A Robust Stackelberg Game Based Uplink Power Control for Device-to-Device Communication with Channel Uncertainty and Outage Probability Constraints

Abstract

This paper studies the problem of robust uplink power control in OFDMA-based Device-to-device (D2D) cellular networks composed of cellular and D2D user equipments (UEs). Both cellular UEs and D2D UEs compete with each other to maximize their own utility considering imperfect channel state information (CSI). We formulate a robust Stackelberg game (RSG) based SINR maximization to model this hierarchical competition, where the cellular UEs and D2D UEs are considered to be leaders and followers, respectively. Then, a comprehensive investigation of the RSG is developed with considering various power constraints, various interference constraints and outage probability constraints under a channel uncertainty model. Moreover, we apply the Lagrange dual decomposition method to solve this problem, and an efficient iterative algorithm is proposed to achieve the robust Stackelberg equilibrium (RSE). Global efficiency of the RSE and its complexity are also analysed. Simulation results show that the proposed RSG solution with imperfect CSI can achieve better performance in terms of sum rate for cellular UEs and D2D pairs, sum transmission power, and outage probability by comparing with other existing Stackelberg game solutions.

Appendix: Proof of Theorem 1

For a market fixed price at the C-UEs, all the D2D pairs aim to maximize their own utility by buying the resource through C-UEs. Thus, we formulate a new objective function for the all D2D pairs, the new utility is the sum utility function of all D2D pairs and can be expressed as follows

$$\tilde{U}_{Sum}^{F} = \sum\limits_{l = 1}^{L} {\sum\limits_{n = 1}^{N} {\rho_{l}^{n} \tilde{\gamma }_{l}^{n} } } - \sum\limits_{l = 1}^{L} {\sum\limits_{n = 1}^{N} {\left( {\sum\limits_{k = 1}^{K} {\rho_{l,k}^{n} c_{k} p_{l}^{n} {\sf {h}}_{lk}^{n} } + \varepsilon_{k}^{n} \sqrt {\sum\limits_{k = 1}^{K} {(\rho_{l,k}^{n} c_{k} p_{l}^{n} )^{2} } } } \right)} }$$

(32)

Therefore, in order to achieve the actable outcome of the proposed RSG, our concern is guarantying the existence and uniqueness of the RSG when maximize (32). The corresponding Lagrangian function for the D2D pair $l$ expressed in and the KTT conditions for the D2D pair l is given by

$$\begin{aligned} \nabla_{{p_{l} }} \tilde{U}_{l} ({\mathbf{p}}_{l} {\mathbf{p}}_{ - l} ,{\mathbf{c}}_{l} ) - \nabla_{{{\mathbf{p}}_{l} }} \left( {\sum\limits_{n = 1}^{N} {\rho_{l}^{n} p_{l}^{n} } - P_{l}^{\hbox{max} } } \right)\upsilon_{l} - \nabla_{{{\mathbf{p}}_{l} }} \left( {\sum\limits_{n = 1}^{N} {\omega_{l}^{n} \left( {\left( {\sum\limits_{j \ne l}^{L + 1} {\rho_{j,l}^{n} } } \right)\log \left( {1 + \frac{{\bar{\gamma }_{l}^{n} X_{l} }}{{p_{l}^{n} }}} \right) - \log \left( {\frac{1}{{1 - \xi_{l}^{n} }}} \right)} \right)} } \right) \hfill \\ \quad \quad - \nabla_{{{\mathbf{p}}_{l} }} \sum\limits_{n = 1}^{N} {\varphi_{l}^{n} } \left( {\sum\nolimits_{l = 1}^{L} {\rho_{l,k}^{n} p_{l}^{n} {\sf {h}}_{lk}^{n} } + \varepsilon_{k}^{n} \sqrt {\sum\nolimits_{l = 1}^{L} {(\rho_{l,k}^{n} p_{l}^{n} )^{2} } } - \tilde{T}_{k}^{n} } \right) \, \hfill \\ \upsilon_{l} \left( {\sum\limits_{n = 1}^{N} {\rho_{l}^{n} p_{l}^{n} } - P_{l}^{\hbox{max} } } \right) = 0 \hfill \\ \sum\limits_{n = 1}^{N} {\omega_{l}^{n} \left( {\left( {\sum\limits_{j \ne l}^{L + 1} {\rho_{j,l}^{n} } } \right)\log \left( {1 + \frac{{\bar{\gamma }_{l}^{n} X_{l} }}{{p_{l}^{n} }}} \right) - \log \left( {\frac{1}{{1 - \xi_{l}^{n} }}} \right)} \right)} = 0 \hfill \\ \sum\limits_{n = 1}^{N} {\varphi_{l}^{n} } \left( {\sum\nolimits_{l = 1}^{L} {\rho_{l,k}^{n} p_{l}^{n} {\sf {h}}_{lk}^{n} } + \varepsilon_{k}^{n} \sqrt {\sum\nolimits_{l = 1}^{L} {(\rho_{l,k}^{n} p_{l}^{n} )^{2} } } - \tilde{T}_{k}^{n} } \right) = 0 \hfill \\ \end{aligned}$$

