Received August 19, 2017, accepted September 13, 2017, date of publication September 26, 2017, date of current version October 25, 2017. Digital Object Identifier 10.1109/ACCESS.2017.2756079
Game-Theory-Based Distributed Power Splitting for Future Wireless Powered MTC Networks KANG KANG 1 , (Student Member, IEEE), RONG YE1 , (Student Member, IEEE), ZHENNI PAN 1 , (Member, IEEE), JIANG LIU1 , (Member, IEEE), AND SHIGERU SHIMAMOTO1,2 , (Member, IEEE) 1 Graduate
School of Fundamental Science and Engineering, Department of Computer Science and Communication Engineering, Waseda University, Tokyo 1698555, Japan 2 Graduate School of Global Information and Telecommunication Studies, Department of Computer Systems and Network Engineering, Waseda University, Tokyo 1890012, Japan
Corresponding author: Kang Kang (
[email protected])
ABSTRACT This paper studies the emerging wireless power transfer for machine type communication (MTC) network, where one hybrid access point (AP) with constant power supply communicates with a set of users (i.e., wearable devices and sensors) without power supply. The information and energy are transferred simultaneously in downlink direction. For MTC networks, most devices only receive several bits control data from AP in downlink transmission. So it is possible to utilize part of the received power to execute energy harvesting provided that the transmission reliability is guaranteed. Since we assume that all devices are without power supply or battery, the power of uplink transmission is entirely from energy harvesting. After converting electromagnetic wave to electricity, the devices are able to transmit their measured and collected data in uplink. Based on these considerations, a non-cooperative game model is formulated and a utility function involving both downlink decoding signal to noise ratio (SNR) and uplink throughput is established. The existence of Nash equilibrium (NE) in the formulated game model is proved. The uniqueness of NE is discussed and the expected NE is selected based on fairness equilibrium selection mechanism. The optimal splitting ratio within the feasible set, which maximizes the utility function, is obtained by an iterative function derived from this utility function. The numerical results show that in addition to ensuring the downlink decoding SNR and maximizing uplink throughput of an individual device, our proposed algorithm outperforms the conventional algorithm in terms of system performance. INDEX TERMS Wireless power transfer, machine type communication, game theory, Nash equilibrium, power splitting. I. INTRODUCTION
In the age of 5G, base station and device densities will be extremely high [1]. More than 1 trillion IoT terminals and 20 billion human-oriented terminals are able to access the networks [2]. Such a large number of terminals bring many challenges. Among them, energy consumption issue has been an unavoidable topic. Some new ideas have been proposed to solve this problem from different dimensions. Utilization of renewable energy (i.e.,solar) has been regarded as a promising approach to prolong the lifetime of energy sensitive terminals, e.g., wearable devices. Compare to the conventional renewable energy source, harvesting energy from ambiance, especially from radio signals radiated by surrounding transmitters, is more reliable and available for the wearable devices. It is reported that at distances of 0.6 and 11 meters, 3.5mW and 1µW can be
20124
harvested from radio signals, respectively [3]. Apart from energy, information is also carried by the radio signals. Thus, simultaneous wireless information and power transfer (SWIPT) [4] has been an ideal approach to be adopted in energy sensitive networks such as MTC networks. However, there are still some challenges including modulation and coding scheme (MCS) selection [5], hardware design [6] (i.e., circuit design, antenna design, rectifier design and receiver architecture design) and security issues [7]. In this paper, we focus on power splitting algorithm applied in MTC networks with embedded SWIPT architecture which is one of the receiver architecture designs. A. RELATED WORKS
In general, wireless power transfer techniques are summarized into three categories [6]: RF energy transfer,
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resonant inductive coupling and magnetic resonance coupling. Compared with the other two coupling related techniques, RF energy transfer outperforms them in terms of effective distance and efficiency. Therefore, we only focus on this RF energy transfer technique in this paper. SWIPT was firstly introduced in [4], where information and energy are transmitted over a single noisy line and the tradeoff between data rate and energy are studied. Inspired by this paper, SWIPT is widely applied in various wireless networks. In [8], SWIPT is used in broadband wireless systems, where singleuser/multi-user systems, downlink/uplink information transfer, and variable/fixed coding rates are considered. SWIPT for three-node relay network, two-way relaying network and multiple relay network are designed and analyzed in [9], [10], and [11], respectively. SWIPT is also involved in cognitive radio in [12], where a wireless power transfer model is proposed. Power outage probability is derived and throughput of a single secondary user and the whole network is maximized. Another paper [13] proposes a protocol that allows secondary users harvest energy from primary users and then help the primary user to transmit its data, in which the data successful probability of both primary user and secondary user is analyzed. In addition, SWIPT for MIMO system in terms of information and energy beamforming and energy efficient resource allocation is developed and evaluated in [14] and [15]. Game theory has been widely applied in wireless communication. It is used as the main approach to address resource allocation problems. In [16], a Multiservice Uplink Power Control game (MSUPC) is formulated and the objective of each user is to maximize its own utility. Furthermore, a more efficient convex pricing power control policy is introduced and a unique Pareto optimal Nash equilibrium is determined for the formulated Multiservice Uplink Power Control game with Convex Pricing (MSUPC-CP). Besides, [17], [18], [19] also propose the optimal power allocation schemes using game theory to maximize their utilities in two-tier femtocell networks, ultra-dense small cell networks and NOMA networks, respectively. In addition to pure power