# On the Efficient Flow Restoration in Spectrally-Spatially Flexible Optical Networks

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. ILP Models

#### 3.1. Network Planning Problem

#### 3.1.1. Node Link (ND) Model

ND approach: sets and indices | |

$v\in V$ | network nodes |

$e\in E$ | network physical links available in a normal state |

$s\in S$ | frequency slices available on each spatial mode in a normal network state |

$k\in K$ | spatial modes |

$d\in D$ | traffic demands to be realized |

$c\in C$ | candidate channels for the set of slices S |

$m\in M$ | available modulation formats |

ND approach: constants | |

${b}_{d}$ | bit-rate (in Gbps) of the demand d |

${s}_{d}$ | source node of the demand d |

${t}_{d}$ | termination (destination) node of the demand d |

${l}_{e}$ | length (in kilometers) of the physical link e |

${a}_{m}$ | lower bound of the distance range supported by the modulation format m |

${n}_{dm}$ | number of slices required to realize the demand d using the modulation m |

${h}_{dm}$ | number of additional slices required for the demand d if the modulation m is applied instead of the modulation m $-1$ |

L | large number |

$A\left(v\right)$ | set of links that originate in the node v |

$B\left(v\right)$ | set of links that terminate in the node v |

${\alpha}_{cks}$ | =1, if the channel c uses the slice s on the spatial mode k; 0, otherwise |

${\beta}_{c}$ | size (the number of the involved slices) of the channel c (excluding a guard-band) |

ND approach: variables | |

${y}_{eks}$ | =1, if the slice s is used on the spatial mode k and the physical link e; 0, otherwise (binary) |

${y}_{s}$ | =1, if slice the s is used in the network (on any link and spatial mode); 0, otherwise (binary) |

${w}_{dc}$ | =1, if the demand d is realized using the channel c; 0, otherwise (binary) |

${x}_{dec}$ | =1, if the light-path selected for the demand d uses the link e and the channel c; 0, otherwise (binary) |

${x}_{de}$ | =1, if the light- path selected for the demand d uses the link e; 0, otherwise (binary) |

${x}_{d}$ | length (in kilometers) of the light-path selected for the demand d (integer) |

${u}_{dm}$ | =1, if any modulation format $i\le m$ can be applied for the demand d; 0, otherwise (binary) |

#### 3.1.2. Link-Path (LP) Model

LP model: sets and indices | |

$e\in E$ | network physical links available in a normal state |

$s\in S$ | frequency slices available on each link in a normal network state |

$k\in K$ | spatial modes |

$d\in D$ | traffic demands to be realized |

$c\in C$ | candidate channels for the set of slices S |

$p\in {P}_{d}$ | candidate routing paths for the demand d considering all network links E |

$l=(p,c)\in {L}_{d}$ | candidate light-paths for the demand d considering the set of links E and the set of channels C |

LP model: constants | |

n | number of routing paths available for each pair of network nodes |

${\alpha}_{cks}$ | =1, if the channel c uses the slice s on the spatial mode k; 0, otherwise |

${\gamma}_{le}$ | =1, if the light-path l uses the link e; 0, otherwise |

LP model: variables | |

${y}_{eks}$ | =1, if the slice s is used on the spatial mode k and the physical link e; 0, otherwise (binary) |

${y}_{s}$ | =1, if the slice s is used in the network (on any link and spatial mode); 0, otherwise (binary) |

${z}_{dl}$ | =1, if the demand d is realized using the light-path l; 0, otherwise (binary) |

**objective**

**subject to**

#### 3.2. Flow Restoration Problem

#### 3.2.1. Node-Link (ND) Model

ND approach: sets and indices (additional) | |

$e\in {E}^{\prime}$ | network physical links available after a single link failure |

$s\in {S}^{\prime}$ | frequency slices available on each spatial mode during a restoration process. Note that $|{S}^{\prime}|=\rho \xb7|S|$ |

$d\in {D}^{\prime}$ | traffic demands to be restored |

$c\in {C}^{\prime}$ | candidate channels for the set of slices ${S}^{\prime}$ |

ND approach: constants (additional) | |

$\rho $ | ratio of the additional spectrum resources available for the purpose of flow restoration |

${\delta}_{eks}$ | considering a particular network state: |

=1, if currently the slice s is occupied on the spatial mode k on the physical link e; 0, otherwise | |

ND approach: variables (additional) | |

${r}_{d}$ | =1, if the demand d is released due to a failure; 0, otherwise (binary) |

