## 4 Dimensioning Analytical Model for the Optical Rings  
We evaluate the number of wavelengths needed in each optical ring under the following assumptions:  
1. A reference area is considered subdivided in a given number of sub-areas where RRUs and RBSs handle the user traffic. We also assume that each Access Switch handles \(D\) sub-areas. The maximum user traffic capacity that the RBS can handle is denoted with \(C_{RBS}\). We also assume that the installed RBSs are always turned on and provides a basic capacity for the coverage of any sub-area. The RRUs provide additional capacity needed in the intervals in which the traffic increases. The RRUs are only used when the basic capacity provides by the RBSs is not sufficient to support the user traffic. 2. Each RRU implements a MIMO technological solution with the possibility of \(Q\) configurations denoted as \(S_{0}\), \(S_{1}\), \(\cdots\), \(S_{Q-1}\). We assume that switching off and switching on of antennas is performed when traffic variation occurs; in particular when the configuration \(S_{i}\) (\(i=0,\cdots,Q-1\)) is activated, \(n_{i}\) (\(i=0,1\cdots,Q-1\)) antennas are turned on with the capacity of handling a user traffic capacity \(C_{i}^{FRU}\) (\(i=0,\cdots,Q-1\)); let \(f_{i}^{CRI}\) (\(i=0,\cdots,Q-1\)) denote the bit rate of the CPRI flow generated /received when the RRU is in the configuration \(S_{i}\) (\(i=0,\cdots,Q-1\)). We order the configurations in increasing order of the bit rate of the CPRI flows. 3. \(R\) two fibers optical rings are considered. Each optical ring is provided with \(S\) ASs. The generic AS is referred to as \(AS_{ij}\) that denotes the \(j\)-th AS (\(j=1,\cdots,S\)) in the \(i\)-th (\(i=1,\cdots,R\)) optical ring. We denote with \(L^{i}\) the set of the links in the \(i\)-th optical ring. 4. The traffic is cycle-stationary with \(N\) stationary periods and our traffic modeling is inherited from [20]; for instance, \(N\) equals 24 when the classical daily traffic variation has to be reproduced. The user peak traffic generated in the sub-areas handled by an AS can follow one of two profiles corresponding to business and residential traffic. The business and residential profiles occur with probability \(p_{b}\) and \(p_{r}\), respectively, with obviously \(p_{b}+p_{r}=1\). The user peak traffic generated in the sub-areas, expressed in Gbps, are independent and identically distributed (i.i.d.) variables; therefore, the upstream (downstream) user peak traffic generated in any business sub-area in the \(k\)-th stationary interval is characterized by a variable \(A_{k}^{u,b}\) (\(A_{k}^{d,b}\)) that, according to [20], we assume log-normal distributed of parameters \((n_{k}^{u,b}(A_{k}^{d,b}),\sigma_{k}^{u,b}(\sigma_{k}^{d,b}))\). For the residential sub-areas, we have the log-normal variables \(A_{k}^{u,r}\) (\(A_{k}^{d,r}\)) of parameters \((n_{k}^{u,r}(\mu_{k}^{d,r}),\sigma_{k}^{u,r}(\sigma_{k}^{d,r}))\). 5. The number of RRUs and RBSs are dimensioned in each sub-area to support the peak offered traffic and by applying the procedure reported in [2]; let us denote with \(n_{RRU}^{b}\) (\(n_{RRU}^{r}\)) and \(n_{RBS}^{b}\) (\(n_{RBS}^{r}\)) the number of RRUs and RBSs installed in a business (residential) sub-area, respectively, when the dimensioning procedure is applied. 6. A routing algorithm characterized by the variables \(x_{l}^{u,i,j}\) (\(x_{l}^{d,i,j}\)) assumes the value 1 if the traffic between the \(AS_{i,j}\) (CO) and the CO (\(AS_{i,j}\)) is routed on the link \(l\in L^{i}\); otherwise, its value is 0. 7. The resources (wavelengths) carrying Ethernet and CPRI flows, respectively, are separated and no wavelength can carry both Ethernet and CPRI flows. 8. OTN/WDM switches do not perform any grooming operation. The meaning of the all parameters and variables used are reported in Tables 1 and 2, respectively.