# Fair Dynamic Spectrum Allocation Using Modified Game Theory for Resource-Constrained Cognitive Wireless Sensor Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Maximum Spectral Efficiency and Fair Spectrum Allocation with Priority

#### 1.2. Minimum Spectrum Handover

#### 1.3. Spectrum Allocation by A Central Authority

^{2}M), where N and M are the numbers of sensors and unused spectrum resources, respectively. In numerical experiments, we show that the proposed algorithm yields fair spectrum allocation reflecting the priority of each sensor and avoiding unnecessary spectrum handover.

## 2. Related Work

## 3. System Model

- For simplicity, the spectrum resources are defined as multiple spectrum units.
- More than two sensors can share a spectrum unit if they do not interfere with one another.
- The spectrum is allocated periodically. That is, the TVWS base station detects unused spectrum units and allocates them to sensors that request transmissions at the start of every predefined epoch.
- Spectrum handover may occur at a receiver as well as at a transmitter. Here, we focus on minimizing spectrum handover only at the transmitter side.

- V: set of transmitting sensors.
- S: set of unused spectrum units.
- L
_{is}(binary): indicates that spectrum unit s has been synchronized with sensor i at its previous transmission epoch. - w
_{i}: the weight that reflects the priority of sensor i. A sensor with higher priority will be allocated more spectrum units. The priority is translated as the demand of each sensor for spectrum resources. - R
_{ij}(binary): indicates that the transmission of sensor i can interfere with sensor j (see Figure 2). Therefore, sensors i and j cannot use the same spectrum unit. We let R_{ij}= 0 if i = j or sensor i and j are transmitting to the same destination. - x
_{is}(binary): indicates that sensor i is synchronized with spectrum unit s during the current transmission epoch.

_{ij}= 1; i.e., if the target of sensor i is within the transmission range of sensor j, they cannot share the same spectrum units.

## 4. Modified Game Theory Approach

#### 4.1. Basics for Modified Game Theory

_{i}(

**X**), the Pareto optimal solution is obtained by solving the following single objective optimization problem:

**c**= [c

_{1}, c

_{2}, …, c

_{r}]

^{T}, and Equations (8) and (9) are the constraints of the original multi-objective problem. Then, the super-criterion, also known as the NBS [24], is constructed as follows:

**X**

_{i}* is the optimal design vector obtained when only f

_{i}is minimized. Then, each objective function f

_{i}(

**X**) is normalized as f

_{ni}(

**X**) by following the normalization procedure that gives 0 and 1 as the minimum and maximum values, respectively:

_{n}= [f

_{n}

_{1}f

_{n}

_{2}… f

_{nr}]

^{T}.

**c***) in a Pareto sense and the final optimal solution (

**X*** =

**X**

_{c}*). Hence, the problem given in Equation (6) and the maximization of S should be computed simultaneously with

**c**and

**X**being the decision variables, which is not feasible in reality.

_{n}, as

_{n}has a value between 0 and 1.

#### 4.2. MGT for the Problem of Spectrum Allocations in WSNs

_{w}is the value of Equation (18) with the optimal solution of the problem given by Equations (19) and (20). In a similar way, H

_{w}is the value of Equation (19) with the optimal solution of the problem defined by Equations (18) and (20). That is, U

_{w}is obtained by solving the optimization problem given by Equations (19) and (20), and H

_{w}is yielded by solving the optimization problem given by Equations (18) and (20). Then, U

_{w}and H

_{w}act as upper bounds of the two objective functions.

_{n}as

## 5. Cooperative Approach

_{w}, and H

_{w}as the upper bounds of the solution. The optimal <H*, U

_{w}> can be computed using the algorithm given in Algorithm 1 in reasonable time, whose complexity is O(N

^{2}M), where N and M are the numbers of transmitting sensors and unused spectrum units, respectively. As shown in the algorithm, the spectrums are synchronized according to L

_{is}(that is defined allocation vector in previous transmission epoch) initially (Step 1). If L

_{is}satisfies the constraint (that is, (x

_{is}+ x

_{js}) R

_{ij}> 1), then there will be no spectrum handover and H* will be optimal. Otherwise, they negotiate the ownership in order not to interfere each other and in the direction of improving U

_{w}(Step 2). That is, start with best solution and then let spectrum handover occur only when the constraint is not satisfied. The solution must be optimal. It is true that there may be other sets of x

_{is}satisfying (x

_{is}+ x

_{js}) R

_{ij}> 1, but there are no other sets that yield better H* than the algorithm does.