**objective**

**subject to**

#### 3.2.2. Link-Path (LP) Model

LP model: sets and indices (additional) | |

$e\in {E}^{\prime}$ | network physical links available after a single link failure |

$s\in {S}^{\prime}$ | frequency slices available on each spatial mode during a restoration process. Note that $|{S}^{\prime}|=\rho \xb7|S|$ |

$d\in {D}^{\prime}$ | traffic demands to be restored |

$c\in {C}^{\prime}$ | candidate channels for the set of slices ${S}^{\prime}$ |

$p\in {P}_{d}^{\prime}$ | candidate routing paths for the demand d considering the set ${E}^{\prime}$ of available links |

$l=(p,c)\in {L}_{d}^{\prime}$ | candidate light-paths for the demand d considering the set of links ${E}^{\prime}$ and the set of channels ${C}^{\prime}$ |

LP model: constants (additional) | |

$\rho $ | ratio of the additional spectrum resources available for the purpose of flow restoration |

${\delta}_{eks}$ | considering a particular network state: |

=1, if currently the slice s is occupied on the spatial mode k on the physical link e; 0, otherwise | |

LP model: variables (additional) | |

${r}_{d}$ | =1, if the demand d is released due to a failure; 0, otherwise (binary) |

**objective**

**subject to**

#### 3.3. Complexity Analysis

## 4. Results and Discussion

#### 4.1. Models Comparison

#### 4.1.1. Network Planning Problem

#### 4.1.2. Flow Restoration Problem

#### 4.2. Efficient Flow Restoration in SS-FON—Case Study

## 5. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

BPSK | binary phase shift keying |

CARG | compound annual growth rate |

EON | elastic optical network |

FMF | few-mode fiber |

FM-MCF | few-mode multi-core fiber |

ILP | integer linear programming |

LP | link-path |

MCF | multic-core fiber |

ND | node-link |

QAM | quadrature amplitude modulation |

QPSK | quadrature phase shift keying |

RSSA | routing space and spectrum allocation |

SDM | space division multiplexing |

SDN | software defined network |

SMFB | single-mode fiber bundle |

SCh | spectral channel |

SS-FON | spectrally-spatially flexible optical network |

SSCh | spatial-spectral channel |

WDM | wavelength division multiplexing |

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**Figure 3.**Ratio of the restored traffic for PL12 and 5 Tbps as a function of the amount of additional resources $\rho $.

**Figure 4.**Ratio of the restored traffic for PL12 and 15 Tbps as a function of the amount of additional resources $\rho $.

**Figure 5.**Ratio of the restored traffic for PL12 and 25 Tbps as a function of the amount of additional resources $\rho $.

**Figure 6.**Ratio of the restored traffic for DT14 and 5 Tbps as a function of the amount of additional resources $\rho $.

**Figure 7.**Ratio of the restored traffic for DT14 and 15 Tbps as a function of the amount of additional resources $\rho $.

**Figure 8.**Ratio of the restored traffic for DT14 and 25 Tbps as a function of the amount of additional resources $\rho $.

**Table 1.**Supported bit-rate and transmission distance for a transponder operating within 37.5 GHz spectrum.

BPSK | QPSK | 8-QAM | 16-QAM | |
---|---|---|---|---|

supported bit-rate [Gbps] | 50 | 100 | 150 | 200 |

transmission reach [km] | 6300 | 3500 | 1200 | 600 |

**Table 2.**Complexity of the proposed ILP models: the number of constraints and variables as a function of the problem characteristics.

Number of Constraints | Number of Variables | |
---|---|---|

network planning problem | ||

ND model | $2\left|E\right|\left|K\right|\left|S\right|+\left|D\right|(3+|E|+|M|+|C\left|\right|V\left|\right)$ | $\left|S\right|\left(\right|E\left|\right|K|+1)+\left|D\right|(1+|M|+|E|+|C|+|C\left|\right|E\left|\right)$ |

LP model | $\left|D\right|+2\left|E\right|\left|K\right|\left|S\right|$ | ${\sum}_{d\in D}|{L}_{d}|+|S\left|\right(\left|E\right|\left|K\right|+1)$ |

flow restoration problem | ||

ND model | $|{E}^{\prime}\left|\right|K\left|\right|{S}^{\prime}|+|{D}^{\prime}\left|\right(3+|{E}^{\prime}|+|M|+|{C}^{\prime}\left|\right|V\left|\right)$ | $|{E}^{\prime}\left|\right|K\left|\right|{S}^{\prime}|+|{D}^{\prime}\left|\right(2+|{C}^{\prime}|+|M|+|{E}^{\prime}|+|{E}^{\prime}\left|\right|{C}^{\prime}\left|\right)$ |

