# Fair and Efficient Rate Allocation for Wireless-Powered Machine-Type Communication Networks

Authors: Nanxing Liao; Guopeng Zhang; Jiansheng Qian; Deqiang Cheng; Kun Yang
Journal: Mobile Information Systems
 (2019)

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## Abstract

This paper proposes a bargaining game theoretic rate allocation scheme for wireless-powered machine-type communications (MTCs). In the considered body area MTC network (MTCN), a battery-powered user equipment (UE) acting as the MTC gateway (MTCG) is responsible for collecting the information uploaded by in/on body wireless-powered MTC devices (MTCDs). By solving the Nash bargaining solution (NBS) of the proposed cooperative game, the minimum rate requirements of the MTCDs are satisfied. In addition, the network resource can be allocated to the MTCDs in afair and efficient manner regarding the difference of their channel qualities. In comparison to other traditional resource allocation methods, the simulation results show that the proposed NBS-based method obtains a good tradeoff between the system efficiency and per-node fairness.


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## Body

## 1. Introduction

Machine-to-machine (M2M) communication is a way of enabling networked devices to exchange information without the assistance of humans. It belongs to an Internet-of-things (IoT) technology. Wireless M2M communication is also referred to as machine-type communication (MTC) in 3GPP LTE-A specifications [1], which has been widely used in smart city, smart grid, wearable networks, and vehicle interconnection. Nowadays, due to the increase of aging population, interests in healthcare monitoring system based on MTC have also grown considerably [2]. Medical devices carried in/on a human body can detect medical events and enable continuous monitoring of patient health remotely during their daily activities [3]. Despite the technological progress, several major challenges should be dealt with in order to support MTC in mobile networks.Firstly, MTC includes a large number of MTC devices (MTCDs) which carry common service features of small data transmission with strict real-time constraints. Connecting these MTCDs directly to base stations (BSs) will lead to an unreasonably low ratio between data payload and required control information [4]. In addition, as in wireless sensor networks (WSNs), battery recharge or replacement is not always feasible for MTCDs [5]. For example, MTCDs deployed in/on a human body for medical care may include wearable and implantable devices. Battery replacement might damage the network and even jeopardize the patient’s health, but increasing battery capacity is not feasible for such a network, since it would lead to an increase in device dimension and weight.Regarding the first issue, a new network element termed as MTC gateway (MTCG) is ordered to be deployed in LTE-A cellular networks [6]. It is used to collect information from a group of MTCDs and forward the information to a BS. An MTCG could be a network infrastructure or a normal user equipment (UE). The data link from an MTCG to a BS is based on LTE-A specifications, while the links from an MTCD to an MTCG can either be via Wi-Fi, Bluetooth, or other short range wireless communication protocols. This leads to a hierarchical network structure for utilizing the cellular spectrum more efficiently [7].As far as the second issue is concerned, the recent technology termed as wireless power transfer (WPT) has been proposed for effective energy provision over middle or short range [8, 9]. Three main WPT methods are magnetic resonance, magnetic induction, and radio frequency (RF) energy harvesting. Magnetic induction is the most popular WPT method because it is easy to implement. Most modern cellphones aim to be compliant with the magnetic induction standard, Qi [10]. However, it does not have the best user experience, as the Qi device and charging table must be precisely aligned. Magnetic resonance offers an improved user experience as it does not require receiver and transmitter alignment, but it can only send power to a device from at most an inch away [10]. In comparison to these two methods, RF energy harvesting receivers can power devices up to 40 feet from the energy transmitter [11], and the receiver and transmitter do not even need line-of-sight to transfer power. The challenge with RF energy harvesting is that the amount of power that can be transferred is limited. Thus, it is best suited to power wireless sensors and small devices of IoT.This paper takes wireless wearable network as an example system of MTC network. As shown in Figure1, the network is deployed in a hospital environment for reliable and real-time health monitoring [12, 13]. Based on the above analysis, we assumed that the RF energy harvesting technology is adopted by the system. Dedicated power beacons (PBs) which can radiate RF energy continuously are preinstalled in the hospital ward, and MTCDs combined with rechargeable storage devices (e.g., supercapacitors) can thus harvest energy from the PBs.Figure 1