allocation algorithm, joint resource allocation algorithm with a game model is also well investigated. In [20], the authors formulate a non-cooperative game model with pricing mechanism involving both utility-based uplink transmission power and rate allocation. The expected equilibrium point is obtained via a distributed, iterative and low-complexity algorithm. A supermodular game-based distributed resource allocation algorithm is proposed in [21], where the utility function represents its perceived satisfaction with respect to its allocated power and rate and user’s utility function is jointly determined by its tier and the requested service. An expected NE is obtained by a distributed and iterative algorithm derived from the formulated game model. In terms of wireless power transfer, game theory is also considered as an efficient methodology to solve the formulated problems. In [11] and [22], game theory is used in relay networks with SWIPT, where the relays not only VOLUME 5, 2017
forward the received signal to the destination but also harvest the radio frequency energy to achieve their own objectives. [11] obtains the maximum utility via optimal power splitting ratio while [22] balances the information transmission efficiency of sourcedestination pairs and the harvested energy of the relay by decomposing the Nash bargaining problem into three subproblems. In [23], multiple sourcedestination pairs with interference channels are considered. Each pair is modeled as a strategic player who aims to minimize its own transmit power under both SINR and energy harvesting constraints at the destination. The sufficient and necessary condition for the existence and uniqueness of NE and the best response strategy of each player are derived. MTC network is characterized by its large number of intelligent devices, low energy consumption devices and high communication reliability [24]. It is widely believed that MTC will take a significant place in the 5G network. However, to the best of our knowledge, no work in the open literature has considered SWIPT in MTC network except [25], where a framework of wireless powered fullduplex M2M communication (WP-FD-M2M) is proposed. The receiver transmits its surplus energy to the transmitter and meanwhile the transmitter sends data to the receiver. The energy transmission and data transmission take place over the same frequency simultaneously. By establishing a game model which is called M2M game, [25] characterizes the interaction between transmitter and receiver. In addition, data transmission and energy transfer are optimized based on the Markov strategy. Note that although [26] also considers wireless power transfer in MTC network, mutual inductance is used as the main approach, which is not in the scope of this paper. In this paper, we also focus on applying SWIPT into MTC network. But different from [11], [25] concerning system model and objective, we consider a cell involving one AP and several MTC devices. These MTC devices have no power supply or battery. The energy used for data collection, data processing and data transmission in uplink is entirely from energy harvesting. Each MTC device splits the signal received from hybrid AP in downlink direction into two streams based on a power splitting ratio: one stream is forwarded to the information processing unit and the other stream is sent to the energy harvesting unit. Later, harvested energy is entirely used for data collection, data processing and uplink transmission. In addition, compared to [11] where the performance of each user is only characterized by its own achievable rate, the achievable uplink throughput and downlink decoding SNR are considered as the performance metrics in this paper. The reason is that based on the features of MTC mentioned above, high transmission reliability has to be guaranteed because the information transmitted in downlink of MTC networks usually contains system control demand data which is used to control a certain functionality of this device. So failed transmission or unreliable transmission may cause outage or misoperation which may cause huge loss and damage to the 20125
K. Kang et al.: Game-Theory-Based Distributed Power Splitting for Future Wireless Powered MTC Networks
device. Besides, 5G is also characterized by extremely high throughput. For MTC network, extremely high throughput is not essential but a moderate throughput is necessary to ensure a smooth uplink transmission. Because of the difference of performance metrics required by MTC network, the proposed utility in this paper is different from that of [11] as well. In [11], the achievable rate of each user is directly considered as its utility while our proposed utility is actually the summation of downlink decoding SNR and uplink throughput with a weighted coefficient for each term. The key to achieving the maximum utility is the power splitting ratio of each MTC device. But this ratio does not only affects its own performance but also affects the performance of other MTC devices because when AP receives signals from one MTC device, there exists interference caused by other MTC devices, which means the performance optimization of one MTC device depends on both its own ratio and others’ ratios. Therefore, to give an optimal power splitting ratio for each MTC device, a well-designed noncooperative game model is adopted to develop a distributed power splitting algorithm for SWIPT in wireless powered MTC network.
presented in Section IV. Finally, this paper is concluded in Section V. II. SYSTEM MODEL
As illustrated in Fig. 1, this paper considers a wireless powered MTC network with SWIPT. The system consists of one hybrid AP and N MTC devices denoted by MTCDi , i = 1, · · · , N . It is assumed that the AP and all devices operate in a half-duplex mode on the same frequency band. Therefore, they interfere with each other in uplink transmission. Different from those MIMO systems, we assume that one single antenna is equipped at both AP and each device. We further assume that all devices have no power supplies. Therefore, the devices have to harvest energy from the received signals transmitted by the AP in downlink direction. After energy harvesting, the devices are capable of using the harvested energy to transmit data to the AP in uplink direction. Note that for the purpose of exposition, the energy consumed by data collection, data processing and harvesting circuit is assumed to be negligible [9] as compared to the energy consumed by data transmission.
