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User association and channel assignment in downlink multicell NOMA networks: A matchingtheoretic approach
EURASIP Journal on Wireless Communications and Networkingvolume 2019, Article number: 220 (2019)
Abstract
This paper studies the problem of stable user association and channel assignment in downlink multicell nonorthogonal multipleaccess (NOMA) networks. To be specific, the goal is to assign network users to the channels at each base station, while accounting for interuser interference and maintaining quality of service (QoS) per user. To that end, a lowcomplexity iterative solution procedure is devised to obtain the optimal power allocation for proportional fairness signaltointerferenceplusnoise ratio (SINR)based maximization, which is then utilized to determine the preferences of network users over the channels available at each base station and the preferences of base stations over the network users. In turn, a manytoone matchingtheoretic model based on the studentproject allocation problem is applied. Particularly, two polynomialtime complexity stable matching algorithms are proposed to associate users with base stations and perform channel assignment, such that no user or base station would deviate and change its association or channel assignment unilaterally. To validate the efficacy of the proposed solution procedure and stable matching algorithms, extensive simulation results are presented to compare them to a centralized joint user association, channel assignment, and power allocation (CJUACAPA) scheme. It is demonstrated that the proposed algorithms efficiently associate users with base stations and assign them to channels as well as efficiently yielding comparable SINR per user to the CJUACAPA scheme, while maximizing proportional fairness and satisfying QoS constraints.
Introduction
Nonorthogonal multiple access (NOMA) has recently attracted so much attention to meet the everincreasing demand for high spectral efficiency, improved fairness, massive connectivity, low transmission latency, and high throughput in future 5G cellular networks [1]. Specifically, the principle of NOMA is based on multiplexing users in the power domain over resource blocks by enforcing power imbalance between the transmitted user signals, which allows successive interference cancelation (SIC) to effectively eliminate interuser interference, and yielding considerable performance gains over conventional orthogonal multiple access (OMA) schemes [2]. To date, the majority of the published literature on resource allocation for NOMA systems has considered on singlecell networks. For example, the problem of sumthroughput maximizing power allocation based on α−fairness in singlecell downlink NOMA networks is studied in [3]. Particularly, the cases of statistical and perfect channel state information at the transmitter (CSIT) are considered. For the former case, fixed target data rates are predefined for all users, while for the latter case, users’ rates are adapted according to the instantaneous CSI. In [4], the authors propose userpairing schemes for capacity maximization in singlecell downlink NOMA networks. To be specific, two userpairing schemes have been proposed so as to provide capacity gains to almost all users by grouping them in pairs. Additionally, the exact sum capacity of a twouser pair has been analytically derived, while taking into account perfect and imperfect SIC. A dynamic power allocation scheme—subject to qualityofservice requirements—for downlink and uplink singlecell NOMA networks with two users has been considered in [5]. Moreover, the exact expressions for the outage probability and average rate resulting from the proposed scheme are obtained. The problem of optimal power allocation with given channel assignment under different performance criteria for singlecell downlink NOMA networks is studied in [6]. Furthermore, the authors proposed a lowcomplexity scheme for joint channel assignment and power allocation via a dynamic/iterative matching algorithm. Although singlecell NOMA networks have received significant attention in the past few years, much less attention has been given to multicell NOMA networks. Therefore, it is of paramount importance to study NOMA in the more realistic multicell scenario, which is much more challenging due to the complex interplay between multiple cells [7].
Recently, a few research works have focused on resource allocation in multicell NOMA networks. For example, in [8], distributed power control for downlink multicell NOMA networks is studied. Particularly, the authors study the problem of total transmit power minimization of all base stations, subject to minimum data rate requirements of network users. Moreover, a distributed algorithm is devised, which has been shown to converge to the optimal solution, if one exists. The problems of sumpower minimization (SPM) and sumrate maximization (SRM) for multicell NOMA networks are studied in [9]. To be specific, the SPM is transformed into a linear programming problem, and closedform solutions to the power allocation of each user are determined. Contrarily, the SRM problem is solved by decomposing it into a power allocation problem for users in a single cell, and a power control problem over multiple cells. After obtaining the optimal sum rate in a single cell, a distributed algorithm is devised to solve the original sumrate maximization. In [10], the authors consider the problem of jointly optimizing power allocation, userpair selection, and timefrequency resource allocation in multicell NOMA networks. In particular, an efficient algorithm for obtaining the equilibrium for resource allocation—while taking into account intercell interference—is proposed, which has also been proven to be the global optimum resource allocation solution. In [11], the authors outline a general framework for coordinated multipoint (CoMP) transmission in downlink multicell NOMA systems with distributed power allocation at each cell. Specifically, the applicability and necessary conditions for different CoMP schemes and network scenarios are investigated, and simulation results are presented to quantify the spectral efficiency gains of CoMPNOMA over CoMPOMA. Efficient power allocation for sum network capacity maximization in downlink multicell multiuser NOMA networks is considered in [12]. Particularly, a local optimal solution iterative scheme is proposed, which has been shown to outperform nonoptimal NOMA and OMA schemes. In [13], the performance of NOMA in multicell downlink millimeterwave (mmWave) networks is investigated. To be specific, closedform outage probability expressions are derived, where it has been demonstrated that NOMA can outperform OMA in multicell mmWave networks.
This paper studies the problem of stable user association and channel assignment in downlink multicell NOMA networks. To be specific, the goal is to assign network users to the channels available at each base station, while accounting for interuser interference, and maintaining quality of service (QoS) per user. To that end, a lowcomplexity iterative solution procedure—that can be executed locally at each base station—is devised to obtain the optimal power allocation for proportional fairness signaltointerferenceplus noise ratio (SINR)based maximization, which is utilized to determine the preferences of network users over the channels available at each base station, and the preferences of base stations over the network users. In turn, a manytoone matchingtheoretic model based on the studentproject allocation (SPA) problem is applied [14, 15]. Particularly, two polynomialtime complexity algorithms are proposed, namely the useroriented stable matching (USM) and the base stationoriented stable matching (BSSM). The proposed SPAbased stable matching algorithms incorporate constraints on the number of users per channel and the number of users to be associated with each base station. Furthermore, in the USM algorithm, a useroptimal stable matching solution is achieved, while in the BSSM algorithm, a base stationoptimal stable matching solution is obtained. More importantly, the USM and BSSM algorithms yield user association and channel assignment stable matchings that are simultaneously best response for all users and all base stations, respectively, such that no user or base station would deviate and change its association or channel assignment unilaterally. Additionally, a centralized joint user association, channel assignment, and power allocation (CJUACAPA) optimization problem is formulated and shown to be computationally expensive. To validate the efficacy of the proposed solution procedure and stable matching algorithms, extensive simulation results are presented to compare them to the CJUACAPA scheme. It is demonstrated that the proposed algorithms efficiently associate users with base stations and assign them to channels as well as yielding comparable SINR per user to the CJUACAPA scheme, while maximizing proportional fairness and satisfying QoS constraints.
Few recent research works have applied matching theory to NOMA networks. For example, in [16], user pairing in a singlecell cognitive radioinspired NOMA network is studied, where the base station allocates power to paired users within a cluster. In particular, a user with poor channel conditions is paired with a user with good channel conditions, while satisfying their rate requirements. To that end, a twosided onetoone distributed matching algorithm is developed, which is shown to yield performance that approaches that of centralized user pairing and power allocation. The authors in [17] investigate channel assignment, power allocation, and scheduling for singlecell downlink NOMA networks. To be specific, the problem of joint channel assignment and sumrate maximizing power allocation with user fairness is considered. Moreover, a manytomany userchannel matching algorithm is proposed, and an iterative solution for joint channel assignment and power allocation based on swap matching is devised. In [18], the authors consider the problem of sumrate maximization for devicetodevice (D2D) communication in uplink singlecell NOMA networks by optimizing channel and power allocation. In particular, a twosided manytoone matching algorithm is proposed to allow D2D groups to reuse the same channel occupied by a cellular user. Power allocation is also studied, where swap matching is applied to jointly allocate channel and power to the D2D groups. In [19], the problem of joint spectrum allocation and power control in single macrocell NOMAenhanced heterogenous networks is considered. In particular, several manytoone matchingtheoretic algorithms with a swap operation are proposed while incorporating users’ fairness in sumrate maximizing power allocation. Clearly, all the aforementioned works focus on singlecell NOMA networks.
To the best of our knowledge, no prior work has applied the SPA matching problem for user association and channel assignment with proportional fairness SINRbased power allocation in downlink multicell NOMA networks. In turn, the main contributions of this work are summarized as follows:

Proposed a lowcomplexity iterative solution procedure for determining the proportional fairness SINRbased maximizing power allocation for different user sets over the channels available at each base station.

