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Device-to-device (D2D) communications and femtocell systems can bring significant benefits to users’ throughput. However, the complicated three-tier interference among macrocell, femtocell, and D2D systems is a challenging issue in heterogeneous networks. As D2D user equipment (UE) can cause interference to cellular UE, scheduling and allocation of channel resources and power of D2D communication need elaborate coordination. In this paper, we propose a joint scheduling and resource allocation scheme to improve the performance of D2D communication. We take UE rate and UE fairness into account by performing interference management. First, we construct a Stackelberg game framework in which we group a macrocellular UE, a femtocellular UE, and a D2D UE to form a two-leader one-follower pair. The cellular UE are leaders, and D2D UE is the follower who buys channel resources from the leaders. We analyze the equilibrium of the game and obtain solutions to the equilibrium. Second, we propose an algorithm for joint scheduling of D2D pairs based on their utility. Finally, we perform computer simulations to study the performance of the proposed scheme.

With the increasing demand for larger system capacity and ubiquitous service quality in wireless communication, device-to-device (D2D) communication is a promising technology which has been considered as an important feature to be integrated into the long term evolution-advanced (LTE-A) system. As a type of proximity communication, D2D communication enables user equipment (UE) to communicate with each other directly without traversing the evolved NodeB (eNB) when UE is in close distance [

On the other hand, network topology has been considered as one of the key issues to make a leap on the performance of networks [

In the literature, studies on resource management of D2D communication focus on power optimization, resource allocation, and mode selection when they perform as an underlay in traditional cellular networks reusing uplink resource [

Generally, as a secondary underlay which reuses the spectrum resources of the primary system, resource allocation of D2D pairs is an important but challenging task to improve the performance of heterogeneous macrocell-femtocell networks [

In this paper, we study resource management for D2D communication in heterogeneous networks utilizing game theory approach. Given D2D’s underlay status in the system, Stackelberg game framework is well suited for the situation. The rest of this paper is organized as follows. In Section

We consider the uplink of a macro/femto/D2D system in a single cell with a macrocell base station (MBS) in the center. One femtocell is randomly located in the same cell. The femtocell is assumed to be round-shaped with a femtocell base station (FBS) deployed at the center of the house. The femtocell serves several indoor users, which are randomly located within the house. Several macrocell UE are distributed out of the femtocell. There are multiple outdoor D2D UE around the considered femtocell. The D2D UE are in pairs, each consisting of one transmitter and one receiver between which the communication distance is

System model.

We assume that the number of macrocell UE is

We use a set of

As D2D communication takes place underlaying the heterogeneous networks, we focus on power control and scheduling of D2D UE, while transmit power of macrocell UE and femtocell UE are assumed to be fixed. D2D communication can utilize the proximity between UE to improve the throughput performance of the system. In the meanwhile, interference from D2D pairs to cellular network should be limited. Thus, transmit power of D2D UE should be properly controlled. Another goal is to guarantee the fairness among D2D UE when scheduling. In this section, we first formulate this problem as a resource allocation method using Stackelberg game based scheme; then we first obtain solutions to the outcomes of the proposed game.

Interactions among selfish cellular UE and D2D UE sharing a channel can be modeled as a noncooperative game using game theory framework. When players choose their strategies independently without any coordination, it usually leads to an inefficient outcome. If we simply model this scenario as a noncooperative game, D2D transmitters will choose to use the maximum transmit power to maximize their own payoffs regardless of other players, whereas cellular UE will choose not to share channel resources with D2D UE. This is an inefficient outcome, as either the interference is too serious or D2D cannot get access into the network.

Therefore, we employ the Stackelberg game to coordinate the scheduling, in which macrocell UE and femtocell UE are leaders and D2D UE are followers. We focus on the behavior of a two-leader one-follower pair, of which a macrocell UE and a femtocell UE are the leaders; a D2D UE is the follower. They share the same channel resource. The leaders own the channel resource and they can charge D2D UE some fees for using the channels. The fees are fictitious money to coordinate the system. Thus, the cellular UE have an incentive to share the channel with D2D UE if it is profitable, and the leaders have the right to decide the price. For D2D UE, under the charging price, they can choose the optimal power to maximize their payoffs. In this way, an equilibrium can be reached.

The D2D pair can be modeled as a buyer and aims to obtain the most benefits, at least possible payments. The utility of the follower can be defined as its own throughput performance minus the cost it pays for using the channel. The fees should be decided according to the leaders’ consideration. Thus, the fee is charged proportionally to the amount of interference the leaders observe, which can be expressed as

The optimization problem of follower-level game can be formulated as

The macrocell UE and femtocell UE can be seen as two seller and aim to not only earn the payment but also gain as many extra profits as possible. The utility of the leaders can be defined as their gain from the follower minus the interference they observe from the D2D pair. The utility function of the leaders can be, respectively, described as

The optimization problem of leader-level is to set a set of charging prices that maximize their utility, that is,

The outcome of the proposed game will be shown in detail in the following section.

In the Stackelberg game, the leaders move first and the follower moves sequentially; that is, the leaders set the prices first, and the follower selects its best transmit power based on the price. The leaders know ex ante that the follower observes their action. Therefore, the game can be solved by backward induction.