The considered WP-MTCN model.It is noted that RF signals can carry energy as well as information at the same time, which is referred to as simultaneous information and power transfer (SWIPT) in literatures [14]. In addition, RF energy sources are inexhaustible but unreliable. These two features change the network designers’ options considerably. In order to ensure that MTCDs will always have enough energy to operate, the time switching between the wireless power transfer and wireless information transfer (WIT) should be carefully designed. In [12], the performance of the WPT-enabled wearable networks is investigated from the system point of view. The MTCDs are assumed to be distributed in a hospital environment according a Poisson cluster process. Each MTCD transmits its messages to an MTCG, and the MTCG can then notify the nearest medical personnel. Based on this operation mode, the probability of correct notification for a clustered network is provided in close form. In [13], the authors studied a dense WPT-enabled wireless sensor network. By examining two different communication scenarios (i.e., the direct communication scenario and the cooperative relay scenario) for data exchange, the theoretical expressions for the probability of successful communication are provided. In contrast to [12, 13], another point of view is to concern with the individual performance of each MTCD in a network. In [15], the authors propose a maximum-rate method to maximize the sum rate of a group of users given a total time constraint. Since the BS has integrated the functions of PB and MTCG, the maximum-rate method can jointly optimize the time allocation for the WPT links (from the BS to the users) and the WIT links (from the users to the BS). It is also indicted in [15] that the maximum-rate method bears the notably unfairness problem, termed as the “doubly near-far problem,” as more transmission time and rate are allocated to the users close to the BS than the users far from the BS. To address the unfairness problem, the authors further proposed in [15] the common-rate method which can assign all users equal rates in the WIT phase, regardless of their distances to the BS. Additionally, the authors in [16] proposed the common-energy method which can equalize the average energy harvested by the users in the WPT phase. However, both the common-rate and common-energy methods protect users with worse channel conditions, but penalize users with better channel conditions; thus, the system efficiency is greatly deteriorated. Besides, both the approaches are not easy to take into account the notion that different MTCDs might require different levels of quality of service (QoS), which can be defined in terms of delay, data rate, or packet loss [17]. The authors in [18] extended the single RF source model to a more complex scenario with multiple RF sources. The fair and efficient issue is address by using the marginal utility theory not the cooperative game theory adopted in this paper. Therefore, the authors did not address the QoS satisfaction issue of the user devices.Different from the above works, this paper addresses thefairness and efficiency issue of resource allocation in a joint manner. Particularly, the resource allocation problem for a group of MTCDs is modeled as an optimization problem from the perspective of cooperative game theory [19]. Considering different MTCDs may require different level of QoS (defined as the minimum transmission rate), the goal of the proposed game problem is to maximize the sum rate of all the MTCDs while fulfilling their minimum rate demands. Although there are many kinds of solution methods for a cooperative game, we select the Nash bargaining solution (NBS) [19] as it provides a Pareto optimal resource allocation for game players. By solving the NBS of the game problem, the desired fair and efficient properties can be finally obtained. The major contributions of this paper are as follows:(1)
A frame structure over time-and-spectrum domain is proposed for SWIPT. This is necessary to analyze the resource allocation problem in a hierarchical SWIPT network.(2)
The utility function for a MTCD is defined based on the achievable data rate. This makes the formulated cooperative game problem be with a clear physical meaning.(3)
By solving the NBS of the proposed game problem, the optimal duty time for the WPT and WIT links is jointly found in close form. As a result, the communication and energy resource are allocated to a group of MTCDs fairly and efficiently in the Pareto optimal sense. It means that after the minimum rate demands of all the MTCDs are satisfied, the remaining resource can be allocated to the MTCDs according to their channel conditions.The rest of this paper is organized as follows. Section2 introduces the considered system model. Section 3 formulates the resource allocation problem as a cooperative game. Section 4 solves the game problem in close form. Section 5 provides the simulation results. Section 6 concludes the paper.


