Enhancing Cellular Coverage Quality by Virtual Access Point and Wireless Power Transfer

The ultradensification deploying for cellular networks is a direct and effective method for the improvement of network capacity. However, the benefit is achieved at the cost of network infrastructure investment and operating overheads, especially when there is big gap between peak-hour Internet traffic and average one. Therefore, we put forward the concept of virtual cellular coverage area, where wireless terminals with high-end configuration are motivated to enhance cellular coverage quality by both providing RF energy compensation and rewarding free traffic access to Internet.This problem is formulated as the Stackelberg game based on three-party circular decision, where a Macro BS (MBS) acts as the leader to offer a charging power to Energy Transferring Relays (ETRs), and the ETRs and their associating Virtual Access Points (VAPs) act as the followers to make their decisions, respectively. According to the feedback from the followers, the leader may readjust its strategy. The circular decision is repeated until the powers converge. Also, the better response algorithm for each game player is proposed to iteratively achieve the Stackelberg-Nash Equilibrium (SNE).Theoretical analysis proves the convergence of the proposed game scheme, and simulation results demonstrate its effectiveness.


Introduction
The proliferation of mobile devices, machine-to-machine (M2M) communication [1], and data collection for smart cities [2,3] leads to the massive growth of Internet traffic.The literature [4] forecasts an increase of thousandfold in wireless traffic in 2020 as compared to the figures in 2010, which can be handled by deploying tremendous amount of small cells resulting in ultradense heterogeneous networks.Although dense networks with small cells for peak-load scenarios can meet soaring capacity and coverage demands in future cellular networks [5,6], these benefits are got at the cost of network infrastructure investment and operating overheads, for example, more access points and higher backhaul costs, more complex interference management, and more power consumption.
Peak-hour Internet traffic increases more rapidly than average Internet traffic.It is reported in [7] that, in 2016, Internet traffic in the busiest 60-minute period in a day increased 51 percent, while average traffic had only 32 percent growth.Moreover, it is predicted by [7] that, busy-hour Internet traffic will increase by a factor of 4.6 between 2016 and 2021, while average Internet traffic will increase by a factor of 3.2.
By reducing transmission range and optimizing transmitting power [8,9], network capacity and reliability can be obviously improved due to effective control of mutual interference.Therefore, a simple and easy solution is that network operators can deploy Small Base Stations (SBSs) enough to support peak traffic.However, they suffer from underutilization of resources at low load conditions.In fact, due to high costs for deployment and maintenance, network operators are unwilling to seamlessly deploy dense small cells to meet the customer needs in peak hour.
To address the above problems, we put forward the concept of virtual cellular coverage area, which is an agile, costeffective, and energy efficient alternative for small cell deployments and thus helpful for reducing operational expenditure to the network operators due to the large number of small cell deployments.The problem of continuous operation of

