## 3 Network Scenario and QoT Model
### Quality of Transmission Model  
To estimate the QoT of lightpaths, we make use of the incoherent noise accumulation model, as expressed by Equation (15) in [28], which is extended for the MCF-based SDM scenario by including the inter-core crosstalk [10]. This model assumes that noise increases linearly along the optical path, where \(\mathrm{XT}\) is one of the sources of noise along with other linear and nonlinear physical layer impairments. According to this model, the QoT of a lightpath is acceptable if its signal-to-noise ratio (SNR) is above the SNR level required by the modulation format applied for this lightpath (denoted as \(SNR^{\mathrm{REG}}\)). The SNR of lightpath \(l\) routed over path \(p\) consists of a number of links (indexed by \(e\)), where each link is composed of a number of fiber spans (indexed by \(i\)) and is calculated as:  
\[SNR_{l}=\left(\sum_{e\in p}\left(\sum_{i\in e}\frac{P_{i}^{\mathrm{ASE}}+P_{i} ^{\mathrm{NL}}}{P_{\mathrm{ch}}}+\mu_{e}^{\mathrm{XT}}\right)\right)^{-1}, \tag{1}\]  
where \(P_{\mathrm{ch}}\), \(P_{i}^{\mathrm{ASE}}\), and \(P_{i}^{\mathrm{NL}}\) denote the signal, EDFA noise, and nonlinear noise powers per channel in fiber span \(i\), respectively, and \(\mu_{e}^{\mathrm{XT}}\) is the crosstalk coefficient in link \(e\). We consider that the channel power is optimized locally for individual links, which leads to the global optimization of QoT in the network according to [29]. The optimal channel power is calculated from \(P_{\mathrm{ASE}}=2P_{\mathrm{NL}}\) constraint [28], which is also valid for the extended model including \(\mathrm{XT}\). In Equation (1), the value of crosstalk coefficient \(\mu_{e}^{\mathrm{XT}}\) is calculated according to [9] as:  
\[\mu_{e}^{\mathrm{XT}}=L_{e}\cdot K_{e}\cdot 10^{(XT^{\mathrm{REF}}+XT^{ \mathrm{MARGN}})/10}, \tag{2}\]  
where \(L_{e}\) is the length of link \(e\), \(K_{e}\) is the number of adjacent cores that cause crosstalk in a given core and in a given spectrum band in link \(e\), \(XT^{\mathrm{REF}}\) denotes the reference level of \(\mathrm{XT}\) between two adjacent cores in a given MCF (expressed in \([\mathrm{dB/km}]\)), and \(XT^{\mathrm{MARGN}}\) is an extra margin with which \(\mathrm{XT}\) is estimated; according to [9], we consider \(XT^{\mathrm{MARGN}}=8\) dB for quantile \(q=0.9999\). In Equation (2), the value of \(K_{e}\) reflects the actual maximum occupancy of cores that are adjacent to a given core assigned to lightpath \(l\), which is in accordance to the precise \(\mathrm{XT}\) methodology used in [14]. Namely, if lightpath \(l\) uses frequency slices \(F=(s_{1},...,s_{S})\) and \(K_{e,s}\) is the number of allocated cores adjacent to a given core in link \(e\) on slice \(s\), then we have \(K_{e}=\max\limits_{s\in F}\{K_{e,s}\}\). In order to apply nonlinear Model (1) in an MIP formulation (in Section 4), we transform it as the following. Firstly, we consider the inverse of SNRs instead of direct SNR values, which leaves us with the summation operator on the right-hand side of Equation (1). Secondly, since the values of \(P_{\mathrm{ch}}\), \(P_{i}^{\mathrm{ASE}}\), and \(P_{i}^{\mathrm{NL}}\) are static and do not depend on the actual load of neighbor MCF cores, we represent the summation of the quotient of these power values by means of static parameter \(\beta(e)\), which is pre-calculated for each network link \(e\). Eventually, we substitute coefficient \(\mu_{e}^{\mathrm{XT}}\) in Equation (1) with Equation (2) and represent the factors that do not depend on MCF link load by means of static parameter \(\gamma(e)\), which is specific for each link \(e\).