In Stackelberg game, the leader has the preferential right of pricing on the follower cost, while the follower has no direct influence on the prices. Therefore, in every step of pricing updating process, since prices have been set by the leaders, they remain constant at this very step from the perspective of the follower. As illustrated in (

The solution of (

From (

Substituting (

We can note that (

From the analysis above, by taking the derivative of

Solving the above equations of

In this section, we prove that the solutions

We first define the SE of the proposed game as follows.

Then, we show that the optimizer

The utility function

The second-order derivative of (

Thus, the solution in (

Due to Property

In the following two properties, we show that the base stations cannot infinitely increase

The optimal power of D2D transmitter

Taking the first-order derivative of the optimal power

Therefore,

Consequently, there is a trade-off for base stations to ask for proper prices, and we can solve the optimal prices by equating

The utility function

We can prove Property

Based on the above properties, we conclude the following theorem.

The

In the next section, we will show that the SE is unique, and the proposed game converges to the unique SE when each base station updates its price according to a simple function.

From the previous section, one base station needs to modify its own price after the other base station changes its price. Consequently, each base station updates

After rearranging (

In order to calculate

We show next the convergence of the iterations in (

A function

The price updating function

Consider the following.

Therefore, to prove the positivity of the second term of RHS of (

After extensive numerical tests for a wide range of parameters when the nodes are randomly located, we observe that the numerator of (

After order-of-magnitude estimation, we can note that

Finally, from the above three parts, we prove that the price updating function

In [

From (

The scheduling process is conducted at each TTI. The D2D UE forms a priority queue for each channel. During each TTI, the MBS selects

In our Stackelberg game framework, the priority is based on the utilities of the followers, which express the satisfaction of the followers. In the design of scheduling scheme, fairness is considered as an important goal. The scheme should take the outcome in the previous TTIs into account. This can be achieved by adjusting prices for using the channel. The follower has to pay an additional fee for using the channel at TTI

Based on the above discussion, during each TTI, every macrocell UE, femtocell UE, and D2D UE form a leader-follower pair and play the Stackelberg game. The optimal price and power can be decided for each pair. The priority for each pair can be calculated and they form a priority queue. Then, the eNB schedules the D2D pairs sequentially according to their order in the queue. If there is a tie, in which one channel has been allocated to another D2D pair, or the D2D pair has been scheduled to another channel, the pair is skipped. When each channel is allocated to one D2D pair, the eNB records the outcome and the scheduling is over. The algorithm is summarized in Algorithm

(1) Given CSI, TTI

(2) Initialize

(3) Update prices according to the iterative function

(4) Calculate the optimal power

(5) Calculate priorities

(6) Sort

(7) while

(8) Select the head of the queue. The pair is

(9) if

(10) Schedule the pair

(11) Set

(12) end if

(13) Delete the head of the queue.

(14) end while

The algorithm has a low complexity, as the optimal strategy for each leader-follower pair is searched in a set with a constant number of elements. To form the priority queue with length

To evaluate the performance of the proposed algorithm, we perform several simulations. We consider a single circular cell environment. The macrocell/femtocell UE and D2D pairs are uniformly distributed in the cell. The two D2D UE in a D2D pair are close enough to satisfy the maximum distance constraint of D2D communication. The received signal power is

Simulation parameters and values.

Parameter | Values |
---|---|

Macrocell radius | 500 m |

Number of cellular UE | 5 |

Number of D2D pairs | 10 |

D2D communication distance | 10 m |

Femtocell radius | 10 m |

Shadow width ( |
5 m |

Cellular UE power | 23 dBm |

D2D transmit power | 0–23 dBm |

Thermal noise power density | −174 dBm/Hz |

Bandwidth | 180 kHz |

Transmission time interval | 1 ms |

As described in previous section, the base stations start increasing their price

We conducted simulations to observe the convergence behavior of the proposed game. In Figure

Prices of MBS and FBS uder different

Power of D2D transmitter under different

In Figure

D2D rate distribution under different

In Figures

Macrocell BS rate distribution.

Femtocell BS rate distribution.

In Figure

UE rate distribution under different

In this paper, we constructed a Stackelberg game framework for joint power control and channel allocation and scheduling of D2D communication in heterogeneous macrocell/femtocell network system. Based on properly designed utility functions, prices for reusing the channel resource and appropriate transmit power of D2D transmitters are adjusted to maximize the utility obtained by base stations and D2D pairs, respectively. Based on the proposed scheme, we analyzed the optimal strategy for every participator (D2D pairs/femtocell users/macrocell users), worked out the solutions for equilibrium state, and proposed an algorithm to allocate resources to schedule D2D UE, where interference management and fairness of the system were considered. Simulation results show that the proposed algorithm can achieve an increase in transmit rate performance for both the cellular and the D2D UE. The D2D UE can be fairly served. The scale factor

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is partly supported by the National Science Foundation of China (Grant nos. NFSC no. 61071083 and no. 61371073) and the National High-Tech Research and Development Program of China (863 Program) no. 2012AA01A506.