Related Work
RF energy harvesting sources are usually divided into two categories [10], where the first category belongs to ambient or unintended RF energy harvesting sources and the second one is the type of dedicated or intended RF ones.Also, there are the three main RF energy harvesting structures at receivingends [11], that is, separated structure (i.e., energy harvesting and information decoding circuitry have their own antennas, respectively), time switching structure (i.e., energy harvesting and information decoding circuitry share the same antenna at different time slots), and power splitting structure (i.e., energy harvesting and information decoding circuitry share the same antenna at the same time but split the input power).A time switching structure is usually used more frequently since it is comparatively compact in form factor and relatively effective in harvesting RF energy.
Furthermore, the literature [12] summarizes Simultaneous Wireless Information and Power Transfer (SWIPT) designing schemes which are usually classified as decoupled SWIPT, closed-loop SWIPT, and integrated SWIPT.The decoupled SWIPT is characterized by the information and the power traffic sent from two separate transmitting sources.In closed-loop SWIPT scenarios [13][14][15][16][17][18], a base station usually powers a wireless device in a downlink and receives the data from the same wireless device in the uplink.In an integrated SWIPT scheme [19], both the information and power are carried by the same source over the same signals.
Concurrent multiple RF source transmission reported in some literatures (e.g., [20,21]) will increase total interference.A harvest-then-transmit protocol allows a receiver to first harvest energy from the downlink broadcast signals and then use the harvested energy to send independent uplink information to the MBS or SBS based on Time Division Multiple Access (TDMA).In order to mitigate total interference, the system presented in [22] employs multiple dedicated RF sources with a harvest-then-transmit protocol to improve the harvested energy and data rate.The achievable data rate can increase by up to 87% when energy is harvested from up to 5 dedicated RF sources, compared to the case when energy is harvested from one source.Some works have focused on energy transfer in multihop manner for wireless networks; for example, [23] proposes a method to inject power into the network as flows, where injected power is carried in multihop manner, while [24] explores the optimization of multihop wireless energy transfer in wireless networks.Some typical noncooperative game models (e.g., Stackelberg game [25] and potential game [26]) are widely used in various network information systems [27][28][29].A Stackelberg game consists of leaders and followers that compete with each other for certain resources, where the leaders first act by considering behaviors of the followers, and the followers act subsequently.
An exact potential game can characterize the set of Nash Equilibria (NE), where a potential function can track the changes in the payoff due to the unilateral deviation of a player, and one or more NE points may exist and coincide with one point that maximizes the potential function.However, it cannot support application scenarios with sequential dependence.
In RF energy harvesting communication networks, some works [30][31][32][33][34][35] employ game theory to investigate different setups.However, they hardly have studied the concerned scenario of virtual cellular coverage area construction for a macrocell based on both RF energy compensation and free traffic access to Internet.
In the latest related work [35], the authors consider a radio frequency energy harvesting based Internet of Things (IoT) system, which consists of a data access point (DAP) and several energy access points (EAPs).The DAP collects information from its associated sensors.EAPs can provide wireless charging services to sensors via the RF energy transfer technique.
In this paper, a MBS firstly offers a charging power to the ETRs and then asks each ETR to determine its charging power to its associating VAP, where the purpose is to encourage the VAP to relay data from its associating UE to the MBS by compensating it in terms of energy.Therefore, the Stackelberg game model is applicable to such application scenario with sequential dependence.

The Game Scheme for Virtual
Ultra-Network Construction 3.1.Network Model.We consider a macrocell with a MBS located at its center as illustrated in Figure 1.Also, there are  UE pieces distributed randomly in Figure 1, where each UE may serve as either an ETR or a VAP and also may not do so.To mitigate interference to UE pieces in neighboring macrocells, the maximum transmission power of MBS is usually limited.If the receiving nodes have a strict requirement for the receiving bit error rate, UE pieces far away from a MBS, especially cellular edge users, hardly meet the requirement of such low receiving bit error rate.In addition, as stated in the introduction, even if the cell-edge UE pieces send data to the MBS by using their maximum transmission powers, it is generally difficult for the MBS to achieve such low receiving bit error rate.There may also be a few SBSs in Figure 1.Therefore, the UE pieces adjacent to the SBSs can access the Internet via the SBSs.However, for the UE pieces far away from SBSs, they may access the Internet via VAPs.Generally, since VAPs do not have backhauls; they forward the data from the cell-edge UE pieces to the MBS.
The hollow triangles in Figure 1 are the reference points used in the election of VAPs, which are usually planned and set by the network operator according to the distribution of the regions without SBSs.Such role as VAP is usually held by the UE pieces that are closer to the reference points and have more energy reserve.However, if the MBS transfers energy to VAPs directly, the power conversion efficiency is usually very low.Therefore, it is necessary to select a few UE pieces to act as ETRs.For example, , , , and  in Figure 1 assume such role.
A set of the algorithms for VAP election, ETR election, and UE association is listed in Algorithms 1, 2, and 3, but this paper does not discuss it since we focus on the design and analysis of game scheme for virtual cellular coverage area construction.By Algorithm 1, the MBS (e.g.,  in Figure 1) will get its set of VAPs (e.g.,   = {,  in Figure 1 }).Also, any VAP (e.g.,  or  in Figure 1) will get its set of ETRs (e.g.,   = {,  in Figure 1 } or   = {,  in Figure 1 }) by Algorithm 2, while it will get its set of associated UE pieces (e.g.,   = {,  in Figure 1 } or   = {, ,  in Figure 1 }) by Algorithm 3.