## 2. System Model

The considered wireless-powered MTCN is deployed in a 3D room space as shown in Figure1. A set K of K MTCDs is implanted on/in a human body, thus constituting a healthcare monitoring system. The MTCDs can sense vital signs (e.g., body temperature, heart rate, blood pressure, and ECG and EEG signals) of the human body. The sensing information is firstly transmitted to the UE which serves as the gateway for the MTCN. Then, the UE forwards the collected information to the core network via the BS. The information is finally conveyed to the healthcare system database for analysis. A low-cost PB is also installed on the ceiling of the room, which is capable of wirelessly broadcasting energy. In this paper, we assume that the MTCDs are battery-less but can harvest the energy broadcasted by the PB, while the UE is equipped with a rechargeable battery as it can be conveniently removed and charged. Therefore, only the energy consumption of the MTCDs is concerned. In addition, we also assume that the PB is with the single antenna setting. Although deploying multiple antennas at an energy transmitter [20] can improve the efficiency of wireless power transfer (note that it is difficult to deploy multiple antennas at an energy receiver due to the limited size of an on/in-body device.), this paper does not adopt this technology because the paper focuses on scheduling the duty time for the WPT phase and the WIT phase in SWIPT, so the WPT efficiency improvement (resulted from adopting multiple antenna or other techniques) does not affect the resource allocation result. In order to facilitate the sequel analysis, the following assumptions are also made:(1)
The cellular uplink (from the UE to the BS) operates in the LTE-A bands, while the WPT links (from the PB to the MTCDs) and the WIT links (from the MTCDs to the UE) operate in the ISM bands.(2)
In order to prevent mutual interference, the WPT and WIT links work on a strict time division multiplexing (TDM) as shown in Figure2. Denote the duration of each time block by T. The first τ0 (0<τ0<1) time in a block is assigned to the WPT link while the remaining 1−τ0 time can then be assigned to the WIT links. During the 1−τ0 time, the K MTCDs can transmit their information independently to the UE by using time division multiple access (TDMA).Figure 2
The time allocation pattern for the WPT and WIT phases.This paper only studies the resource allocation problem among the WPT and the WIT links. For easy analysis, the unit block time,T, is normalized to 1. Denote the transmission time assigned to the ith (∀i∈K) MTCD in a block by τi (0≤τi≤1−τ0). The following time constraint should be satisfied in the resource allocation:(1)τ0+∑i=1Kτi=1.Assume both the WPT and the WIT links are with quasistatic flat-fading channels. Denote the channel power gain from the PB to theith MTCD by gPB,i and from the ith MTCD to the UE by gi,UE. Hence, gPB,i and gi,UE for ∀i∈K remain constant during one block but can vary from one block to another. In addition, we also assume that the MTCDs adopt the harvest-then-transmit (HT) protocol as in [14–16], and the average power harvested at the end of the energy harvesting (EH) circuitry is a linear function of the average input power. Denote the transmission power at the PB in the WPT phase by PPB. The amount of energy harvested by the ith MTCD can be given by(2)Ei=ξiPPBgPB,iτ0,∀i∈K,where ξi (0<ξi<1) is the EH efficiency at the receiver of the ith MTCD. It is noted that recent works [21, 22] show that the usage of linear model may not properly model the power dependent EH efficiency, but this ideal model can well reveal the mechanism of the proposed cooperative game theoretic approach on resource allocation. It is left for our future work to select a suitable nonlinear power conversion model and obtain the optimum performance of an SWIPT system.In the WPT phase, the MTCDs replenish their energy. In the subsequent WIT phase, the MTCDs can transmit their information to the UE in the allocated time period. As in [14–16], we assume that the ith MTCD uses a fixed portion αi (0<αi<1) of the harvested energy for WIT. The average transmission power at the ith MTCD can be given by(3)Pi=αiEiτi,∀i∈K.Denote the noise power at the receiver of the UE byσ2. Using equations (2) and (3), we can calculate the achievable data rate of the ith MTCD by(4)fτ0,τi=τilog21+Ai⋅τ0τi,∀i∈K,where Ai=αiξiPPBgPB,igi,UE/σ2 is a constant during the resource allocation in one block. From equation (4), we note that the parameters τ0 and τi (∀i∈K) should be jointly optimized and thus can lead to a fairness rate allocation among the MTCDs.