_{w}serves as the upper bound of U in the algorithm, it is essential to improve U

_{w}to obtain a better U without compromising H*.

**X**) and <U*, H

_{w}> is intractable. Thus, we envisage a cooperative approach to determine the approximate minimum F(

**X**) and <U*, H

_{w}> in reasonable time.

Algorithm 1. The algorithm for determining optimal U_{W} and H^{*}. | |

Step 1. Initialization | for each i $\in $ V and s $\in $ S set x _{is} = L_{is} |

Step 2. Call Negotiation | for each i, j $\in $ V and s $\in $ S {if ((x_{is} + x_{js})R_{ij} > 1) call Negotiate(i, j, s) } |

Step 3. Define U_{w} and H* | Define U_{w} and H* using Equations (18) and (19). |

Subroutine Negotiate(i, j, s) {x _{is} = 1, x_{js} = 0Compute U using Equation (18) and store it in temp _{1}.x _{is} = 0, x_{js} = 1 Compute U using Equation (18) and store it in temp_{2}.if temp_{1} < temp_{2}x _{is} = 1, x_{js} = 0} |

Algorithm 2. The cooperative game-based algorithm without transmission power control. | |

Step 1. Initialization | For the minimum F(X),-ϕ( X) = F(X)-U* = $-1\times {\displaystyle \sum}_{i\in V}{w}_{i}\left|S\right|$, H*= $-1\times {\displaystyle \sum}_{i\in V}{\displaystyle \sum}_{s\in S}{L}_{is.}$ For finding <U*, H _{w}>, -ϕ( X) = $-1\text{}\times \text{}{\displaystyle \sum}_{i\in V}{w}_{i}\mathrm{ln}\left({\displaystyle \sum}_{s\in S}{x}_{is}\right)$.For all, -minϕ( X) = ∞ -Assign all idle spectrum units to sensors. |

Step 2. Winner selection on each spectrum | for each s $\in $ S {for each i $\in $ V {for each j (≠ i) $\in $ Vx _{js} = 0x _{is} = 1Compute ϕ( X).if (ϕ(X) < minϕ(X)) {minϕ( X) = ϕ(X)winner = i } x _{is} = 0.} Assign spectrum unit s to winner. } |

Step 3. U* and H_{w} | For finding <U*, H_{w}>, U* = ϕ(X) and H_{w} = $-1\times {\displaystyle \sum}_{i\in V}{\displaystyle \sum}_{s\in S}{x}_{is}{L}_{is.}$. |

^{2}M), where N and M are the numbers of transmitting sensors and unused spectrum units, respectively. However, using this algorithm, it is impossible to improve H

_{w}without compromising U*. Therefore, we get H

_{w}using the computation result of U*.

Algorithm 3. The cooperative game-based algorithm with transmission power control. | |

Step 1. Initialization | For the minimum F(X), -ϕ( X) = F(X)-U* =$\text{}-1\times {\displaystyle \sum}_{i\in V}{w}_{i}\left|S\right|$, H* = $-1\times {\displaystyle \sum}_{i\in V}{\displaystyle \sum}_{s\in S}{L}_{is.}$ For finding <U*, H _{w}>, -ϕ( X) = $-1\text{}\times \text{}{\displaystyle \sum}_{i\in V}{w}_{i}\mathrm{ln}\left({\displaystyle \sum}_{s\in S}{x}_{is}\right)$.For all, -Assign all idle spectrum units to sensors. -negotiated[i] = 0 for all i $\in $ V. |

Step 2. Forming coalitions | Copy X to Y.Sort i $\in $ V in accordance with w _{i} and arrange the index k = 1, …, |V| from the largest to smallest.for k = 1, …, |V| {if (!negotiated[k]) {minϕ( Y) = ∞for each j (≠ i) $\in $ V {if (!negotiated[j]) {Negotiate(k, j, Y) and compute ϕ(Y).if (ϕ(Y) < minϕ(Y)) {minϕ( Y) = ϕ(Y)coalition[k] = j } } } negotiated[k] = 1, negotiated[coalition[k]] = 1 } } |

