Max–Min Fairness Optimization for D2D Communications Coexisting with Cellular Networks

: This paper considers a system consisting of a nonorthogonal multiple access (NOMA)-based device-to-device (D2D) communication system within a cellular network, in which the time and frequency resources are shared by everyone. In particular, D2D groups that constitute different pairs are assigned to the subchannels that the cellular users occupy. A max–min fairness optimization problem with power budget constraints is formulated and solved in this paper to reduce the mutual interference between the cellular users and D2D devices that substantially impacts that with the worst channel condition. For a low computational complexity solution, we propose the use of the bisection method together with the solution of a system of linear equalities. The proposed algorithm can provide uniformly good service to all of the cellular users and D2D devices in the coverage area by utilizing the minimal total transmit power. The simulation results indicate the effectiveness of the proposed algorithm in the improvement of the spectral efﬁciency of the worst user under the different widely used subchannel assignments and pairing techniques.


Introduction
The 5G-and-beyond networks must cope with the drastic increases in mobile data traffic and the number of wireless devices from different services foreseen in the near future [1,2]. When considering next-generation wireless access technologies, a major concern is achieving a thousandfold increase in the spectral efficiency as compared with that of contemporary wireless networks while providing new applications with uniform coverage and high reliability [3,4]. This is indeed a vital goal that is considered by both academia and industry. Specifically, in recent years, wireless local area networks have become increasingly popular, requiring massive connections not only between many devices and a base station, but also between devices, which is called device-to-device communication (D2D) [5]. Spectrum resources that are allocated for cellular users are limited and require more local facilities. As a solution, incorporating D2D links into an existing cellular infrastructure can potentially enhance the spectrum and energy efficiency, extend the network coverage, and improve the communication reliability [6]. Spectrum sharing and frequency selection have attracted much research attention, for example, [7] and the references therein.
For better performance in future wireless networks, various techniques have been integrated into a D2D system, such as cell-free massive multiple-input multiple-output (MIMO) [8,9], to reuse the bandwidth from that dedicated to the cellular users, which is controlled by a base station (BS). For a given QoS level, we then transfer the max-min fairness optimization problem into a minimum total transmit power consumption problem under the QoS requirements and the power constraints. • We observe the quasilinear convexity of the proposed max-min fairness optimization problem and utilize the bisection method to obtain the global optimum. We rigorously find an upper bound on the QoSs that the system can provide to the cellular users and D2D devices. This upper bound offers a reduction of the computational complexity and also confirms that the proposed algorithm can be implemented in polynomial time. Our proposed algorithm balances between maximizing the worst spectral efficiency performance and minimizing the total transmit power consumption.

•
The numerical results demonstrate the effectiveness of the proposed optimization framework in providing equally good QoS to all cellular users and D2D devices under the different subchannel assignments (either random assignment or a heuristic assignment that minimizes the mutual interference between cellular users and D2D devices), pairing techniques (either random paring or a heuristic pairing that reduces the total transmit power), and channel conditions.
The rest of this paper is organized, as follows: Section 2 presents the considered network model and the spectral efficiency analysis for each user. Section 3 formulates and solves the max-min QoS optimization problem subject to the power constraints. Finally, the numerical results are analyzed in Section 4, while the main conclusions are given in Section 5.
Notations: we use the upper and lower bold letters to denote matrices and vectors, respectively. The circularly symmetric Gaussian distribution is CN (·, ·), while E{·} is the expectation of a random variable. The Hermitian transpose is denoted (·) H and C is the complex field. Finally, I M is the identity matrix of size M × M.

System Model and Performance Analysis
This section presents the considered system model and the computation of the performance for each user under uncorrelated Rayleigh fading channels.

System Model and Assumptions
We consider a system where a number of downlink communication D2D groups operate within the uplink cellular network, as schematically illustrated in Figure 1. The base station is equipped with M antennas, managing L subchannels and serving M l single-antenna cellular users on each subchannel. The system admits K D2D groups in the same geographical area to reuse the spectrum resources. All of the D2D devices, i.e., D2D transmitters and D2D receivers, are equipped with a single antenna. In each D2D group, the devices are assigned in pairs and they utilize successive interference cancellation in the scheme of NOMA.
The cellular users are assumed to be randomly distributed in a disc, with radius r C . Similarly, the D2D devices communicate in an area with radius r D , and the D2D transmitters are located at the center of each group. In the uplink scenario of the cellular network, the channel between user m using subchannel l and the base station is denoted by the vector h l,m ∈ C M . The channel between the k-th D2D transmitter and the BS on the l-th is g l,k ∈ C M . At the downlink D2D communication, the channels between the D2D transmitter and the two receivers in the n-th pair are denoted by g l,k,n,1 , g l,k,n,2 ∈ C, respectively. In addition, the channels between the cellular user m using subchannel l for the two users in the n-th pair are h l,m,k,n,1 ∈ C and h l,m,k,n,2 ∈ C.