B. CONTRIBUTIONS
In this paper, a distributed power splitting algorithm is developed based on a non-cooperative game model. To the best of our knowledge, this is the first game theory based distributed power splitting algorithm designed for MTC network employing SWIPT. Our main contributions are summarized as follows: • In order to satisfy the requirement of reliability and throughput for MTC, we develop an innovative utility function which combines both the downlink decoding SNR and uplink throughput together. Each device aims at maximizing its own utility by selecting an optimal power splitting ratio. • We develop a distributed power splitting algorithm of SWIPT in wireless powered MTC networks using game theory. In particular, each MTC device is modeled as a rational player in a non-cooperative game model. And each player selects its own power splitting ratio to maximize its individual utility. • We prove the existence of Nash equilibrium (NE) for the formulated non-cooperative game model and discuss the uniqueness of NE. Different from [11], there are multiple NEs in our game model. So we propose an equilibrium selection mechanism where the selected NE can achieve better fairness than any other NE. • All the theoretical analysis is verified by extensive numerical simulations. And the simulation results show that this proposed algorithm can achieve an optimal performance for both system and individual MTC device. The rest of this paper is organized as follows: Section II shows the system model of this SWIPT MTC networks. In Section III, the distributed power splitting algorithm is derived based on game theory. The numerical results are 20126
FIGURE 1. System model.
The downlink and uplink channels are assumed to be quasistatic flat-fading. hi and gi are denoted as the channel gain of downlink and uplink, respectively. Since no battery is embedded in the devices, the harvest-then-transmit protocol [27] is adopted in this paper. This means the MTC device will exhaust its harvested energy to implement the uplink transmission. A. DOWNLINK TRANSMISSION
In downlink direction, the AP broadcasts the control signal to all MTC devices and the signal received by the MTCDi is written as p (1) yDi = Pt hi x + ni where Pt is the transmission power of the AP and x is the transmitted information with E[|x|2 ] = 1. Note that here we assume that the transmitted information for each MTC device is identical, which means all MTC devices receive the same control demand and will implement their corresponding VOLUME 5, 2017
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functionality. ni ∼ CN (0, σA2i ) is the additive Gaussian noise introduced by the received antenna at the MTCDi .
noise at the AP. Then, the signal-to-interference plus noise ratio (SINR) of signal transmitted by MTCDi at AP can be written as γTi = PN
Qi |gi |2
2 j=1,j6=i Qj |gj |
= PN
+ σTi2
ηθi Pt |hi |2 |gi |2
2 2 j=1,j6=i ηθj Pt |hj | |gj |
Subsequently, the received signal at MTCDi goes through the architecture showed in Fig. 2. Note that this architecture is different from the one in [8] since our system is half-duplex. Then the received signal is split into √ two streams with the power splitting ratio θi . The fraction θi of the received signal is √ sent to the energy harvesting unit and the rest fraction 1 − θi is forwarded to the information processing unit. In this case, the harvested energy at MTCDi is written as Qi = ηθi Pt |hi |2
(2)
where η ∈ (0, 1] is the harvesting conversion efficiency which depends on the energy harvesting circuit. Note that we actually neglect the term ηθi σi2 in (2) for simplicity [28]. For the other stream, the signal received by the information processing unit at MTCDi is given by p p p yIDi = 1 − θi yDi + nIi = 1 − θi Pt hi x + nIi (3) where nIi ∼ CN (0, σI2i ) is the additive Gaussian noise introduced by the signal processing circuit at the MTCDi . Then the signal-to-noise ratio (SNR) for information processing at MTCDi can be expressed as Pt |hi |2 (1 − θi ) σi2
(4)
= σA2i +σI2i . After these procedures, the MTCDi has received the control information yIDi which is processed by the information processing unit and harvested energy Qi . Later, the MTCDi will utilize the harvested energy Qi to transmit in uplink. B. UPLINK TRANSMISSION
In uplink direction, since harvest-then-transmit protocol is adopted, all MTC devices transmit their data to the AP with the power of Qi simultaneously and the received signal at the AP can be expressed as p
Qi gi xi +
N X p
Qj gj xj + nTi
(5)
j=1,j6=i
where xi is the transmitted information of MTCDi with E{|xi |2 } = 1. nTi ∼ CN (0, σTi2 ) is the additive Gaussian VOLUME 5, 2017
Ri (2) = Blog(1 + γTi )
(7)
where B is the uplink bandwidth shared by all MTC devices. For simplicity, we assume a normalized bandwidth (i.e., the bandwidth is set to one). 2 is defined as 2 := [θ1 , θ2 , · · · , θi , · · · , θN ]. In this paper, we consider that the performance of each MTC device is characterized by its corresponding utility function involving both downlink decoding SNR and uplink throughput. Therefore, the sum of utility functions of all MTC devices is regarded as the system performance metric. In the following sections, a distributed power splitting algorithm is developed to achieve a global optimal performance. III. DISTRIBUTED POWER SPLITTING ALGORITHM
In this section, we focus on the design of distributed power splitting algorithm so that 2, a set of power splitting ratio, is selected to achieve optimal system performance. By taking transmission reliability and uplink transmission of each MTC device into consideration, a utility function is established as follows if γiI < γit 0, if Ri = 0 ui (2) = 0, (8) q t I bi Ri + 2ai γ − γ , else i
where σi2
yTi =
(6)
The second term on the right-hand side of (6) is obtained by substituting (2) into the first term on the right-hand side of (6). Therefore, the corresponding uplink throughput of MTCDi can be written as
FIGURE 2. SWIPT architecture [8].