## 3. Problem Formulation

It has been shown in [15, 16] that the unfairness problem of resource allocation is resulted from the large difference of the distance from the end users to the BS. Although the MTCDs installed in/on the human body have almost the same distance to the PB as well as to the UE, clothes on body, human tissues, and human posture changes can cause significant signal attenuation with shadowing duration of 80 ms [23]. The different shadowing for the WPT and the WIT links may incur the unfairness problem again, as it is an inherent problem with respect to the channel quality.In this paper, we assume that different MTCDs are associated with different applications. The QoS demand of theith MTCD is defined as its minimum rate requirement, which is denoted by Rimin (Rimin≥0). The major concern is how to fairly allocate the transmission rate among the MTCDs (i.e., satisfy their minimum rate requirements) while maximizing their sum rate. To this end, we use the NBS method borrowed from the cooperative game theory [5].


### 3.1. Basic Concepts and Theorems of Cooperative Game Theory

The cooperative game theory can be described as follows [19]. Let K=1,2,…,K be the set of K players. Let Ω be a closed and convex subset of ℝK to represent the set of feasible payoff allocations that the players can get if they all work cooperatively. Let Rimin be the minimal payoff that the ith player would expect; otherwise, the player will not cooperate. Let R=R1,R2,…,RK and Rmin=R1min,R2min,…,RKmin. Suppose R∈Ω∣Ri≥Rimin,∀i∈K is a nonempty bounded set. The pair K,Rmin is called a K-person cooperative game. Within the feasible set Ω, the notion of Pareto optimal as a selection criterion for the game solutions can be formally defined as follows.Definition 1.
The pointR=R1,R2,…,RK is said to be Pareto optimal, if and only if there is no other allocation R′=R1′,R2′,…,RK′ such that Ri′≥Ri for ∀i∈K and Ri′≥Ri, ∃i∈K.
The solution of a cooperative game can be obtained by several methods, e.g., Nash bargaining solution (NBS) [20] and Raiffa–Kalai–Smorodinsky bargaining solution (RBS). Among them, NBS provides a unique and Pareto optimal operation point, which encapsulates the requirement of yielding Pareto optimal in a payoff sense. Besides, the NBS should satisfy other five axioms, i.e., individual rationality, feasibility, independence of irrelevant alternatives, independence of linear transformations, and symmetry. The detailed explanation for these axioms can be found in [19]. The following theorem shows that there is exactly one NBS that satisfies the above axioms [19].Theorem 1.
(existence and uniqueness ofNBS). Let the payoff functions Ri, ∀i∈K be convex, upper-bounded defined on Ω. Let K be the set of indices of users who are able to achieve a performance strictly superior to their minimal payoff. Then, there exists a unique NBS that verifies R=R1,R2,…,RK∈Ω∣Ri≥Rimin,∀i∈K by solving the unique solution of the maximization problem:(5)argmaxR∈Ω,Ri≥Rimin,∀i∈K∏i=1KRi−Rimin.