(33)

We note that the robust followers’ game shows a jointly convex generalized RSE problem, therefore, the solution of the RSE problem with (2, 16, 18) can is a variational inequality ${\text{VI}}I\left( {{\mathbf{P}},{\mathbf{F}}} \right)$, where ${\mathbf{P}}$ is the set of joint convexity. It is important to determine a vector $z^{*} \in {\mathbf{P}} \subset {{R}}^{n}$, such that $\langle {\mathbf{F}}({\mathbf{z}}^{*} ),{\mathbf{z}} - {\mathbf{z}}^{*} \rangle \ge 0$, for all $z \in {\mathbf{P}}$ and ${\mathbf{F}}({\mathbf{p}}) = - (\nabla_{p} \tilde{U}_{l} ({\mathbf{p}}_{l}^{{}} ))_{l = 1}^{L}$ [34]. Then, the solution of ${\text{VI}}\left( {{\mathbf{P}},{\mathbf{F}}} \right)$ is a variational RSE.

In this paper, we only focus on the power control in D2D cellular networks by assuming the channel assignment has already been done. Then, we can divide the variational inequality ${\text{VI}}\left( {{\mathbf{P}},{\mathbf{F}}} \right)$ into $N$ subprolems, each subproblem denotes ${\text{VI}}\left( {{\mathbf{P}}_{n} ,{\mathbf{F}}_{n} } \right)$ on the sub-channel $n$ and they are independent. Therefore, on the sub-channel $n$, the KKT conditions can be expressed as [33]

$$\begin{aligned} & {\mathbf{F}}_{n} ({\mathbf{p}}) + \upsilon_{n} \nabla_{p} \left( {\sum\limits_{l = 1}^{L} {\rho_{l}^{n} p_{l}^{n} \rho_{l}^{n} p_{l}^{n} } - \sum\limits_{l = 1}^{L} {\rho_{l}^{n} P_{l}^{\hbox{max} } } } \right) + \omega_{n} \nabla_{p} \left( {\sum\limits_{l = 1}^{L} {\left( {\left( {\sum\limits_{j \ne l}^{L + 1} {\rho_{j,l}^{n} } } \right)\log \left( {1 + \frac{{\bar{\gamma }_{l}^{n} X_{l} }}{{p_{l}^{n} }}} \right)} \right)} - \sum\limits_{l = 1}^{L} {\log \left( {\frac{1}{{1 - \xi_{l}^{n} }}} \right)} } \right) \\ & \quad \quad + \varphi_{n} \nabla_{p} \left( {\sum\limits_{l = 1}^{L} {\rho_{l,k}^{n} p_{l}^{n} {\sf {h}}_{lk}^{n} } + \varepsilon_{k}^{n} \sqrt {\sum\nolimits_{l = 1}^{L} {(\rho_{l,k}^{n} p_{l}^{n} )^{2} } } - \sum\limits_{n = 1}^{N} {\rho_{l,k}^{n} \tilde{T}_{k}^{n} } } \right) \, \\ \end{aligned}$$

(34)