γiI =
+ σTi2
i
where 2 = [θ1 , . . . , θN ] denotes the power splitting ratio vector of all MTCDs. ai and bi are two positive coefficients that ai , bi ∈ (0, 1). γit is the SNR threshold which guarantees the transmission reliability. The natural question arises is ‘‘why the utility function is designed in this form?’’. Firstly, once γiI < γit holds, mathematically the value of the corresponding utility will be a complex number which is meaningless and in practical, γiI < γit means the downlink signal is not able to be decoded correctly. Therefore, although the entire harvested energy is still used for supporting the uplink transmission, the subsequent actions executed based on this unreliable downlink commend are meaningless. So the utility is set to zero if γiI < γit . Secondly, the utility is set to zero when Ri = 0. This means the device does nothing after decoding. So the actual utility it produces is zero as well. And thirdly, in the ‘‘else’’ condition, the reason that we design the utility function in this form is that we expect to 20127
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involve both transmission reliability and uplink throughput into one equation. These two metrics are crucial for MTC networks as we state in previous sections. It is very clear that the first term in (8) is the uplink throughput and the second term in (8) represents the gap between decoding SNR and SNR threshold. These two terms contribute to the utility simultaneously. Note that different from conventional SNR constraint that making the SNR no less than the required threshold is enough, the downlink decoding SNR is expected to be much larger than the required threshold in large user quantity cases so that the utility can achieve the maximum. Moreover, this form of ui also guarantees the concavity which will be proved in the next section. Hence, by maximizing the utility function ui , our objective can be achieved. Naturally, the global optimization problem can be expressed as max 2
N X
ui (2)
i=1
s.t. 2 ∈ A
(9)
where A = {2|0 ≤ θi ≤ 1} is the feasible set of 2. However, in order to address this global optimization problem, a centralized processing unit needs to be employed in practical, which increases the cost. In addition, the signaling overhead used to report the relevant information of each device required by the centralized method is huge, especially for these low-power MTC devices. The energy used for reporting is a huge burden. Motivated by these considerations, we develop a distributed algorithm that all MTCDs are rational and strategic with objectives of maximizing their individual utility by selecting their own power splitting ratios. This optimization problem can be express as max ui (θi , 2−i ) θi
s.t. θi ∈ Ai
(10)
where 2−i = [θ1 , . . . , θi−1 , θi+1 , . . . , θN ] denotes the power splitting ratio vector including all MTCDs except the ith device. It can be observed from (8) and (10) that this optimization problem of each MTCD is coupled together due to the mutual interference which affected by 2−i . So this power splitting problem is modeled to be a non-cooperative game [29], which is represented as follows: • Players: N MTC devices. • Actions: All possible power splitting ratios determined by each individual MTC device, • Utilities: The utility function defined in (8). This formulated game can be denoted as G = hN , {Ai }, {ui (θi , 2−i )}i
(11)
In the following subsections, we focus on proving the existence and discussing the uniqueness of the solution for this formulated non-cooperative game model. 20128
A. PROOF OF EXISTENCE FOR NASH EQUILIBRIUM
Nash Equilibrium (NE) is the most well-known solution to the non-cooperative games. The definition of Nash Equilibrium indicates that no one can increase its utility by deviating from the Nash Equilibrium solution [29]. For the formulated game G, the NE satisfies the following condition, ui (θi∗ , 2∗−i ) ≥ ui (θi , 2∗−i ),
∀θi ∈ Ai , ∀2−i ∈ A−i
(12)
To prove the existence of NE, the following theorem proposed in [30] is adopted: Theorem 1: A NE exists in a non-cooperative game hN , {Pi }, {ui (2)}i if ∀i ∈ N , {Pi } is a compact and convex set. ui (2) is continuous in 2 and quasi-concave in 2i , where 2 = (θi , 2−i ). After checking the properties of the action sets Ai and utility function ui (θi , 2−i ), we have the following proposition in terms of NE existence: Proposition 1: The action sets Ai are all compact and convex. The utility function ui (θi , 2−i ) is quasi-concave in θi , ∀i ∈ N . Proof: See Appendix A. Based on the above proof, it can be concluded that the formulated game G possesses at least one NE. B. ANALYSIS OF UNIQUENESS FOR NASH EQUILIBRIUM