### 3.2. Resource Allocation Game Formulation

Based on the cooperative game theory [19], the resource allocation for the considered SWIPT system can be formulated as the following optimization problem:(6)maxQQ=∑i=1KlnRi−Rimin,(6.1)s.t.τ0+∑i=1Kτi≤1,∀i∈K,(6.2)Ri≤fτ0,τi,(6.3)Ri−Rimin≥0,∀i∈K.It is noted that the standard objective function of the cooperative game problem (6) was to be Q¯=∏i=1KRi−Rimin [24]. Since Q¯ is nonconcave, we thus transform Q¯ as a concave function by taking advantage of the strictly increasing property of ln⋅ function.The objective of the cooperative game problem (6) is to achieve a Pareto optimal rate allocation among the MTCDs. In detail, the set of K MTCDs can be taken as the bargainers in a rate allocation game. R1min,…,RKmin and R1,…,RK consist of the disagreement point and an agreement point for the bargainers, respectively. The goal of the cooperative game problem (6) is to find a feasible agreement R1∗,…,RK∗; thus, Ri∗≥Rimin for ∀i∈K, and there is no other rate allocation R1′,…,RK′ which can lead to superior performance for some MTCDs without causing performance deterioration for some other MTCDs. The property of the problem (6) is summarized in Proposition 1.Proposition 1.
The optimization problem (6) is a convex optimization problem.Proof.
It is noted that the objective functionQ of the problem (6) is a strictly concave function. The constraints (6.1) and (6.2) are linear functions and thus are convex. Next, we can just focus on the property of the constraint (6.3).
To verify that the constraint (6.3) is convex, we first transform the constraint (6.3) as(7)xi=Ri−fτ0,τi≤0.
Letxi=xi,1+xi,2, where xi,1=Ri and xi,2=−fτ0,τi. We note that xi is the sum of a linear function xi,1 which is convex, and a nonlinear function xi,2. If xi,1 and xi,2 are both convex functions, so is their sum xi. Hence, we just need to verify that for ∀i∈K, xi,2 is a convex function over τ0,τi.
From [24], we know that if the Hessian matrix of xi,2 is positive semidefinite, xi,2 is a convex function. The Hessian matrix of xi,2 can be given by(8)∇2xi,2=dm,ni,0≤m,n≤K,∀i∈K,where m and n represents the row and the column of the matrix, respectively. Let Bi=1+Ai⋅τ0/τi; the diagonal entries and the off-diagonal entries of ∇2xi,2 can be given by(9)dm,ni=Ai2τi−1Bi−2ln2,m=0,Ai2τi−3Bi−2τ0−2ln2,m=i,m=n,0,otherwise,dm,ni=−Ai2τi−2Bi−2τ0ln2,m=0,n=i,0,otherwise,respectively. In order to verify that ∇2xi,2 is a positive semidefinite matrix, we must show that for all real vectors VT=v0,v1,v2,…,vK≠0, VT∇2xi,2V>0.
After some algebraic operation, we indeed have(10)VT∇2xi,2V=1ln2v0v0Ai2Bi2τi−viAi2τ0Bi2τi2−viv0Ai2τ0Bi2τi2−τ02Ai2viBi2τi3=1ln2Ai2Bi2τi3viτ0−v0τi2≥0,which indicates that ∇2xi,2 is positive semidefinite.
We can then conclude that the objective functionQ of the problem (6) is strictly concave and the constraint set (6.1)∼(6.3) is convex. Problem (6) is a convex problem [24].


## 3.1. Basic Concepts and Theorems of Cooperative Game Theory

The cooperative game theory can be described as follows [19]. Let K=1,2,…,K be the set of K players. Let Ω be a closed and convex subset of ℝK to represent the set of feasible payoff allocations that the players can get if they all work cooperatively. Let Rimin be the minimal payoff that the ith player would expect; otherwise, the player will not cooperate. Let R=R1,R2,…,RK and Rmin=R1min,R2min,…,RKmin. Suppose R∈Ω∣Ri≥Rimin,∀i∈K is a nonempty bounded set. The pair K,Rmin is called a K-person cooperative game. Within the feasible set Ω, the notion of Pareto optimal as a selection criterion for the game solutions can be formally defined as follows.Definition 1.
The pointR=R1,R2,…,RK is said to be Pareto optimal, if and only if there is no other allocation R′=R1′,R2′,…,RK′ such that Ri′≥Ri for ∀i∈K and Ri′≥Ri, ∃i∈K.
The solution of a cooperative game can be obtained by several methods, e.g., Nash bargaining solution (NBS) [20] and Raiffa–Kalai–Smorodinsky bargaining solution (RBS). Among them, NBS provides a unique and Pareto optimal operation point, which encapsulates the requirement of yielding Pareto optimal in a payoff sense. Besides, the NBS should satisfy other five axioms, i.e., individual rationality, feasibility, independence of irrelevant alternatives, independence of linear transformations, and symmetry. The detailed explanation for these axioms can be found in [19]. The following theorem shows that there is exactly one NBS that satisfies the above axioms [19].Theorem 1.
(existence and uniqueness ofNBS). Let the payoff functions Ri, ∀i∈K be convex, upper-bounded defined on Ω. Let K be the set of indices of users who are able to achieve a performance strictly superior to their minimal payoff. Then, there exists a unique NBS that verifies R=R1,R2,…,RK∈Ω∣Ri≥Rimin,∀i∈K by solving the unique solution of the maximization problem:(5)argmaxR∈Ω,Ri≥Rimin,∀i∈K∏i=1KRi−Rimin.


