NonOrthogonal Multiple Access for Unicast and Multicast D2D: Channel Assignment, Power Allocation and Energy Efficiency
Abstract
:1. Introduction
2. Related Work
 1.
 The investigated system models in the literature assume a single/group of D2D pairs can share CUs resource blocks. In this unicast communication model, a transmitter sends data to a single receiver. However, in our system model, devices with a common interest form a group, where a single transmitter multicasts data to a set of receivers (the MD2D group). Note that the MD2D mode also includes the particular case of unicast D2D communications (when the groups just include one receiver).
 2.
 In multicast communications, group data rate is determined according to the receiver with the poorest channel quality (CQ). Therefore, we assume that the receivers in a MD2D group as well as the CUs are able to apply SIC to the stronger interference signal. This would reduce the received interference to a minimum value leading to enhancements in MD2D communication quality.
 3.
 We model the resource allocation subproblem in underlay MD2D using matching theory and overlapping coalition formation game to minimize harmful mutual interference as the means to maximize energy efficiency for both the system as a whole and for individual users, metrics not considered in other related works.
 4.
 The resource allocation approach involves two design parameters (the reuse degree and the split factor) which allow considering all the possible RB sharing variants when analyzing the system behavior.
 5.
 The power control subproblem is optimally solved using fractional programming. In this work, we assume that a central entity is in charge of assigning transmission power to each transmitter in the network.
 6.
 We evaluate the performance of NOMAbased systems under a broad range of resource sharing scenarios. As performance metrics, we analyze both EE and transmission rate for the whole system (global EE and sumrate). A potential drawback of this approach is that global performance measures do not properly capture the service fairness among users. Thus, we also formulate and analyze the MMF EE and the rate for individual users.
 Scenario 1
 ($r=s=1$): Each CU shares its communication channel with a single D2D pair/MD2D group. Similarly, each D2D pair/MD2D group uses only one channel.
 Scenario 2
 ($s>1$, $r=1$): A D2D pair/MD2D group takes advantage of multiple cellular channels and distributes its message over them. Here, a D2D pair/MD2D group may disperse its transmission power budget or rate among the occupied channels. However, a channel cannot support more than one D2D pair/MD2D group.
 Scenario 3
 ($s=1$, $r>1$): Each cellular communication channel can support up to r D2D pairs/MD2D groups, which are allowed to use one cellular channel, at most. Compared to the previous scenarios, having r devices using the same channel leads to mutual interference accumulation over the CU and each D2D pair/MD2D group per channel.
 Scenario 4
 ($s>1$, $r>1$): Each D2D pair/MD2D group uses up to s different cellular channels. Moreover, each channel can support r D2D pairs/MD2D groups. As in Scenario 3, the accumulated mutual interference among the channel’s devices may negatively affect both type of communications and limit the benefits of D2D pairs/MD2D groups in term of spectral and energy efficiency.
3. Problem Formulation
3.1. System Model
 The CU interference is caused by the BS transmissions to the CU(s) over the channel(s) used by the MD2D group at the same time.
 The intergroups interference is caused by the transmitters on those MD2D groups that are reusing the channel(s) the MD2D group is also accessing.
3.2. Channel Model
3.3. Resource Allocation Problem
4. Resource Allocation: Power Control and Channel Assignment
4.1. Power Control Algorithm
Algorithm 1: Optimal power control for GEE and MMFEE (Dinkelbach’s algorithm). 