Modeled the user association and channel assignment problem in downlink multicell NOMA networks as a SPA matching problem.

Devised two polynomialtime complexity stable matching algorithms based on the SPA matching problem, which associate network users with base stations and channels, such that user optimal and base stationoptimal stable matching solutions are obtained.

Compared the proposed algorithms to the CJUACAPA scheme and demonstrated that the proposed algorithms yield comparable SINR per user as well as efficiently assigning channels to celledge users, while maximizing proportional fairness and satisfying QoS constraints.
The proposed algorithmic designs in this work aim at filling the gap for effective resource allocation solutions in multicell NOMAbased 5G cellular networks. In particular, the proposed solution procedure can be executed locally at each base station to efficiently determine the proportional fairness SINRbased maximizing power allocation of cellular users over each channel and within each base station, and with minimal computational complexity, while taking into account interuser interference, QoS requirements, and SIC decoding constraints. In fact, the proposed solution procedure can incorporate other power allocation strategies and performance criteria (e.g., sum rate, energy efficiency, etc.) to establish the preferences of users over channels and base stations over users. Furthermore, the proposed stable matching algorithms can efficiently be executed among the base stations and without the need for a centralized controller. In summary, by optimizing the SINR of the network users via proportional fairnessbased power allocation, and efficiently associating users with base stations and assigning channels to them over multiple cells, this work fulfills some of the requirements of NOMAbased 5G cellular networks.
The rest of this paper is organized as follows. Section 2 presents the network model, while Section 3 presents the centralized joint user association, channel assignment, and power allocation problem formulation. Section 4 discusses the proposed solution procedure for proportional fairness SINRbased maximizing power allocation. In Section 5, the SPAbased stable matching algorithms are devised, whereas in Section 6, the simulation results are presented. Finally, Section 7 draws the conclusions.
Network model
Consider a downlink multicell NOMA network with a set of Q base stations (BSs), denoted \(\mathcal {B} = \left \{\mathrm {BS_{1}, {BS}_{2}, \ldots, {BS}}_{Q}\right \}\), a set of N users \(\mathcal {U} = \left \{U_{1}, U_{2}, \ldots, U_{N} \right \}\), and a set of K channels \(\mathcal {C} = \left \{C_{1}, C_{2}, \ldots, C_{K} \right \}\). Particularly, the network bandwidth is divided into K nonoverlapping equal bandwidth channels, such that each base station \({\text {BS}}_{q} \in \mathcal {B}\) is allocated a nonempty set of channels \(\mathcal {C}_{q} \subset \mathcal {C}\), where \(\mathcal {C}_{1}, \mathcal {C}_{2}, \ldots, \mathcal {C}_{q}\) partition \(\mathcal {C}\). In other words, \(\mathcal {C}_{q} \cap \mathcal {C}_{w} = \phi \) for w≠q, and \(\bigcup ^{Q}_{q=1} \mathcal {C}_{q} = \mathcal {C}\). Furthermore, let \(\mathcal {U}_{q} \subset \mathcal {U}\) be the subset of network users within the coverage area of each base station \({\text {BS}}_{q} \in \mathcal {B}\) (see Fig. 1). Also, let ζ_{q, k} be the maximum number of users that can be assigned to channel \(C_{q,k} \in \mathcal {C}_{q}\). That is, each channel \(C_{q,k} \in \mathcal {C}_{q}\) can be occupied by at most ζ_{q, k} users, \(\forall {BS}_{q} \in \mathcal {B}\)^{Footnote 1}. Moreover, note that some network users may fall within the overlapping region of two or more cells. Therefore, the maximum number of users that can be associated with each base station BS_{q} is set to ξ_{q} (i.e., a quota per base station). Additionally, each user may be associated with at most one base station and assigned to one channel. Our network model mimics that of a multicell orthogonal frequencydivision multiple access (OFDMA)NOMA network, where each channel is allocated to only one user, as per conventional OFDMA networks; while multiple users within a cell can share a channel via NOMA^{Footnote 2}.
In downlink NOMA networks, each base station BS_{q} sends data (over each assigned channel \(C_{q,k} \in \mathcal {C}_{q}\)) to its users simultaneously via powerdomain superposition coding. The instantaneous channel between each BS and the network users within its coverage area follows narrowband Rayleigh fading with zeromean N_{0}variance additive white Gaussian noise^{Footnote 3}. Particularly, \(h^{k}_{q,n} \sim \mathcal {C}\mathcal {N}\left (0, \sigma ^{2}_{q,n} \right)\) is the channel coefficient between BS_{q} and user \(U_{n} \in \mathcal {U}_{q}\) over channel \(C_{q,k} \in \mathcal {C}_{q}\), and \(\sigma ^{2}_{q,n} = d^{\nu }_{q,n}\) is the channel variance, with ν and d_{q, n} being the path loss exponent and distance, respectively. Moreover, the instantaneous channel gain is defined as \(g^{k}_{q,n} \triangleq \left h^{k}_{q,n}\right ^{2}\). Thus, without loss of generality, let the channel gains between base station BS_{q} and the users within its vicinity over each channel C_{q, k} be ordered as \(g^{k}_{q,1} \leq g^{k}_{q,2} \leq \cdots \leq g^{k}_{q,\mathcal {U}_{q}}\). In turn, the power allocation coefficients are ordered as \(a^{k}_{q,1} \geq a^{k}_{q,2} \geq \cdots \geq a^{k}_{q,\mathcal {U}_{q}}\). Additionally, the total transmit power at each base station BS_{q} is set as \(P_{\text {BS}}, \forall {\text {BS}}_{q} \in \mathcal {B}\). More importantly, let \(P_{q,k} = P_{BS}/\mathcal {C}_{q} \triangleq P\) be the transmit power per allocated channel, \(\forall C_{q,k} \in \mathcal {C}_{q}\) and \(\forall {\text {BS}}_{q} \in \mathcal {B}\), where · denotes the cardinality of the parameter set.
In this work, perfect SIC is assumed^{Footnote 4}, and thus, the signaltointerferenceplusnoise ratio (SINR) of user \(U_{n} \in \mathcal {U}_{q}\) over channel \(C_{q,k} \in \mathcal {C}_{q}\) is written as
with \(\rho \triangleq P/N_{0}, \bar {a}^{k}_{q,n} \triangleq \sum ^{\mathcal {U}_{q}}_{m = n+1} a^{k}_{q,m}\), and \(\bar {a}^{k}_{q,\mathcal {U}_{q}} = 0\).
Remark 1
The SINR function \(\gamma ^{k}_{q,n}, \forall n \neq \mathcal {U}_{q}\), is linearfractional (LF) function, with \(\rho g^{k}_{q,n} a^{k}_{q,n}\) and \(\rho g^{k}_{q,n}\bar {a}^{k}_{q,n} + 1 > 0\) being linear functions in \(a^{k}_{q,n}\) and \(\bar {a}^{k}_{q,n}\), respectively [21]. However, the SNR function \(\gamma ^{k}_{q,\mathcal {U}_{q}}\) is a linear function in \(a^{k}_{q,\mathcal {U}_{q}}\).
Centralized joint user association, channel assignment, and power allocation
In this work, the aim is to jointly associate users with base stations and perform channel assignment, along with proportional fairness SINRbased maximizing power allocation. Additionally, each user must satisfy a target minimum SINR \(\gamma _{T_{n}}, \forall U_{n} \in \mathcal {U}_{q}\), and \(\forall {BS}_{q} \in \mathcal {B}\) (i.e., QoS constraints). To this aim, define the binary decision variable \(\mathcal {I}^{k}_{q,n}\) as
Therefore, the centralized joint user association, channel assignment, and power allocation (CJUACAPA) problem is formulated as
