# Rearrangeable Nonblocking Conditions for Four Elastic Optical Data Center Networks
## 3 RNB DCN1 and DCN3 Networks
### RNB Sufficient Conditions  
A sufficient value of \(k\) for a DCN1(\(r\), \(q\), \(k\)) network in the sense of being RNB is derived in Property 2.  
**Property 2.** A DCN1(\(r\), \(q\), \(k\)) network for \(m\)-slot connections with \(1\leq m\leq m_{\max}\) is RNB if  
\[k\geq k_{\mathrm{RNB}}=q\cdot m_{\max} \tag{3}\]  
**Proof.** This property holds if Routing Algorithm A is feasible, and Routing Algorithm A is feasible if each \(m\)-slot connection can be carried by the corresponding windows. Since \(m\leq m_{\max}\), each \(m\)-slot connection can be carried by any window \(W_{I}\) if \(|W_{I}|=m_{\max}\) for \(1\leq l\leq q\), which implies that each fiber has \(k=q\cdot m_{\max}\) FSUs. Therefore, when Routing Algorithm A is applied, a DCN1(\(r\), \(q\), \(k\)) network with \(k\geq q\cdot m_{\max}\) is RNB. \(\square\)  
Comparing Equation (3) with Equation (1), we have \(k_{\mathrm{RNB}}\)/\(k_{\mathrm{SNB}}\leq 1/2\) for \(m_{\max}\geq 2\) and \(q\geq 3\). Property 2 implies that \(k_{\mathrm{RNB}}\) reduces the SNB DCN1 result given in [13], namely, \(k_{\mathrm{SNB}}\), down to at least half for most cases. In addition, numerical results are given in Table 1 which show that \(k_{\mathrm{RNB}}\) can reduce \(k_{\mathrm{SNB}}\) down to as low as one third, for example, the cases with \(m_{\max}\geq 6\) and \(q=4\), and the cases with \(m_{\max}\geq 4\) and \(q\geq 8\).  
The sufficient condition for being an RNB DCN1(\(r\), \(q\), \(k\)) network (Property 2) is also the necessary condition if only one connection rate \(m_{\max}\) is considered (Property 3).  
**Property 3.** Suppose only one connection rate, \(m_{\max}\), is considered. Then, the DCN1(\(r\), \(q\), \(k\)) network is RNB if and only if \(k\geq k_{\mathrm{RNB}}=q\cdot m_{\max}\).  
**Proof.** The sufficient condition of this property is true since it is a special case with one connection rate of Property 2. In addition, the necessary condition holds when \(q\) connections (\(u\), \(u\), \(m_{\max}\)) for \(1\leq u\leq r\) are generated from each ToR-T \(u\). \(\square\)  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(m_{\max}\)} & \multicolumn{2}{c}{\(q=4\)} & \multicolumn{2}{c}{\(q=8\)} & \multicolumn{2}{c}{\(q=10\)} \\ \cline{2-7}  & \(k_{\mathrm{SNB}}\) & \(k_{\mathrm{RNB}}\) & \(k_{\mathrm{SNB}}\) & \(k_{\mathrm{RNB}}\) & \(k_{\mathrm{SNB}}\) & \(k_{\mathrm{RNB}}\) \\ \hline
2 & 20 & 8 & 44 & 16 & 56 & 20 \\ \hline
4 & 46 & 16 & 102 & 32 & 130 & 40 \\ \hline
6 & 72 & 24 & 160 & 48 & 204 & 60 \\ \hline
8 & 98 & 32 & 218 & 64 & 278 & 80 \\ \hline
10 & 124 & 40 & 276 & 80 & 352 & 100 \\ \hline \end{tabular}
\end{table}
Table 1: Numerical results of \(k\) required for being an SNB or RNB DCN1(\(r\), \(q\), \(k\)) network for \(m\)-slot connections with \(q=4\), \(8\), \(10\) and \(1\leq m\leq m_{\max}\), where \(k_{\mathrm{SNB}}\) and \(k_{\mathrm{RNB}}\) are given in Equations (1) and (3), respectively.  
From the architectures of the DCN1 and DCN3 networks (see Figures 2a and 3a), we can see that a DCN1(\(r\), \(q\), \(k\)) network for \(k\geq q\cdot m_{\max}\) functions the same as a DCN3(\(r\), \(q\), \(k\), \(p\)) network with \(p=1\). Thus, we derive Property 4 immediately.  
A DCN3(\(r\), \(q\), \(k\), \(p\)) network for \(m\)-slot connections with \(1\leq m\leq m_{\max}\) and \(k\geq q\cdot m_{\max}\) is RNB if \(p\geq 1\).  
This property is true for two reasons: i) a DCN3(\(r\), \(q\), \(k\), \(p\)) network with \(p=1\) functions as well as a DCN1(\(r\), \(q\), \(k\)) network, and ii) a DCN1(\(r\), \(q\), \(k\)) network for \(k\geq q\cdot m_{\max}\) is RNB (Property 2).  
For a DCN1(\(r\), \(q\), \(k\)) (or DCN3(\(r\), \(q\), \(k\), \(p\))) network, recall that the resource of FSUs per fiber in practical systems is limited, namely, \(k\leq 350\). This implies that to have an RNB DCN1(\(r\), \(q\), \(k\)) (or DCN3(\(r\), \(q\), \(k\), \(p\))) network for \(m\)-slot connections with \(1\leq m\leq m_{\max}\) in the real word, we also need \(q\cdot m_{\max}\leq 350\) due to Property 2 (or Property 4).