Time-Slotted Scheme for Energy and Information Transference.
In general, the MBS adopts OFDMA and communicates with each UE piece over a few subcarriers in Run at the macro base station  Input: Determine the number of virtual access points (e.g., 6) (2) Set their reference coordinates (e.g., the positions of dotted triangle shown in Figure 1) For each reference coordinate point (e.g.,V) do (4) Broadcast a request packet at its maximum transmission power (5) Set  and  V as 0 and a very large value respectively (6) Set the timer  Δ as Δ (7) while the timer  Δ does not expire do (8) If receive a responding packet from node  then (9) If  V >  V then  =  and  V =  V End if (10) End if (11) End while (12) Add  to   (13) Send a confirmation package to node  (14) End for Run at any node  Input: null If receive a requesting packet with a reference point (e.g.,V) then (2) If  V <  th and   >  th then (3) Send a response packet to MBS  (4) End if (5) End if (6) If receive a confirmation with a reference point (e.g.,V) then (7) Mark itself as the virtual access point with respect to the reference coordinate point V (8)
Run at any virtual access point (VAP)  ∈   Input: Broadcast a requesting packet for charging node election at its maximum transmission power (2) Set the timer  Δ as Δ (3) while the timer  Δ does not expire do (4) If receive a respond packet for charging node election from any node (e.g., ) then (5) If   <  th then (6) add  to   (7) Send a confirmation package to node  (8) End if (9) End if (10)  End if (5) End if (6) If receive a confirmation package from any VAP (e.g., ) then (7) Mark itself as the charging node of the VAP  (8)

End if
Algorithm 2: ETR election for VAP.
Run at any virtual access point (VAP)  ∈   Input: Broadcast a requesting packet for member association at its maximum transmission power (2) Set the timer  Δ as Δ (3) while the timer  Δ does not expire do (4) If receive a respond packet for member association from any node (e.g., ) then (5) Add  to   (6) Send a confirmation package to node  (7) End if (8) End while Run at any node  Input: null Output: null (1) If receive a requesting packet for member association from any VAP (e.g., ) then (2) =  and    =   (3) Set the timer  Δ as Δ (4) while the timer  Δ does not expire do (5) If receive a requesting packet for member association from any VAP (e.g., ) then (6) If    >   then   =  and    =   End if (7) End if (8) End while (9) Send a respond packet for member association to the VAP   (10) If receive a confirmation package from the VAP   then (11) mark itself as the member of the VAP   (12) End if (13) End if Algorithm 3: UE association for VAP.
Energy transfer from MBS to ETR Energy transfer from ETR to VAP Information transmission the downlink, while each UE piece adopts SC-FDMA and communicates with the MBS over the same number of subcarriers in the uplink.Without loss of generality, we assume that the fixed number of orthogonal subcarriers is used by the MBS to communicate with each UE piece, where the total channel frequency band for any UE is denoted as .Meanwhile, it is assumed that the MBS is furnished with |  |+ 1 antennas at least so that it can receive the data from |  | VAPs and broadcast energy to |  |⋅|  | ETRs simultaneously, while all UE pieces, including ETRs and VAPs, are equipped with one single antenna each.In this paper, | ⋅ | denotes the number of members in a set (e.g., |  | is the number of the VAPs in   , while |  | is the number of the ETRs in   ).
The MBS divides the communication time into time frames with the same size (e.g., ).For each virtual cellular coverage area with a VAP (e.g.,  in   ), the MBS specifies the structure of each time frame shown in Figure 2, which includes two time slots: (1) energy transfer time slot, in which ETRs harvest the energy from the MBS over frequency band  during the  ⋅  ⋅  time and the VAP (i.e., ) harvests the energy from the ETRs (i.e., the members in   ) over frequency band  during the (1 − ) ⋅  ⋅  time, with  being the fraction of the energy transfer time in T time and  being the fraction of the energy transfer time from the MBS to ETRs in  ⋅  time, 0 <  < 1, and 0 <  < 1; (2) information transmission time slot, in which a transmitting node sends the information signal to the corresponding receiver by using the harvested energy during the  ⋅  time, together with its remaining battery energy.