Constraint (3a) ensures that if a user U_{n} is paired to a channel C_{q, k}, then it must satisfy the target minimum SINR \(\gamma _{T_{n}}\), while constraint (3b) ensures that the total number of users paired to channels in any base station BS_{q} must not exceed ξ_{q}. Moreover, constraint (3c) ensures that the number of users paired to each channel \(C_{q,k} \in \mathcal {C}_{q}\) is at most ζ_{q, k}. Constraint (3d) ensures that a user is associated with at most one base station, while constraint (3e) ensures that the sum of power allocation coefficients of the assigned users over any channel does not exceed one. Constraint (3f) ensures that the decoding order of the SIC is preserved. Constraint (3g) defines the range of values each power allocation coefficients can take. Particularly, if \(\mathcal {I}^{k}_{q,n} = 1\), then \(0 \leq a^{k}_{q,n} \leq 1\); otherwise, \(a^{k}_{q,n} = 0\). Finally, the last constraint defines the range of values the binary decision variables can take.
Remark 2
The CJUACAPA problem comprises an upperbound total of \(N \cdot \sum _{BS_{q} \in \mathcal {B}} \mathcal {C}_{q}\) continuous decision variables (i.e., \(a^{k}_{q,n}\)) and a similar number of binary decision variables (i.e., \(\mathcal {I}^{k}_{q,n}\)). In addition, it can be verified that there is an upperbound total of \(N + Q + 2\left (\sum _{BS_{q} \in \mathcal {B}}\mathcal {C}_{q} + \sum _{BS_{q} \in \mathcal {B}} \mathcal {C}_{q} \cdot \mathcal {U}_{q} \right) + \sum _{BS_{q} \in \mathcal {B}} \mathcal {C}_{q} \cdot \left (\mathcal {U}_{q}  1\right)\) constraints.
Remark 3
Problem CJUACAPA is a classified as a mixedinteger nonlinear programming problem (MINLP), which is NPhard (i.e., extremely computationally expensive [22,23]). Thus, it can only be solved via a global optimization package.
Based on Remark 3, problem CJUACAPA can be decomposed in two subproblems: (1) proportional fairness SINRbased maximizing power allocation and (2) stable matchingtheoretic user association and channel assignment. Particularly, the aim is to optimally solve the SINRmaximizing power allocation per base station and over each channel, while accounting for interuser interference and QoS constraints. This will be achieved via a lowcomplexity iterative solution procedure, which also determines the preferences of users over channels, and the preferences of base stations over users. After that, basestation association and channel assignment is performed via the stable matching algorithms.
Proportional fairness SINRbased maximizing power allocation
This section focuses on proportional fairness SINRbased maximizing (PRSINRMAX) power allocation, subject to target minimum SINR \(\gamma _{T_{n}}\) per user \(U_{n} \in \mathcal {U}_{q}\). Specifically, the objective function of base station BS_{q} over channel \(C_{q,k} \in \mathcal {C}_{q}\) be given by
which can be reexpressed as
where \(\mathbf {a}^{k}_{q} = \left [a^{k}_{q,1}, a^{k}_{q,2}, \ldots, a^{k}_{q,\mathcal {U}_{q}} \right ]\) is the power allocation vector, while \({\mathcal {I}}^{k}_{q} = \left [\mathcal {I}^{k}_{q,1}, \mathcal {I}^{k}_{q,1}, \ldots, \mathcal {I}^{k}_{q,\mathcal {U}_{q}}\right ]\) is the channel assignment vector. Since each channel \(C_{q,k} \in \mathcal {C}_{q}\) has a quota ζ_{q, k}, then \(\sum _{U_{n} \in \mathcal {U}_{q}} \mathcal {I}^{k}_{q,n} \leq \zeta _{q,k}\). The objective function in (5) stipulates that if user U_{n} is assigned channel C_{q, k} (i.e., \(\mathcal {I}^{k}_{q,n} = 1\)), then its SINR function is maximized; otherwise, it is set to one. Therefore, the PRSINRMAX optimization problem can be formulated as
Constraint (6a) ensures that if a user is selected, then it must satisfy the target minimum SINR, while constraint (6b) ensures that the sum of power allocation coefficients of the selected users does not exceed one. Constraint (6c) ensures that the maximum number of users per channel \(C_{q,k} \in \mathcal {C}_{q}\) does not exceed ζ_{q, k}, while constraint (6d) maintains the SIC decoding order. Constraint (6e) ensures that if channel C_{q, k} is assigned to user U_{n} (i.e., \(\mathcal {I}^{k}_{q,n} = 1\)), then the power allocation coefficient \(a^{k}_{q,n}\) must not exceed one; otherwise, \(a^{k}_{q,n} = 0\) (since \(\mathcal {I}^{k}_{q,n} = 0\)). The last constraint defines the range of values the binary decision variables take.
Remark 4
Problem PRSINRMAX is classified as a mixedinteger linearfractional programming (MILFP) problem. More importantly, it is nonconvex (and NPhard [24]); and thus, it is still computationally expensive.
In turn, a simple solution procedure is devised to efficiently solve problem PRSINRMAX. To that end, let \({\Omega }_{q,k} = \left \{{\Omega }_{q,k,1}, \ldots, {\Omega }_{q,k,\zeta _{q,k}} \right \}\) be the set of all combinations of assigning u_{q} users in \(\mathcal {U}_{q}\) (for 1≤u_{q}≤ζ_{q, k}) over each channel \(C_{q,k} \in \mathcal {C}_{q}\), where \({\Omega }_{q,k,u_{q}} = \left \{{\omega }_{q,k,1}, {\omega }_{q,k,2}, \ldots, {\omega }_{q,k,\varpi _{u_{q}}} \right \}\), and
with \(\varpi _{u_{q}} = {\Omega }_{q,k,u_{q}}\). Each combination is defined as \({{\omega }_{q,k,\varsigma } = \left [\!\mathcal {J}^{k}_{q,\varsigma,1}, \mathcal {J}^{k}_{q,\varsigma,2}, \ldots, \mathcal {J}^{k}_{q,\varsigma,\mathcal {U}_{q}} \!\right ]}\), for \(\varsigma = 1,2,\ldots, \varpi _{u_{q}}\). Specifically, \(\mathcal {J}^{k}_{q,\varsigma,n}\) is defined as
For example, if \(\mathcal {U}_{q} = 3\) and ζ_{q, k}=2, then Ω_{q, k}={Ω_{q, k,1},Ω_{q, k,2}}, where Ω_{q, k,1} includes the combinations [1,0,0],[0,1,0], and [0,0,1], while Ω_{q, k,2} contains the combinations [1,1,0],[1,0,1], and [0,1,1]. In other words, in the first set of combinations Ω_{q, k,1}, only one user is assumed to be assigned to channel C_{q, k}, while in Ω_{q, k,2}, different combinations of two users are assigned to that channel.
Remark 5
The total number of all possible user combinations over each channel \(C_{q,k} \in \mathcal {C}_{q}, \forall {\text {BS}}_{q} \in \mathcal {B}\) is obtained as
For notational convenience, let \(\mathcal {U}_{q,k,\varsigma } \subseteq \mathcal {U}_{q}\) be the subset of users in \(\mathcal {U}_{q}\) with \(\mathcal {J}^{k}_{q,\varsigma,n} = 1\) in \({\omega }_{q,k,\varsigma } \in {\Omega }_{q,k,u_{q}}\phantom {\dot {i}\!}\). Thus, the SINR of each user \(U_{n} \in \mathcal {U}_{q,k,\varsigma }\) is expressed as
It can be seen that for each possible combination \(\phantom {\dot {i}\!}{\omega }_{q,k,\varsigma } \in {\Omega }_{q,k,u_{q}}\), each user \(U_{n} \in \mathcal {U}_{q,k,\varsigma }\) will have a different SINR value, which depends on the other users sharing the same channel in that combination.
Remark 6
The SINR function of each user is one with peer effects (also known as negative network externality), which is due to the fact that each user’s SINR \(\gamma ^{k}_{q,n,\varsigma }\) is influenced by the allocated power to all the other users utilizing the same channel.
Remark 7
For u_{q}=1, only one user is assigned to each channel \(C_{q,k} \in \mathcal {C}_{q}\), in any combination ω_{q, k,ς} in \({\Omega }_{q,k,1}, \forall \varsigma = 1,2,\ldots,\mathcal {U}_{q}\). Consequently, the power allocation coefficient for each user U_{n} with \(\mathcal {J}^{k}_{q,\varsigma,n} = 1\) in ω_{q, k,ς} is \(a^{k}_{q,n,\varsigma } = 1\), whereas \(\bar {a}^{k}_{q,n,\varsigma } = 0\), and hence, \(\gamma ^{k}_{q,n,\varsigma } = \rho g^{k}_{q,n,\varsigma }\).
Now, for each possible combination \(\phantom {\dot {i}\!}{\omega }_{q,k,\varsigma } \in {\Omega }_{q,k,u_{q}}\), for 2≤u_{q}≤ζ_{q, k}, problem PRSINRMAX can be rewritten as
\(\hspace {0mm} \max \quad \gamma ^{k}_{q,\varsigma }\left (\mathbf {a}^{k}_{q,\varsigma }, {\omega }_{q,k,\varsigma }\right) = \prod _{U_{n} \in \mathcal {U}_{q,k,\varsigma }} \frac {\rho g^{k}_{q,n,\varsigma } a^{k}_{q,n,\varsigma }}{\rho g^{k}_{q,n,\varsigma }\bar {a}^{k}_{q,n,\varsigma } + 1}\)
which can be verified to be nonconvex, although all the constraints are linear [25]. By utilizing the fact that the ln(·) function is concave and strictly monotonically increasing, then the objective function of problem PRSINRMAX (ω_{q, k,ς}) can be reexpressed as
