# A Simple Approach to Dynamic Optimisation of Flexible Optical Networks with Practical Application
## 7 Extensions and Generalisations  
DRP-FON has been defined, in this paper, for simplicity of the initial analysis, solution, and testing. There are a number of respects in which it may be too simple for widespread application. In particular, the following real-world extensions may be considered:  
Ext1: Relays are only permitted at certain nodes: This is a likely extension to the basic model, reflecting restrictions on equipment installation at some locations or resulting from network management strategies, such as only installing regenerators at some (half, say) of all locations. It is \begin{table}
\begin{tabular}{c c c} \hline \(m\) & **US NSF a** & **COST-266 a** \\ \hline
3 & 1 & 2 \\
4 & 0 & 1 \\
5 & 1 & 0 \\
6 & - & 1 \\
7 & - & 0 \\
8 & - & - \\ \hline \end{tabular}
\end{table} 
Table 1: Comparing AlgR with known optima. Figure 4: The COST-266 European network (topologically, not geographically, accurate). easily dealt with by a simple restriction on the \(\underline{r}\) array or, if necessary, a separate \(\underline{p}\) 'permitted' array. The adaptation of AlgV and AlgR is then trivial. Ext2: Use physical distances instead of hop counts: If regeneration limits are to be measured by cumulative distance, rather than merely hop count, then reachability becomes an issue of both fixed physical edge weight, \(W=(w_{ij})\) (the Euclidean distance between \(i\) and \(j\), for example), and the dynamically-changing limit, \(m\), as transmission parameters vary. As a result, the viability matrix, \(V\), is calculated by natural extension to AlgH and AlgV, and AlgR remains essentially unaltered. Ext3: Allow different transmission limits for each node pair: The basic model assumes the same changes in transmission parameters and, hence, \(m\), across the FON. However, if transmission parameters are allowed to vary locally (independently) then the single value of \(m\) will need to extend to a matrix of limits, \(M\). Nevertheless, individual values, \(v_{ij}\), of the viability matrix, \(V\), can be calculated, without difficulty, using the appropriate element, \(m_{ij}\): AlgV and AlgR otherwise work as before. Taking these three classes of extension together gives a new optimisation problem to be considered. Firstly, we define the _permitted/present vector_, \(\underline{p}\), as \(\underline{p}=(p_{k})\) where \(\forall k,p_{k}=T\) if a relay is allowed/installed at node \(k\); \(p_{k}=F\) otherwise. \(\underline{p}\) is a physical network constraint for any node set. Then, as before, for any given solution, the _relay vector_, \(\underline{L}\) is defined as: \(\underline{r}=(r_{k})\) where \(\forall k,r_{k}=T\) if the relay at node \(k\) is switched on (powered); \(r_{k}=F\) otherwise. Next, we include the physical distance (or any appropriate metric) by introducing the weight matrix, \(W=(w_{ij})\), where \(w_{ij}\) represents the 'weight' of the link between \(i\) and \(j\) (if present). In the simplest case, \(w_{ij}\) represents the Euclidean distance between the two nodes (Figure 5). However, other uses are possible to discourage the use of inappropriate paths, etc. [12]. Finally, we extend our transmission limit concept to allow local variation across the FON. On this basis, we define the maximum hop matrix, \(M=(m_{ij})\), where \(m_{ij}\) is the maximum number of hops for which an optical signal can be transmitted, between nodes \(i\) and \(j\), without the need for regeneration by a relay. As before, the _cost_ of a solution, \(\underline{r}\), is then the _number of relays switched on_ (the cardinality of the set of powered relays) in any given configuration (now defined by both \(M\) and \(W\)). \(C(\underline{r})=\left\lvert\left\lvert k:r_{k}=T\right\rvert\right\rvert.\)  
Figure 5: Euclidean (geographically accurate) COST-266 European and US NSF networks [7]. Note the additional links since the publication of the library: adapted from [25].