# On the Performance of Underlay Device-to-Device Communications
## 2 State-of-the-Art  
One advantage of D2D communications is that it can easily combine with other advanced techniques to further enhance the performance of wireless communications systems. In particular, the work in [6; 7; 8] investigated the performance of massive multiple-input multiple-output (MIMO) with D2D-enabled communications. It derived the lower bound of the capacity of cellular users by assuming that all cells share the same set of pilots [7]. [PERSON] et al. [8], proposed a dynamic power control policy for mitigating the inter-interference in underlay D2D massive MIMO cellular networks. The work in [9; 10] addressed the performance of the D2D-enabled relaying networks. More specifically, [PERSON] et al. in [10] studied bit error rate (BER) and spectral efficiency where base stations (BSs) applied spatial modulation (SM). The joint optimization of the relay selection, spectrum allocation, and power control was investigated in dual-hop communications by employing the matching theory in [9]. Their findings unveiled that the performance in the presence of a relay was far better than the case without relays. [PERSON] et al. [11], on the other hand, investigated the security problem of the D2D relay-assisted networks. In particular, their proposed scheme, which was suitable for any infrastructure, was significantly enhanced the security of any D2D settings. The works in [12; 13; 14] applied tools from stochastic geometry (SG) in order to better capture the randomness of both the D2D and the cellular users. More specifically, the closed-form expressions of average queue length, average throughput, average delay, and dropping probability were evaluated in [12]. On the other hand, authors in [13] derived coverage probability (Pcov) of the cellular users. The performance of the D2D-assisted mmWave communications was provided in [15; 16; 17; 18]. In particular, the offloading gain of the D2D mmWave communications was studied in both half- and full-duplex in [18], while [15] discussed the potential of D2D-assisted mmWave communications. The combination of the D2D and Long-Range (LoRa) networks was studied in [19; 20], while the throughput-outage analysis of D2D caching networks was addressed in [21]. The work in [22], meanwhile, addressed the sum rate of the NOMA-based cellular networks with imperfect self-interference cancellation. This work, nonetheless, computed the sum rate via numerical results in place of the rigorous mathematical framework. Authors in [23] studied the outage probability of the cognitive radio D2D transmission. The amount of fading (AoF) of the D2D networks was investigated in [24; 25], however, it was derived based on the Fox-H function, which is not stable and mathematically intractable. [PERSON] et al. [26] derived the system throughput under the assumption of interference limitation, while the subchannel and power allocation were addressed in [27; 28]. The performance of the D2D networks itself was studied widely in [29; 30; 31; 32; 33; 34; 35; 36; 37; 38]. The work in [29] investigated the fairness scheduling policies in multi-user diversity D2D cellular networks. In particular, two fairness scheduling policies were considered in that work, i.e., cellular fairness scheduling (CFS) and D2D fairness scheduling (DFS). The outcomes illustrated that the DFS scheme is far better than the CFS scheme in terms of the effective rate and the access probability. A heuristic algorithm for proportional fair scheduling (PPS) was proposed in [30] to maximize the logarithm sum of average rate. The proposed algorithm achieved a high sum of average rate while maintaining low computational complexity. [PERSON] et al. devised a social-aware virtual medium access control (SV-MAC) to integrate virtualization and social-awareness to D2D cellular networks [31]. Under the proposed protocol, the considered network is unified in a single protocol stack, giving a common abstraction for end-users and mobile network operators (MNOs) to share their network resources.