# On the Performance of Underlay Device-to-Device Communications
## 2 State-of-the-Art
The results reveal that the proposed protocol outperforms both the conventional networks and the social-aware no virtualization model in terms of the energy efficiency. The resource allocation of the uplink heterogeneous cellular networks was addressed in [32], where the Kuhn-Munkres (KM) algorithm was employed to maximize the total capacity of D2D networks. Both power allocation and link scheduling in D2D-assisted wireless cachingnetworks were addressed in [33]. The emergency route selection (ERS) of D2D cellular networks during an urban terrorist attack were studied. More precisely, different wireless routing algorithms, namely shortest-path-routing (SPR), interference-aware-routing (IAR), and broadcast-routing (BR) were studied in [36]. Simulation results depicted that the IAR was the best routing scheme, the SPR scheme was the second best, and BR was the worst. An adaptive IAR scheme was also proposed in [37] which improved the network capacity by over 50%. A novel routing algorithm that can achieve the highest trusted connectivity probability (TCP) of any pair of D2D devices was proposed in [38]. Computer-based results presented that the proposed algorithm attained almost the same performance as the exhaustive search. The most closely related research works available in the literature were [39; 40; 41; 42; 43]. However, authors in [39] merely studied the outage probability by assuming that the system was interference-limited, thus ignoring the impact of background noise. However, it is obvious that under the sparsely loaded scenario, the D2D transmission is dominated by background noise. Additionally, from the mathematical framework point of view, without the presence of noise, the complexity of the derived framework significantly reduces. Furthermore, they merely considered the equal power allocation rather than optimizing the transmit power in order to substantially enhance the system performance. The average rate and AoF are considered the most important metrics in wireless communications; nevertheless, they did not study these essential metrics. Regarding [40], the authors addressed the channel allocation problem in the D2D networks. Although both interference and noise were taken into consideration in their work, they did not compute the outage probability in the closed-form expression. In fact, a numerical algorithm was proposed to solve the considered problem; however, this did not allow one to study the trend/behavior of the system metric with respect to an important metric, e.g., the transmit power, via the rigorous mathematical framework. Moreover, as in [39], neither the average rate or amount of fading are taken into account. As for [41], the author also investigated the OP. Nevertheless, it considered a simplified scenario that there is only a pair of D2D users; thus, no intra-D2D interference is taken into consideration. Additionally, they also ignored the background noise when addressing the power control problem that obviously simplifies the framework. Finally, as in [39; 40] both average rate and amount of fading are skipped in [41]. Authors in [42] studied the outage probability and the moment generating function (MGF) of the interference in millimeter-wave with the absence of noise. This work, nonetheless, did not study the impact of the transmit power on the performance of the outage probability and the average rate and the amount of fading as in [39; 40; 41]. Similarly, the work in [43] likewise skipped the impact of noise, nor was the power allocation considered. In the present paper, different from the above-mentioned works, we study not only the outage probability but also the average rate and amount of fading of D2D networks and the coverage probability of the cellular networks. It is obvious that the derivation of the average rate is more challenging than the OP due to the additional integration from the probability density function multiplied with the logarithm function. Additionally, we take into consideration both the interference and noise.