Problem Formulation and Theoretical Analysis.
In this subsection, we first describe wireless energy transfer model and wireless information transmission model and then develop a game-theoretic framework for virtual cellular coverage area construction.
After the MBS (as a leader) offers an initial charging power to all the ETRs, each ETR (as a follower) determines its charging power to its associating VAP.Once each VAP (as a follower) receives the charging powers from all the ETRs associated with it, it determines its transmission power for relaying data from its associating UE pieces to the MBS.Since an ETR charging power is subject to the MBS, it is more appropriate that the MBS acts as the leader for the VAPs.
We will analyze the proposed Stackelberg game based on three-party circular decision and obtain its SNE of this game.Based on our analysis, we know that each party's strategy affects other parties' strategies.Therefore, a reverse induction method can be used to analyze the proposed game, because it can capture the sequential dependence of the decisions in all the game stages.

Wireless Signal Propagation Model.
When a receiver  works in the energy harvesting mode, the power harvested from source  can be calculated as follows: In ( 1),  , denotes the power conversion efficiency factor from node  to node ,   , is the transmission power at source ,   , is the harvested power at receiver , and ℎ , denotes the channel power gain between source  and receiver .Here, let  , = |ℎ , | 2 , which is calculated by using the following formula [22,36]: In (2),  is path loss exponent; PL ref is the path loss in dB at the reference distance;  , is the body shadowing loss margin between source  and receiver  in dB,  , is the distance between source  and receiver ;  , is the smallscale fading power gain between source  and receiver .For simplicity, the paper assumes  , =  and  , = .
Let  and  2 denote the transmission bandwidth and noise power, respectively.When the receiver  works in the information decoding mode, the information decoding rate  , from source  to receiver  is as follows: (3)

Game Formulation and Theoretical Analysis for ETRs.
In the subsection, we consider a noncooperative scenario in which the ETRs are all rational and self-interested such that they only want to minimize their charging powers and maximize their transmitting powers.Therefore, we formulate the noncooperative game for the ETRs, where the game players are the |  | ETRs, the game actions are that each ETR determines the charging power and transmitting power under the constraint of its maximum transmitting power, and the game utility is the data rate for each ETR.For any ETR (e.g., ), when the MBS (i.e., ) broadcasts energy at   , , its harvested power   , can be calculated according to formula (1) and specified as   , =  ⋅   , ⋅  , .Also, when the maximum transmission power of the ETR  is set as  max  , the power used in transferring energy from  to  ∈   (i.e.,   , ) and that used in transmitting data from  to  (i.e.,   , ) have to meet the following relation.
On the one hand, the ETR  should employ the more value of   , to transfer energy to the VAP in order to obtain the more free traffic access to Internet; on the other hand, it has to use more value of   , to transmit its own data such that the free traffic share is fully utilized.Therefore, the ETR  must determine optimal transmission power splitting ratio under the constraint of  max  .When the ETR  adopts  max  and its battery energy storage and harvested power are   and   , , respectively, its continuous emission time (i.e.,  max

