# A Novel Spectrum Scheduling Scheme with Ant Colony Optimization Algorithm

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## Abstract

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## 1. Introduction

## 2. System Model and Problem Formulation

#### 2.1. Matrices for Spectrum Allocation

**Matrices Definition**(See Table 1).

**Explanation for Matrices**

- Available matrix L. The matrix represents the availability of licensed bands for cognitive users. If ${l}_{n,m}=1$, user n can access spectrum m without interference to primary users, otherwise ${l}_{n,m}=0$. As shown in Figure 1, spectrum channel B is available for $SU1$, then ${l}_{1,2}=1$.
- Benefit matrix B. The matrix indicates the benefit that a cognitive user gets by successful access to a licensed spectrum band, where ${b}_{n,m}>0$ only if ${l}_{n,m}=1$.
- Interference matrix C. The three-axis matrix describes the interference relationship of any two vertices n and k when they access spectrum m. As shown in Figure 1, $SU3$ and $SU4$ overlap in some area, then ${c}_{3,4,1}=1$, ${c}_{3,4,3}=1$, ${c}_{3,4,4}=1$.
- Allocation matrix A. The matrix is a spectrum allocation result which is interference free. If ${a}_{n,m}=1$, cognitive user n can access spectrum m and transmission data in this band. A conflict free allocation needs to satisfy the interference constraints: ${a}_{n,m}+{a}_{k,m}\le 1,\mathrm{if}\phantom{\rule{4pt}{0ex}}{c}_{n,k,m}=1,\forall n,k<N,m<M$.
- Degree matrix for cognitive users Z. The matrix represents the available spectrum number for each cognitive users. In Figure 1, ${z}_{1}=2,{z}_{2}=4,{z}_{3}=4,{z}_{4}=3$.
- Degree ascending matrix K. The matrix is another representation of the available matrix, which incrementally orders the rows according to the degree matrix Z.

#### 2.2. Problem Formulation and Measure Functions

- (1)
- Max-Sum-Reward-Mean (MSRM): This function is used to measure the average of total spectrum utilization in the system, which is the average of the sum user rewards.$${U}_{mean}=\frac{1}{n}\sum _{n=1}^{N}\sum _{m=1}^{M}{a}_{n,m}\times {b}_{n,m}$$
- (2)
- Max-Proportional-Fair (MPF): The function is to measure the fairness among cognitive users accessing the spectrum in the system, which is driven by ${\sum}_{m=1}^{M}{a}_{n,m}\times {b}_{n,m}$.$${U}_{fair}={(\prod _{n=1}^{N}\sum _{m=1}^{M}{a}_{n,m}\times {b}_{n,m}+{10}^{-4})}^{\frac{1}{N}}$$
- (3)
- Max-Min-Reward (MMR): The function is to maximize the spectrum utilization at the bottleneck cognitive users who receive the lowest reward, which is a simple notion of fairness.$${U}_{min}=\underset{1\le n\le N}{\mathrm{min}}(\sum _{m=1}^{M}{a}_{n,m}\times {b}_{n,m})$$

## 3. The IACO-Based Spectrum Allocation Method

#### 3.1. The Basic Idea

#### 3.2. Transform for the Spectrum Allocation Problem

#### 3.3. Differential Evolution Process in IACO

#### 3.4. Variable Neighborhood Search Process in IACO

#### 3.5. The Process and Description of IACO

**Step 1:**Initialization. Generate the initial vector ${P}_{initial}$ and initialize pheromone of each path with ${\tau}_{0}$ (the edge between vertices is connected) or ${\tau}_{1}$ (the edge between vertices is available). Set the maximum evolution time ${E}_{max}$, the maximum evolution time with low convergence speed ${E}_{convergence}$, the number of population $Num$.

**Step 2:**Interference removal. The spectrum allocation scheme A must be interference free, and therefore A needs to satisfy the interference constraints from matrix C. Based on this, we would remove the interference-path to correct the solution. While A does not satisfy constraints defined by C, it is necessary to equiprobably set one value to 0.

**Step 3:**Fitness evaluation. Calculating fitness value is a way to convert binary sequence solutions into real space R, which can be expressed as: $f\to {R}^{+}$. Get the best path sequence ${P}_{best}$ and the highest fitness value ${f}_{max}$.

**Step 4:**Monitor convergence rate. The monitoring mechanism is designed in IACO to detect the rate-of-change in fitness. If the growth rate stays slow and meets Equation (8) during $\Delta t$, then turn to the 7th step to accelerate convergence speed of IACO by employing DE, otherwise, go to the 5th step for ACO traversing.

**Step 5:**Ants traversal. All ants move back and forth between the starting point and the ending point. As described below in Figure 3, spectrum allocation can be seen as row traversal, where row and column represent cognitive users and bands, respectively. In this row traversal, if the available channel number ${z}_{i}>0$ and ${k}_{i,j}=1$, then band j is available for user i, that is to say node i in column j is visitable for ants, otherwise the ants skip the node. Then the evolution time $E=E+1$.

