# Traffic-Based Heuristics for Regenerator Placement in Translucent Dynamic Optical Networks
## Appendix A Model Extension for Multi-Rate (Elastic) Networks  
The formulation is shown in Sections 3 and 4 stands for networks under single bit-rate demands. We can also extend the formulation for a scenario with heterogeneous bit-rate demands. This is the case with elastic optical networks (EON) [11; 34; 35]. The extension starts by including an extra \(b\) sub-index to represent each possible transmission rate in the following symbols: \(\omega_{ijb}\), \(\gamma_{ijb}\), \(D_{ijb}\), \(C_{ijb}\), \(K_{ijb}\), \(\Gamma_{nb}\). The set \(U\) of allowable \(B\) bit rates is \(U=\{u_{1},u_{2},\ldots,u_{b},\ldots,u_{B}\}\). For instance, \(\omega_{ijb}\) represents the offered load of requests between the nodes \(i\) and \(j\) that requires a \(u_{b}\) transmission rate. To simplify the notation, we assume that the \(b\)-indexed transmission bit rate requires \(b\) regenerators to perform its regeneration. This is a valid consideration assuming the virtualized shared pool of elastic regenerators strategy proposed by [PERSON] et al. [34]. The elastic regenerators consist of an array of fixed-rate spectrum-selective subchannel regenerators (SSRs) that act as a pool of virtualized regeneration resources [34]. Each SSR operates under a fixed transmission bit rate.  
Then, we use the theory of erlang loss systems under a multi-slot traffic [31] to evaluate the counterpart of the erlang-b formula under this multi-rate traffic scenario that occurs in EONs. Let us assume a node with \(s\) virtualized regenerators. A state \(e=(I_{1},I_{2},\ldots,I_{B})\) represents the number of active \(u_{b}\)-rate connections using SSRs in such a node, where \(I_{b}\) is the number of active \(u_{b}\)-rate connections, each of which, as explained before, using \(b\) regeneration resources. All possible states when \(s\) SSRs are installed in a node form the set \(\mathcal{E}_{s}\), which includes any occurrence of values for \((I_{1},I_{2},\ldots,I_{B})\) so that \(\sum_{b=1}^{B}b\cdot I_{b}\leq s\). Let \(\mathcal{E}_{s}(b)\) be the set of states that cannot admit a \(u_{b}\)-rate demand, i.e., \(\mathcal{E}_{s}(b)=\left\{\forall e|\sum_{x=1}^{B}x\cdot I_{x}>s-b\right\}\), as the summation accounts for the number of SSRs in use in the analyzed node and \(s-b\) accounts for the maximum number of SSRs in use in the node that still admits a regeneration of a \(u_{b}\)-rate demand in such node. In such scenario, we can replace (3) and (4) by  
\[\Gamma_{nb}=\sum_{i=1}^{N}\sum_{j=1}^{N}E_{ijb}(n), \tag{1}\]  
in which \(E_{ijb}\) is a place holder for either \(G_{ijb}(n)=\gamma_{ijb}\) if \(\exists\ m\ /\ n\in c_{ijb}^{(m)}\) and 0 otherwise or for \(F_{ijb}\):\[F_{ijb}(n)=\left\{\begin{array}{cl}\frac{\gamma_{ijb}}{M_{ij}^{M}}\sum_{m=1}^{M_{ ij}^{t}}P_{ijm}(n),&\text{ if }n\in\pi_{ij}\\ 0,&\text{otherwise},\end{array}\right. \tag{10}\]  
with \(P_{ijm}(n)=1\) if \(n\in d_{ijb}^{(m)}\) and 0 otherwise.  
In a node \(n\) with \(s\) SSRs, the probability of a state \(e\in\mathcal{E}_{s}\) is given by [31]  
\[p_{n}(e,s)=Q_{s}\prod_{b=1}^{B}\frac{\Gamma_{nb}^{t_{b}}}{l_{b}!}, \tag{11}\]  
in which \(\Gamma_{nb}\) is the total offered load estimate from \(u_{b}\)-rate demands on node \(n\) and \(Q_{s}\) is the normalization coefficient given by [31]  
\[Q_{s}=\frac{1}{\sum\limits_{e\in\mathcal{E}_{s}}p_{n}(e,s)}. \tag{12}\]  
Then, the blocking probability of \(u_{b}\)-rate demands in node \(n\) may be evaluated by  
\[BP_{n}(s,b)=\sum\limits_{e\in\mathcal{E}_{s}(b)}p_{n}(e,s), \tag{13}\]  
and (6) can be replaced by  
\[BP_{N}(n,s)=\frac{\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{b=1}^{B}PB_{n}(s,b)\cdot E _{ijb}(n)}{\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{b=1}^{B}\omega_{ijb}}. \tag{14}\]  
For a node \(n\) with a total of \(r(n)\) installed regenerators, it is possible to replace (8) in a multi-rate scenario using  
\[\Delta BP_{N}(n,r(n))=BP_{N}(n,r(n))-BP_{N}(n,r(n)+1). \tag{15}\]