𝑟
) can be estimated by the following formula: During  max  , the utility of the ETR  is represented as follows: In ( 6), 0  is seen as the contribution from a fictitious selfless node and usually set to a constant (e.g., 10 −7 ), where the aim is to prevent game participants from colluding to cheat.We notice that each player's strategy (e.g.,   , of the ETR ) only depends on the strategies of the other players (i.e.,  − = {    , | ∀  ∈   \   , }).Thus, given the strategies of the other players  − , the best response strategy of the ETR  is the solution to the following optimization problem.That is, the ETR  needs to find the optimal transmission power   , to offer to the VAP  to maximize its individual profit, which can be obtained by solving the following optimization problem: max We denote by   * , the optimal solution to optimization problem (7).It is straightforward to verify that (7) always has a feasible solution.A SNE of the formulated game is a feasible profile   * , that satisfies (7).Now, we calculate the best response strategy   * , for the ETR  by solving optimization problem (7), which is described in the following theorem.
In fact, this power   * , cannot be solved directly from formula (8) since it is included in  and  of formula (8).However, due to the concave characteristic stated in Theorem 1, a new approximating algorithm (i.e., Algorithms 4) is proposed in the following text to obtain its approximate optimal value.

Game Formulation and Theoretical Analysis for VAPs.
In the subsection, we consider a noncooperative scenario in which the VAPs are all rational and self-interested such that they only want to minimize their forwarding powers and maximize their transmitting powers.Therefore, we formulate the noncooperative game for the VAPs, where the game players are |  | VAPs, the game actions are that each VAP determines the forwarding power and transmitting power under the constraint of its maximum transmitting power, and the game utility is the data rate for each VAP.For any VAP (e.g., ), its harvesting energy from the |  | ETRs can be estimated by the following formula: Also, when the maximum transmission power of the VAP  is set as  max  , the power used in forwarding data from UE to the MBS  (i.e.,   , ) and that used in transmitting its own data to the MBS  (i.e.,   , ) have to meet the following relation: On the one hand, the VAP  should employ more value of   , in forwarding data to the MBS  in order to obtain the more free traffic access to Internet; on the other hand, it has to use more value of   , to transmit its own data such that the free traffic share is fully utilized.
Therefore, the VAP  must determine optimal transmission power splitting ratio under the constraint of  max  .When the VAP  adopts  max  and its battery energy storage and harvested energy from the ETRs are   and   , respectively, its continuous emission time (i.e.,  max

𝑠
) can be estimated by the following formula: During  max  , the utility of the VAP  is represented as follows: In (13), the meaning of 0  is similar to 0  and also set to a constant (e.g., 3).We notice that each player's strategy (e.g.,   , of the VAP ) only depends on the strategies of the other players (i.e.,  − = {    , | ∀  ∈   \ {  , }}).Thus, given the strategies of the other players  − , the best response strategy of the VAP  is the solution to the following optimization problem.That is, the node  needs to find the optimal forwarding power   , to offer data forwarding service for its associating UE pieces in order to maximize its individual profit, which can be obtained by solving the following optimization problem: max We denote by   * , the optimal solution to optimization problem (14).It is straightforward to verify that ( 14) always has a feasible solution.A SNE of the formulated game is a feasible profile   * , that satisfies (14).Now, we calculate the best response strategy   * , for the VAP  by solving optimization problem (14), which is described in the following theorem.Theorem 2. Given  − = {    , | ∀  ∈   \ {  , }}, the optimal response strategy   * , of the VAP  can be expressed as follows: where ) . ( Proof.To maximize its net utility, each VAP hopes to determine an optimal transmission power splitting ratio.For an arbitrary VAP , its net utility function defined in ( 13) is a concave function of   , since  2   (  )/(  , ) 2 < 0.
Similar to the discussion and solution for   * , , we also provide a new approximate solution (i.e., Algorithms 5) in the following text to solve   * , .

