Abstract
This paper studies the joint relay selection and spectrum allocation problem for multi-user and multi-relay cellular networks, and per-user fairness and system efficiency are both emphasized. First, we propose a new data-frame structure for relaying resource allocation. Considering each relay can support multiple users, a \(K\)-person Nash bargaining game is formulated to distribute the relaying resource among the users in a fair and efficient manner. To solve the Nash bargaining solution (NBS) of the game, an iterative algorithm is developed based on the dual decomposition method. Then, in view of the selection cooperation (SC) rule could help users achieve cooperation diversity with minimum network overhead, the SC rule is applied for the user-relay association which restricts relaying for a user to only one relay. By using the Langrangian relaxation and the Karush–Kuhn–Tucker condition, we prove that the NBS result of the proposed game just complies with the SC rule. Finally, to guarantee the minimum rate requirements of the users, an admission control scheme is proposed and is integrated with the proposed game. By comparing with other resource allocation schemes, the theoretical analysis and the simulation results testify the effectiveness of the proposed game scheme for efficient and fair relaying resource allocation.
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Acknowledgments
The work of G. Zhang and P. Liu was supported by the China Fundamental Research Funds for the Central Universities (2014QNA82), the Natural Science Foundation of Jiangsu Province of China (BK2012141), and the Post Doctoral Fellowship Program of the China Scholarship Council. The work of Kun Yang and Dongdai Zhou was supported by the UKEPSRC Project DANCER (EP/K002643/1), the EU FP7 Project MONICA (GA-2011-295222) and CLIMBER (GA-2012-318939).
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Appendices
Appendix 1
Proof of Proposition 1
Based on the fixed transmission power and full CSI assumptions made in Sect. 2, the values of \(\gamma _{k,\mathrm{bs}} \) in (3), \(\gamma _{k,m} \) and \(\gamma _{m,\mathrm{bs}} \) in (11), and, hence, the values of \(\Gamma _{k,m,\mathrm{bs}}^{\mathrm{HD+AF+MRC}} \) in (13), \(R_{k,m,\mathrm{bs}}^{\mathrm{HD+AF+MRC}} \) in (14), \(R_k^{\mathrm{DT(HD)}} \) in (20) and \(R_k^{\mathrm{AVG}} \) in (19) can be regarded as constants within a cycle of optimization.
To prove set \(\mathcal{R}\) is convex, according to [25], we need to prove that for any \(\mathbf{R}^{\mathrm{AVG}(a)}=\left( {R_1^{\hbox {AVG(}a\hbox {)}} ,\ldots ,R_K^{\mathrm{AVG}(a)} } \right) \in \mathcal{R}\) and \(\mathbf{R}^{\mathrm{AVG}(b)}=\left( {R_1^{\mathrm{AVG}(b)} ,\ldots ,R_K^{\mathrm{AVG}(b)} } \right) \in \mathcal{R}\), and, any \(\theta \) with \(0\le \theta \le 1\), we have \(\theta \mathbf{R}^{\mathrm{AVG}(a)}+\left( {1-\theta } \right) \mathbf{R}^{\mathrm{AVG}(b)}\in \mathcal{R}\).
For \(\forall k\in \mathcal{K}\), after some simple derivation, we can get
Since \(0\le t_{m,k}^{(a)} \le N, 0\le t_{m,k}^{(b)} \le N\), and \(0\le \theta \le 1\), it is easy to derive
Now, comparing (61) with (19), we know that, for \(\forall k\in \mathcal{K}, \theta R_k^{\mathrm{AVG}(a)} +(1-\theta )R_k^{\mathrm{AVG}(b)} \) has the same value space as \(R_k^{\mathrm{AVG}} \), and, hence, \(\theta \mathbf{R}^{\mathrm{AVG}(a)}+\left( {1-\theta } \right) \mathbf{R}^{\mathrm{AVG}(b)}\in \mathcal{R}\). Therefore, we can conclude that \(\mathcal{R}\) is a convex set, and, this completes the proof.
Appendix 2
Proof of Proposition 2
For the strict concave function \(U\left( {\mathbf{R}^{\mathrm{AVG}}} \right) =\ln \left( {1+\mathbf{R}^{\mathrm{AVG}}} \right) , {\hat{\mathbf{R}}}^{\mathrm{AVG}}\) is optimal if and only if
where “Tr” is the matrix transpose operator, and \(\nabla U\left( {\mathbf{R}^{\mathrm{AVG}}} \right) =\left[ {{U}'_1 (t_{m,1} ),\ldots ,{U}'_K (t_{m,K} )} \right] ^{\mathrm{Tr}}\). As logarithmic function \(U_k =\ln \left( {1+R_k^{\mathrm{AVG}} } \right) , \forall k\in \mathcal{K}\), is used, so we have
Considering \(R_k^{\mathrm{AVG}} >0\), for \(\forall k\in \mathcal{K}\), hence, (63) is identical to (24). It indicates that the PF condition is guaranteed, and this completes the proof.
Appendix 3
Proof of Proposition 3
Since the problem (23) is convex and the inequality constraint functions, i.e., the first and the third constraints in problem (23) are affine, according to [27], we only need to prove that there exists a point \(\mathbf{{S}'}=[{t}'_{m,k} ]_{M\times K} \in \mathbf{relint}\mathcal{S}\), where \(\mathcal{S}\) denotes the feasible set for the resource allocation matrix and \(\mathbf{relint}\mathcal{S}\) denotes the relative interior of set \(\mathcal{S}\), such that
For the second constraint in problem (23), we consider \({t}'_{m,k} =1/N\) for \(\forall m\in \mathcal{M}\). Then, the second and the third trivial conditions, i.e., \(-{t}'_{m,k} <0\) and \(\sum _{k=1}^K {{t}'_{m,k} } -N=0\) in (23) are satisfied. And, since the number of users, \(K\), will be much larger than the number of relays, \(M\) in practical relay based cellular networks, we then have
It means that the last trivial condition in (23) is also satisfied. So, we can conclude that the Slater’s condition, and, hence, the strong duality is satisfied by the primal problem (23). This completes the proof.
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Zhang, G., Yang, K., Liu, P. et al. Fair and Efficient Spectrum Resource Allocation and Admission Control for Multi-user and Multi-relay Cellular Networks. Wireless Pers Commun 78, 347–373 (2014). https://doi.org/10.1007/s11277-014-1757-4
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DOI: https://doi.org/10.1007/s11277-014-1757-4