## 1 Introduction  
In the latest research studies, relating to wireless network environments, one of the most significant and, at the same time, critical issues is represented by the management of energy consumption of nodes that could highly limit the overall network performance with reference to protocols and application fields. In this regard, several aspects should be considered in order to address the problems implied by the main features of these kinds of networks that significantly affect the behaviors at physical, MAC, and network layers. For instance, let us consider mobile ad hoc networks (MANETs); usually, in these kinds of environments, omnidirectional antennas are used for communication among nodes both for transmission as well as for reception; this approach results in very limited performance relating to physical, link, and routing layer statistics [1; 2; 3; 4]. Omnidirectional antennas have the ability to transmit and receive signals in the same way in all directions. Main advantages of omnidirectional antennas include ease of configuration and implementation, low designing cost, very and simple architecture (hardware-less) [5; 6]. Nevertheless, despite these few benefits they introduce a considerable number of drawbacks such as limited range and coverage (implied by low gain), high energy consumption, high interference probability (especially in dense networks),and very high performance dependency on the environments in which they are employed (indoor or outdoor) [7, 8]; nonetheless, omnidirectional antennas cannot exploit the benefits of cross-polarization because they are vertically polarized [9]. More specifically, this last issue helps to increase the probability of interference between communicating nodes in the channel [10]. The majority of these issues could be partially mitigated using directional antennas. Directional antennas offer a great number of benefits with respect to the classic omnidirectional antennas. In fact, because they focus the radiating power only toward a specific direction, directional antennas are able to improve the spatial reuse as well as the coverage range of the signal. Higher spatial reuse and longer communication range is reflected in greater network capacity and improved network connectivity [11, 12, 13]. Most researchers, as we presented in Section 2, in order to mitigate these undesired phenomena in directional contexts, demonstrated that, through the employment of smart antenna systems (SASs) in place of the classic directional antennas, it is possible to create an efficient system, exploiting the Spatial Division Multiplexing (SDMA) technique that this kind of device well provides. Using an SAS, efficient spatial reusing and high link robustness may be obtained [14, 15]. In contexts in which directional smart antenna systems are used, the beamforming issue has to be deeply investigated. The use of DRTS and DCTS messages along with directional network allocator vector (DNAV) aims to reduce the excessive burden of collisions occurring when omnidirectional antennas are used; however, when SASs are employed, it is not sufficient to adopt these expedients [16, 17, 18, 19]. The most common drawback in this field is represented by the fact that, in literature, there is an important lack relating to directional MAC proposals that could be synthesized by two main aspects: Primarily, most directional MAC works are designed for classical directional antennas and do not address issues involved by the use of an SAS. This aspect represents a huge limit for latest wireless mobile network developments that, however, allow for the use of more complex antennas technologies such as SASs and massive MIMO systems. Secondarily, addressing directional MAC and, at the same time, energy consumption issues when using SAS technologies represents a very critical task; indeed, in case of high gain directive communication beams, it is hard to limit the overall waste of energy of nodes, precisely because it is an implication induced by the employment of high gains.