Game Formulation and Theoretical Analysis for MBS.
In the subsection, we mainly focus on how the decision in the MBS is affected by the game results of ETRs and VAPs.Based on the conditions described in the previous text, the Stackelberg game theory, including leaders and followers competing with each other for certain resources, applies to this scenario.In this paper, the MBS acts as the leader and offers an initial charging power to the ETRs with its maximum transmission power.The purpose is to encourage all UE pieces to participate actively.Following this, the ETRs and their associating VAP act as the followers and make their decisions according to their utility functions, respectively.According to the feedback from the followers, the leader may readjust its strategy.
It is assumed that the MBS broadcasts energy to |  | ⋅ |  | ETRs with omnidirectional antenna.Therefore, for an arbitrary ETR (e.g., ), the charging power   , is simplified as    .The MBS hopes to obtain the maximum forwarding traffic from the |  | VAPs but just wants to pay the minimum price of energy.Therefore, the utility of the MBS  is represented as follows: In (17),   is traffic price per bit, while   is energy price per joule.According to formulas (1), ( 5), (10), and ( 12),  max  can be rewritten as follows: Thus, given   and   from the ETRs and VAPs, respectively, the best response strategy of the MBS  is the solution to the following optimization problem: max Now, we calculate the best response strategy   *  for the MBS  by solving optimization problem (19), which is described in the following theorem.Theorem 3. Given   and   from the ETRs and VAPs, respectively, when the following relation is met, the optimal response strategy   *  of the MBS  can be expressed as follows: where Proof.The optimization problem with respect to (19) , and it will be less than 0 according to conditions ( 23) and ( 24) when expression (20) is met.
By solving the first-order derivative of   (   ,   ,   ) with respect to    , we can obtain the following expression:

ETR game ETRs
Each player hopes to find the optimal charging power offered to its associating VAP to maximize its individual profit.
The utility of any player is computed by formula (6).

VAP game VAPs
Each player hopes to find the optimal forwarding power to offer data forwarding service for its associating UEs to maximize its individual profit.
The utility of any player is computed by formula (13).
Stackelberg game

MBS ETRs VAPs
The MBS hopes to obtain the maximum forwarding traffic from the VAPs but just wants to pay the minimum price of energy.
Let expression (25) (i.e., the first-order derivative of   (   ,   ,   ) with respect to    ) be equal to 0; we can obtain expression (21) with some algebra manipulations, which completes the proof.
From Theorem 3, we know that expression (19) satisfies the concavity under a given condition.Therefore, if    that satisfies expression (20) is less than MBS maximum power  max  , we can design an approximate algorithm to search the approximate value for   *  in the interval from [   ] + to  max  .
To facilitate readers' understanding, Table 1 summarizes the game scenarios, actors (or players), strategies (or actions), and payoffs (or utilities) in this paper.

Distributed Iterative Algorithms for Obtaining SNE.
Since we have derived the optimal transmission power splitting levels for ETRs and VAPs to achieve their desired data rates (i.e., from their own to the MBS), as well as allowing the MBS to achieve its maximum utility by adjusting its charging power, we now propose the three efficient and practical algorithms that find the approximately optimal power levels, which are based on the results of theoretical analysis in Section 3.3.
Based on a reverse induction mode, the system interaction is as follows.Given an initial charging power offered by the MBS to the ETRs, (1) each ETR firstly determines its charging power to its associating VAP by running Algorithm 4; (2) then each VAP determines its transmission End if (16) Send (  , ,   , ) to the MBS  (17) End if (18) If receive "end" from the MBS  then return End if (19) End while (20) End if Algorithm 5: VAP-level game.
for relaying data from its associating UE pieces to the MBS according to the results of Algorithm 5; (3) finally, the MBS updates its charging power to the ETRs by executing Algorithm 6.The steps (1), (2), and (3) are repeated until the powers converge.
Parameter  is set to a small value (e.g., 0.1%∼3.5%).In any of the three algorithms, the better response of any player in the current round is the better transmission power splitting levels or charging power offered to maximize its total revenue, given those offered by the other players at the previous round of the iteration process.