The uniqueness of the Nash Equilibrium is crucial for predicting the network state and the convergence issues. However, there are important scenarios where the NE is not unique. This is because that the choice of actions of different players is dependent (i.e., non-cooperative games with correlated constraints) [21]. In this section, the multi-NE scenario is proved by showing the contradiction with the uniqueness definition. The methodologies for analyzing equilibrium in wireless games have been summarized in [31]. However, since our formulated game G is concave and most of the methodologies are suitable for convex function, the standard function approach [32] is adopted to show the contradiction. First of all, we derive the response function for a given action set, which can be described as the following lemma: Lemma 1: Given a power splitting ratio profile (action set 2−i ) of all MTC devices except the ith device in game G, the response of the i th device is expressed as q Di 4Ai Ci + 4Ai + D2i − 4rit − (2Ai Ci + D2i ) RE i (2−i ) = 2Ai (13) where Ai = Ci =
PT |hi |2 σi2 PN
2 2 j=1,j6=i ηθj Pt |hj | |gj | ηPt |hi |2 |gi |2
bi ai ln2 Proof: See Appendix B. Di =
(14) + σTi2
(15) (16)
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Then we define the response vector function RE(2) = [RE 1 (2−1 ), . . . , RE N (2−N )]. And based on the fix point theorem [31], the action set 2∗ is a NE of the formulated game G if and only if it is the fixed point of RE(2). Therefore, to prove the uniqueness of NE for the formulated game G is equivalent to prove the uniqueness of the fixed point of function RE(2). Furthermore, according to [32], the fixed point of RE(2) is unique if RE(2) is a standard function, which is defined as follows: Definition 1: A function g(x) is a standard function if the following three properties are satisfied for all x ≥ 0: • Positivity: g(x) > 0. • Monotonicity: If x ≥ x 0 , then g(x) ≥ g(x 0 ). • Scalability: αg(x) > g(αx), ∀α > 1. After checking the properties of the response function RE i (2−i ), the following proposition in terms of the uniqueness of the NE is concluded. Proposition 2: There exists multiple NEs in the formulated non-cooperative game G. Proof: See Appendix C. Note that based on the analyses in Appendix A, Appendix B and Appendix C, it can be shown that the value of Di is crucial for the convergence and system state. Recall (16), Di is determined by bi /ai . So in the simulations, each bi /ai is carefully determined so that (22) (25) can be satisfied. Furthermore, since we assume that Di is identical for all devices, the value of Di is actually the maximum one among all candidate values. C. EQUILIBRIUM SELECTION
Since there are multiple NEs in the formulated game G, a natural question is how to select an appropriate equilibrium. Thanks to [33], several equilibrium selection schemes are introduced. In our paper, the fairness equilibrium selection scheme is adopted, which means the selected NE is supposed to achieve a fairer state than other NEs. The reason is that for MTC networks, in most cases the outputs of these devices need to be coordinated so that a certain system output is able to be produced. This may require fairness among all the devices. In this paper, the output is the utility. Therefore, the Jain’s index [34] is used to evaluate the fairness and select the appropriate NE which is expressed as RE(2) = arg max(F(2))
Algorithm 1 Distributed Power Splitting Algorithm 1: initial t = 1 and 2(t) = rand(N , 1); 2: for t = t + 1 do 3: θi (t + 1) = RE i (2−i (t)); 4: if (θi (t + 1) − θi (t) ≤ ) then 5: Output: θi 6: end if 7: end for
to update its own power splitting ratio so that the individual utility is maximized. Regarding the practical issues, it really depends on the necessary information needed to be collected and exchanged to implement the proposed algorithm. Based on the response function given by (13), each MTCDi has to obtain the values of parameters Ai , Ci , Di as well as γit . Note that according to the type of service it provides, γtt of each device can be different. So here are several steps that MTCDi needs to follow to perform this algorithm. • Step 1: Obtain the transmission power Pi from the AP, measure the channel gains hi and gi , and then calculate the value of Ai . • Step 2: Set γit by considering both its own transmission requirement and channel condition. • Step 3: Obtain the power of the received signal at the AP and then calculate the value of Ci . • Step 4: Set Di based on γit , Ai and Ci to ensure the constrains can be satisfied. • Step 5: Send the calculated NEs to AP. • Step 6: Obtain the selected NE sent from AP. It can be observed from this process that in order to implement this algorithm the data and overheads needed are (1) value of transmission power in the AP, (2) pilots to estimate the channel state information (CSI) in both downlink and uplink, (3) the received signal power in the AP and (4) power used for uploading all NEs. From this description, the MTC device only has to transmit several bits containing the candidate NEs and receive data from the AP which extremely saves energy for these low power devices. Furthermore, the first two data and the fourth power are just needed for one time and the third one has to be collected in each iteration.
