## 2 Related Work
### User Mobility Models  
User mobility models are crucial for time-dependent analysis, as they specify the rationale for beam misalignment. In general, we distinguish between user micro-mobility and macro-mobility models.  
User macro-mobility has been a hot research topic over the last three decades in the context of cellular systems. Most of the utilized models are based on the random walks theory, describing the user movement either algorithmically or using closed-form stochastic equations. The most celebrated model is the random waypoint (RWP) mobility model that has been utilized already in the 1990s of the previous century [35]. This model assumes that the user moves in a certain compartment by randomly and uniformly selecting the waypoint and moving towards it at a randomly and uniformly chosen speed. For example, in [36], the authors explore this model to estimate the rate of handovers in heterogeneous networks. The research shows that the standard RWP model overestimates the rate of handover and the handover rate does not increase linearly with the number of users. Another well-known model is a random direction model (RDM, [37]). According to this model, a user chooses the directional randomly and uniformly in \((0,2\pi)\) and then moves in a selected direction for exponentially distributed time at the constant speed \(v\). In addition to these two simple models, there are more comprehensive ones accounting for large displacements in the mobility process, such as Levy flights and Brownian motion, with jumps and models capturing the dependence in the user trajectories; see [38] for an exhaustive review.  
The important property of mobility models for applied mathematical modeling is the ability to characterize the distribution of the user location as a function of time. Unfortunately, such results are available for simple models only, such as Brownian motion. Such models are barely utilized in practice due to their inability to capture the main mobility characteristics of users. However, the performance comparison of mobility models by the authors in [39] demonstrated that performance metrics expressed as functions of models do not differ by more than 10%. This implies that even the simplest models such as Brownian motion can be utilized for a first-order performance assessment in mmWave/sub-THz cellular systems.  
Micro-mobility refers to the spontaneous displacement and rotation of the UE in the hands of the user during an active connection. This effect was first characterized by sub-THz systems having extremely directional antenna radiation patterns in [40] and was observed in practice already for 5G mmWave systems [41]. In practice, small displacements barely affect the beam alignment, while rotations over the vertical (yaw) and transverse (pitch) axes have a drastic impact, as shown in [22].  
To model micro-mobility, the authors in [22] utilized two independent Brownian motions over the respective rotation axes. The authors in [6] confirmed that for a large class of applications affecting user micro-mobility, the micro-mobility components over these axes are indeed independent. Further, in [42], based on the measurements reported in [6], the authors compared the accuracy of three models: a two-dimensional Markov chain model inherently capturing the dependence between mobility over the vertical and transverse axes, independent one-dimensional Markov models neglecting this dependence, and decomposed Brownian motion that also does not account for the above-mentioned dependence. They demonstrated that none of those models outperform their counterparts, implying that the simplest Brownian motion models can serve as a first-order approximation for the micro-mobility process.