Now from the definition of [33], we have

$${\mathbf{F}}_{1} = \left[ \begin{aligned} {{\uprho}}_{1,k}^{1} \left( {c_{k} ({\sf {h}}_{1k}^{1} + \varepsilon_{k}^{1} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - 1}^{1} )}}} \right) \hfill \\ {{\uprho}}_{2,k}^{1} \left( {c_{k} ({\sf {h}}_{2k}^{1} + \varepsilon_{k}^{1} )} \right) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - 2}^{1} )}} \hfill \\ \, \vdots \hfill \\ {{\uprho}}_{L,k}^{1} \left( {c_{k} ({\sf {h}}_{Lk}^{1} + \varepsilon_{k}^{1} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - L}^{1} )}}} \right) \hfill \\ \end{aligned} \right],{\mathbf{F}}_{2} = \left[ \begin{aligned} {{\uprho}}_{1,k}^{2} \left( {c_{k} ({\sf {h}}_{1k}^{2} + \varepsilon_{k}^{2} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - 1}^{2} )}}} \right) \hfill \\ {{\uprho}}_{2,k}^{2} \left( {c_{k} ({\sf {h}}_{2k}^{2} + \varepsilon_{k}^{2} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - 2}^{2} )}}} \right) \hfill \\ \, \vdots \hfill \\ {{\uprho}}_{L,k}^{2} \left( {c_{k} ({\sf {h}}_{Lk}^{2} + \varepsilon_{k}^{2} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - L}^{2} )}}} \right) \hfill \\ \end{aligned} \right], \cdots ,{\mathbf{F}}_{n} = \left[ \begin{aligned} {{\uprho}}_{1,k}^{n} \left( {c_{k} ({\sf {h}}_{1k}^{n} + \varepsilon_{k}^{n} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - 1}^{n} )}}} \right) \hfill \\ {{\uprho}}_{2,k}^{n} \left( {c_{k} ({\sf {h}}_{2k}^{n} + \varepsilon_{k}^{n} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - 2}^{n} )}}} \right) \hfill \\ \, \vdots \hfill \\ {{\uprho}}_{L,k}^{n} \left( {c_{k} ({\sf {h}}_{Lk}^{n} + \varepsilon_{k}^{n} ) - \frac{1}{{\tilde{I}({\mathbf {p}}_{ - L}^{n} )}}} \right) \hfill \\ \end{aligned} \right]$$

(35)

Therefore, the Jacobian of is

$${\mathbf{J}}_{1} = \left[ \begin{aligned} \rho_{1,k}^{1} {\sf {h}}_{1k}^{1} { 0 } \cdots { 0 } \hfill \\ { 0 }\rho_{2,k}^{1} {\sf {h}}_{2k}^{1} \, \cdots { 0 } \hfill \\ \, \vdots \hfill \\ { 0 0 } \cdots \, \rho_{L,k}^{1} {\sf {h}}_{Lk}^{1} \hfill \\ \end{aligned} \right],{\mathbf{J}}_{2} = \left[ \begin{aligned} \rho_{1,k}^{2} {\sf {h}}_{1k}^{2} { 0 } \cdots { 0 } \hfill \\ { 0 }\rho_{2,k}^{2} {\sf {h}}_{2k}^{2} \, \cdots { 0 } \hfill \\ \, \vdots \hfill \\ { 0 0 } \cdots \, \rho_{L,k}^{2} {\sf {h}}_{Lk}^{2} \hfill \\ \end{aligned} \right], \cdots ,{\mathbf{J}}_{n} = \left[ \begin{aligned} \rho_{1,k}^{n} {\sf {h}}_{1k}^{n} { 0 } \cdots { 0 } \hfill \\ { 0 }\rho_{2,k}^{n} {\sf {h}}_{2k}^{n} \, \cdots { 0 } \hfill \\ \, \vdots \hfill \\ { 0 0 } \cdots \, \rho_{L,k}^{n} {\sf {h}}_{Lk}^{n} \hfill \\ \end{aligned} \right]$$

(36)

Each ${\mathbf{F}}_{n} {\mathbf{J}}_{n}$ is a diagonal matrix and all the diagonal elements are positive. Therefore, ${\mathbf{F}}_{n} {\mathbf{J}}_{n}$ is positive definition on ${\mathbf{P}}_{n}$, and so, ${\mathbf{F}}_{n}$ is strictly monotone. Hence, the global RSG problem admits a unique global variational equilibrium solution [33]. Due to the jointly convex nature of the global RSE problem, the variational equilibrium is the unique global maximizer of (32) [33], which completes the proof in the literature [33].