# Provision of Energy- and Wavelength-Efficient Traffic Grooming for Sparse WDM-Enabled Distributed Satellite Cluster Networks

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## Abstract

**:**

## 1. Introduction

## 2. Scenario and Models of Traffic Grooming in the DSCNs

#### 2.1. Scenario Description

#### 2.1.1. Basic Node Pair Representation

(m, n) | Originating and terminating satellite nodes in the physical topology G, which also serves as the endpoints of an ISL. |

(i, j) | Originating and terminating nodes of a virtual lightpath, which traverses through several ISLs in the physical topology. Each virtual lightpath is allocated a wavelength. |

(s, d) | Source and destination nodes of an end-to-end connection request, or aggregated request, which will be routed on the virtual lightpaths. |

#### 2.1.2. Sets and Parameters

${Q}_{m}$ | Set of visible satellites for satellite m. |

${\alpha}_{m}$ | The node degree of satellite m; $2\le {\alpha}_{m}\le \left|{Q}_{m}\right|$ and ${\alpha}_{m}\in {N}^{+}$. This implies that each satellite in the DSCNs should be connected by at least two and at most $\left|{Q}_{m}\right|$ ISLs. |

$\mathrm{\Omega}$ | Capacity of a wavelength and $\mathrm{\Omega}=\delta {\mathrm{\Omega}}_{s}$, where ${\mathrm{\Omega}}_{s}$ is the capacity of a sub-wavelength and $\delta $ is the number of sub-wavelength-level aggregated flows that a wavelength can carry. |

${\mathrm{\Delta}}_{m}^{agg}$ | Number of ports used for aggregating the low-speed traffic requests in satellite m. |

${\mathrm{\Delta}}_{m}^{oe}$ | Number of ports used for optical/electric conversion in satellite m. |

${\nabla}_{m}^{o}$ | Total number of optical ports in satellite m. |

${\nabla}_{m}^{e}$ | Total number of electric ports in satellite m. |

${I}_{mn}$ | Number of ISLs from m to n; If $m\in {Q}_{n}$ or $n\in {Q}_{m}$, ${I}_{mn}=1$, otherwise, ${I}_{mn}=0$. |

${L}_{ij}$ | Number of virtual lightpaths from i to j; ${L}_{ij}={L}_{ij}^{w}$, where ${L}_{ij}^{w}$ is the number of lightpaths from i to j on wavelength w. |

${\Gamma}_{ij}^{sd}$ | Number of connection requests between node pair (s, d) which passes through virtual lightpath (i, j). |

${\mathrm{\Theta}}_{mn}^{ij}$ | Number of virtual lightpaths from i to j that traverse ISL (m, n). |

${\mathrm{\Psi}}_{ij}^{f}$ | Real capacity of lightpath (i, j) which is occupied by the f-th connection request and ${\mathrm{\Psi}}_{ij}^{f}\le {\mathrm{\Omega}}_{s}$, $\forall 1\le f\le {F}_{s,d},i,j\in V$. |

#### 2.1.3. Decision Variables

${\mu}_{ij}^{w}$ | Binary variable related to wavelength assignment, which equals 1 if there is a lightpath from i to j on wavelength w; otherwise, 0; ${\mathrm{\Theta}}_{mn}^{ij}={\displaystyle {\sum}_{w}{\mu}_{ij}^{w}}$, $\forall m,n$. |

${\xi}_{f,p}$ | Binary variable equals to 1 if the f-th connection request is aggregated into the p-th sub-wavelength; otherwise, 0. |

${\vartheta}_{p,w}$ | Binary variable equals to 1 if the p-th sub-wavelength is groomed onto wavelength w; otherwise, 0; $\delta ={\displaystyle {\sum}_{p}{\vartheta}_{p,w}}$, $\forall w$. |

${Z}_{ij}^{f,p,w}$ | The value is 1 when the f-th connection request is aggregated into the p-th sub-wavelength which traverses lightpath (i, j) on wavelength w; otherwise, 0. |

${R}_{mn}^{ij,w}$ | The value is 1 when the lightpath (i, j) traverses ISL (m, n) on wavelength w; otherwise, 0. |

#### 2.2. Optimization Models

#### 2.2.1. Minimize the Energy Consumption

_{1}~C

_{3}, where C

_{1}means that the occupied ports cannot exceed the maximum number of ports for traffic aggregation, C

_{2}represents that the optical/electric and electric/optical conversions of traffic requests must be completed within the limited ${\mathrm{\Delta}}_{k}^{oe}$. Apart from ports used for traffic transformation, other ports are responsible for lightpath bypass, which is depicted as C

_{3}.