## 3.2. Resource Allocation Game Formulation

Based on the cooperative game theory [19], the resource allocation for the considered SWIPT system can be formulated as the following optimization problem:(6)maxQQ=∑i=1KlnRi−Rimin,(6.1)s.t.τ0+∑i=1Kτi≤1,∀i∈K,(6.2)Ri≤fτ0,τi,(6.3)Ri−Rimin≥0,∀i∈K.It is noted that the standard objective function of the cooperative game problem (6) was to be Q¯=∏i=1KRi−Rimin [24]. Since Q¯ is nonconcave, we thus transform Q¯ as a concave function by taking advantage of the strictly increasing property of ln⋅ function.The objective of the cooperative game problem (6) is to achieve a Pareto optimal rate allocation among the MTCDs. In detail, the set of K MTCDs can be taken as the bargainers in a rate allocation game. R1min,…,RKmin and R1,…,RK consist of the disagreement point and an agreement point for the bargainers, respectively. The goal of the cooperative game problem (6) is to find a feasible agreement R1∗,…,RK∗; thus, Ri∗≥Rimin for ∀i∈K, and there is no other rate allocation R1′,…,RK′ which can lead to superior performance for some MTCDs without causing performance deterioration for some other MTCDs. The property of the problem (6) is summarized in Proposition 1.Proposition 1.
The optimization problem (6) is a convex optimization problem.Proof.
It is noted that the objective functionQ of the problem (6) is a strictly concave function. The constraints (6.1) and (6.2) are linear functions and thus are convex. Next, we can just focus on the property of the constraint (6.3).
To verify that the constraint (6.3) is convex, we first transform the constraint (6.3) as(7)xi=Ri−fτ0,τi≤0.
Letxi=xi,1+xi,2, where xi,1=Ri and xi,2=−fτ0,τi. We note that xi is the sum of a linear function xi,1 which is convex, and a nonlinear function xi,2. If xi,1 and xi,2 are both convex functions, so is their sum xi. Hence, we just need to verify that for ∀i∈K, xi,2 is a convex function over τ0,τi.
From [24], we know that if the Hessian matrix of xi,2 is positive semidefinite, xi,2 is a convex function. The Hessian matrix of xi,2 can be given by(8)∇2xi,2=dm,ni,0≤m,n≤K,∀i∈K,where m and n represents the row and the column of the matrix, respectively. Let Bi=1+Ai⋅τ0/τi; the diagonal entries and the off-diagonal entries of ∇2xi,2 can be given by(9)dm,ni=Ai2τi−1Bi−2ln2,m=0,Ai2τi−3Bi−2τ0−2ln2,m=i,m=n,0,otherwise,dm,ni=−Ai2τi−2Bi−2τ0ln2,m=0,n=i,0,otherwise,respectively. In order to verify that ∇2xi,2 is a positive semidefinite matrix, we must show that for all real vectors VT=v0,v1,v2,…,vK≠0, VT∇2xi,2V>0.
After some algebraic operation, we indeed have(10)VT∇2xi,2V=1ln2v0v0Ai2Bi2τi−viAi2τ0Bi2τi2−viv0Ai2τ0Bi2τi2−τ02Ai2viBi2τi3=1ln2Ai2Bi2τi3viτ0−v0τi2≥0,which indicates that ∇2xi,2 is positive semidefinite.
We can then conclude that the objective functionQ of the problem (6) is strictly concave and the constraint set (6.1)∼(6.3) is convex. Problem (6) is a convex problem [24].