with \(f\left (\mathbf {a}^{k}_{q,\varsigma } \right)\) being defined as
while
It is noteworthy that \(f\left (\mathbf {a}^{k}_{q,\varsigma } \right)\) and \(g\left (\mathbf {a}^{k}_{q,\varsigma } \right)\) are concave functions, which are also twice continuously differentiable in \(a^{k}_{q,n,\varsigma }\) and \(\bar {a}^{k}_{q,n,\varsigma }\), respectively. Hence, \(\ln \gamma ^{k}_{q,\varsigma }\left (\mathbf {a}^{k}_{q,\varsigma }, {\omega }_{q,k,\varsigma }\right)\) is a difference convex function [26,27]. As a result, problem PRSINRMAX (ω_{q, k,ς}) can be reformulated as [28,29]
with μ being an auxiliary variable. It can be easily verified that the objective function and the first constraint are concave functions in \(a^{k}_{q,n,\varsigma }, \forall U_{n} \in \mathcal {U}_{q,k,\varsigma }\), whereas the remaining constraints are linear.
Remark 8
RPRSINRMAX (ω_{q, k,ς}) is a concave minimization problem [30] and thus is solved efficiently with minimal complexity via any standard convex optimization package [31].
Remark 9
The optimal solution of problem RPRSINRMAX (ω_{q, k,ς}) is also the optimal solution of the original problem PRSINRMAX (ω_{q, k,ς}) [30].
The following solution procedure determines the resulting SINR per user U_{n} for each channel C_{q, k} of each base station BS_{q}. The goal is to iterate over all possible user combinations in Ω_{q, k}. Specifically, the solution procedure starts by considering singleuser combinations (i.e., when u_{q}=1 and for each ω_{q, k,ς}∈Ω_{q, k,1}) over each channel \(C_{q,k} \in \mathcal {C}_{q}\) and determines whether the target minimum SINR \(\gamma _{T_{n}}\) is satisfied by calculating \(\gamma ^{k}_{q,n,\varsigma }, \forall U_{n} \in \mathcal {U}_{q,k,\varsigma }\). If \(\gamma ^{k}_{q,n,\varsigma } < \gamma _{T_{n}}\) for any \(\forall U_{n} \in \mathcal {U}_{q,k,\varsigma }\), then set \(\gamma ^{k}_{q,n,\varsigma } = 0\); otherwise, keep the value of \(\gamma ^{k}_{q,n,\varsigma }\) for later use to determine the preference lists. After that, the solution procedure considers user combinations of sizes 2≤u_{q}≤ζ_{q, k}, and solves problem RPRSINRMAX (ω_{q, k,ς}) for each combination, such that the optimal SINR of each user in each combination over each channel is evaluated to determine whether the target minimum SINR is satisfied, as stated earlier. Upon completion of the solution procedure, each user \(U_{n} \in \mathcal {U}_{q}\) calculates
for each channel \(C_{q,k} \in \mathcal {C}_{q}\), which is considered as a weight of how valuable that channel is to that user. Contrarily, each base station \({\text {BS}}_{q} \in \mathcal {B}\) calculates
for each user \(U_{n} \in \mathcal {U}_{q}\), which is used as a weight for how valuable that user is to that base station.
The proposed solution procedure for proportional fairness SINRbased maximization (SPPFSINRMAX) is outlined in Algorithm 1, which can be performed locally at each base station.
Remark 10
For the case when u_{q}=1 (i.e., only combinations ω_{q, k,ς}∈Ω_{q, k,1}, for ς=1,2,…,ϖ_{1}) in which only a single user is considered over each channel \(C_{q,k} \in \mathcal {C}_{q} \left (\text {i.e.}\ \mathcal {J}^{k}_{q,1,n} = 1\right)\), if a user \(U_{n} \in \mathcal {U}_{q,k,\varsigma }\) cannot satisfy the target minimum SINR \(\gamma _{T_{n}}\), then it cannot satisfy it when a greater number of users are considered (i.e., when 2≤u_{q}≤ζ_{q, k}) over that channel (due to negative network externality). Hence, \(\chi ^{k}_{q,n} = 0\) for user \(U_{n} \in \mathcal {U}_{q}\) over channel \(C_{q,k} \in \mathcal {C}_{q}\).
Remark 11
The greater the value of \(\chi ^{k}_{q,n}\) is, the more preferred channel \(C_{q,k} \in \mathcal {C}_{q}\) is to user \(U_{n} \in \mathcal {U}_{q}\). In a similar manner, the greater the value of ψ_{q, n} is, the more preferred user \(U_{n} \in \mathcal {U}_{q}\) is to base station \({BS}_{q} \in \mathcal {B}\).
Remark 12
If for any user \(U_{n} \in \mathcal {U}_{q}\), the value of \(\chi ^{k}_{q,n} = 0\), then that channel is considered unacceptable to that user. In a similar manner, if, for a base station \({BS}_{q} \in \mathcal {B}, \psi _{q,n} = 0\), then user U_{n} is deemed unacceptable.
Remark 13
At the outset, the proposed solution procedure may seem to be excessively complex. However, for each base station \({BS}_{q} \in \mathcal {B}\), there is a total of \(\mathcal {C}_{q} \cdot \Sigma \left (\mathcal {U}_{q}, \zeta _{q,k} \right)\) iterations, and a convex optimization problem is efficiently solved in only \(\mathcal {C}_{q} \cdot \Sigma \left (\mathcal {U}_{q}, \zeta _{q,k} \right)  \mathcal {U}_{q}\) iterations and thus is guaranteed to converge^{Footnote 5}.
It should be noted that in multicell NOMA networks, a small number of users are to be multiplexed into a channel, which is mainly to minimize interuser interference, reduce SIC hardware complexity and error propagation, and leverage capacity gains [7], while satisfying the target minimum SINR constraint per user (i.e., ζ_{q, k} must be kept reasonably small, \(\forall C_{q,k} \in \mathcal {C}_{q}\)). Lastly, the proposed solution procedure eliminates the need for swap operations, which have been utilized in [17–19]. Specifically, in the aforementioned references, a swap operation is utilized in an iterative manner to ensure stability after power allocation. However, in the solution procedure, the optimal power allocation per user over each channel is already determined, which is then used to determine the preference lists in the proposed matching algorithms (discussed in the following section). Consequently, the stable matching solutions resulting from the proposed stable matching algorithms will already have obtained the optimal power allocation for the users sharing each channel within each base station.
Stable matching algorithms
In this section, the stable matching algorithms based on the SPA problem are devised.
Description
The classical SPA problem involves a set of students (users), projects (channels), and lecturers (base stations). Each project is offered by a specific lecturer, and each lecturer has quotas (i.e., a maximum number of students per project and per lecturer, respectively). Moreover, students have preferences over the projects that are acceptable (i.e., in which they would like to be involved), while each lecturer has preferences over the acceptable students^{Footnote 6}. Typically, a project is assigned to at most one student, while in some other cases, a project may be undertaken by more than one student to work on. More importantly, there should be a number of projects that coincides with the number of potential students, and each lecturer is responsible for offering a range of projects, which are not necessarily all taken up. By utilizing the students’ and lecturers’ preferences, the goal is to find a stable matching that pairs students to projects offered by each lecturer, while satisfying the quotas [14,15]. In an analogous manner, the goal is to assign users to channels available at each base station, such that a stable matching solution is obtained, while satisfying the quotas of the channels and base stations.
To that end, a few definitions must first be given.
Definitions
Definition 1
(Acceptability) A channel \(C_{q,k} \in \mathcal {C}_{q}, \forall {BS}_{q} \in \mathcal {B}\), is said to be acceptable to user \(U_{n} \in \mathcal {U}_{q}\) if \(\chi ^{k}_{q,n} > 0\). Thus, let \(\mathcal {A}_{U_{n}}\) be the list of acceptable channels by user U_{n}. In a similar fashion, \(\mathcal {A}_{BS_{q}}\) is the list of acceptable users \(U_{n} \in \mathcal {U}_{q}\) to base station BS_{q} (i.e., for which ψ_{q, n}>0).
Definition 2
(Assignment) An assignment \(\mathcal {M}\) is defined as a subset of \(\mathcal {U} \times \mathcal {C}\), such that:

