## 4 Proposed RCSA Method
### Definition of Weight Parameters  
To efficiently reduce the blocking probability of lightpath establishment, it is important to appropriately determine the weight parameter \(W(R_{p})\) for lightpaths requiring \(R_{p}\) FSs, which is utilized in (9). We expect to reduce the blocking probability of lightpath establishment by allocating lightpaths that require the same number of FSs to the same core, as discussed in policy (b). However, the number of FSs required by lightpaths takes various values depending on the traffic demand and path length, while the number of available cores is limited. Therefore, it is difficult to exclusively allocate lightpaths requiring the same number of FSs to dedicated cores. In order to solve this problem, the proposed method groups lightpath-setup requests based on their required number of FSs. Specifically, in this paper, we form three groups according to the procedure outlined below. **Grouping procedure:**  
1. Requests that require a number of FSs divisible by 3 (i.e., \(R_{p}=3,6,9,\ldots\)) are assigned to Group \(g_{1}\). 2. Requests, other than those in Group 1, that require a number of FSs divisible by 2 (i.e., \(R_{p}=2,4,8,\ldots\)) are assigned to Group \(g_{2}\). 3. The remaining requests (i.e., \(R_{p}=1,5,7,\ldots\)) are assigned to Group \(g_{3}\). The proposed method determines the weight parameter \(W(R_{p})\) based on these groups. As illustrated in Figure 4, the proposed method aims to allocate lightpaths belonging to the same group to the same core. Furthermore, in each core, the proposed method assigns FS indices with priority to lightpaths based on their required number of FSs. By doing so, we expect to prevent spectrum fragmentation. In order to achieve this, the weight parameter \(W(R_{p})\) is given by the sum of the parameters derived from these two strategies, i.e., \(W(R_{p})=W_{c}(R_{p})+W_{f}(R_{p})\). In the first strategy, we allocate these groups to specific cores. Here, we assume 3-, 7-, and 13-core fibers, as illustrated in Figure 5. For lightpath-setup requests in the same group, the preferential use of the allocated cores is expected to reduce spectrum fragmentation. This is because by grouping requests whose required number of FSs is divisible by a specific number, frequency allocation can be performed more efficiently. Furthermore, in order to suppress inter-core crosstalk, it is necessary to avoid the simultaneous use of adjacent cores. Therefore, we assign usage priorities to the cores for each group. Specifically, we define a weight parameter \(W_{c}(R_{p})\) for each group, as shown in Tables 2-4, to preferentially  
The second strategy introduces a weight parameter \(W_{f}(R_{p})\) based on the FSs used for lightpath establishment to further reduce the blocking probability of lightpath establishment. This strategy aims to exclusively assign the prioritized FS range to lightpath-setup requests according to their required number \(R_{p}\) of FSs in each group. Specifically, for each \(R_{p}\), we set \(W_{f}(R_{p})=0.0\) for FS \(f\) within the prioritized FS range and \(W_{f}(R_{p})=1.0\) for all other FSs. The width of the prioritized FS range is approximately adjusted in proportion to the product of the value of \(R_{p}\) and the intended number of lightpaths requiring \(R_{p}\) FSs to be established. As an example, we assume the case where we intend to preferentially establish 20 lightpaths with \(R_{p}=3\) belonging to Group \(g_{1}\). In this case, the assigned FS width is \(3\times 20=60\). Note that we ignore the guard band in this example. Accordingly, we set \(W_{f}(3)=0.0\) for FSs \(f_{1}\leq f\leq f_{60}\) as the prioritized FS range, while we set \(W_{f}(3)=1.0\) for all other FSs. Additionally, we assume the case where we intend to preferentially establish five lightpaths with \(R_{p}=6\), also belonging to Group \(g_{1}\). In this case, the assigned FS width is \(6\times 5=30\), so we set \(W_{f}(6)=0.0\) for FSs \(f_{61}\leq f\leq f_{90}\), while we set \(W_{f}(6)=1.0\) for all other FSs.