_{4}~C

_{7}. Wavelength constraint C

_{4}is utilized to guarantee that the number of wavelengths used in every ISL cannot exceed W. It is assumed that lightpaths in an ISL comprise both the forward and the reverse ones. C

_{5}ensures that each traffic flow can traverse at most one virtual lightpath on any wavelengths, which implies that the traffic flows cannot be split in the process of forwarding. The flow conservation constraint is described as C

_{6}, where connection requests originating from the same nodes terminate at different destinations via several virtual lightpaths. Capacity constraint is provisioned in C

_{7}to assure that massive traffic flows can be aggregated as long as the maximum capacity of sub-wavelength is not exceeded.

#### 2.2.2. Minimize the Number of Wavelengths

_{1}~C

_{5}, C

_{7}.

## 3. Traffic Grooming Algorithm Design

#### 3.1. Traffic Aggregation and Sub-Wavelength Assignment Algorithm

_{1}, l

_{2}, and l

_{3}represent the shortest, the second shortest, and the third shortest path, respectively. The requests can be divided into M nonoverlapped groups, i.e., $\mathcal{U}=\{{\mathcal{U}}_{1},{\mathcal{U}}_{2},\cdots ,{\mathcal{U}}_{M}\}$ where ${\mathcal{U}}_{m}=\{{\mathcal{S}}_{m,1},{\mathcal{S}}_{m,2},\cdots ,{\mathcal{S}}_{m,{N}_{m}}\}$ and the corresponding paths satisfy ${L}_{{\mathcal{S}}_{m,1}}\subseteq {L}_{{\mathcal{S}}_{m,2}}\subseteq \cdots {L}_{{\mathcal{S}}_{m,{N}_{m}}}$. The requests within a group will be aggregated into a sub-wavelength. For any two groups ${\mathcal{U}}_{i}$ and ${\mathcal{U}}_{j}$, where ${\mathcal{U}}_{i}\cap {\mathcal{U}}_{j}=\varnothing $, the preference ${\mathcal{U}}_{i}\prec {}_{{\mathcal{S}}_{n}}\mathcal{U}{}_{j}$ indicates that the request ${\mathcal{S}}_{n}$ is willing to be a part of group ${\mathcal{U}}_{j}$, rather than group ${\mathcal{U}}_{i}$.

_{1}~C

_{7}. Therefore, the preference of the request improves when the matching utility increases. Requests can swap among different groups if less energy consumption can be actualized. The swap operation of any request is decided by the strict preference. By compare-and-swap operations, the preference of all requests will reach and keep a final equilibrium state where the minimum energy consumption is achieved. For any request ${\mathcal{S}}_{n}$ in group ${\mathcal{U}}_{m}$, the preference is defined as below.

Algorithm 1: TAASA algorithm |

Input: Sets of connection requests $\mathcal{S}$ and its corresponding paths ${L}_{\mathcal{S}}$, sub-wavelengths (groups) $\mathcal{U}$, and the matching utility function $\mathrm{\Xi}(\mathcal{S},\mathcal{U})$. |

Output: Matching pairs between connection requests and sub-wavelength. |

1. Initialization |

2. Allocate all connection requests initially to groups according to the path affiliation. |

3. Proposing and rejecting |

4. while at least one connection request is unmatched do |

5. Sub-wavelength ${\mathcal{U}}_{m}$ proposes to connection request ${\mathcal{S}}_{n}$ according to (12). |

6. if connection request ${\mathcal{S}}_{n}$ is proposed by more than one sub-wavelength then. |

7. Connection request ${\mathcal{S}}_{n}$ selects the sub-wavelength ${\mathcal{U}}_{m}$ with the minimum sum energy consumption $\mathrm{\Xi}(\mathcal{S},\mathcal{U})$ from the candidates and rejects other proposals. |

