# Optimal User Association Strategy for Large-Scale IoT Sensor Networks with Mobility on Cloud RANs
## 5 Discussion on Time Complexity  
As shown in the previous section, the speed of IoT devices does not affect the time complexity significantly. The number of IoT devices in the network, on the other hand, increases the CPU time for solving the proposed optimization problems. This is because the increased number of IoT devices also increases the dimension of the decision variable **X** in the proposed problem formulation, hence increasing the problem complexity. We expect the same to happen if we increase the number of RRHs in the network. Although all different configurations tested in this study did not disturb the correct functioning of the proposed methods, it may not be the same in the case when there are much more IoT devices, e.g., thousands of IoT devices, working with much shorter time slots, e.g., \(\ll\)1 s.  
One simple, yet effective solution we propose to overcome the possible limitation of the proposed schemes is _sectorization_. In a nutshell, the network divides the coverage area into multiple disjoint sectors, and then, runs the proposed methods for each sector separately in parallel. Although it is not always guaranteed to retrieve the location of IoT devices in real time, the location of RRHs are known in general. Therefore, we can easily identify which RRH belongs to which section. As each sector is separately handled from the rest, there should not be any IoT devices belonging to multiple sectors at the same time. Otherwise, an IoT device belonging to two sectors, for example, may end up having two associations at the same time, which is not allowed in this work.  
To resolve this issue, we propose a probabilistic approach by using the number of RRHs nearby. As an example, let us consider the network as described in Figure 11. The area is 1000 m by 1000 m, and there are five RRHs marked as 1-5 and six IoT devices \(a\)-\(f\). The coverage area of each RRH is shown by dashed lines. The location of each RRH is assumed to be known, while it is not the case for IoT devices. We divide the network into two sectors: upper half (denoted as \(U\)) and lower half (denoted as \(L\)). Given the location of RRHs, RRHs 1-3 are grouped into \(U\), while RRHs 4 and 5 belong to \(L\). IoT devices \(a\) and \(b\) belong to \(U\), and \(c\) and \(d\) to \(L\). However, it is not clear for \(e\) and \(f\). Again, we do not assume their location to be known. For \(e\), it is nearby RRH 1 in \(U\) and 4 in \(L\). Using the proposed probabilistic algorithm, information on its proximity to \(U\) (i.e., to RRH 1) or \(L\) (i.e., to RRH 4) is replaced with probability 0.5 each. On the other hand, for \(f\), information on its proximity to \(U\) (i.e., to RRH 2 and 3) or \(L\) (i.e., to RRH 5) is replaced with probability 1/3 or 2/3, respectively.  
Figure 11: An example network with five RRHs from 1 to 5 and six IoT devices from \(a\) to \(f\). The network is divided into two sectors, i.e., the upper and lower sectors, where RRHs 1–3 and 4–5 belong to the upper and lower sector, respectively.  
In this way, the dimension of the decision variable **X** is reduced from 5-by-6 to 3-by-4 for \(U\) and 2-by-2 for \(L\). As the proposed algorithms optimize each sector separately in parallel, the time required to optimize the association for the entire network equals the CPU time to optimize the most complex sector. However, it is worth mentioning that the sectorization method does not yield the optimal solution, although there is a chance to achieve this. However, it is an effective way of reducing the time complexity of the proposed schemes, which might be essential for some applications where real-time operation is more important than yielding the optimal solution. We leave the further discussion on this as a future research topic.