Step 3. Negotiation in each coalition | for each i $\in $ VNegotiate(i, coalition[i], X)if X satisfies Equation (20), go to Step 4; else go to Step 2. |

Step 4. U* and H_{w} | For finding <U*, H_{w}>, U* = ϕ(X) and H_{w} = $-1\times {\displaystyle \sum}_{i\in V}{\displaystyle \sum}_{s\in S}{x}_{is}{L}_{is.}$. |

Subroutine Negotiate(a, b, Z) {for each s $\in $ S { if ((z _{as} + z_{bs})R_{ij} > 1)z _{as} = 1, z_{bs} = 0Compute ϕ( Z) and store it in ϕ_{1}(Z).z _{as} = 0, z_{bs} = 1Compute ϕ( Z) and store it in ϕ_{2}(Z).if (ϕ _{1}(Z) < ϕ_{2}(Z))z _{as} = 1, z_{bs} = 0} } |

**X**), as shown in Step 1. Instead, we use their lower bounds to avoid computing the best U* and H* in every coalition and negotiation step. Nonetheless, the algorithm yields satisfactory results (i.e., <U, H>).

**X**) or maximum proportional fairness.

## 6. Numerical Experiments

_{w}> indicates the results obtained when we optimize only U using the algorithms in Algorithms 2 and 3. The column <-U

_{w}, -H*> indicates the results obtained when we optimize only H using the algorithm in Algorithm 1. The column <-U, -H> shows the results of MGT using the algorithms in Algorithms 2 and 3. The results of LP relaxation are shown in the last column of Table 2.

_{w}> and <-U

_{w}, -H*>, which implies the solutions of the multi-objective problem are between the solutions of the two single objective problems.

_{i}= 1 for all sensors.

## 7. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**An illustration of interference. If the target of sensor i lies within the transmission range of j, then i and j cannot share the same spectrum bands.

Experimental Parameter | Value |
---|---|

Topology size (n) | 20, 40 sensors |

Number of idle spectrum units | 176 (for N = 20), 271 (for N = 40) |

Transmission power control | With or without |

P_{hold} ^{1} | 0.1 |

Weight | Assign a random weight between 0.1 and 100 to each sensor |

q (in (24)) | 2 |

^{1}P

_{hold}: probability that each sensor holds a spectrum unit in its previous transmission phase.

Algorithm | <-U*, -H_{w}> | <-U_{w}, -H*> | <-U, -H> | LP Bound of -U* |
---|---|---|---|---|

Without transmission power control | <2517.99, 14> | <2432.99, 152> | <2581.39, 152> | 5178.38 |

With transmission power control | <3777.29, 44> | <3645.67, 235> | <3713.06, 232> | 5178.38 |

Algorithm | <-U*, -H_{w}> | <-U_{w}, -H*> | <-U, -H> | LP bound of -U* |
---|---|---|---|---|

Without transmission power control | <3784.31, 26> | <4133.67, 274> | <4080.36, 272> | 8982.25 |

With transmission power control | <7679.41, 152> | <6220.94, 684> | <7622.12, 678> | 8982.25 |

**Table 4.**Fairness index of each optimization problem: max-weighted-log-sum problem vs. max-weighted-sum problem.

Algorithm | Max-Weighted-Log-Sum Problem | Max-Weighted-Sum Problem |
---|---|---|

Without transmission power control | 0.8512 | 0.4301 |

With transmission power control | 0.8733 | 0.5613 |

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**MDPI and ACS Style**

Byun, S.-S.; Gil, J.-M.
Fair Dynamic Spectrum Allocation Using Modified Game Theory for Resource-Constrained Cognitive Wireless Sensor Networks. *Symmetry* **2017**, *9*, 73.
https://doi.org/10.3390/sym9050073

**AMA Style**

Byun S-S, Gil J-M.
Fair Dynamic Spectrum Allocation Using Modified Game Theory for Resource-Constrained Cognitive Wireless Sensor Networks. *Symmetry*. 2017; 9(5):73.
https://doi.org/10.3390/sym9050073

**Chicago/Turabian Style**

Byun, Sang-Seon, and Joon-Min Gil.
2017. "Fair Dynamic Spectrum Allocation Using Modified Game Theory for Resource-Constrained Cognitive Wireless Sensor Networks" *Symmetry* 9, no. 5: 73.
https://doi.org/10.3390/sym9050073