LP model | $|{D}^{\prime}|+|{E}^{\prime}\left|\right|K\left|\right|{S}^{\prime}|$ | $|{D}^{\prime}|+{\sum}_{d\in {D}^{\prime}}|{L}_{d}^{\prime}|+|{E}^{\prime}\left|\right|K\left|\right|{S}^{\prime}|$ |

PL12 | DT14 | |
---|---|---|

number of nodes | 12 | 14 |

number of links | 36 | 46 |

avg nodal degree | 2.9 | 3.3 |

min link length [km] | 70 | 37 |

max link length [km] | 360 | 353 |

avg link length [km] | 185 | 182 |

**Table 4.**Comparison of the node-link and the link-path models for the network planning problem and PL12—the average gap to global optimum [%] and the processing time [s] (for each $\left|K\right|$ value and traffic volume, the best result is bold).

Traffic | $\left|\mathit{K}\right|=2$ | $\left|\mathit{K}\right|=3$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

[Tbps] | ND | LP(5) | LP(10) | LP(15) | LP(20) | LP(25) | LP(30) | ND | LP(5) | LP(10) | LP(15) | LP(20) | LP(25) | LP(30) |

gap to global optimum | ||||||||||||||

2.0 | 0.00% | 0.48% | 0.48% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

2.5 | 0.00% | 4.70% | 3.75% | 3.75% | 3.75% | 2.50% | 1.25% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

3.0 | 0.00% | 4.23% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 1.43% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

3.5 | 0.00% | 6.25% | 3.75% | 3.75% | 3.75% | 3.75% | 2.50% | 0.00% | 0.48% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

4.0 | 0.00% | 5.36% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

4.5 | 0.00% | 10.23% | 6.48% | 6.48% | 6.48% | 6.48% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% |

processing time [s] | ||||||||||||||

2.0 | 5490.2 | 0.61 | 0.92 | 1.39 | 2.10 | 2.49 | 3.06 | 11,331.94 | 0.76 | 1.29 | 2.23 | 2.96 | 3.50 | 3.84 |

2.5 | 16,898.9 | 0.61 | 0.89 | 1.25 | 1.76 | 3.90 | 7.70 | 17,485.22 | 0.55 | 0.72 | 0.91 | 1.09 | 1.80 | 3.73 |

3.0 | 31,939.4 | 2.13 | 4.20 | 9.75 | 11.89 | 12.47 | 14.10 | 24,378.62 | 1.87 | 2.57 | 3.49 | 4.05 | 4.23 | 5.14 |

3.5 | 47,150.3 | 2.73 | 7.61 | 17.62 | 20.08 | 22.25 | 25.46 | 30,960.82 | 1.67 | 2.96 | 4.07 | 4.72 | 5.01 | 6.01 |

4.0 | 125,781.8 | 9.70 | 21.13 | 32.14 | 30.66 | 35.05 | 52.03 | 100,357.6 | 3.38 | 3.94 | 5.23 | 9.79 | 13.35 | 13.31 |

4.5 | 313,770.7 | 13.01 | 25.82 | 40.00 | 45.11 | 53.61 | 77.40 | 446,313.9 | 6.06 | 8.80 | 11.89 | 19.41 | 20.19 | 33.57 |

**Table 5.**Comparison of the node-link and the link-path models for the traffic restoration problem for $\rho =1.0$: the average ratio of the restored traffic [%] and the processing time [s] (for each topology and traffic volume, the best result is bold).

PL12 Topology | DT14 Topology | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Traffic | Node-Link Model | Link-Path Model | Node-Link Model | Link-Path Model | ||||||||