(a)
\((U_{n}, C_{q,k}) \in \mathcal {M}\) (i.e., user U_{n} finds channel C_{q, k} acceptable).

(b)
A user U_{n} is assigned at most one channel (i.e., \( \left \{(U_{n}, C_{q,k}) \in \mathcal {M} \text { for}\, C_{q,k} \in \mathcal {A}_{U_{n}} \right \}  \leq 1\)).
In turn, if \((U_{n}, C_{q,k}) \in \mathcal {M}\), then user U_{n} is considered to be assigned to channel C_{q, k}, and vice versa (i.e., C_{q, k} is assigned to U_{n}). Also, let \(\mathcal {M}\left (U_{n} \right) = C_{q,k}\) indicate that channel C_{q, k} is assigned to user U_{n} in \(\mathcal {M}\), whereas \(\mathcal {M}\left (C_{q,k} \right) = U_{n}\) indicates that user U_{n} is assigned to channel C_{q, k}.
Definition 3
(Preference) If user U_{n} prefers channel C_{q, k} to C_{w, l} (i.e., \(\chi ^{k}_{q,n} > \chi ^{l}_{w,n}\) for k≠l)^{Footnote 7}, then \(C_{q,k} \succ _{U_{n}} C_{w,l}\phantom {\dot {i}\!}\). In a similar fashion, if base station BS_{q} prefers user U_{n} to U_{m} (i.e., ψ_{q, n}>ψ_{q, m} for n≠m), then \(\phantom {\dot {i}\!}U_{n} \succ _{BS_{q}} U_{m}\).
Definition 4
(Preference list) Let \(\mathbb {P}_{U_{n}} = \left \{C^{(1)}_{q,k}, \ldots,\right.\hspace *{3pt} \hspace *{1pt}\left.C^{(\mathcal {A}_{U_{n}})}_{w,l} \right \}\) denote the preference list of user U_{n}, where \(C^{(1)}_{q,k} \left (C^{(\mathcal {A}_{U_{n}})}_{w,l} \right)\) refers to the most (least) preferred channel to U_{n}. Similarly, \(\mathbb {P}_{BS_{q}} = \left \{U^{(1)}_{n}, \ldots, U^{(\mathcal {A}_{BS_{q}})}_{m} \right \}\) denotes the preference list of base station BS_{q}, where \(U^{(1)}_{n} \left (U^{(\mathcal {A}_{BS_{q}})}_{m} \right)\) refers to the most (least) preferred user to BS_{q}.
Definition 5
(Projected preference list) Let \(\overline {\mathbb {P}}^{k}_{BS_{q}},\! \forall \! {BS}_{q}\! \in \! \mathcal {B}\), be the projected preference list of base station BS_{q}, which is obtained from \(\mathbb {P}_{BS_{q}}\) by eliminating the users who find channel \(C_{q,k} \in \mathcal {C}_{q}\) unacceptable.
Definition 6
(Subscription) A channel \(C_{q,k} \in \mathcal {C}_{q}\) is said to be undersubscribed, full, or oversubscribed if \(\mathcal {M}\left (C_{q,k} \right)\) is less than, equal to, or greater than ζ_{q, k}, respectively. In a similar manner, for any base station \({BS}_{q} \in \mathcal {B}\) under assignment \(\mathcal {M}\) (i.e., \(\mathcal {M}\left ({BS}_{q} \right)\)), BS_{q} is considered to be undersubscribed, full, or oversubscribed if \(\mathcal {M}\left ({BS}_{q} \right)\) is less, equal to, or greater than ξ_{q}, respectively.
Definition 7
(Matching) A matching\(\mathcal {M}\) is an assignment, such that [14]:

(a)
For each channel \(C_{q,k} \in \mathcal {C}_{q}, \mathcal {M}\left (C_{q,k} \right) \leq \zeta _{q,k}\).

(b)
For each base station \({BS}_{q} \in \mathcal {B}, \mathcal {M}\left ({BS}_{q} \right) \leq \xi _{q}\).
In other words, under matching \(\mathcal {M}\), no channel \(C_{q,k} \in \mathcal {C}_{q}\) is assigned to more than ζ_{q, k} users and that each base station BS_{q} is assigned at most ξ_{q} users.
Definition 8
(Blocking) The pair \(\left (U_{n}, C_{q,k} \right) \in \left (\mathcal {U} \times \mathcal {C} \right) \setminus \mathcal {M}\) is said to block a matching \(\mathcal {M}\) if [15]:

(a)
\(C_{q,k} \in \mathcal {A}_{U_{n}}\) (i.e., user U_{n} finds C_{q, k} acceptable).

(b)
Either U_{n} is unassigned in \(\mathcal {M}\) or U_{n} prefers C_{w, l} to \(\mathcal {M}\left (U_{n} \right)\).

(c)
Either

(c1)
C_{q, k} is undersubscribed, and BS_{q} is undersubscribed, or

(c2)
C_{q, k} is undersubscribed, BS_{q} is full, and either \(U_{n} \in \mathcal {M}\left ({BS}_{q} \right)\) or BS_{q} prefers U_{n} to the worst user in \(\mathcal {M}\left ({{BS}}_{q} \right)\), or

(c3)
C_{q, k} is full and BS_{q} prefers U_{n} to the worst user in \(\mathcal {M}\left (C_{q,k} \right)\).