4.2. Centralized Channel Assignment: Matching Theory
4.2.1. Channel Assignment Algorithm
 OnetoOne Matching: A onetoone match $\mu $ is a mapping from $\mathcal{D}\cup \mathcal{C}$ to itself such that, for any ${\mathcal{D}}_{k}\in \mathcal{D}$, if $\mu \left({\mathcal{D}}_{k}\right)\ne {\mathcal{D}}_{k}$, then $\mu \left({\mathcal{D}}_{k}\right)\in \mathcal{C}$, and, if $\mu \left({C}_{m}\right)\ne {C}_{m}$ for some ${C}_{m}\in \mathcal{C}$, then $\mu \left({C}_{m}\right)\in \mathcal{D}$. The partner ${\mathcal{D}}_{k}$ is referred to as $\mu \left({C}_{m}\right)$ if $\mu \left({C}_{m}\right)={\mathcal{D}}_{k}$. The preference function for setting the matching uses the received aggregate interference on each MD2D group ${\mathcal{D}}_{k}$ given by$${\alpha}_{k}^{\left(m\right)}=\underset{j\in {\mathcal{D}}_{k}}{max}{I}_{k:j,m},\forall {\mathcal{D}}_{k}\in \mathcal{D},$$$${\Gamma}_{m}=\sum _{k\in \mathcal{D}}{c}_{k,m}{h}_{k,m}{p}_{k,m},\forall {C}_{m}\in \mathcal{C}.$$Since ${\Gamma}_{m}$ is additive, we isolate the contribution of the MD2D group ${\mathcal{D}}_{i}$ by denoting ${\Gamma}_{m}^{{\mathcal{D}}_{i}}={h}_{i,m}{p}_{i,m}$, or, equivalently, setting ${c}_{i,m}=1$ and ${c}_{k,m}=0,\forall k\ne i$, in (12). Then, the preference relationship is defined as follows: (i) group ${\mathcal{D}}_{k}$ prefers channel ${C}_{i}$ to ${C}_{j}$ if ${\alpha}_{k}^{\left(i\right)}<{\alpha}_{k}^{\left(j\right)}$; and (ii) user ${C}_{m}$ prefers group ${\mathcal{D}}_{i}$ to ${\mathcal{D}}_{j}$ if ${\Gamma}_{m}^{{\mathcal{D}}_{i}}<{\Gamma}_{m}^{{\mathcal{D}}_{j}}$. Note that, if a channel m is empty, then $\mu \left({C}_{m}\right)={C}_{m}$, and, when group ${\mathcal{D}}_{k}$ is forbidden to transmit on any channel, $\mu \left({\mathcal{D}}_{k}\right)={\mathcal{D}}_{k}$.
 ManytoOne Matching: A manytoone match $\mu $ is a mapping from $\mathcal{D}\cup \mathcal{C}$ to itself such that, for each ${\mathcal{D}}_{k}\in \mathcal{D}$, if $\mu \left({\mathcal{D}}_{k}\right)\ne {\mathcal{D}}_{k}$, then $\mu \left({\mathcal{D}}_{k}\right)\in \mathcal{C}$, and, if $\mu \left({C}_{m}\right)\ne {C}_{m}$ for some ${C}_{m}\in \mathcal{C}$, then $\mu \left({C}_{m}\right)\in \mathcal{D}$. The partner ${\mathcal{D}}_{k}$ is referred to as $\mu \left({C}_{m}\right)$ if $\mu \left({C}_{m}\right)={\mathcal{D}}_{k}$. For manytoone matches, the preference relationship is similarly defined as follows: (i) transmitter in ${\mathcal{D}}_{k}$ prefers channel ${C}_{i}$ to channel ${C}_{j}$ if ${\alpha}_{k}^{\left(i\right)}<{\alpha}_{k}^{\left(j\right)}$; and (ii) user ${C}_{m}$ prefers ${\mathcal{D}}_{i}$ to ${\mathcal{D}}_{j}$ if ${\Gamma}_{m}^{{\mathcal{D}}_{i}}<{\Gamma}_{m}^{{\mathcal{D}}_{j}}$; (iii) $\left\mu \right({C}_{m}\left)\right$$\le r$, where r is channel m’s reuse factor.
 ManytoMany Matching: For matches between arbitrary subsets of $\mathcal{D}$ and $\mathcal{C}$, the following preference relationship is defined by: (i) ${\mathcal{D}}_{k}$ prefers channel ${C}_{i}$ to ${C}_{j}$ if ${\alpha}_{k}^{\left(i\right)}<{\alpha}_{k}^{\left(j\right)}$; (ii) user ${C}_{m}$ prefers ${\mathcal{D}}_{i}$ to ${\mathcal{D}}_{j}$ if ${\Gamma}_{m}^{{\mathcal{D}}_{i}}<{\Gamma}_{m}^{{\mathcal{D}}_{j}}$; (iii) $\left\mu \right({C}_{m}\left)\right$$\le r$, where r is channel m’s reuse factor; and (iv) $\left\mu \right({\mathcal{D}}_{k}\left)\right$$\le s$, where s is group ${\mathcal{D}}_{k}$’s split factor.
Algorithm 2: NOMA interferencebased matching algorithm. 