8. else |

9. Connection request ${\mathcal{S}}_{n}$ is matched with the proposing sub-wavelength. |

10. Connection request ${\mathcal{S}}_{n}$ is removed from $\mathcal{S}$. |

11. end if |

12. end while |

13. Compare-and-swap operations |

14. repeat |

15. for connection request $n=1\to N$, where ${\mathcal{S}}_{n}\in \mathcal{S}$ |

16. for group ${\mathcal{U}}_{m}={\mathcal{U}}_{1}\to {\mathcal{U}}_{M}$,where ${\mathcal{U}}_{k}\cap {\mathcal{U}}_{l}=\varnothing ,1\le k,l\le M$ |

17. Calculate the sum energy consumption for group ${\mathcal{U}}_{k}$ and ${\mathcal{U}}_{l}$. |

18. Connection request ${\mathcal{S}}_{n}$ moves from group ${\mathcal{U}}_{k}$ to group ${\mathcal{U}}_{l}$. |

19. Calculate the sum energy consumption for the updated group ${\mathcal{U}}_{k}$ and ${\mathcal{U}}_{l}$, respectively. |

20. Compare the change of the sum energy consumption. |

21. if the sum energy consumption decreases by the swap operation |

22. Connection request n stays in group ${\mathcal{U}}_{l}$. |

23. else connection request n moves back to ${\mathcal{U}}_{k}$. |

24. end if |

25. end for |

26. end for |

27. until no connection request is willing to be a part of other groups. |

28. return the sets of matching pairs. |

#### 3.2. Sub-Wavelength Grooming Algorithm

Algorithm 2: SG algorithm |

Input: Sets of connection requests $\mathcal{S}$ and its corresponding paths ${L}_{\mathcal{S}}$, sub-wavelengths (groups) $\mathcal{U}$, and wavelength $\mathcal{W}$. |

Output: A stable matching ${\Re}^{*}$. |

1. Initialization |

2. Perform Algorithm 1 to obtain the optimal sub-wavelength unit ${\mathcal{U}}^{*}$. |

3. Reconstruct the path affiliation ${L}_{{\mathcal{U}}_{1}^{*}}\subseteq {L}_{{\mathcal{U}}_{2}^{*}}\subseteq \cdots \subseteq {L}_{{\mathcal{U}}_{k}^{*}}$. |

4. Calculate $\mathrm{\Xi}(\mathcal{S},\mathcal{U})$ and construct the preference matrix ${{\rm Z}}_{n,j}$. |

5. Denote an initial sub-wavelength unit and wavelength as ${\mathcal{U}}_{0}^{*}$ and ${\mathcal{W}}_{0}$, respectively. |

6. Swap operation |

7. Repeat |

8. Select a matching pair $({\mathcal{W}}_{\phi},({\mathcal{S}}_{n},{\mathcal{U}}^{*}))\notin \Re $ satisfying the path affiliation. |

9. if $|\Re ({\mathcal{W}}_{\phi})|\le \mathcal{I}$, and ${\Re}_{({\mathcal{W}}_{\phi},({\mathcal{S}}_{n},{\mathcal{U}}^{*}))}^{({\mathcal{W}}_{0},({\mathcal{S}}_{n},{\mathcal{U}}_{0}^{*}))}$ is feasible, then |

10. Execute the swap matching and set $\Re ={\Re}_{({\mathcal{W}}_{\phi},({\mathcal{S}}_{n},{\mathcal{U}}^{*}))}^{({\mathcal{W}}_{0},({\mathcal{S}}_{n},{\mathcal{U}}_{0}^{*}))}$. |

11. else keep looking for a feasible matching. |

12. Select two pairs $({\mathcal{W}}_{0},({\mathcal{S}}_{n},{\mathcal{U}}_{0}^{*}))\in \Re $ and $({\mathcal{W}}_{1},({\mathcal{S}}_{n},{\mathcal{U}}_{1}^{*}))\in \Re $ |