[Tbps] | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ |

ratio of restored traffic [%] | ||||||||||||

5T | 99.1% | 100.0% | 100.0% | 97.1% | 97.1% | 97.1% | 97.9% | 100.0% | 100.0% | 96.0% | 96.0% | 96.0% |

10T | 100.0% | 100.0% | 100.0% | 97.1% | 97.1% | 97.1% | 94.8% | 98.3% | 98.5% | 86.6% | 86.6% | 86.6% |

15T | 97.3% | 97.6% | 97.8% | 76.4% | 76.4% | 76.4% | 90.4% | 93.3% | 93.3% | 65.3% | 65.3% | 65.3% |

20T | 94.9% | 95.3% | 95.4% | 48.0% | 48.0% | 48.0% | 83.7% | 86.1% | 86.7% | 37.6% | 37.6% | 37.6% |

25T | 92.1% | 93.0% | 93.3% | 28.0% | 28.0% | 28.0% | 82.4% | 84.6% | 85.1% | 31.9% | 31.9% | 31.9% |

processing time [s] | ||||||||||||

5T | 3.31 | 6.87 | 21.46 | 33.79 | 34.49 | 33.44 | 3.73 | 9.29 | 31.95 | 27.21 | 44.91 | 42.77 |

10T | 6.06 | 17.13 | 29.27 | 51.65 | 51.04 | 48.24 | 9.76 | 22.07 | 49.94 | 33.18 | 54.13 | 51.90 |

15T | 13.89 | 24.98 | 38.25 | 52.98 | 51.67 | 48.85 | 11.02 | 21.00 | 31.29 | 27.34 | 46.03 | 45.39 |

20T | 13.15 | 18.47 | 24.58 | 38.86 | 38.66 | 37.85 | 10.04 | 17.09 | 27.47 | 41.25 | 42.47 | 38.59 |

25T | 9.70 | 6.98 | 20.81 | 29.79 | 29.33 | 29.19 | 10.56 | 21.62 | 45.60 | 34.14 | 35.26 | 32.98 |

**Table 6.**Comparison of the node-link and the link-path models for the traffic restoration problem for $\rho =1.4$: the average ratio of the restored traffic [%] and the processing time [s] (for each topology and traffic volume, the best result is bold).

PL12 Topology | DT14 Topology | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Traffic | Node-Link Model | Link-Path Model | Node-Link Model | Link-Path Model | ||||||||

[Tbps] | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{1}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{2}$ | ${\mathit{K}}_{\mathit{C}}=\mathbf{3}$ |

ratio of restored traffic [%] | ||||||||||||

5T | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% | 100.0% |

10T | 100.0% | 100.0% | 100.0% | 99.9% | 99.9% | 99.9% | 99.4% | 99.6% | 100.0% | 96.7% | 96.7% | 96.7% |

15T | 99.5% | 100.0% | 100.0% | 95.0% | 95.0% | 95.0% | 98.9% | 99.4% | 99.9% | 88.2% | 88.2% | 88.2% |

20T | 98.4% | 99.8% | 99.9% | 84.4% | 84.4% | 84.4% | 97.6% | 99.4% | 100.0% | 75.0% | 75.0% | 75.0% |

25T | 96.0% | 97.3% | 99.3% | 59.0% | 59.0% | 59.0% | 95.2% | 97.5% | 99.3% | 62.9% | 62.9% | 62.9% |

processing time [s] | ||||||||||||

5T | 8.70 | 23.98 | 172.86 | 73.47 | 73.42 | 73.27 | 8.79 | 21.12 | 219.47 | 92.15 | 91.72 | 91.80 |

10T | 19.13 | 88.98 | 127.84 | 112.11 | 112.26 | 112.17 | 24.03 | 63.88 | 328.26 | 126.14 | 126.82 | 128.52 |

15T | 43.34 | 94.17 | 150.27 | 150.94 | 150.54 | 151.40 | 29.82 | 67.50 | 228.65 | 113.69 | 112.92 | 112.62 |

20T | 42.24 | 105.74 | 226.65 | 130.23 | 130.54 | 130.61 | 24.22 | 59.87 | 96.93 | 122.93 | 122.90 | 122.95 |

25T | 31.12 | 96.15 | 346.50 | 107.32 | 107.63 | 107.56 | 30.50 | 61.35 | 203.84 | 98.89 | 99.41 | 98.99 |

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**MDPI and ACS Style**

Goścień, R.
On the Efficient Flow Restoration in Spectrally-Spatially Flexible Optical Networks. *Electronics* **2021**, *10*, 1468.
https://doi.org/10.3390/electronics10121468

**AMA Style**

Goścień R.
On the Efficient Flow Restoration in Spectrally-Spatially Flexible Optical Networks. *Electronics*. 2021; 10(12):1468.
https://doi.org/10.3390/electronics10121468

**Chicago/Turabian Style**

Goścień, Róża.
2021. "On the Efficient Flow Restoration in Spectrally-Spatially Flexible Optical Networks" *Electronics* 10, no. 12: 1468.
https://doi.org/10.3390/electronics10121468