4.2.2. Stability
4.3. Distributed Channel Assignment: Coalition Formation
 1.
 $\mathcal{N}=\mathcal{C}\cup \mathcal{D}$ is the set of players, with $\mathcal{C}$ and $\mathcal{D}$ denoting the sets of CUs and MD2D groups.
 2.
 $\mathrm{v}$ is the valuation function that gives the value of a coalition in a game. This is a set function that maps each ${\mathcal{S}}_{i}\subseteq \mathcal{N}$ to real nonnegative number interpreted as the absolute value of the coalition.
 3.
 $\mathcal{S}=\{{\mathcal{S}}_{1},{\mathcal{S}}_{2},\cdots ,{\mathcal{S}}_{n}\}$ is the set of formed coalitions namely the coalition structure. Here, each coalition ${\mathcal{S}}_{i}$ is a subset of $\mathcal{N}$ (${\mathcal{S}}_{i}\subseteq \mathcal{N}$, for $i=1,\cdots ,n$).
 Merge coalitions
 Any subset of coalitions $\{{\mathcal{S}}_{1},\cdots ,{\mathcal{S}}_{l}\}$ may be merged whenever the merged form is preferred by the players. The preference relationship is the same defined in Section 4.2.1 for the centralized solution approach.
 Split coalitions
 Any coalition ${\bigcup}_{j=1}^{l}{\mathcal{S}}_{j}$ may be split whenever the split form is preferred by the players.
 1.
 The weakest receiver in group ${\mathcal{D}}_{k}$ suffers less individual interference if the group is moved to ${\mathcal{S}}_{j}$ (recall the the receiver with the poorest channel sets the transmission rate for the group).
 2.
 The total mutual interference of coalition ${\mathcal{S}}_{j}\cup {\mathcal{D}}_{k}$ does not increase: $\mathrm{v}({\mathcal{S}}_{j}\cup {\mathcal{D}}_{k})\le \mathrm{v}\left({\mathcal{S}}_{j}\right)$. This condition is essential for guaranteeing convergence of the algorithm. It simply states that movements among coalitions are allowed only if they improve the local value of the coalition.
 3.
 The new coalition structure ${\mathcal{S}}^{\prime}$ results in less total interference than the current one: $\mathrm{v}\left({\mathcal{S}}^{\prime}\right)\le \mathrm{v}\left(\mathcal{S}\right)\Rightarrow {\mathcal{S}}^{\prime}$ is preferred to $\mathcal{S}$. In other words, only movements that also improve the total value of the network are approved.
4.4. Algorithmic Complexity
Algorithm 3: NOMA based mergeandsplit for the coalitional game. 

5. Numerical Results
5.1. Distributed Resource Allocation
5.2. Comparison to Optimal Resource Allocation
6. Discussion
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Ref.  Scenario  Approach  Model  Problem  Objective  NOMA 

[17]  Dedicated CUs  Optimization  D2D pair  PC,RA  D2D sum rate  CU 
[22]  Distributed Groups  Matching Theory  D2D group  PC,RA  Energy consumption, delay  CU 
[25]  Distributed Groups  Optimization  D2D pair  RA,PC  Min transmission power  D2D + CUs 
[26]  Dedicated CUs  Graph Theory  D2D group  PC,RA  D2D EE  D2D 
[24]  Distributed Groups  Match Theory  D2D group  RA  Network sum rate  D2D + CUs 
[28]  Dedicated CUs  Optimization  D2D pair  RA +  System sum rate  CU 
Mode Selection  
[29]  Dedicated CUs  Matching Theory  D2D group  RA  System sum rate  D2D 
[30]  Dedicated CUs  Game Theory  D2D group  RA  CUs throughput  D2D 
[18]  Dedicated CUs  Matching Theory  D2D group  PC,RA  Maximum users SINR  D2D 
[27]  Dedicated CUs  Hungarian Algorithm  D2D pairs  PC,RA  D2D energy  D2D 
Parameter  Value 

Cell radius  500$\mathrm{m}$ 
Reuse factor (r)  $\{2,3\}$ 
Network density  $250$ devices/cell 
Split factor (s)  $\{3,4\}$ 
Path loss exponent  $2.5$ 
Minimum transmission rate  $\{0.1,0.5\}$$\mathrm{bit}/\mathrm{s}/\mathrm{Hz}$ 
Number of CU users (M)  $\{5,6,8\}$ 
Maximum transmission powers  $[5,25]$$\mathrm{dBm}$ 
Number of MD2D groups (K)  $\{4,10,15\}$ 
Number of receivers ${\mathcal{D}}_{j}$  $\{3,4,5\}$ 
Circuit power  10$\mathrm{dBm}$ 
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Hmila, M.; FernándezVeiga, M.; RodríguezPérez, M.; HerreríaAlonso, S. NonOrthogonal Multiple Access for Unicast and Multicast D2D: Channel Assignment, Power Allocation and Energy Efficiency. Sensors 2021, 21, 3436. https://doi.org/10.3390/s21103436
Hmila M, FernándezVeiga M, RodríguezPérez M, HerreríaAlonso S. NonOrthogonal Multiple Access for Unicast and Multicast D2D: Channel Assignment, Power Allocation and Energy Efficiency. Sensors. 2021; 21(10):3436. https://doi.org/10.3390/s21103436
Chicago/Turabian StyleHmila, Mariem, Manuel FernándezVeiga, Miguel RodríguezPérez, and Sergio HerreríaAlonso. 2021. "NonOrthogonal Multiple Access for Unicast and Multicast D2D: Channel Assignment, Power Allocation and Energy Efficiency" Sensors 21, no. 10: 3436. https://doi.org/10.3390/s21103436