# Multi-Tier Heterogeneous Beam Management for Future Indoor FSO Networks
## 4 Analysis Framework  
In this paper, we highlight a specific focus on the optimization of the beam refinement process. We aim to allow UDs to connect to the highest resolution/maximum throughput tier while minimizing access delay -- both during initial access and upon occurrence of an outage. The heterogeneous beam refinement allows the Tier 2 search to occur while other UDs use Tier 3 for communication. Furthermore, a variety of search techniques (RSSI, ToF, AoA, etc.) may be used, and the protocol may avoid connecting to the middle tiers during beam refinement in order to speed up access to the higher tier. Our preliminary analysis focuses on the ideal relationship between range (i.e., potential coverage), resolution, and steering velocity at each tier. To demonstrate the optimization problem, consider a three-tier beam refinement strategy where each tier narrows the search space to the area covered by a single configuration, and the next tier searches over the  
Figure 4: MTH radio resource control for beam management in the TCP/IP 5-layer model. Figure 5: Potential coverage, instantaneous coverage, and associated multiple access strategies for single element (**top left**), multi-element (**top right**), and sectored multi-element (**bottom left**) steerable FSO links as well as a sectored multi-element zone with distributed elements (**bottom right**). space covered by a single configuration of the previous tier. Assuming non-overlapping configurations, the worst case search time is  
\[T_{s}=T_{d}+t_{2}n_{2}+t_{3}n_{3}=T_{d}+t_{2}\alpha_{2}\beta_{2}+t_{3}\alpha_{3} \beta 3 \tag{1}\]  
where \(T_{d}\) is the RFSC's device discovery time, \(t_{i}\) is the time to switch between configurations for tier \(i\), and \(n_{i}\) is the number of configurations in the \(i\)th tier. With the assumption of non-overlapping configurations, we can define \(n_{i}=\alpha_{i}\beta_{i}\) where \(\alpha_{i}\) and \(\beta_{i}\) are the resolution and range of tier \(i\), respectively. We define the tier resolution as the number of configurations per unit area and the range as the total area covered by the tier. These parameters are summarized in Table 1.  
Assuming that the search space of tier \(i+1\) is equal to the instantaneous coverage (i.e., area covered by a single configuration) of tier \(i\), then \(\beta_{i+1}=1/\alpha_{i}\). In other words, tier \(i\) narrows the search space to an area covered by a single configuration and tier \(i+1\) searches this space. Accordingly, Equation (1) can be rewritten as either of the following:  
\[T_{s}=T_{d}+t_{2}\alpha_{2}\beta_{2}+t_{3}\frac{\alpha_{3}}{\alpha_{2}} \tag{2}\]  
\[T_{s}=T_{d}+t_{2}\frac{\beta_{2}}{\beta_{3}}+t_{3}\alpha_{3}\beta_{3} \tag{3}\]  
The key observation in this analysis is the impact of \(\alpha_{2}\) (or, similarly, \(\beta_{3}\)). Specifically, there is an ideal resolution of Tier 2 that minimizes \(T_{s}\) and this resolution is dependent on the range of Tier 2 and resolution of Tier 3. Furthermore, the relative importance of \(\beta_{2}\) and \(\alpha_{3}\) is dependent on the relative switching speeds of Tiers 2 and 3. To determine the optimal Tier 2 resolution, we must determine \(\alpha_{2}\) that minimizes \(T_{s}\) for given \(T_{d}\), \(t_{2}\), \(t_{3}\), \(\beta_{2}\) and \(\alpha_{3}\). Observing Equations (2) and (3), we can find the rate of change of \(T_{s}\) with respect to \(\alpha_{2}\) and \(\beta_{3}\), respectively. \[\frac{d}{d\alpha_{2}}T_{s}=t_{2}\beta_{2}-t_{3}\frac{\alpha_{3}}{\alpha_{2}^{ 2}} \tag{4}\]  
\[\frac{d}{d\beta_{3}}T_{s}=-t_{2}\frac{\beta_{2}}{\beta_{3}^{2}}+t_{3}\alpha_{3} \tag{5}\]  
Given that \(T_{s}\) is a convex function for \(\alpha_{2}\in\mathbb{R}_{>0}\), we can set the derivative from Equation (4) to zero and solve for the positive value of \(\alpha_{2}\) (i.e., Tier 2 resolution) in order to find the value of \(\alpha_{2}\) that minimizes \(T_{s}\). We can similarly observe Equation (5) in order to view the impact in terms of \(\beta_{3}\) (i.e., Tier 3 range or, similarly, Tier 2 instantaneous coverage).