## 4. Optimization via Dual Decomposition

In this section, we try to solve the optimization problem (6) in close form. It is noted that the problem (6) satisfies Slater’s condition. So, the optimal solution to the problem (6) can be recovered by solving its dual problem. In what follows, we give a detailed solution process by employing the dual decomposition method.


### 4.1. Dual Problem Formulation

In order to formulate the dual problem for the primal problem (6), we introduce the Lagrangian multipliers λ≥0 and μ=μ0,μ1,…,μK≥0 to the constraints (6.1) and (6.2), respectively. The partial Lagrangian is expressed by(11)LR,τ,λ,μ=∑i=1KlnRi−Rimin−μiRi+∑i=1Kμiτilog21+Ai⋅τ0τi−λτi+λ1−τ0,where R=R1,…,RK and τ=τ0,τ1,…,τK are the feasible rate allocation vector and time allocation vector for the problem (6), respectively. Then, the dual function is given by(12)gλ,μ=maxR,τLR,τ,λ,μ.The dual problem of the primal problem (6) can be formulated as(13)minλ≥0,μ≥0gλ,μ,(13.1)s.t.Ri−Rimin≥0,∀i∈K.As the dual gap (i.e., the difference between the optimal solution to the primal problem (6) and that to the dual problem (13)) is zero [24]. The optimal solution to the primal problem (6) can be recovered by solving the dual problem (13).


### 4.2. Solution to the Dual Problem

It is noted that for a givenλ¯≥0 and μ¯=μ¯0,μ¯1,…,μ¯K≥0, the Lagrangian (11) consists of two sets of variables, i.e., the rate allocation vector R and the time allocation vector τ. So, we can decompose the dual problem (13) into the following rate allocation problem:(14)maxRQ1,Q1=∑i=1KlnRi−Rimin−μ¯iRi,(14.1)s.t.Ri−Rimin≥0,∀i∈K,and the time allocation problem(15)maxτQ2Q2=∑i=1Kμ¯iτilog21+Ai⋅τ0τi−λ¯τi+λ¯1−τ0.By solving the problem (14), the optimal rate allocation vector R∗=R1∗,…,RK∗ can be easily obtained as(16)Ri∗=1μ¯i+RiminRiminRimax,∀i∈K,where the function zba denotes the projection of z on [a, b]. In equation (16), Rimax (Ri∗≤Rimax for ∀i∈K) is introduced to limit the maximum rate that can be allocated to the ith MTCD. Rimax is set as a constant and does not change the optimal solution to the primal problem (6) [25].Now, we need to deal with the problem (15). By setting the first-order derivative of Q2 with respect to τ0 and τ1,…,τK equal to zero, respectively, we can get(17)∑i=1Kμ¯iAi1+Ai⋅τ0/τi=λ¯ln2,(18)μ¯iln1+Ai⋅τ0τi−μ¯iAi/τi1+Ai⋅τ0/τi=λ¯ln2,i∈K.From equation (17), we can derive(19)A1τ1=A2τ2=⋯=AKτK=C,where C (C>0) is a constant. By substituting equation (19) into equation (18), we can rewrite equation (18) as(20)ylny=y−D+1,where D=∑i=1Nμ¯iAi/μ¯i is a constant and y=1+Cτ0.Then, by solving equation (20), we can get the optimal solution of τ0 as(21)τ0∗=y∗−1D+y∗−1.It is noted that the optimal time allocationτ∗=τ0∗,τ1∗,…,τK∗ should satisfy the equality in the constraint (6.1). That means(22)τ0∗+∑i=1Kτi∗=1.Otherwise, the remaining1−τ0∗+∑i=1Kτi∗ time can be allocated to the PB or any one of the MTCDs, which will increase the value of Q without decreasing the achievable rates of the other MTCDs.Finally, we substitute equations (19) and (21) into equation (22) and can get the optimal solution of τi as(23)τi∗=AiD+y∗−1,∀i∈K.Up to this point, we have found the optimal solution, i.e.,τ0∗, τi∗ and Ri∗ for ∀i∈K of the problem (6) in close form.