Channel Capacity of D2D Devices and Cellular Users
This subsection computes the uplink channel capacity for each cellular user, exploiting a detection technique to decode the desired signals. We also compute the downlink channel capacity for each D2D device under the exploitation of the successive interference cancellation technique.

Uplink Cellular Network Transmission
The base station sends a message x l,m to user m while using the l-th subchannel with E{|x l,m | 2 } = 1. The received signal at the l-th subchannel is given by η l,k g l,k P l,k,n,1 t l,k,n,1 + P l,k,n,2 t l,k,n,2 + ζ ζ ζ l , where P l,m denotes the allocated power from the base station to user m using the l-th subchannel. At the n-th pair in the k-th group, the two complex signals t l,k,n,1 and t l,k,n,2 are transmitted from the transmitter with E{|t l,k,n,1 | 2 } = 1 and E{|t l,k,n,2 | 2 } = 1. The corresponding power levels P l,k,n,1 and P l,k,n,2 are allocated to these signals. ζ ζ ζ l is additive noise, which is distributed as ζ ζ ζ l ∼ CN (0, σ 2 UL I M ). Here, η l,k is a binary variable and it is defined as For the practical power constraints, the transmitted power levels from the base station, and the D2D transmitters satisfy 0 ≤ P l,k,n,1 ≤ P max,l,k,n,1 , 0 ≤ P l,k,n,2 ≤ P max,l,k,n,2 .
To decode the desired signal from user m using subchannel l, we assume that the base station utilizes a detection vector w l,m ∈ C M aŝ η l,k w H l,m g l,k P l,k,n,1 t l,k,n,1 + P l,k,n,2 t l,k,n,2

D2D Interference
In Equation (19), the first term contains the desired signal and the second term denotes mutual interference from other cellular users. The last terms are mutual interference from the D2D devices and thermal noise. Therefore, the spectral efficiency of user m in the l-th subchannel is computed as where the signal-to-interference-and-noise (SINR) ratio, denoted by γ l,m , is expressed as The spectral efficiency expression that is obtained in Equation (20) is generally applied to all of the channel models and detection techniques. In particular, the numerator in Equation (21) manifests the effectiveness of the utilized detection method by increasing the received power at the base station. Additionally, the denominator contains both the mutual interference and noise that degrades the spectral efficiency of any cellular user m, ∀m.

Downlink D2D Transmission
For D2D communications at the n-th pair in the k-th group, the device with smaller channel gain is called the weaker device. Without loss of generality, we state that in the n-th pair, device 1 has the weaker channel gain (let us call the weaker device), and device 2 has the stronger channel gain (let us call the stronger device), for which it is shown that |g l,k,n,1 | 2 ≤ |g l,k,n,2 | 2 .
In each pair, we assume that the perfect successive interference cancellation only applies to the stronger device. This suggests that the weaker device employs a conventional decoding method. We then model the received signal for each of the two D2D devices while using the l-th subchannel. In particular, the received signal of device 1 in the n-th pair is formulated as y l,k,n,1 = g l,k,n,1 P l,k,n,1 t l,k,n,1 Desired Signal + g l,k,n,1 P l,k,n,2 t l,k,n,2

Intra Pair Interference
η l,k g l,k ,n,1 P l,k ,n,1 t l,k ,n,1 + P l,k ,n,2 t l,k ,n,2 Inter Pair Interference where ζ l,k,n,1 ∼ CN (0, σ 2 DL ). In Equation (23), the first term denotes the desired signal and the second term refers to the intrapair interference. The interference from the cellular users and other D2D transmitter devices using the l-th subchannel are displayed by the third and fourth terms. The last term represents thermal noise. The spectral efficiency of this device is given as where the SINR value, denoted by γ k,n,1 , us γ l,k,n,1 = P l,k,n,1 |g l,k,n,1 | 2 P l,k,n,2 |g l,k,n, In Equation (25), the numerator indicates the essential level of the power control and how much signal is depleted over the dissemination environment. In contrast, device 2 in the n-th pair utilizes perfect successive interference cancellation for subtracting intrapair interference; therefore, we obtain the spectral efficiency as R l,k,n,2 = log 2 (1 + γ l,k,n,2 where the SINR value of the device 2 of the n-th pair in the k-th group is computed as γ l,k,n,2 = P l,k,n,2 |g l,k,n,2 | 2 Equation (27) illustrates that the intrapair interference from device 1 is completely mitigated using perfect successive interference cancellation. Hereby, we use these obtained spectral efficiencies to formulate and solve an optimization, which provides uniformly good service for every user in the system, regardless of the instantaneous CSI conditions.

