## 3 System Model  
### System Description  
We consider a highway downlink communication scenario with a single RSU and \(K\) vehicles, where the vehicles move along a multi-lane road at speed \(v\) in the positive direction of the x-axis, as shown in Figure 1. The communication range of the RSU is \([0,D]\), where the communication range denotes the distance traveled by vehicles from when they start receiving service from the RSU on the road until they hand off to another RSU. We assume that there is no interference among \(K\) V2I links [33; 34]. Since the vehicle moves, the beam alignment must be performed periodically or aperiodically between the RSU and the moving vehicles. We assume that the RSU and vehicles are equipped with a directional antenna and are enabled to rotate the antenna bore-sight toward the desired direction. The RSU selects a beamwidth from the available set of beamwidths [35; 36]. The directional antenna gain in the 3 GPP Gaussian antenna model is given by [17; 37]:  
\[G(\theta)=\left\{ \begin{array}{ll}G_{m}e^{-\rho\theta^{2}},&|\theta|\leq \varphi\\ G_{s},&\text{otherwise},\end{array} \right. \tag{1}\]  
where \(\rho=2.028\ln(10)/\varphi^{2}\), \(2\varphi\) is the main-lobe beamwidth, and \(\theta\) denotes an alignment error relative to the antenna's bore-sight direction. \(G_{m}=\pi 10^{2.028}/(42.6443\varphi+\pi)\) and \(G_{s}=G_{m}e^{-\rho\varphi^{2}}=10^{-2.028}G_{m}\) represent the maximum main-lobe gain and the side-lobe gain, respectively. Several pilot transmissions and receptions are required to align the beam between the RSU and the vehicle. We assume a two-step hierarchical beam alignment method, where the first stage finds a sector-level beamwidth via an exhaustive search, and the second stage refines the beam search using narrower beam-level beamwidths in a subspace of the selectable sector-level beamwidths for all possible combinations [26; 38]. For the \(k\)th V2I link, when \(\psi^{k}\) denotes the sector-level beamwidth and \(\varphi^{k}\) denotes the beam-level beamwidth, the number of possible combinations to be searched is \([\psi^{k}/\varphi^{k}]^{2}\). Without loss of generality, assuming the sector-level beamwidth alignment has been searched, the beam alignment time required to find the beam-level beamwidth can be expressed as follows [38]:  
\[t_{a}^{k}(\varphi^{k})=\left\lceil\frac{\psi^{k}}{\varphi^{k}}\right\rceil^{2 }\cdot T_{p}, \tag{2}\]  
where \(T_{p}\) is the pilot transmission time. From (2), when the sector-level beamwidth is fixed, the narrower the beam-level beamwidth, the longer it takes exponentially to perform the beam alignment, which may result in the lower throughput due to the decrease in the data transmission time. Figure 1: A system model. In a phased array antenna, the beamwidth can be adjusted via the control of the phase and the amplitude of the signals fed to each individual antenna element. By changing the relative phase of the signals fed to each antenna element, we can steer the beam in a particular direction. The steering changes the effective aperture of the array in the direction of interest, which in turn affects the beamwidth. Moreover, by applying specific amplitude weights to the individual elements, we can shape the radiation pattern and adjust the beamwidth. The number of antenna elements and the spacing between the elements also affect the beamwidth. Increasing the element spacing typically narrows the beamwidth, while decreasing the spacing widens it [39]. The authors of [40] proposed two types of beamwidth control methods based on conventional beamforming and Dolph-Chebyshev beamforming, for the cases with constant-modulus phase shifters and amplitude-adjustable phase shifters, respectively. The authors of [41] presented a beamforming technology utilizing transmission-line transformers and balanced impedance phase shifters. The impedance may vary while controlling the beamwidth due to changes in array configuration.