Performance Evaluation
4.1.Simulation Setting.In our simulations, the macrocell network consists of one MBS and a set of UE pieces (denoted as ⌀), where the number of members in ⌀ is denoted as |⌀|.We assume that |⌀| is 1000.For any UE , the initial battery capacity   is randomly distributed from 0.05 to 0.2 Joule, where  can take different values ranging from 20% to 80%.Therefore, the average initial battery capacity  av is obtained by the following formula.
We assume that the coverage radius of the MBS is 500 m, and the UE pieces are randomly located in the macrocell.Each ETR or VAP elected from UE pieces has more battery energy reserve, which can improve the network performance.The UE pieces associated with each VAP are basically located near the edge of the macrocell, since such UE pieces need the help of VAPs more.We set the coordinates of reference points as shown in Figure 5, where  ℎ is valued at 100 m.It is assumed that channel noise power  2 (at the receivers of all UE pieces, including ETRs and VAPs) is valued at a fixed value, which is randomly distributed from −120 to −60 dBm/MHz.The other simulation parameters are listed in Table 2.

Simulation Schemes and Metrics.
In this subsection, we evaluate the performance of the scheme in this paper via simulations under a few typical values with respect to the time slot parameters (i.e., , , , , and ).For any UE , we set a time allocation strategy for   as follows: For any ETR , the adaptive time allocation for   is determined by the following formula: Also, the fixed time allocation for   is determined by the following formula: Our simulation metrics include average data rate for ETRs, average data rate for VAPs, and average data rate relayed by VAPs for associated UE pieces.Here, the first metric is denoted as DR av, and estimated by the following formulas.
The second metric is denoted as DR av, and estimated by the following formulas: The third metric is denoted as DR av, and estimated by the following formulas: In (32), BER is bit error ratio and takes different values (e.g., 10 −8 ).When the set number of UE pieces (i.e., |  |) must meet the following relational expression, the relational expressions in (32) are true.
In this paper, the meaning of ⌈⌉ is the integer that is not less than .

Simulation Results and Analysis
. Firstly, we consider the two groups of simulations, which adopt adaptive time allocation for   (i.e., formula (28)).In the first group of simulations, the channel noise power  2 at all the receivingends is fixed as −90 dBm/MHz, and the parameter  is set as 0.1%.We evaluate the performance in terms of the three metrics under the change of average UE battery capacity level (i.e., the value of ), as shown in Figure 6.
From Figures 6(a) and 6(b), we see that the change of average UE battery capacity has little impact on the average data rates of ETRs and VAPs.The main reason can be found from formula (30) and formula (31).Without loss of generality, we take formula (30) as an example, where, on the one hand, the continuous emission time   max gets longer with increase of average UE battery capacity, and thus the amount of emitted data is larger during   max ; on the other hand, the number of time frames proportionately increases with the increase of   max .Therefore, according to formula (30), the average data rate for ETRs has been hardly changed.
As shown in Figure 6(c), the change of average data rate for associated UE pieces with increase of average UE battery capacity is different from those of ETRs and VAPs.For the associated UE pieces, when the data generated in a time frame is greater than the forwarding amount of their associating VAP in the same time frame, more time frames  explanation in Figure 6 can be used to explain the situation in Figure 7.
Then, our simulations use the fixed time allocation for   (i.e., formula ( 29)) instead of the adaptive time allocation for   (i.e., formula (28)), where the purpose is to compare the average data rates of ETRs under the two types of time allocation schemes, since the average data rates of VAPs and associated UE pieces are independent of the parameter   .The other configuration of simulation parameters is the same as those in the above two groups of simulations.The simulation results are shown in Figures 8 and 9.
When comparing Figure 8 with Figure 6(a), we observe that the difference between the two time allocation schemes is very small.It may be that the advantage of adaptive time allocation is not obvious due to the adoption of relatively short time frame.Also, when comparing Figure 9 with Figure 7(a), we can find similar result.
Finally, we evaluate the game convergence rates of ETR, VAP, and MBS under the variation of step size (i.e., the parameter ) through the simulation result shown in Figure 10.Also, we evaluate the average date rates of ETR, VAP, and associated UE pieces under the variation of step size through the simulation result shown in Figure 11.
From Figure 10, we see that a larger step size leads to a faster convergence.The reason behind this phenomenon is straightforward from the description of Algorithm 4-6.Meanwhile, from Figure 11, we find that the variation of step size has little impact on the average date rate.Therefore, the step size can be appropriately increased to speed up the convergence of the proposed game scheme.
In addition, Figure 10 shows the game rounds of a MBS are obviously less than those of ETRs and VAPs.From the description of Algorithm 6, we can easily find the reason.Also, from Figure 11, we see that a VAP is highly efficient in terms of its own data transmission.Based on balance between contribution and return in terms of data transmission,