13. if $|{\varsigma}_{({\mathcal{W}}_{1},({\mathcal{S}}_{n},{\mathcal{U}}_{0}))}^{({\mathcal{S}}_{n},{\mathcal{U}}_{i})}+{\varsigma}_{({\mathcal{W}}_{0},({\mathcal{S}}_{n},{\mathcal{U}}_{1}))}^{({\mathcal{S}}_{n},{\mathcal{U}}_{i})}|>|{\varsigma}_{({\mathcal{W}}_{1},({\mathcal{S}}_{n},{\mathcal{U}}_{1}))}^{({\mathcal{S}}_{n},{\mathcal{U}}_{i})}+{\varsigma}_{({\mathcal{W}}_{0},({\mathcal{S}}_{n},{\mathcal{U}}_{0}))}^{({\mathcal{S}}_{n},{\mathcal{U}}_{i})}|$, then |

14. Execute the swap matching and update $\Re ={\Re}_{({\mathcal{W}}_{1},({\mathcal{S}}_{n},{\mathcal{U}}_{0}^{*}))}^{({\mathcal{W}}_{0},({\mathcal{S}}_{n},{\mathcal{U}}_{1}^{*}))}$. |

15. else implement the swap matching and update $\Re ={\Re}_{({\mathcal{W}}_{1},({\mathcal{S}}_{n},{\mathcal{U}}_{1}^{*}))}^{({\mathcal{W}}_{0},({\mathcal{S}}_{n},{\mathcal{U}}_{0}^{*}))}$ |

16. end if |

17. end if |

18. Update $\mathrm{\Xi}(\mathcal{S},\mathcal{U})$, ${{\rm Z}}_{n,j}$, and $\mathcal{W}$ |

19. Until the total number of wavelengths remains unchanged by the swap operation. |

20. Return the stable matching ${\Re}^{*}$. |

#### 3.3. Property Analysis

## 4. Simulation and Performance Evaluation

#### 4.1. Analysis of the AWUR

#### 4.2. Analysis of the ECS

#### 4.3. Analysis of the Number of Wavelengths and Hops

#### 4.4. Analysis of Blocking Probability

#### 4.5. Analysis of Convergence Property

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A group of U satellites is interconnected by sparse WDM-enabled wireless laser links, providing services for users within its coverage, where three data flows traverse from the sources to destinations.

**Figure 3.**Comparisons of the AWUR between the DLG, TPTG, DLG_GA, and TPTG_MA under (

**a**) 6-node topology, (

**b**) 12-node topology, and (

**c**) 22-node topology.

**Figure 4.**Comparisons of the ECS between the DLG, TPTG, DLG_GA, and TPTG_MA under (

**a**) 6-node topology, (

**b**) 12-node topology, and (

**c**) 22-node topology.

**Figure 5.**Comparison of the number of wavelengths per node between the DLG, TPTG, DLG_GA, and TPTG_MA.

Parameters | Value |
---|---|

Bandwidth range of a traffic request | 20 Mbps–300 Mbps |

Capacity of a sub-wavelength | 2 Gbps |

Capacity of a wavelength | 10 Gbps |

Ports for traffic aggregation | 40 |

Ports for optical/electric conversion | 20 |

Total optical/electric ports | 20/60 |

Energy consumption per port ${E}_{oe}/{E}_{eo}/{E}_{agg}/{E}_{edfa}/{E}_{TX}$ | 15/15/5/10/20 W |

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**MDPI and ACS Style**

Peng, C.; He, Y.; Yan, D.; Fu, H.; Zhao, S. Provision of Energy- and Wavelength-Efficient Traffic Grooming for Sparse WDM-Enabled Distributed Satellite Cluster Networks. *Photonics* **2022**, *9*, 494.
https://doi.org/10.3390/photonics9070494

**AMA Style**

Peng C, He Y, Yan D, Fu H, Zhao S. Provision of Energy- and Wavelength-Efficient Traffic Grooming for Sparse WDM-Enabled Distributed Satellite Cluster Networks. *Photonics*. 2022; 9(7):494.
https://doi.org/10.3390/photonics9070494

**Chicago/Turabian Style**

Peng, Cong, Yuanzhi He, Di Yan, Huajun Fu, and Shanghong Zhao. 2022. "Provision of Energy- and Wavelength-Efficient Traffic Grooming for Sparse WDM-Enabled Distributed Satellite Cluster Networks" *Photonics* 9, no. 7: 494.
https://doi.org/10.3390/photonics9070494