, define user association index variable xkj as

xkj =

{ 1, if user k is connected to BS j

0, otherwise. (2)

Let the achievable rate of user k connecting to BS j be Rkj . We define ηkj = xkjRkj and the overall data rate of user k to be ηk, where

ηk = ∑ j

ηkj = ∑ j

xkjRkj . (3)

For users connecting to a massive MIMO BS j (i.e., the MBS), their achievable rate can be approximated with the following deterministic rate [12]:

Rkj = log

( 1 +

Mj − Lj + 1 Lj

Pj lj,k 1 +

∑ j′ �=j Pj′ lj′,k

) (4)

where Mj is the number of antennas at the BS, Lj is the prefixed load parameter of the BS indicating how many users it could serve, and Pj is the transmit power from the MBS. Note that there is no small-scale fading factor in (4). This approximation has been proven to be accurate in [12].

As shown in [17], there are various sources of interferences in a HetNet. However, among the PBSs, intercell interference coordination could be effectively performed according to [18].

We hereby ignore the intercell interference and only consider the intracell interference. The achievable rate of a user k connecting to PBS j can be then represented as follows:

R̃kj = log

⎛⎜⎝1 + Pj ∣∣∣hHj,kwj,k∣∣∣2

1 + ∑

k′ �=k Pj

∣∣∣hHj,kwj,k′ ∣∣∣2 ⎞⎟⎠ (5)

where wj,k is the kth column of BS j’s precoding matrix Wj , and | · | represents the absolute value. There are many precoding designs for convention