Conclusion
This paper explores a charging power decision and distributed transmission power splitting problem based on game theory for constructing virtual cellular coverage area in overlay cellular networks.To characterize rational behaviors of wireless terminal users, we formulate a noncooperative game for the considered system, where each terminal user is modeled as a strategic player that aims to maximize its own data rate under the constraint for its maximum transmission power.Step size (mW) Theoretical analysis proves the convergence of the proposed game scheme.Simulation results indicate that, under the certain conditions, the game-theoretic charging power decision and transmission power splitting mechanism can speed up the convergence on the premise of QoS requirement in terms of average date rate.However, we do not consider the optimizing problem for time allocation, since we focus on the design and analysis of game scheme for virtual cellular coverage area construction.Therefore, we plan to address it in our future work.The power conversion efficiency factor from sender  to receiver  ℎ , :

Notations Used in Our Work
The channel power gain from sender  to receiver  : Path loss exponent PL ref : The path loss in dB at the reference distance  , : Th eb o d ys h a d o w i n gl o s sm a r g i nb e t w e e n sender  and receiver  in dB  , : The small-scale fading power gain between sender  and receiver   , : The distance between sender  and receiver  : Transmission bandwidth for each UE   : Transmission bandwidth for ETRs   : Transmission bandwidth for VAPs  2 : N o i s ep o w e r  , : The information decoding rate in link  →    max : The maximum transmission power of ETR    max : The maximum transmission power of VAP    : The battery energy storage of ETR   max  : The continuous emission time of ETR   max  : The continuous emission time of VAP    (  ): The utility of ETR    (  ): The utility of VAP    (   ,   ,   ): The utility of MBS   , : The fraction of bandwidth for ETR  sending its data to MBS   , : The fraction of bandwidth for VAP  sending its data to MBS  0  , 0  : The contribution from a fictitious selfless node   * , : Th eE T R's optimal charging power to VAP    *  : Th eM B S's optimal charging power to ETRs   : Th eh a r v e s t i n ge n e r g yo fV A P from the |  | ETRs   :

Figure 2 :
Figure 2: The structure of time frame.

Figure 5 :
Figure 5: The coordinates of reference points.

Figure 8 :
Figure 8: Average date rate versus UE battery capacity under fixed time allocation.

 = 0 Figure 9 :
Figure 9: Average data rate versus channel noise power under fixed time allocation.

Figure 10 :
Figure 10: Average convergence time versus step size.

Figure 11 :
Figure 11: Average data rate versus step size.

𝑆
: The set of VAPs   : The set of ETRs associated with a VAP (e.g., )   : The set of UEs associated with a VAP (e.g., ) | ⋅ |: The number of members in a set (e.g., |  | is the number of the VAPs in   ) : The time frame for communication : The fraction of energy transfer time in  time : The fraction of energy transfer time from the MBS to ETRs in  ⋅  time : The fraction of information transmission time from a VAP to MBS in  ⋅  ⋅  time for the data received by the VAP from its associating UEs   : The ratio of time when an ETR transfers energy to its associating VAP   : The ratio of time when UE transmits its data to its associating VAP   , : The transmission power at sender  in link  →    , : The receiving power at receiver  in link  →   , :

Table 1 :
The game scenarios in our work.