Evolving optical interconnection topology: from survivable rings to resilient meshes

Abstract

Physical topologies are evolving from elementary survivable rings into complex mesh networks. Nevertheless, no topology model is known to provide an economic, systematic, and flexible interconnection paradigm for ensuring that those meshes bear resilience features. This paper argues that intrinsic resilience can be brought by twin graph topologies, as they satisfy equal length disjoint path property with minimal number of physical links. Also, they benefit from property preserving recursive methods to graciously scale up/down and merge/split topologies. An exhaustive investigation is performed across twin graph families composing networks from 4 to 17 nodes, whereas diverse real-world topologies and ring networks are used as benchmarks. First, we illustrate the growing and the merging processes, and discuss the topology diversity of twin graphs. We analyze the impact of single cable cuts between neighbouring nodes, then we stress topologies with 2, 3, and 4 simultaneous cable cuts. Improved resiliency is seen for neighbor nodes and also reduction of cut sets able to disconnect the twin topologies in comparison with real-world networks. At transport layer, we derive and validate an upper bound for additional capacity required to implement $1+1$ path dedicated protection. As networks grow larger, this protection cost is consistently reduced compared to benchmark topologies. We also test the suitability of our approach at optical layer regarding transponders consumption. Finally, we present as a use case the redesign of CESNET into a resilient network.

Appendix

In this Appendix we present the proof of Eq. (4), which is an upper bound for the link protection coefficient of twin graphs depending only on the number of nodes.

The link protection coefficient $K_{p}^{l}$ is defined as the additional capacity required to implement $1 + 1$ path dedicated protection against single link failures [17, 20]. Based on the transmission H of a graph [25], in this Appendix we compute the link protection coefficient of rings, and an upper bound for the link protection coefficient of twin graphs.

The link protection coefficient $K_{p}^{l}$ is given by [17, 20]:

$$\begin{aligned} K_{p}^{l} = \frac{h^{l}}{h}, \end{aligned}$$

(7)

where h is the average distance for working paths (shortest paths), and $h^{l}$ is the average distance for backup paths used in case of a link failure (second link-disjoint shortest paths). The average distance for working paths, considering unit demands is:

$$\begin{aligned} h = \frac{1}{D} \frac{H}{2} = \frac{H}{n(n-1)}, \end{aligned}$$

(8)

where $D=n(n-1)/2$ is the number of bidirectional demands, considering all-to-all communication; and the total distance for working paths is the transmission H.

The average distance for backup paths used in case of a link failure is given by:

$$\begin{aligned} h^{l} = \frac{H^{l}}{n(n-1)}, \end{aligned}$$

(9)

where $H^{l}$ is the total distance for backup paths used in case of a link failure. Then, the link protection coefficient is:

$$\begin{aligned} K_{p}^{l} = \frac{h^{l}}{h} = \frac{\frac{H^{l}}{n(n-1)}}{\frac{H}{n(n-1)}} = \frac{H^{l}}{H}. \end{aligned}$$

(10)

The transmission H of a ring with n nodes is given by [25]:

$$\begin{aligned} H = n \left\lfloor \frac{n^2}{4} \right\rfloor . \end{aligned}$$

(11)

Considering that, for each path with k hops, the backup path has $n-k$ hops, we obtain for ring topologies:

$$\begin{aligned} K_{p}^{l} = {\left\{ \begin{array}{ll} \frac{3n-4}{n}, &{} {\text {if }}n{\text { is even}};\\ \frac{3n-1}{n+1}, &{} \text {if }n{\text { is odd}}. \end{array}\right. } \end{aligned}$$

(12)

In order to compute an upper bound for the link protection coefficient of twin graphs, firstly we notice that H can be decomposed as $H = H_{1} + H_{2+}$, where the term $H_{1}$ is due to pairs at unit distance, i.e., adjacent pairs, and $H_{2+}$ is due to pairs at distance 2 or more. Since $H_{1} = 2m$, we can write:

$$\begin{aligned} H = 2m + H_{2+}. \end{aligned}$$

(13)

In a twin graph, the distance between a node pair (u, v) can be changed only if the link connecting u and v fails, and in this case a distance 1 becomes 3, since each node belongs to (at least) one cycle of order 4 [26]. Then:

$$\begin{aligned} H^{l}_{1}= & {} 3H_{1} \end{aligned}$$

(14)

$$\begin{aligned} H^{l}_{2+}= & {} H_{2+} \end{aligned}$$

(15)

$$\begin{aligned} K_{p}^{l}= & {} \frac{H^{l}}{H} = \frac{H^{l}_{1} + H^{l}_{2+}}{H_{1} + H_{2+}} = \frac{6m + H - 2m}{2m + H -2m} = \frac{4m}{H} + 1. \end{aligned}$$

(16)

From [25], if a graph G has order n and size m, then:

$$\begin{aligned} H \ge 2n(n-1)-m. \end{aligned}$$

(17)

Using $m=2n-4$, we have for twin graphs:

$$\begin{aligned} H \ge 2n(n-3)+8. \end{aligned}$$

(18)

Then,

$$\begin{aligned} K_{p}^{l} \le \frac{n^{2}+n-4}{n^{2}-3n+4}. \end{aligned}$$

(19)