Abstract
In this paper we study the joint rate-and-power allocation for multi-channel spectrum sharing networks with balanced QoS provisioning and power saving. We formulate this cross-layer optimization problem as a non-cooperative game G JRPA in which each user has a coupled two-tuple strategy, i.e., simultaneous rate and multi-channel power allocations. A multi-objective cost function is proposed to represent user’s awareness of both QoS provisioning and power saving. We analyze the properties of Nash equilibrium (N.E.) for our G JRPA , including its existence, and properties of QoS provisioning and power saving. Furthermore, we derive a layered structure by applying the Lagrangian dual decomposition to G JRPA and design a distributed algorithm to find the N.E.. Simulation results are presented to show the validity of our game theoretic formulation and the performance of our proposed algorithm. Detailed studies on the performance tradeoff between QoS provisioning and power saving are also carried out.
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Notes
Zhao et al. [1] classified the DSA mechanisms into three categories, i.e., exclusive usage mechanism, hierarchical access mechanism, and spectrum sharing mechanism.
Practically, \(R_{i}^{min}\) can be set as user i’s minimum data rate and \(R_{i}^{max}\) can be determined by user i’s maximum achievable data rate without considering the interference from other users. We assume in this paper that \(R_{i}^{min}<R_{i}^{tar}<R_{i}^{max},\forall i\in\mathcal {N},\) holds. \(pl_{i}^{k}\) and \(pu_{i}^{k}\) can be considered as user i’s spectrum mask bounds on channel k, which are specific to cognitive radio system [12].
We consider a relatively static network in this paper, i.e., the set of channel power gains \(g_{ji}^{k},\forall i,j\in \mathcal {N}, \forall k\in\mathcal {K}\) keeps unchanged during the time interval of interest.
In this paper, QoS provisioning is referred to achieving the target rate as exactly as possible.
In [11] the capacity region \(\mathcal {R}\) is defined as \(\mathcal {R}=\{(r_{1},...,r_{N})|r_{i}\leq \sum_{k\in\mathcal {K}}\mathit{\Phi}_{i}^{k}(p_{i}^{k},\mathbf{p}_{-i}^{k}),\forall i\in\mathcal {N};\mathbf{p}_{i}\in \mathit{\Omega}_{i}^{P},\forall i\in\mathcal {N}\}\).
\([x]_{a}^{b}\) denotes the projection of a real number x into the interval [a,b]. Similarly, [x] + denotes the projection of x into [0, ∞ ).
\(\|(A)\|_{2}\triangleq \max\{\sqrt{\lambda}:\lambda\) is an eigenvalue of A ∗ A} according to [19], where A ∗ denotes the conjugate transpose of matrix A.
According to [16], the mapping T(x) from a subset \(\mathcal {X}\) of ℝn to itself is contractive if it has the property that \(\|T(x)-T(y)\|\leq \varrho\|x-y\|,\forall x,y\in \mathcal {X}\) and the decay modulus \(\varrho\) belongs to [0,1). Besides, the contractive mapping x(t + 1) = T(x(t)) has a geometric convergence as \(\|x(t)-x^{*}\|\leq \varrho^{t}\|x(0)-x^{*}\|,\forall t\geq0\) assuming \(x^{*}\in\mathcal {X}\) is a fixed point.
Specifically, \(g_{ij}^{k}=d_{ij}^{-\tau}|\xi_{ij}^{k}|^{2}\), where d ij is the distance between TX i and RX j and we set τ = 2. \(\xi_{ij}^{k}\) is a complex Gaussian random variable modeling the frequency selective fading across the channels with distribution \(\mathcal {C}\mathcal {N}(0,1)\).
Relative error of power allocation is defined as \(\log_{10}\left(\frac{\|\mathbf{p}_{i}-\mathbf{p}_{i}^{\ast}\|_{2}}{\|\mathbf{p}_{i}^{\ast}\|_{2}}\right)\) and relative error of dual price is defined as \(\log_{10}\left(\frac{|z_{i}-z_{i}^{\ast}|}{z_{i}^{\ast}}\right)\). Both values are averaged over all the users.
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Wu, Y., Tsang, D.H.K. Joint Rate-and-Power Allocation for Multi-channel Spectrum Sharing Networks with Balanced QoS Provisioning and Power Saving. Mobile Netw Appl 14, 198–209 (2009). https://doi.org/10.1007/s11036-008-0136-3
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DOI: https://doi.org/10.1007/s11036-008-0136-3