(17) IV. NUMERICAL RESULTS
where F(·) is denoted as the fairness index function. D. DISTRIBUTED ALGORITHM
In previous subsections, we have proved the existence and discussed the uniqueness of NE as well as NE selection scheme for our formulated game G. However, this selected NE is practical and meaningful only if an algorithm which is capable of driving the system from any non-equilibrium state to the equilibrium state. In this subsection, we propose an iterative and distributed power splitting algorithm described in Algorithm 1, upon which each player (MTC device) is able VOLUME 5, 2017
So far, we have complete all theoretical analyses and in this section, several numerical results are presented to illustrate and validate the aforementioned derivations and propositions. In order to obtain meaningful insights of the system performance, the transmission power of the AP is restricted to 0 dBW, i.e., Pt = 1 W. The channels of downlink and uplink are assumed to be symmetric and subject to independent Rayleigh fading. Furthermore, by taking the impact of path loss into consideration, the channel model is E{|hi |2 } = E{|gi |2 } = di−α , where di is denoted as the distance between AP and MTCDi and the path loss factor, α, is set to be 2 [35]. 20129
K. Kang et al.: Game-Theory-Based Distributed Power Splitting for Future Wireless Powered MTC Networks
The inter distance between two neighbour MTC devices, da , is assumed to be identical. So di and da have such relation: q di = d12 + ((i − 1)da )2 , ∀i ∈ N . Without loss of generality, we set η = 0.8 in all simulations. Note that we assume the distance between two neighbour access points is unit length, e.g., daa = 1. First of all, we consider a simple scenario consisting of two MTC devices with d1 = 0.3 and da = 0.3. So d2 = 0.4243 and the corresponding channel gains of both MTC devices are |h1 |2 = |g1 |2 = 11.11 and |h2 |2 = |g2 |2 = 5.56. And γit = 0 dB. In Fig. 3, the response functions θ1 (θ2 ) and θ2 (θ1 ) of these two devices in the formulated game G are plotted, respectively. According to [31], the intersection of the response functions represents the Nash Equilibrium of the formulated game. As it is shown in Fig. 3, the two curves cross at two points, which indicate that our formulated game G in this wireless powered MTC network possesses two NEs. And after compare the fairness index of these two NEs (i.e., F(θ1∗ , θ2∗ ) = 0.5 and F(θ1NE , θ2NE ) = 0.6227), the NE (θ1NE , θ2NE ) is selected as the appropriate NE. Fig. 4 shows the convergence process of our proposed algorithm, in which the power splitting ratios of two MTC devices achieve their selected NEs from randomly initial points. Combined with Fig. 3 and Fig .4, it is very clear to conclude that the selected NE obtained in Fig. 3 is achieved by the proposed algorithm via an iterative process in Fig. 4 with random initial points. Moreover, the corresponding downlink decoding SNRs of these two devices are (γ1I , γ2I ) = (1.81, 4.611) dB which are greater than γit = 0 dB. In order to show that our proposed algorithm is also applicable for multi-device scenarios, the convergence processes of five devices are shown in Fig. 5 with randomly generated initial points, where d1 = 0.3, da = 0.1, γit = 0 dB and bi ai = 30. It can be observed from Fig. 5 that our proposed algorithm is able to converge to the same NE from two different initial points, which again validate our aforementioned theoretical analyses. The the corresponding downlink
FIGURE 4. Convergence of response functions.
FIGURE 5. Convergence of proposed algorithm for five devices with randomly initial points.
decoding SNRs of these five devices are (γ1I , γ2I , γ3I , γ4I , γ5I ) = (1.43, 1.51, 1.72, 2.12, 2.92)
FIGURE 3. The response functions of game G. 20130
which are greater than γit = 0 dB. Note that due to the space limitation in Fig. 5, the convergence value is not highlighted as it is done in Fig. 4. Fig. 6 shows the individual utility of these five MTC devices. It can be observed that there exist an extreme point for each utility and the utility is non-decreasing before this point and non-increasing after this point. Once we combine Fig. 5 and Fig. 6, it is very clear to observe that each MTC device obtains its maximum individual utility when it achieves the selected NE. Note that there is a rapid decline in each utility curve and the reason is that once γiI − γit < 0 the corresponding ui is set to be 0. Because it is believed that if downlink decoding SNR is less than the minimum threshold (i.e., γiI < γit ), it means this MTC device is not able to decode the control information sent by the AP correctly. Furthermore, the uplink data are meaningless once the incorrect functionality is executed due to the failed decoding. VOLUME 5, 2017
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cases. And the Game-T algorithm is always superior to the Equal-ST. When these MTC devices stay close to the AP, the advantage of our proposed algorithm is huge compared with the other two algorithms. However, the proposed algorithm experiences performance loss when these devices move far away to the AP, which is due to the poor channel condition. And it can be predicted that our proposed Game-ST algorithm will finally be decay to the Game-T algorithm with the increase of di since the SNR gap gain decreases with worse channel condition. But in MTC networks, all devices are lowpowered (i.e., IoT devices). Therefore, in order to ensure the transmission reliability and rate, the distance between AP and device is not too long.
FIGURE 6. The individual utility of five devices with different power splitting ratios.
Next, we investigate the average sum-utility of the wireless powered MTC network, where each MTC device executes the proposed power splitting algorithm. A five-device scenario with dynamic positions and a multi-device with fixed positions are considered. Note that in the following figures, each curve is obtained by averaging over 10000 independent simulations. And the system performance is compared between our proposed algorithm, the algorithm from [11] where θi = arg(γiI = γit ) and equal allocation algorithm (i.e., θi = 0.5). Note that our proposed algorithm is named as ‘‘Game-ST ’’ because we consider both SNR and throughput with game theory. The algorithm from [11] is named as ‘‘Game-T ’’ because only throughput is maximized using game theory. The equal allocation is named as ‘‘Equal-ST ’’ because the received signal is equally forwarded to information processing unit and energy harvesting unit Fig. 7 illustrates the average sum-rate of wireless powered MTC network consisting of 5 devices, where da = 0.3 and bi ai = 200. It can be observed that our proposed Game-ST algorithm always outperforms Game-T and Equal-ST in all
FIGURE 7. The impact of AP-device distance on average sum-utility. VOLUME 5, 2017
FIGURE 8. The impact of device number on average sum-utility.
