Research on multi-service slice resource allocation over licensed and unlicensed bands

Abstract

Cognitive radio is known to be an important technology to overcome the shortage of spectrum resources, and the resource allocation problem of multi-service multi-carrier slice in 5G system has remained a challenge. In this paper, a multi-leader multi-follower Stackelberg game is designed to analyze the interaction between operators and users over licensed and unlicensed bands. Further, we verify the impact of users’ dynamic throughput allocation ratio on Nash equilibrium (NE) in the game. The game is divided into two independent subgames: licensed band and unlicensed band games. A simplified Vickrey-Clarke-Groves algorithm is designed in the former sub-game to ensure the fairness of user transmission. In the latter, an interference price is set to protect the transmission performance of primary users in unlicensed band. By predicting the actions of other players, optimal prices or spectrum and power demands are set in a non-cooperative way. We prove the existence of NE solution in the Stackelberg game and design dynamic distributed algorithms for operators and users to achieve NE. Simulation results show the effectiveness of our proposed resource management schemes based on Stackelberg game. Compared to other Stackelberg models, our scheme provides users with high-quality services and provides a guidance on pricing strategies for operators.

Appendices

Appendix A: The Proof of Theorem 1

Proof

The pure policy set of users is concave, closed and bounded. The second partial derivative is as follows:

$$\begin{aligned} \frac{{\partial U_{j,a}^2}}{{{\partial ^2}a_{i,j}^{\left( l \right) }}} = \frac{{ - 2{\gamma _j}{a_{ - j}}}}{{{{\left( {{a_j} + {a_{ - j}}} \right) }^3}}}B_i^{\left( l \right) }{R_{i,j}} < 0 \end{aligned}$$

(A1)

where ${a_{ - j}} = \mathop \sum \limits _{z \in {N_{i,l}}} a_{i,z}^{\left( l \right) } - a_{i,j}^{\left( l \right) }$. The function is always less than zero in the domain of definition, which satisfies the definition of concave function. There is the second order partial derivative:

$$\begin{aligned} \frac{{\partial U_{j,a}^2}}{{{\partial ^2}{a_{ - j}}}} = \frac{{2{\gamma _j}B_i^{\left( l \right) }{R_{i,j}}}}{{{{\left( {{a_j} + {a_{ - j}}} \right) }^3}}} > 0 \end{aligned}$$

(A2)

The function is always greater than zero in the domain of definition, which satisfies the definition of convex function. Theorem 1 shows that there is at least one pure strategy Nash equilibrium in sub-game 1.

From $\frac{{\partial {U_{j,a}}}}{{\partial {a_j}}} = 0$, we get $a_{i,j}^{\left( l \right) *} = \sqrt{\frac{{{\gamma _j}{R_{i,j}}{a_{ - j}}}}{{\rho _i^{\left( l \right) }}}} - {a_{ - j}}$. Substitute $a_{i,j}^{\left( l \right) *}$ into the function of Leader1,

$$\begin{aligned} {W_{i,a}} = \;\mathop \sum \limits _{l = 1}^L \mathop \sum \limits _{j \in {N_{i,l}}} \rho _i^{\left( l \right) }\left( {\sqrt{\frac{{{\gamma _j}{R_{i,j}}{a_{ - j}}}}{{\rho _i^{\left( l \right) }}}} - {a_{ - j}}} \right) B_i^{\left( l \right) } \end{aligned}$$

(A3)

There is the second order partial derivative:

$$\begin{aligned} \frac{{\partial W_{i,a}^2}}{{{\partial ^2}\rho _i^{\left( l \right) }}} = \;\mathop \sum \limits _{j \in {N_{i,l}}} \frac{{\sqrt{{\gamma _j}{R_{i,j}}{a_{ - j}}} }}{{ - 4\sqrt{\rho {{_i^{\left( l \right) }}^3}} }} < 0 \end{aligned}$$

(A4)

The function is always less than zero in the domain of definition, which satisfies the definition of concave function. From $\frac{{\partial {W_{i,a}}}}{{\partial \rho _i^{\left( l \right) }}}$, we get $\rho _i^{\left( l \right) *} = \frac{{{{(\mathop \sum \nolimits _{j \in {N_{i,l}}} \sqrt{{\gamma _j}{R_{i,j}}{a_{ - j}}} )}^2}}}{{4{{\left( {{N_{i,l}} - 1} \right) }^2}{{\left( {{a_j} + {a_{ - j}}} \right) }^2}}}$.

Lemma 1 shows the basic conditions for the existence of Nash equilibrium. We have proved that sub-game 1 satisfies the conditions of Lemma 1and obtained a unique set of equilibrium solutions. A set of Nash equilibrium solutions is obtained as:

$$\begin{aligned} \;\rho _i^{\left( l \right) *}= & {} \frac{{{{(\mathop \sum \nolimits _{j \in {N_{i,l}}} \sqrt{{\gamma _j}{R_{i,j}}{a_{ - j}}} )}^2}}}{{4{{\left( {{N_{i,l}} - 1} \right) }^2}{{\left( {{a_j} + {a_{ - j}}} \right) }^2}}}\;,\;\forall i \in \varvec{N},\forall j \in \varvec{N_{i,l}}; \end{aligned}$$