- (1)
- Path selection. The transfer probability between node i and s in the choice process is presented as follows:$$s=\left\{\begin{array}{cc}arg{\mathrm{max}}_{j\in allowe{d}_{l}}\left\{{\tau}_{i,j}^{\alpha}{\mu}_{i,j}^{\beta}\right\},\hfill & q\le {q}_{0}\hfill \\ J,\hfill & q>{q}_{0}\hfill \end{array}\right.$$$${p}_{i,s}=\left\{\begin{array}{cc}\frac{{\tau}_{i,s}^{\alpha}{\mu}_{i,s}^{\beta}}{{\sum}_{{\mathrm{max}}_{j\in allowe{d}_{l}}}{\tau}_{i,j}^{\alpha}{\mu}_{i,j}^{\beta}},\hfill & s\in allowe{d}_{l}\hfill \\ 0,\hfill & s\notin allowe{d}_{l}\hfill \end{array}\right.$$
- (2)
- Update pheromone. Using an elitist strategy to update pheromone. The pheromone concentration on path$(i,j)$ is updated as the following rules:$${\tau}_{i,j}=(1-\rho )\times {\tau}_{i,j}+\Delta {\tau}_{i,j}$$$$\Delta {\tau}_{i,j}=\sum _{k=1}^{m}\Delta {\tau}_{i,j}^{k}$$$$\Delta {\tau}_{i,j}^{k}=\frac{Q}{{f}_{min}}$$

**Step 6:**Termination judgment. If reach the maximum iteration times ${E}_{max}$, the solving process is over. Then map the best solution ${P}_{best}$ to allocation matrix A. If $E<{E}_{max}$ and the convergence speed is slow and and satisfy ${E}_{convergence}$, then go to the 7th step, otherwise, go to the 8th.

**Step 7:**Accelerate the convergence speed. Adopting DE to accelerate the global convergence speed by optimizing the searching mode of the ants and guaranteeing the diversity of the population. Initialize DE with the subpopulation of ACO, ${E}_{convergence}={E}_{convergence}+1$.

**Step 8:**Local search. Employing VNS to improve the local searchability of the improved algorithm by searching around the initial solution.

#### 3.6. Pseudocode of IACO

Algorithm 1: An Improved Ant Colony Optimization Algorithm |

## 4. Simulation Results and Discussion

**Experiment 1.**

**Experiment 2.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Name of Matrix | Definition of Matrix |
---|---|

Available matrix | $L=\left\{{l}_{n,m}\right|{l}_{n,m}\in \{0,1\}{\}}_{N\times M},\phantom{\rule{1.em}{0ex}}1\le n\le N,1\le m\le M$ |

Benefit matrix | $B={\left\{{b}_{n,m}|{b}_{n,m}\ge 0\right\}}_{N\times M},\phantom{\rule{1.em}{0ex}}1\le n\le N,1\le m\le M$ |

Interference matrix | $C={\left\{{c}_{n,n,m}|{c}_{n,n,m}\in \{0,1\}\right\}}_{N\times N\times M},\phantom{\rule{1.em}{0ex}}1\le n\le N,1\le m\le M$ |

Allocation matrix | $A=\left\{{a}_{n,m}\right|{a}_{n,m}\in \{0,1\}{\}}_{N\times M},\phantom{\rule{1.em}{0ex}}1\le n\le N,1\le m\le M$ |

Degree matrix for cognitive users | $Z=\left\{{z}_{n}\right|{z}_{n}=\{0,1,\dots ,M\}{\}}_{N},\phantom{\rule{1.em}{0ex}}1\le n\le N$ |

Degree ascending matrix | $K=\left\{{k}_{n,m}\right|{k}_{n,m}\in \{0,1\}{\}}_{N\times M},\phantom{\rule{1.em}{0ex}}1\le n\le N,1\le m\le M$ |

Iteration | Algorithm | Relative Difference (%) | ||
---|---|---|---|---|

MSRM | MMR | MPF | ||

30 | IACO | 0.366 | 0.447 | 1.711 |

ACO | 1.144 | 1.676 | 3.017 | |

PSO | 0.324 | 1.275 | 2.083 | |

GA | 1.033 | 2.876 | 3.496 | |

100 | IACO | 0 | 0 | 0.013 |

ACO | 0 | 1.514 | 2.504 | |

PSO | 0 | 1.309 | 0.952 | |

GA | 0.472 | 2.666 | 3.224 | |

200 | IACO | 0 | 0 | 0.012 |

ACO | 0 | 1.177 | 2.299 | |

PSO | 0 | 0.616 | 0.564 | |

GA | 0.063 | 2.282 | 2.71 |

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**MDPI and ACS Style**

Liu, L.; Wang, N.; Chen, Z.; Guo, L.
A Novel Spectrum Scheduling Scheme with Ant Colony Optimization Algorithm. *Algorithms* **2018**, *11*, 16.
https://doi.org/10.3390/a11020016

**AMA Style**

Liu L, Wang N, Chen Z, Guo L.
A Novel Spectrum Scheduling Scheme with Ant Colony Optimization Algorithm. *Algorithms*. 2018; 11(2):16.
https://doi.org/10.3390/a11020016

**Chicago/Turabian Style**

Liu, Liping, Ning Wang, Zhigang Chen, and Lin Guo.
2018. "A Novel Spectrum Scheduling Scheme with Ant Colony Optimization Algorithm" *Algorithms* 11, no. 2: 16.
https://doi.org/10.3390/a11020016