(c1)
Accordingly, a (student, project) pair satisfying the above conditions should not be included in the matching solution, as such matching would not be stable.
Definition 9
(Stable matching) A matching \(\mathcal {M}\) is considered stable if \(\mathcal {M}\) contains no blocking pairs.
Algorithmic design
In this subsection, the useroriented stable matching (USM) and base stationoriented stable matching (BSSM) algorithms are described [15].
Useroriented stable matching
Initially, all users are assumed to be free, and all channels and base stations are completely unsubscribed. The USM algorithm is based on a sequence of proposals by each user to the acceptable channels in its preference list. Such proposals lead to provisional assignments between users, channels, and base stations, which may be modified during the execution of the algorithm. That is, entries from the preference lists of users and the preference lists of base stations may be eliminated. To be more specific, deleting a pair (U_{n},C_{q, k}) corresponds to eliminating channel C_{q, k} from user U_{n}’s preference list \(\mathbb {P}_{U_{n}}\) and also removing U_{n} from the projected list \(\overline {\mathbb {P}}^{k}_{\text {BS}_{q}}\) of base station BS_{q}, which offers channel C_{q, k}. The algorithm iterates as long as there is some user with a nonempty preference list and who is still unpaired to any channel under any base station. If some channel C_{q, k} is oversubscribed, then its base station rejects the worst user U_{m} assigned to that channel (i.e., the pair (U_{m},C_{q, k}) is deleted). Also, if some base station BS_{q} is oversubscribed, then it rejects its worst assigned user U_{m} and deletes the pair (U_{m},C_{q, l}), where C_{q, l} is the channel previously assigned to user U_{m}. On the other hand, if a channel C_{q, k} is full, then for the worst user U_{m} paired to the channel (as per \(\overline {\mathbb {P}}^{k}_{\text {BS}_{q}}\)), delete the pair of each successive user U_{p} with that channel (i.e., (U_{p},C_{q, k})). Lastly, if some base station BS_{q} is full, then for the worst user U_{m} associated with BS_{q}, delete the pair (U_{p},C_{q, l}) of each successive user U_{p} (in \(\mathbb {P}_{\text {BS}_{q}}\)) with each channel C_{q, l} that U_{p} finds acceptable by base station BS_{q}. The USM algorithm is outlined in Algorithm 2.
Base stationoriented stable matching
In a similar manner to the USM algorithm, all users are initially assumed to be free, and all channels and base stations are totally unsubscribed. Now, the BSSM algorithm iterates over each base station BS_{q} that is undersubscribed and offers a channel C_{q, k} to a user U_{n}, which is the first on BS_{q}’s preference list \(\mathbb {P}_{\text {BS}_{q}}\), and C_{q, k} must be the first undersubscribed channel on U_{n}’s preference list \(\mathbb {P}_{U_{n}}\), such that \(U_{n} \in \overline {\mathbb {P}}^{k}_{\text {BS}_{q}}\). After breaking any existing assignment of user U_{n}, then it is provisionally assigned to channel C_{q, k} under base station BS_{q}. In turn, any pair (U_{n},C_{w, l}) to which user U_{n} prefers C_{q, k} to C_{w, l} is deleted, and hence, C_{w, l} is removed from U_{n}’s preference list, and U_{n} is eliminated from the projected list \(\overline {\mathbb {P}}^{l}_{\text {BS}_{w}}\) of base station BS_{w} offering channel C_{w, l}. This process is repeated until convergence, as given in Algorithm 3.
Remark 14
Backhaul links can be utilized to efficiently execute the USM and BSSM algorithms among the base stations with minimal overhead [32,33] and without the need for a centralized controller.
Properties
In the following subsections, the properties of the USM and BSSM algorithms are discussed. To be concised, the properties of the USM algorithm are discussed, which are also applicable to the BSSM algorithm.
Convergence to a stable matching solution
Lemma 1
The USM algorithm converges in a finite number of iterations to a stable matching solution.
Proof
In the USM algorithm, a free user applies to the first channel on its preference list in each iteration. In particular, no user can apply to the same channel more than once (i.e., each userchannel pair can occur at most once). This can be verified from the fact that once a pair (U_{m},C_{q, k}) is deleted, then user U_{m} is freed and cannot apply to that channel again. This also implies that no pair deleted during the execution of the USM algorithm can block the matching under construction [15]. If this had not been the case (i.e., the pair (U_{m},C_{q, k}) is not deleted), then user U_{m} must be assigned to some channel \(\mathcal {M}_{U}\left (U_{m} \right) \neq C_{q,k}\); otherwise, user U_{m} remains free with a nonempty preference list containing channel C_{q, k} (i.e., a contradiction). Therefore, the pair (U_{m},C_{q, k}) must be deleted, since if U_{m} prefers C_{q, k} to \(\mathcal {M}_{U}\left (U_{m}\right)\), then (U_{m},C_{q, k}) must block \(\mathcal {M}_{U}\). More importantly, the USM algorithm never deletes a stable pair during its execution [15]. Lastly, since the lengths of the user preferences lists are bounded, then the total number of iterations is also bounded. Hence, the USM algorithm converges in a finite number of iterations to a stable matching solution. □
Complexity
Lemma 2
The USM algorithm converges with polynomialtime complexity of \(\mathcal {O}\left (\mathcal {U}\cdot \mathcal {C} \right)\), where \(\mathcal {U}\) and \(\mathcal {C}\) are the total number of users and channels, respectively [15].
Proof
It is straightforward to verify that in the worstcase scenario of the USM algorithm, each free user \(U_{n} \in \mathcal {U}\)–with a nonempty preference list—applies to at least one channel out of the possible \(\mathcal {C}\) channels. Moreover, throughout the execution of the algorithm, some userchannel pairs may be deleted, and their corresponding entries are deleted from the preference lists of users and from the projected preference lists of base stations. Hence, the overall worstcase complexity of the USM algorithm is \(\mathcal {O}\left (\mathcal {U}\cdot \mathcal {C} \right)\). □
Optimality
Lemma 3
The stable matching resulting from the USM algorithm is optimal with respect to each assigned user (i.e., useroptimal stable matching).
Proof
Since each user is assigned to its first preferred channel on its reduced preference list, and no stable pair is deleted during the execution of the USM algorithm [15], then each user is simultaneously assigned to the best channel it can get in any stable matching. In addition, and due to the proposed solution procedure SPPFSINRMAX, each user is not only assigned to its best channel, but also is allocated power optimally over its assigned channel. □
By similar arguments, the BSSM algorithm can straightforwardly be verified to converge to a stable matching solution within a finite number of iterations, and with polynomialtime complexity of \(\mathcal {O}\left (\mathcal {U}\cdot \mathcal {C} \right)\) [15]. It should be noted that the complexity of both the USM and BSSM algorithms may be much lower than \(\mathcal {O}\left (\mathcal {U}\cdot \mathcal {C} \right)\), which is due to the fact that each user \(U_{n} \in \mathcal {U}_{q}\) only has preferences over the channels within its cell (i.e., \(\mathcal {C}_{q}\)). In other words, it is only the users within the overlapping region of all cells that may have preferences over all channels. Lastbutnotleast, the stable matching resulting from the BSSM algorithm is simultaneously optimal with respect to each base station. That is, each base station is associated with the best set of users it can get.
Despite the optimality of the proportional fairness SINRbased maximizing power allocation (and hence the solution procedure SPPFSINRMAX), and also the optimality of the proposed stable matching algorithms (i.e., USM and BSSM), the resulting basestation association, channel assignment, and power allocation solutions are not necessarily global optimal. This is due to the following two reasons. Firstly, decomposing problem CJUACAPA into two subproblems does not necessarily guarantee the global optimal solution. Secondly, enforcing stability in terms of the user association and channel assignment via the proposed stable matching algorithms may lead to a suboptimal solution. Nevertheless, our algorithmic solutions pose a tradeoff between complexity, optimality, and stability.
Remark 15
Problem CJUACAPA globally optimally maximizes proportional fairness, but does not necessarily ensure that the resulting user association and channel assignment are stable.
The intuition behind Remark 15 is that for some network instance, a certain user may be paired to a certain channel in the solution of problem CJUACAPA. However, that (user, channel) pair forms a blocking pair and thus cannot be included in the stable matching solution of any of the stable matching algorithms. In other words, that (user, channel) pair may be formed under problem CJUACAPA so as to obtain the global optimal solution, however would be excluded from the matching solution to ensure stability. Specifically, stability is particularly important from a network’s perspective, as it ensures that no user or base station would unilaterally want to change its channel assignment or association. This in turn minimizes communications overheads and signaling.
Simulation results
This section evaluates the performance of the proposed solution procedure and stable matching algorithms. The simulations assume a network consisting of Q=3 base stations/cells, with N=12 users, K=9 channels in a 200×200 m^{2} area and cell radius of 50 m per base station (see Fig. 2). Also, each base station \({\text {BS}}_{q} \in \mathcal {B}\) for q∈{1,2,3} is allocated the channel sets \(\mathcal {C}_{1} = \{C_{1}, C_{2}, C_{3}\}, \mathcal {C}_{2} = \{C_{4}, C_{5}, C_{6}\}\), and \(\mathcal {C}_{3} = \{C_{7}, C_{8}, C_{9}\}\), respectively. Moreover, the transmit power per allocated channel is set to P=1 W, the noise variance is N_{0}=10^{−7} W, and the path loss exponent is ν=3. The target minimum SINR per user is set to \(\gamma _{T_{n}} = \gamma _{T} = 15\) dB, \(\forall U_{n} \in \mathcal {U}\). Furthermore, the simulations are averaged over 10^{3} independent instances of randomly generated channel coefficients, where the channel coefficients are assumed to be quasistatic during each network instance, but vary from one network instance to another^{Footnote 8}.
In the simulations, the following scenarios are compared: Scenario 1: ξ_{q} = 4 and \(\zeta _{q,k} = 2, \forall C_{q,k} \in \mathcal {C}_{q}\) and \(\forall {\text {BS}}_{q} \in \mathcal {B}\). Scenario 2: ξ_{q} = 4 and \(\zeta _{q,k} = 3, \forall C_{q,k} \in \mathcal {C}_{q}\) and \(\forall {\text {BS}}_{q} \in \mathcal {B}\).
These scenarios aim at investigating the effect of assigning different numbers of users to each channel. In addition, in the aforementioned scenarios, the following two deterministic power allocation schemes are compared: Equal power allocation (EPA)In this scheme, the power is equally allocated across all users over each channel. Specifically, for Scenario 1 (i.e., ζ_{q, k}=2), \(a^{k}_{q,n} = 1/2\), while in Scenario 2 (i.e., ζ_{q, k}=3), \(a^{k}_{q,n} = 1/3, \forall U_{n} \in \mathcal {U}_{q}, \forall C_{q,k} \in \mathcal {C}_{q}\), and \(\forall {BS}_{q} \in \mathcal {B}\). Conventional power allocation (CPA)For this scheme, the power allocation coefficients of the ordered network users (for 1≤n≤ζ_{q, k}) are set as \(a^{k}_{q,n} = \frac {3n}{3}\) in Scenario 1, while in Scenario 2, \(a^{k}_{q,n} = \frac {4n}{6}, \forall U_{n} \in \mathcal {U}_{q}, \forall C_{q,k} \in \mathcal {C}_{q}\), and \(\forall {BS}_{q} \in \mathcal {B}\) [34]. That is, for the ordered users under Scenario 1, \(a^{k}_{q,1} = \frac {2}{3}\) and \(a^{k}_{q,2} = \frac {1}{3} \left (\text {i.e.,}\ a^{k}_{q,1} > a^{k}_{q,2}\right)\), while under Scenario 2, \(a^{k}_{q,1} = \frac {3}{6}, a^{k}_{q,2} = \frac {2}{6}\), and \(a^{k}_{q,3} = \frac {1}{6}\ \left (\text {i.e.,}\ a^{k}_{q,1} > a^{k}_{q,2} > a^{k}_{q,3}\right)\).