Fig. 8 shows the impact of the number of MTC devices on the average sum-utility in a multi-device scenario, where d1 = 0.3, da = 0.05, abii = 5000 and γit = 0 dB, ∀i ∈ N . It can be observed that the performance of proposed Game-ST algorithm is always better than that of the other two algorithms. But the advantage is relatively small when the number of devices is small and the advantage rockets with the increase of the device quantity. In addition, when the number of devices is less than 40, the performances of the proposed Game-ST and Game-T algorithm are almost the same. This is because the SNR gap term dominates the utility when the device quantity is huge and the uplink throughput term dominates the utility when the device quantity is small due to less device making less interference. The Equal-ST algorithm outperforms Game-T when the device quantity exceeds 50. This is because SNR gap is always equal to zero in the Game-T algorithm. In Fig. 9, we plot the average utility curves of different SNR thresholds with γit = −5 dB, γit = −2 dB and γit = 1 dB, respectively. The system is configured as d1 = 0.3, da = 0.1 and baii = 100000. It can be observed that with lower SNR threshold, the MTC devices are able to achieve higher average utility. This is because it is relatively easier to achieve a lower SNR threshold compared to a higher one (i.e., using less received power to decode). When the 20131
K. Kang et al.: Game-Theory-Based Distributed Power Splitting for Future Wireless Powered MTC Networks
both convex and compact and the utility function ui (θi , 2−i ) is continuous in θi . Then, in order to prove the quasiconcavity of the utility function, a very straightforward way is to derive the first-order derivative of the utility function and check its positivity and negativity. Therefore, the firstorder derivative of the utility function ui (θi , 2−i ) with respect to (w.r.t) θi is derived as D2 Xi − A2i Yi2 ∂ui (θi , 2−i ) = √ √i ∂θi ln2Yi Xi (bi Xi + ai Ai Yi ln2)/(ai ln2)2 (18)
FIGURE 9. The impact of device number on average utility.
where Xi = Ai (1 − θi ) − γit , Yi = θi + Ci are defined for simplicity of notations. Ai , Ci , Di are defined in (14), (15) and (16), respectively. It is observed that the sign of (18) is determined by the numerator 0i (θi ) = D2i Xi − A2i Yi2
number of devices is less than 5, the gaps between these three curves are not huge, which is because the first term, throughput term, dominates the performance of the algorithm as it is analyzed above. With the increase of the number of devices, the multi-user diversity gain contributed by the second term is significant, which also follows the aforementioned analysis. However, once the number of devices exceeds a certain value (i.e., N = 46 for γit = −5 dB, N = 47 for γit = −2 dB, N = 32 for γit = 1 dB), the average utility decreases, which is because the diversity gain of the second term is not enough to compensate the throughput loss due to the huge mutual interference among these MTC devices. V. CONCLUSION
In this paper, a game theory based algorithm that solves the power splitting problem via a distributed manner for simultaneous wireless information and power transfer (SWIPT) in wireless powered MTC networks is developed. A noncooperative game is formulated and each device is modeled as a rational and strategic game player with its own objective of maximizing its individual utility through a careful selection of the received power splitting ratio. Then we prove Nash Equilibriums (NEs) do exist in the formulated game by a series of mathematical derivations. And in order to select the appropriate NE among candidate NEs, a fairness based selection scheme is adopted. Moreover, the power splitting algorithm is developed based on the response function, which is proved to converge to the selected NEs. The numerical results show that starting from any randomly initial points, the proposed algorithm is able to converge to the selected NEs which maximize the utility. Furthermore, the proposed algorithm also achieves the system-level optimum compared with conventional algorithms. APPENDIX A PROOF OF PROPOSITION 1
First of all, based on the definition of convex set [36] and compact set [37], it is straightforward to observe that 2i is 20132
= D2i (Ai (1 − θi ) − γit ) − A2i (θi + Ci )2 = −A2i θi2 −(2A2i Ci + D2i Ai )θi − A2i Ci2 − D2i γit +D2i Ai (19) It can be observed that 0i (θi ) is a quadratic function w.r.t θi . In order to determine the sign of (18), we further investigate the properties of (19) in the feasible set θi ∈ [0, 1]. First of all, we have 0i (1) = −A2i − 2A2i Ci − A2i Ci2 − D2i γit 0i (0) =
−A2i Ci2
− D2i γit
+ D2i Ai
(20) (21)
where (20) always holds negative value 0i (1) < 0 and this requires that (21) should always holds positive value 0i (0) > 0. So we have the first constrain on Di , which is expressed as Di > q
Ai Ci Ai − γit
(22)
Thus, based on the anlyses above, we can conclude that there exists a point λi ∈ [0, 1] that makes 0i (λi ) = 0, 0i (θi ) > 0 for θi < λi and 0i (θi ) < 0 for θi > λi . This can be further concluded that on the feasible domain (i.e., θi ∈ [0, 1]), the utility function ui (θi , 2−i ) is increasing when θi < λi and decreasing when θi > λi . Hence, the utility function ui (θi , 2−i ) can be claimed to be quasi-concave in θi . Additionally, according to Theorem 1, our formulated game G possesses at least one NE. APPENDIX B PROOF OF LEMMA 1
Based on the analysis in Appendix A, for a given power splitting ratio profile 2−i , the expression of λi is actually the response function of MTCDi . Therefore, we give the solution of 0i (λi ) = 0 which is expressed as q Di 4Ai Ci + 4Ai + D2i − 4rit − (2Ai Ci + D2i ) (23) λi = 2Ai VOLUME 5, 2017