Max-Min QoS Optimization
This section aims at maximizing the QoS level for all cellular users and D2D devices by utilizing the minimal total transmit power consumption in the condition of the radio channels that are impacted by a propagation environment. Indeed, the provision of a concrete level to every cell user and D2D device is hardly standardized. Therefore, we propose a method to determine the common good QoS level for both D2D devices and cellular users. In this model, we assume that all D2D groups have been assigned into specific subchannels by the diversity-based method in [11]; therefore, the set of η l,k has been defined. In the following, we consider the optimization problem of maximizing the lowest QoS level, corresponding to that with the worst channel condition in the network, as maximize {P l,k,n,1 ,P l,k,n,2 ,P l,m ≥0} min (l,k,n,1),(l,k,n,2),(l,m) R l,k,n,1 w l,k,n,1 , R l,k,n,2 w l,k,n,2 , R l,m w l,m Subject to P l,k,n,1 ≤ P max,l,k,n,1 ∀l, k, n, P l,k,n,2 ≤ P max,l,k,n,2 , ∀l, k, n, P l,m ≤ P max,l,m , ∀l, m, where w l,k,n,1 , w l,k,n,2 , w l,m > 0 are the weights that are associated with each corresponding user and they demonstrate the users' priority. By introducing a new optimization ξ as a consequence of utilizing the epigraph representation [32,33], we convert problem (28) into the equivalent form of maximize {P l,k,n,1 ,P l,k,n,2 ,P l,m ,ξ≥0} ξ subject to R k,n,1 ≥ ξw l,k,n,1 , ∀l, k, n, R l,k,n,2 ≥ ξw l,k,n,2 , ∀l, k, n, R l,m ≥ ξw l,m , ∀l, m, P l,k,n,1 ≤ P max,l,k,n,1 ∀l, k, n, P l,k,n,1 ≤ P max,l,k,n,2 , ∀l, k, n , P l,m ≤ P max,l,m , ∀l, m.
We stress that jointly optimizing the power coefficients and the max-min SINR level makes problem (30) a geometric program, since the SINR constraints can be reformulated as posynomials, while the power budget constraints are monomials. Because a geometric program is nonconvex in standard form, the global optimum to problem (30) is obtained by exploiting a hidden convex structure, but this structure has high computational complexity. Consequently, as a main contribution of this paper, we propose an algorithm to solve Equation (30) with lower cost. For this purpose, the following observations are attained for a given ξ, as in Lemma 1.
We have proven that the global solution to problem (31) can be obtained for a fixed ξ by solving a linear program. However, this problem does not minimize the total transmit power consumption, which is a goal of beyond 5G networks for improving energy efficiency. Therefore, we recast Equation (31) into a new optimization problem that can also provide an energy-efficient solution while guaranteeing the SINR requirements, as minimize {P l,k,n,1 ,P l,k,n,2 ,P l,m ≥0} (P l,k,n,1 + P l,k,n,2 ) + L ∑ l=1 M l ∑ m=1 P l,m subject to γ l,m ≥ ξ l,m , ∀l, m, γ l,k,n,1 ≥ ξ l,k,n,1 , ∀l, k, n, γ l,k,n,2 ≥ ξ l,k,n,2 , ∀l, k, n, P l,k,n,1 ≤ P max,l,k,n,1 ∀l, k, n, P l,k,n,2 ≤ P max,l,k,n,2 , ∀l, k, n, P l,m ≤ P max,l,m , ∀l, m.
Following a methodology that is similar to that in Lemma 1, problem (33) is proven to be a linear program; therefore, the global optimum can be obtained by utilizing a general optimization toolbox with the interior-point method [34]. From the constraints about the SINR requirements of all users and devices, we observe that the solution of all SINR constraints when they are equal is the solution of problem (33). Consequently, the ∑ L l=1 M m SINR constraints of the cellular users can be rewritten, as: which contain the ∑ L l=1 M m power