For notational convenience, let the proposed USM and BSSM matching algorithm when combined with the SPPFSINRMAX be denoted SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) for short. Similarly, let the power allocation schemes when combined with the proposed stable matching algorithms be denoted EPA (USM), EPA (BSSM), CPA (USM), and CPA (BSSM). Lastly, the proposed algorithms are compared to the CJUACAPA scheme^{Footnote 9}.
In Fig. 3a, the average SINR per user for Scenario 1 is illustrated. In particular, it can be seen that for the SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) algorithms, all network users satisfy the target minimum SINR per user of γ_{T}=15 dB. On the other hand, for the proposed stable matching algorithms with the EPA and CPA schemes, not all users satisfy γ_{T}. This is due to the fact that power is deterministically allocated to the paired users and without ensuring that the target minimum SINR constraint per user over each channel is satisfied. This proves that the SPPFSINRMAX is effective in guaranteeing that γ_{T} is met for all network users. It is also evident that the highest average SINR among all users is achieved by users U_{2},U_{7}, and U_{10}, which is due to their locations being closest to their respective base stations, as can be seen from Fig. 2. Moreover, the CJUACAPA scheme marginally outperforms the proposed SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) algorithms, while guaranteeing that γ_{T} is achieved. This is because the CJUACAPA scheme maximizes proportional fairness of the SINR of all network users via user association and channel assignment without any bearing on the network stability (as per Remark 15). Figure 3b shows the average power allocation coefficient per user. Evidently, the SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) algorithms allocate power very similarly to all network users. However, the power allocated per user resulting from the CJUACAPA scheme is lower than the aforementioned algorithms. As before, this is attributed to the fact that the CJUACAPA scheme aims at maximizing proportional fairness without necessarily enforcing stability. Generally speaking, the CJUACAPA scheme achieves only marginally higher average SINR per network user with relatively lower transmit power per user. Lastly, it is observed that the users U_{2},U_{7}, and U_{10} are allocated relatively lower power than the other network users, since they are relatively closer to their respective base stations than the other users (i.e., in agreement with the concept of NOMA).
Similar observations to Fig. 3 can be made for Scenario 2 (see Fig. 4a and b). However, the average SINR per user in Scenario 2 is relatively lower than in Scenario 1. This is attributed to the fact that in Scenario 2, \(\zeta _{q,k} = 3, \forall C_{q,k} \in \mathcal {C}_{q}, \forall {BS}_{q} \in \mathcal {B}\), which implies more users can be assigned to each channel. This in turn means that the available transmit power over each channel is distributed over a potentially larger number of users, and hence, each user’s share of the allocated power is less than that of Scenario 1. As before, it can be seen from Fig. 4a that γ_{T} is satisfied for all network users under both proposed SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) algorithms as well as the CJUACAPA scheme, but this is not the case for the EPA and CPA schemes when combined with the USM and BSSM matching algorithms.
Figure 5 illustrates Jain’s fairness index based on the average SINR of each network user under both scenarios [36]. Evidently, under both scenarios, the CJUACAPA scheme achieves the greatest fairness, which is followed by the proposed SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) algorithms, respectively. Contrarily, the EPA and CPA schemes when combined with the USM and BSSM matching algorithms achieve the lowest fairness among all schemes. In addition, the different schemes achieve relatively greater fairness in Scenario 2 than in Scenario 1. This is because Scenario 2 considers a potentially greater number of user combinations over each channel, which in turn maximizes proportional fairness across a greater number of users per channel. Hence, this confirms that the SPPFSINRMAX maximizes proportional fairness among all network users.
In Fig. 6, the average number of associated users per base station of the SPPFSINRMAX (USM) algorithm is illustrated. One can see that for Scenario 1 (2), base station BS_{3} is associated with 4 users about 90.79% (93.47%) of the time, which is greater than base stations BS_{1} and BS_{2}. More importantly, BS_{3} is always associated with at least 3 users under both scenarios. In Fig. 7, the average number of associated users per base station of the SPPFSINRMAX (BSSM) is illustrated. As before, for BS_{3}, the average numbers of associated users for Scenarios 1 and 2 are 82.88% and 84.07%, respectively, which are greater than those of BS_{1} and BS_{2}. By comparing Figs. 6 and 7, base stations BS_{1} and BS_{2} under the SPPFSINRMAX (BSSM) algorithm are associated with four users more often than the SPPFSINRMAX (USM) algorithm, under both scenarios. An opposite observation can be made for base station BS_{3}.
Figure 8 illustrates the percentage of user unassignment under the USM and BSSM algorithms when combined with the SPPFSINRMAX for both scenarios. Evidently, U_{1} and U_{8} are the two users with the highest percentage of unassignment, and this is because they are farthest from their respective base stations as well as not being in the overlapping region of other base stations. It is also noteworthy that although user U_{6} is located at the cell edge of all three base stations, the percentage of it being unassigned is at most 2%, under both scenarios. This is due to the fact that it can be allocated a channel by any of the three base stations, since it falls within the overlapping region of all three base stations. Additionally, all users except U_{1} and U_{8} are assigned to a channel and associated with a base station at least 90% of the time, under both scenarios.
Figure 9 demonstrates the percentage of channel assignment of users U_{3},U_{4},U_{5},U_{6}, and U_{9} under both matching algorithms (with SPPFSINRMAX) under Scenario 1. Specifically, one can see that user U_{3} is paired only with the channels of base stations BS_{1} and BS_{3} (i.e., in \(\mathcal {C}_{1}\) and \(\mathcal {C}_{3}\)). As for user U_{6}, it is paired with all channels, since it falls within the overlapping region of all three base stations. Similar observations can be made to the other users. To summarize, it has been verified that all users can only be paired with the channels corresponding to cells where they are located. Similar observations are made for Scenario 2 in Fig. 10.
Figure 11a and b illustrate the average number of iterations of each stable matching algorithms (with SPPFSINRMAX) under Scenarios 1 and 2. It is evident that both algorithms require almost the same number of iterations under the two scenarios. More importantly, the BSSM algorithm requires more iterations than its USM counterpart algorithm. In addition, for our simulated network scenario in Fig. 2, \(\mathcal {C}_{1} = \mathcal {C}_{2} = \mathcal {C}_{3} = 3\), and \(\mathcal {U}_{1} = \mathcal {U}_{2} = \mathcal {U}_{3} = 6\). Based on Remarks 5, 8, and 13, and for Scenario 1, the execution of the solution procedure locally at each base station requires a total of 63 iterations, of which 45 iterations involve the solution of a convex optimization problem, which can be solved within polynomialtime complexity. As for scenario 2, each base station involves at total of 123 iterations of the solution procedure, of which 105 iterations involve the solution a convex optimization problem. In turn, both SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) algorithms can be executed efficiently and with lower computational complexity than the CJUACAPA scheme. On the other hand, in Fig. 11c, the percentage of identical stable matchings of the proposed stable matching algorithms is illustrated. Particularly, it can be seen that for Scenario 2, the percentage of identical matchings is slightly less than that of Scenario 1. This is due to the fact in Scenario 2, there are more user combinations than in the case of Scenario 1, which results in identical matchings occurring slightly less often. In all, increasing the maximum number of users that can be assigned a channel reduces the possibility of having identical matchings for both stable matching algorithms. This also explains the discrepancies in the resulting average SINR, average number of associated users with each base station, and percentages of user unassignment and channel assignment of the proposed stable matching algorithms.
In summary, the proposed algorithmic solutions have been shown to yield comparable performance to the CJUACAPA scheme in terms of average SINR per user and network SINRbased fairness. Additionally, the proposed algorithms have been shown to efficiently associate users with base stations and assign them to channels. More importantly, the proposed solution procedure can be executed locally at each base station to determine the proportional fairness maximizing power allocation and preference lists, while the stable matching algorithms can be efficiently executed among the base stations to perform user association and channel assignment with minimal communication overheads and signaling, and with the added merit of network stability. In other words, the proposed SPPFSINRMAX (USM) and SPPFSINRMAX (BSSM) algorithms offer a reasonable tradeoff between average SINR, fairness, complexity, and stability.
Conclusions
In this paper, the problem of joint user association and channel assignment with proportional fairness SINRbased power allocation in downlink multicell NOMA networks has been studied. Particularly, a lowcomplexity iterative solution procedure has been devised to determine the optimal power allocation for proportional fairness SINRbased maximization over each channel within each cell. Moreover, two manytoone polynomialtime complexity matching algorithms have been proposed to associate users with base stations and perform channel assignment. To validate the efficacy of the proposed solution procedure and stable matching algorithms, extensive simulation results have been presented, which illustrate that the proposed algorithms efficiently yield comparable SINR per user to the CJUACAPA scheme as well as maximizing proportional fairness and satisfying QoS constraints. Finally, the proposed algorithms have been shown to efficiently assign channels to celledge users and especially those within the overlapping region of multiple cells.
Notes
 1.
The SIC receiver complexity is \(\mathcal {O}\left (\zeta ^{3}_{q,k} \right), \forall C_{q,k} \in \mathcal {C}_{q}\), and \(\forall {BS}_{q} \in \mathcal {B}\) [2].
 2.
In other words, no frequency reuse is assumed in this work, and thus, intercell interference is not considered.
 3.
Perfect CSI knowledge is assumed at the base stations.
 4.
The case of imperfect SIC is beyond the scope of this work; however, practical solutions for mitigating SIC errors can be found in [20].
 5.
Each convex optimization problem is solved efficiently within polynomialtime complexity [31].
 6.
The preferences are ordered in a descending order, from most preferred to the least.
 7.
It is noteworthy that C_{w, l} refers to a channel different from C_{q, k} under a possibly different base station BS_{w}.
 8.
Our algorithmic designs are applicable to arbitrary network topologies and sets of parameters, provided that the selected parameters yield feasible solutions.
 9.
The CJUACAPA problem is solved via MIDACO [35], with tolerance set to 0.001. Moreover, problem CJUACAPA involves a total of 648 decision variables and 144 inequality constraints.
Abbreviations
 BS:

Base station
 BSSM:

Base stationoriented stable matching
 CJUACAPA:

Centralized joint user association, channel assignment, and power allocation
 CPA:

Conventional power allocation
 CoMP:

Coordinated multipoint
 CSI:

Channel state information
 CSIT:

Channel state information at transmitter
 D2D:

Devicetodevice
 EPA:

Equal power allocation
 LF:

Linearfractional
 mmWave:

Millimeterwave
 NOMA:

Nonorthogonal multiple access
 OFDMA:

Orthogonal frequencydivision multiple access
 OMA:

Orthogonal multiple access
 PRSINRMAX:

Proportional fairness SINRbased maximization
 SIC:

Successive interference cancelation
 SINR:

Signaltointerferenceplusnoise ratio
 SPPFSINRMAX:

Solution procedure for proportional fairness SINR maximization
 SPA:

Studentproject allocation
 SPM:

Sumpower minimization
 SRM:

Sumrate maximization
 USM:

Useroriented stable matching
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Acknowledgements
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Funding
This work is supported partially by the Kuwait Foundation for the Advancement of Sciences (KFAS), under project code PN1715EE02.
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All authors have contributed to the work presented in this paper. Particularly, the contributions can be stated as follows: MWB contributed to the problem formulation, mathematical analysis, writing, and simulation. ZB contributed to the mathematical analysis and simulation. EA contributed to the problem formulation, mathematical analysis, and writing. Lastly, all authors read and approved the final manuscript.
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Correspondence to Mohammed W. Baidas.
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Keywords
 Channel assignment
 Matching
 Multicell
 Nonorthogonal multiple access
 Proportional fairness
 User association