(A5)

$$\begin{aligned} \;a_{i,j}^{\left( l \right) **}= & {} min\left\{ {max}\left\{ {a_{i,j}^{\left( l \right) *},\frac{{{\theta _j}{\eta _j}{a_{ - j}}}}{{B_i^{\left( l \right) }{R_{i,j}} - {\theta _j}{\eta _j}}}} \right\} ,1 \right\} \end{aligned}$$

(A6)

$\square$

Appendix B: The Proof of Theorem 2

Proof

The pure policy set of users is concave, closed and bounded. The second partial derivative is as follows:

$$\begin{aligned} \mathrm{{\;}}\frac{{\partial U_{j,b}^2}}{{{\partial ^2}{p_j}}} = \frac{{ - {\gamma _j}g_j^2}}{{{{\left( {{z_j} + \mathop \sum \nolimits _{k = 1}^M \mathop \sum \nolimits _{z \in {N_{k,l}}} {p_z}{g_z}} \right) }^2}}}B_u^{\left( l \right) } < 0 \end{aligned}$$

(B7)

The function is always greater than zero in the domain of definition, which satisfies the definition of convex function. From $\frac{{\partial {U_{j,b}}}}{{\partial {p_j}}} = 0$, we get $p_j^\mathrm{{*}} = \frac{{{\gamma _j}B_u^{\left( l \right) }}}{{{g_j}\mathop \sum \nolimits _{k = 1}^M r_k^{\left( l \right) }}} - \frac{{{z_j} + {p_{ - j}}{g_{ - j}}}}{{{g_j}}}$. Substitute $p_j^\mathrm{{*}}$ into the function of Leader2:

$$\begin{aligned} {W_{i,b}} = \;\mathop \sum \limits _{l = 1}^L r_i^{\left( l \right) }\left[ {\frac{{B_u^{\left( l \right) }\mathop \sum \nolimits _{k = 1}^M \;\mathop \sum \nolimits _{j \in {N_{k,l}}} {\gamma _j}}}{{\mathop \sum \nolimits _{k = 1}^M r_k^{\left( l \right) }}} - \mathop \sum \limits _{k = 1}^M \;\mathop \sum \limits _{j \in {N_{k,l}}} \left( {{z_j} + {p_{ - j}}{g_{ - j}}} \right) } \right] \end{aligned}$$

(B8)

There is the second order partial derivative:

$$\begin{aligned} \frac{{\partial W_{i,b}^2}}{{{\partial ^2}r_i^{\left( l \right) }}} = \frac{{ - r_{ - i}^{\left( l \right) }\mathop \sum \nolimits _{k = 1}^M \;\mathop \sum \nolimits _{j \in {N_{k,l}}} {\gamma _j}B_u^{\left( l \right) }}}{{{{\left( {r_i^{\left( l \right) } + r_{ - i}^{\left( l \right) }} \right) }^3}}} < 0 \end{aligned}$$

(B9)

where $r_{ - i}^{\left( l \right) } = \mathop \sum \limits _{k = 1}^M r_k^{\left( l \right) } - r_i^{\left( l \right) }$. The function is always less than zero in the domain of definition, which satisfies the definition of concave function. From $\frac{{\partial {W_{i,b}}}}{{\partial r_i^{\left( l \right) }}} = 0$, we get $r_i^{\left( l \right) *} = \sqrt{\frac{{r_{ - i}^{\left( l \right) }\mathop \sum \nolimits _{k = 1}^M \;\mathop \sum \nolimits _{j \in {N_{k,l}}} {\gamma _j}B_u^{\left( l \right) }}}{{\mathop \sum \nolimits _{k = 1}^M \;\mathop \sum \nolimits _{j \in {N_{k,l}}} \left( {{z_j} + {p_{ - j}}{g_{ - j}}} \right) }}} - r_{ - i}^{\left( l \right) }$.

Lemma 1 shows the basic conditions for the existence of Nash equilibrium. We have proved that sub-game 2 satisfies the conditions of Lemma 1 and obtained a unique set of equilibrium solutions. A set of Nash equilibrium solutions is obtained as:

$$\begin{aligned} r_i^{\left( l \right) *}= & {} max\left\{ {\sqrt{\frac{{r_{ - i}^{\left( l \right) }\mathop \sum \nolimits _{k = 1}^M \;\mathop \sum \nolimits _{j \in {N_{k,l}}} {\gamma _j}B_u^{\left( l \right) }}}{{\mathop \sum \nolimits _{k = 1}^M \;\mathop \sum \nolimits _{j \in {N_{k,l}}} \left( {{z_j} + {p_{ - j}}{g_{ - j}}} \right) }}} - r_{ - i}^{\left( l \right) },0} \right\} \end{aligned}$$

(B10)

$$\begin{aligned} \;p_j^{**}= & {} max\left\{ p_j^{*},\frac{{{z_j} + {p_{ - j}}{g_{ - j}}}}{{{g_j}}}{e^{\frac{{\left( {1 - {\theta _j}} \right) {\eta _j}}}{{B_u^{\left( l \right) }}}}} - \frac{{{z_j} + {p_{ - j}}{g_{ - j}}}}{{{g_j}}} \right\} ^{-} \end{aligned}$$

(B11)

where ${\left( x \right) ^ - } = \mathrm{{min}}\left\{ {x,p^{max}} \right\}$. $\square$