Joint Resource Block and Power Allocation for Interference Management in Device to Device Underlay Cellular Networks: A Game Theoretic Approach

Abstract

This paper addresses the problem of joint resource block (RB) and uplink transmission power allocation in a Device to Device (D2D) underlay cellular network via a game theoretic approach. In contrast to the majority of previous research works in this area, the proposed framework aims mainly at interference mitigation and energy efficiency. We do not restrict a priori the number of D2D pairs which can reuse a RB allocated to a cellular link. However, this is determined dynamically by the objective of interference minimization. To address the resource allocation problem under consideration and deal with its inherent complexity, a two-step distributed approach is proposed. At first the RB allocation process is modeled as an exact potential game which is shown to minimize the total interference in the network. Second a non-cooperative game theoretic model for the uplink transmission power allocation process is proposed. For both stages of the proposed framework, efficient and distributed algorithms for the computation of the desired Nash Equilibrium (NE) point of each game are introduced. The efficiency of the overall proposed framework is evaluated through modeling and simulation, while comparative numerical results are presented that demonstrate the superiority of our methodology against other state of the art approaches.

Appendix I

1.1 Proof of Lemma 1

The potential function of game G can be written as:

$$ \begin{array}{l} Pot\left({s}_i,{s}_{-i}\right)={\displaystyle \sum_{i=1}^N\left[a{T}_i^1\left({s}_i,{s}_{-i}\right)+\left(1-a\right){T}_i^2\left({s}_i,{s}_{-i}\right)\right]} = \\ {}={\displaystyle \sum_{i=1}^N\left[-a{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-\left(1-a\right){\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f}\left.\left({c}_i,{c}_j\right)\right)\right]}=\\ {}-a{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-\left(1-a\right){\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i,{c}_j\right)}\\ {}{\displaystyle \sum_{k\ne i,k=1}^N\left[-a{\displaystyle \sum_{j\ne k,j=1}^N{p}_j{G}_{jk}f\left({c}_j,{c}_k\right)}-\left(1-a\right){\displaystyle \sum_{j\ne k,j=1}^N{p}_k{G}_{kj}f\left({c}_k,{c}_j\right)}\right]}=\\ {}=-a{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-\left(1-a\right){\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i,{c}_j\right)}\\ {}+{\displaystyle \sum_{k\ne i,k=1}^N}\Big[-a{p}_i{G}_{ik}f\left({c}_i,{c}_k\right)-a{\displaystyle \sum_{j\ne k,j\ne i,j=1}^N{p}_j{G}_{jk}f\left({c}_j,{c}_k\right)}-\\ {}-\left(1-a\right){p}_k{G}_{ki}f\left({c}_k,{c}_i\right)-\left(1-a\right){\displaystyle \sum_{j\ne i,j\ne k,j=1}^N{p}_k{G}_{kj}f\left.\left({c}_k,{c}_j\right)\right)\Big]}\\ {} Pot\left({s}_i,{s}_{-i}\right)=-a{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-\left(1-a\right){\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i,{c}_j\right)}\\ {}+{\displaystyle \sum_{k\ne i,k=1}^N\Big[-a{p}_i{G}_{ik}f\left({c}_i,{c}_k\right)-}\left(1-a\right){p}_k{G}_{ki}f\left({c}_k,{c}_i\right)\Big]+\\ {}{\displaystyle \sum_{k\ne i,k=1}^N\Big[-a{\displaystyle \sum_{j\ne k,j\ne i,j=1}^N{p}_j{G}_{jk}f\left({c}_j,{c}_k\right)}-}\left(1-a\right){\displaystyle \sum_{j\ne i,j\ne k,j=1}^N{p}_k{G}_{kj}f\left({c}_k,{c}_j\right)\left)\right]}\end{array} $$

Let $Q\left({s}_{-i}\right)={\displaystyle \sum_{k\ne i,k=1}^N\Big[-a{\displaystyle \sum_{j\ne k,j\ne i,j=1}^N{p}_j{G}_{jk}f\left({c}_j,{c}_k\right)}-}\left(1-a\right){\displaystyle \sum_{j\ne i,j\ne k,j=1}^N{p}_k{G}_{kj}f\left({c}_k,{c}_j\right)\left)\right]}$ Then: $\begin{array}{l} Pot\left({s}_i,{s}_{-i}\right)=-a{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-\left(1-a\right){\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i,{c}_j\right)\sum_{k\ne i,k=1}^N\Big[-a{p}_i{G}_{ik}f\left({c}_i,{c}_k\right)-}\left(1-a\right){p}_k{G}_{ki}f\left({c}_k,{c}_i\right)\Big]+Q\left({s}_{-i}\right)=-\left(a+\left(1-a\right)\right){\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-\left(\left(1-a\right)+a\right){\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i,{c}_j\right)}\\ {}+Q\left({s}_{-i}\right)=-{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-{\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i,{c}_j\right)}+Q\left({s}_{-i}\right)\end{array}$

If link i changes strategy from s _i to s ^' _i then from (9) we have

$$ \begin{array}{l} Pot\left({s}_i^{\prime },{s}_{-i}\right)=-{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i^{\prime}\right)}-{\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i^{\prime },{c}_j\right)}\\ {}+Q\left({s}_{-i}\right)\end{array} $$

Then follows:

$$ \begin{array}{l} Pot\left({s}_i,{s}_{-i}\right)- Pot\left({s}_i^{\prime },{s}_{-i}\right)=\\ {}=-{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i\right)}-{\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i,{c}_j\right)}+Q\left({s}_{-i}\right)\\ {}-\left(-{\displaystyle \sum_{j\ne i,j=1}^N{p}_j{G}_{ji}f\left({c}_j,{c}_i^{\prime}\right)}-{\displaystyle \sum_{j\ne i,j=1}^N{p}_i{G}_{ij}f\left({c}_i^{\prime },{c}_j\right)+Q\left({s}_{-i}\right)}\right)\\ {}={U}_i\left({s}_i,{s}_{-i}\right)-{U}_i\left({s}_i^{\prime },{s}_{-i}\right)\\ {}\forall i\in N,\ \forall {s}_i,s{\hbox{'}}_i\in {S}_i\end{array} $$

The above formulated game G is an exact potential game with the exact potential function Pot(s _i , s _− i ). ■