variables that are associated with all of the cellular users. Similarly, the 2 ∑ K k=1 N k SINR constraints of D2D devices are reformulated as P l,k,n,1 |g l,k,n,1 | 2 − ξ k,n,1 P l,k,n,2 |g l,k,n,1 | 2 − M l ∑ m=1 ξ l,k,n,1 P l,m |h l,m,k,n,1 ξ l,k,n,1 η l,k,n,2 |g k ,n,1 | 2 × (P l,k ,n,1 + P l,k ,n,2 ) − ξ l,k,n,1 σ 2 DL = 0, ∀k, n, P l,k,n,2 |g l,k,n,2 | 2 − ξ l,k,n,2 ξ l,k,n,2 η l,k |g l,k ,n,2 | 2 (P l,k ,n,1 + P l,k ,n,2 ) − ξ k,n,2 σ 2 DL = 0, ∀k, n, which contain the power variables for the D2D devices. With a nonempty feasible set, we obtain the optimal solution to problem (33) by solving the system of linear equations in (34) and (35). Otherwise, problem (33) is infeasible if the solution contains one power coefficient that does not satisfy the power budget constraints. In addition, the solution has been obtained for a given set of {ξ k,n,1 , ξ k,n,2 , ξ l,m }. To find the optimal max-min SINR values for all cellular users and D2D devices, we propose the use of the bisection method with the search range that is defined in Theorem 1.
With the optimistic assumption that there is no thermal noise or mutual interference from the cellular users or from the other D2D devices, the upper bound of the SINR value of the weak D2D device in pair k γ l,k,n,1 ≤ P l,k,n,1 |g l,k,n,1 | 2 P l,k,n,2 |g l,k,n,1 | 2 = P l,k,n,1 P l,k,n,2 . (39) For the strong D2D device in pair k, we ignore all the mutual interference to obtain an upper bound on the SINR value of γ l,k,n,2 ≤ P l,k,n,2 |g l,k,n,2 | 2 Combining Equations (37)-(40) and obtaining an upper bound that is applied for all users, we obtain the result that is shown in the theorem.
From Theorem 1, problem (33) is solved in an iterative manner by reducing the search range, and then the maximum QoS level can be obtained. One such line search procedure is the well-known bisection method [35], which we exploit in this paper. For more detail, we define two parameters, ξ candidate and δ. While ξ candidate represents the selected QoS value at each iteration search, δ is the line-search accuracy value. After determining the upper and lower bounds, ξ min , ξ max , of the SINR requirement, the feasibility of Equation (33) is demonstrated for the value ξ candidate = (ξ max + ξ min )/2.
If Equation (33) is feasible, then its solution to the power variables is saved in P save , ξ min is changed to the middle value ξ candidate , and the value ∆, which is the gap between the upper and lower bound of ξ, is compared with δ. The algorithm is terminated when the gap ∆ is smaller than the line-search accuracy δ. Otherwise, if problem (33) is infeasible, ξ max is labeled by ξ candidate . The search range declines by half after each iteration, since either its lower or upper bound is assigned to ξ candidate . Algorithm 1 summarizes the proposed max-min QoS optimization. In each iteration, the main cost of Algorithm 1 spends on solving problem (33) or alternatively solving a system of linear equations in (34) and (35). This is dominated my a matrix inverse computation. By exploiting the Cholesky decomposition [36], the computational complexity of matrix inversion is in the order of Moreover, the number of iteration in need for the bisection method is log 2 ξ upper 0 /δ , where · is the ceiling function. As a consequence, the total computational complexity of Algorithm 1 is which can be performed in the polynomial time and also demonstrate the effectiveness of selecting the initial value ξ upper 0 as shown in Theorem 1.