K. Kang et al.: Game-Theory-Based Distributed Power Splitting for Future Wireless Powered MTC Networks
since λi is the response function of the corresponding given 2−i , we set RE i (2−i ) = λi . Then the expected result in (13) is obtained. This completes the proof. APPENDIX C PROOF OF PROPOSITION 2
In order to prove the response function RE i (2−i ) is standard, we have to show RE i (2−i ) follows the three properties given in Definition 1. (1) Positivity: It can be observed from (13) that the positivity of RE i (2−i ) is determined by numerator, which is defined as q ωi = Di 4Ai Ci + 4Ai + D2i − 4rit − (2Ai Ci + D2i ) (24) In order to satisfy the positivity, ωi has to be greater than 0, ωi > 0. After some mathematical manipulations, we obtain the second contrain on Di that ensures the positivity, which is expressed as Ai Ci (25) Di > q Ai − γit which is just exactly the same as (22). (2) Monotonicity: Recall the definitions of Ai , Ci and Di in (14), (15) and (16). It can be observed that only Ci is related to 2−i . Assume 2−i and 20−i are two different action profiles and hold 2−i > 20−i . Therefore, the corresponding response funtion of 2−i and 20−i can be written as RE i (2−i ) =
q Di 4Ai Ci + 4Ai + D2i − 4rit − (2Ai Ci + D2i ) 2Ai (26)
RE i (20−i ) =
q Di 4Ai Ci0 + 4Ai + D2i − 4rit − (2Ai Ci0 + D2i ) 2Ai (27)
where Ci0 = (ηPt |hi |2 |gi |2 ) and
PN
( j=1,j6=i ηθj0 Pt |hj |2 |gj |2 + σTi2 )/ Ci > Ci0 is held for 2−i > 20−i .
It can be seen that the only difference between (26) and (27) lies in Ci and Ci0 . Thus, proving RE i (2−i ) > RE i (20−i ) is equivalent to proving RE i (Ci ) > RE i (Ci0 ). Furthermore, we just need to prove that RE i (Ci ) is non-decreasing w.r.t Ci . So we have Di ∂RE i (Ci ) =q −1 (28) ∂Ci 4A C + 4A + D2 − 4γ t i i
since
Ai >qγit
i
i
i
and Ci > 0, then Ai (1 + Ci ) > γit holds.
Therefore, D2i + 4Ai (1 + Ci ) − 4γit > Di , which means ∂RE i (Ci )/∂Ci < 0. This contradicts the Monotonicity in Definition 1. So we can conclude that there exist multiple NEs in the formulated game G. VOLUME 5, 2017
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KANG KANG received the B.S. degree in electrical engineering from the Qingdao University of Science and Technology, Qingdao, China in 2012, and the M.S. degree in telecommunication from the University of New South Wales, Sydney, Australia, in 2014. He is currently pursuing the Ph.D. degree in wireless communication engineering with the Waseda University, Tokyo, Japan. His research interests include radio resource management, scheduling, game theory in wireless communication, wireless power transfer, machine type communication, and cognitive radio. RONG YE received the B.S. degree from the Nanjing University of Posts and Telecommunications, China, in 2010. She is currently pursuing the master’s degree in wireless communication engineering with Waseda University, Tokyo, Japan. Her current research interests include wireless information and power transfer, wireless powered networks, full-duplex wireless communication, and relay-based multi-hop communication.
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ZHENNI PAN received the B.S. degree in computer science engineering from China Agricultural University, Beijing, China, in 2007, and the M.S. degree in information and telecommunications from the Graduate School of Global Information and Telecommunication Studies, Waseda University, Japan, in 2011, where she is currently pursuing the Ph.D. degree. She has been a Research Associate with the Graduate School of Global Information and Telecommunication Studies, Waseda University, since 2013. Her research interests include mobile communications, wireless sensor networks, intelligent transportation systems, and optical communication.
JIANG LIU received the B.S. degree in electronic engineering from the Chongqing University of Technology, China, in 2001, and the M.S. degree and the Ph.D. degree in information and telecommunications from the Graduate School of Global Information and Telecommunication Studies, Waseda University, Japan, in 2006 and 2012, respectively. She was a Research Associate with the Graduate School of Global Information and Telecommunication Studies, Waseda University, from 2009 to 2012. She is currently an Assistant Professor with the International Center for Science and Engineering Programs, Faculty of Science and Engineering, Waseda University. Her research interests include optical wireless communications, intelligent transportation systems, and smart grid systems.
SHIGERU SHIMAMOTO received the B.E. and M.E. degrees in communications engineering from the University of Electro Communication, Tokyo, Japan, in 1985 and 1987, respectively, and the Ph.D. degree from Tohoku University, Japan, in 1992. From 1987 to 1991, he was with NEC Corporation. From 1991 to 1992 and from 1992 to 1993, he was an Assistant Professor with the University of Elector Communication and the Department of Computer Science, Gumma University, respectively. From 1994 to 2000, he was an Associate Professor with the Department of Computer Science, Gumma University. Since 2002, he has been a Professor with GITS, Waseda University. In 2008, he also served as a Visiting Professor with Stanford University, USA. Since 2014, he has been a Professor with the Department of Communications and Computer Engineering, Waseda University. His research interests include satellite and mobile communications, access scheme, and unmanned aerial vehicle for disaster networks.
VOLUME 5, 2017
