## 4 Proposed RCSA Method
### Detailed Procedure for Proposed RCSA Method
#### 4.2.2 Lightpath Establishment in Proposed Method  
For the establishment of each new lightpath, the proposed method uses the establishment cost \(U_{p,f}\) for each FS \(f\) of each routing path \(p\) to select a combination of routing path, FSs, and cores. We assume that the establishment cost becomes small in the case where the three above-mentioned policies are satisfied. The proposed method selects a combination with the smallest establishment cost. To do so, we define the establishment cost \(U_{p,f}\) for a lightpath requiring \(R_{p}\) FSs as  
\[U_{p,f}=\sum_{e\in p}\left\{\min_{c\in\mathcal{C}}\sum_{f^{\prime}=f}^{f+R_{p }-1}S^{[R_{p}]}_{e,c,f^{\prime}}\right\}, \tag{7}\]  
where \(S^{[R_{p}]}_{e,c,f}\) represents the sub-cost of FS \(f\) in core \(c\) of link \(e\), which is defined later. In (7), \(\sum_{f^{\prime}=f}^{f+R_{p}-1}S^{[R_{p}]}_{e,c,f^{\prime}}\) assumes the case where the successive FSs \(f\) to \(f+R_{p}-1\) are allocated to the lightpath. Note that, as defined in (1), the required number \(R_{p}\) of FSs is calculated based on the traffic demand and modulation level. The establishment cost \(U_{p,f}\) is calculated, assuming that the core with the minimum sub-cost is selected in each link. Specifically, the proposed method uses core \(c_{[\text{min}]}\) at the link, given by  
\[c_{[\text{min}]}=\min_{c\in\mathcal{C}}\sum_{f^{\prime}=f}^{f+R_{p}-1}S^{[R_{ p}]}_{e,c,f^{\prime}}, \tag{8}\]  
when the link is included in the selected routing path. The sub-cost \(S^{[R_{p}]}_{e,c,f}\) is defined as follows so as to include the three above-mentioned policies. Let \(s_{e,c,f}\) be \(1\) if FS \(f\) in core \(c\) of link \(e\) is already used by existing lightpaths; otherwise (i.e., it is available), it is \(0\). The sub-cost of FS \(f\) of core \(c\) in link \(e\) is defined as  
\[S^{[R_{p}]}_{e,c,f}= \begin{cases}W(R_{p})\times(1+\sum_{f^{\prime}\in\mathcal{ F}:f\
eq f}s_{e,c,f^{\prime}})+\gamma\times\sum_{c^{\prime}\in\mathcal{C}:c\
eq c ^{\prime}}u_{c,c^{\prime}}s_{e,c^{\prime},f},&\text{if }s_{e,c,f}=0\\ \infty&\text{if }s_{e,c,f}=1,\end{cases}  \tag{9}\]  
where \(\gamma\) is a constant parameter. \(W(R_{p})\) is a weight parameter for lightpaths requiring \(R_{p}\) FSs, the value of which is defined in Section 4.3. \(u_{c,c^{\prime}}\) is an indicator function that is equal to \(1\) if core \(c^{\prime}\) is an adjacent core of core \(c\); otherwise, it is \(0\). In (9), the first term reflects policies (a) and (b) discussed in Section 4.1. Specifically, the sub-cost \(S^{[R_{p}]}_{e,c,f}\) grows as the number of FSs already used by existing lightpaths in the same core increases. Furthermore, by appropriately selecting \(W(R_{p})\), the proposed method reduces the cost \(S^{[R_{p}]}_{e,c,f}\) of specific cores for each value of \(R_{p}\). Additionally, the second term corresponds to policy (c), where \(S^{[R_{p}]}_{e,c,f}\) increases in proportion to the number of occupied FSs in adjacent cores. The influence of these policies can be tuned via the parameter \(\gamma\). In (9), if \(s_{e,c,f}=1\), i.e., the FS is unavailable, the cost is set to infinity because the lightpath cannot be established using this FS in the core. When a new lightpath-setup request with traffic demand \(D\) arrives from sender \(i\) to receiver \(j\), the proposed method establishes a lightpath using the procedure described below. **Lightpath establishment procedure:**  
1. The required number \(R_{p}\) of FSs for each routing path \(p\in\mathcal{P}_{i,j}\) is calculated using (1). 2. The establishment cost \(U_{p,f}\) for each FS \(f\in\mathcal{F}\) of each routing path \(p\in\mathcal{P}_{i,j}\) is calculated using (7). 3. If there are no available combinations of a routing path \(p\) and FSs \(f\) to \(f+R_{p}-1\), the lightpath-setup request is blocked, and the procedure terminates. 4. An available combination of a routing path \(p\) and FSs \(f\) to \(f+R_{p}-1\) with the smallest \(U_{p,f}\) is selected according to the following: 1.