Algorithm 1:
Max-min quality of service (QoS) based on the bisection method.

Numerical Results
In this section, we analyze the system performance by utilizing the bisection method in order to maximize the worst spectral efficiency in the network subject for the given power budgets. To clarify the effectiveness of the proposed approach, we compare the four scenarios of the subchannel assignment and D2D device pairing that have been proposed in [11], consisting of (1) The diversity-based method (DBM) combined with heuristic pairing (denoted DBM + Heuristic Pairing in the figures): All cellular users have already been assigned to the subchannels. Subsequently, these subchannels are allocated to the D2D devices via channel diversity. In each D2D group, the devices are paired by the heuristic algorithm. (2) The DBM using random pairing (denoted DBM + Random Pairing in the figures): the system utilizes the heuristic algorithm to allocate the subchannels for the D2D groups; however, the devices in each group are paired randomly. The general parameters of the system setting and propagation channels are listed in Table 1, while several specific parameters are presented in each figure. Figure 2 depicts the relationship between the number of D2D groups and the max-min fairness QoS (bit/s/Hz) of each cellular user and D2D device. Generally, when the number of D2D groups in the system increases, it leads to an increase in interference, and thus, the achievable rate obviously declines as the number of groups increases. However, the system utilizing the bisection method still demonstrates the good performance of the heuristic algorithm. In particular, the highest QoS rate of all users and devices is obtained when the system only allows one D2D group. In the case where the heuristic algorithm is used to pair the D2D devices, the max-min QoS is approximately 3.8 bit/s/Hz. Additionally, the QoS of the system that does not use the heuristic assignment algorithm and uses random pairing only reaches 3.4 bit/s/Hz. Subsequently, all four scenarios show downward trends when the number of D2D groups is increased due to the growth of mutual interference. When the system accepts 10 D2D groups reusing the subchannels, the QoS provided to everyone in the case where the heuristic algorithm and heuristic pairing are used still reaches 2.1 bit/s/Hz. This spectral efficiency is significantly higher than that of the system that uses random pairing only, with approximately 1.8 bit/s/Hz.   Figure 3 manifests the efficiency of the bisection method for the system. The simulated scenario includes 5 cellular users, which have been assigned to the corresponding subchannels, and the 6 D2D groups are allowed to access the system and reuse the subchannels. With the heuristic algorithm, the system can provide up to 80% of all users and devices with a medium rate over 1.5 bit/s/Hz. In contrast, for the system without the heuristic algorithm, only 65% users reach the data rate of 1.5 bit/s/Hz. The effectiveness of the network that applies the bisection method and the heuristic algorithm for assigning the D2D devices yields better QoS than that with random pairing since approximately 18% users and devices can transmit signals at the pace of 3 bit/s/Hz, while that of the users without the heuristic algorithm is approximately 8%.  Figure 4 compares the total transmit power of all the D2D devices as the number of groups increases. The total transmit power of the D2D devices is directly proportional to the number of groups. The figure clearly illustrates the efficiency of the network in four cases. First, in the system with the heuristic algorithm, the total transmit power of all the devices is approximately 7-20% lower when the proposed pairing is used. Particularly, the total transmit power with the heuristic algorithm and proposed pairing is 100 mW, while that with random pairing is 80 mW for the case where only one group accesses the network. Subsequently, the total transmit power values of both the DBM with the proposed pairing and random pairing increase substantially, especially when the number of groups exceeds the number of subchannels, up to 280 and 305 mW, respectively. Second, without the heuristic algorithm, the system reaches power consumption values as high as 190 mW when using random pairing and reaches the peak level of 560 mW with 10 groups accessing the network. The figure also shows that, when the number of groups exceeds the number of subchannels, the total transmit power increases more sharply. However, from Figure 2, the spectral efficiency of each user when the heuristic algorithm is used is still higher than that when this algorithm is not used. In a similar trend, the total transmit power of all the D2D devices when the heuristic algorithm is used is lower than that when this algorithm is not used.
In contrast with the trend of the power allocation for D2D devices, Figure 5 indicates that the total transmit power (measured in mW) of all the cellular users tends to decline over the number of groups allowed to reuse spectrum resources in all four studied schemes. The rationale is that the base station does not apply the NOMA technique and, thus, when the maximum spectral efficiency decreases, the total transmit power decreases as well. A comparison of the four scenarios shows that the bisection method perfectly conjugates with the model, since the system that uses the heuristic algorithm consumes more power than that without this algorithm and allows all of the users and devices to connect to the network at a higher data rate. Particularly, the total transmit power of all the cellular users with the heuristic algorithm and heuristic pairing uses 3900 mW to 2600 mW when the system serves 2 to 10 groups. However, the total transmit power of the heuristic algorithm and random pairing decreases to 3500 mW to 2300 mW.   Figure 6 shows the feasible probability that computes the fraction of the number of cellular users/D2D devices who are served by the QoS not less than 1.5 bit/s/Hz and the total number of existing cellular users and D2D devices among the 2000 realizations of different locations. The fixed QoS was proposed in [11,29]. Those results demonstrate the significant improvements of the proposed max-min fairness optimization than a fixed QoS assignment, which provides the better feasible probability. For example, the utilization of max-min fairness for the DBM combined with heuristic pairing gives 78% of cellular users and D2D devices with a higher rate than 1.5 bit/s/Hz. This is 28% better than the corresponding method in [29] with the fixed QoS.

Conclusions
We investigated a NOMA-based D2D communication system coexisting with a cellular network, where the power resource needs to be carefully allocated due to the shortage in practice. We formulated a max-min fairness optimization problem with energy budget constraints to provide a good QoS to all devices and cellular users in the network while still limiting the total transmit power of the whole system. The bisection method combined with solving linear equalities is proposed in this model for the sake of reducing the computational complexity. Consequently, the numerical results demonstrate that all cellular users and devices acquire better spectral efficiency with the limited power budget.