### 4.3. Centralized Solution Method

It is noted that the problem (6) is solved at the UE by using a centralized manner. Thus, the channel state information (CSI) of the MTCDs is required by the UE, and thereafter, the scheduled transmission time should be broadcasted to the MTCDs by the UE. These messages can be conveyed between the UE and the MTCDs through a dedicated feedback channel.


## 4.1. Dual Problem Formulation

In order to formulate the dual problem for the primal problem (6), we introduce the Lagrangian multipliers λ≥0 and μ=μ0,μ1,…,μK≥0 to the constraints (6.1) and (6.2), respectively. The partial Lagrangian is expressed by(11)LR,τ,λ,μ=∑i=1KlnRi−Rimin−μiRi+∑i=1Kμiτilog21+Ai⋅τ0τi−λτi+λ1−τ0,where R=R1,…,RK and τ=τ0,τ1,…,τK are the feasible rate allocation vector and time allocation vector for the problem (6), respectively. Then, the dual function is given by(12)gλ,μ=maxR,τLR,τ,λ,μ.The dual problem of the primal problem (6) can be formulated as(13)minλ≥0,μ≥0gλ,μ,(13.1)s.t.Ri−Rimin≥0,∀i∈K.As the dual gap (i.e., the difference between the optimal solution to the primal problem (6) and that to the dual problem (13)) is zero [24]. The optimal solution to the primal problem (6) can be recovered by solving the dual problem (13).


## 4.2. Solution to the Dual Problem

It is noted that for a givenλ¯≥0 and μ¯=μ¯0,μ¯1,…,μ¯K≥0, the Lagrangian (11) consists of two sets of variables, i.e., the rate allocation vector R and the time allocation vector τ. So, we can decompose the dual problem (13) into the following rate allocation problem:(14)maxRQ1,Q1=∑i=1KlnRi−Rimin−μ¯iRi,(14.1)s.t.Ri−Rimin≥0,∀i∈K,and the time allocation problem(15)maxτQ2Q2=∑i=1Kμ¯iτilog21+Ai⋅τ0τi−λ¯τi+λ¯1−τ0.By solving the problem (14), the optimal rate allocation vector R∗=R1∗,…,RK∗ can be easily obtained as(16)Ri∗=1μ¯i+RiminRiminRimax,∀i∈K,where the function zba denotes the projection of z on [a, b]. In equation (16), Rimax (Ri∗≤Rimax for ∀i∈K) is introduced to limit the maximum rate that can be allocated to the ith MTCD. Rimax is set as a constant and does not change the optimal solution to the primal problem (6) [25].Now, we need to deal with the problem (15). By setting the first-order derivative of Q2 with respect to τ0 and τ1,…,τK equal to zero, respectively, we can get(17)∑i=1Kμ¯iAi1+Ai⋅τ0/τi=λ¯ln2,(18)μ¯iln1+Ai⋅τ0τi−μ¯iAi/τi1+Ai⋅τ0/τi=λ¯ln2,i∈K.From equation (17), we can derive(19)A1τ1=A2τ2=⋯=AKτK=C,where C (C>0) is a constant. By substituting equation (19) into equation (18), we can rewrite equation (18) as(20)ylny=y−D+1,where D=∑i=1Nμ¯iAi/μ¯i is a constant and y=1+Cτ0.Then, by solving equation (20), we can get the optimal solution of τ0 as(21)τ0∗=y∗−1D+y∗−1.It is noted that the optimal time allocationτ∗=τ0∗,τ1∗,…,τK∗ should satisfy the equality in the constraint (6.1). That means(22)τ0∗+∑i=1Kτi∗=1.Otherwise, the remaining1−τ0∗+∑i=1Kτi∗ time can be allocated to the PB or any one of the MTCDs, which will increase the value of Q without decreasing the achievable rates of the other MTCDs.Finally, we substitute equations (19) and (21) into equation (22) and can get the optimal solution of τi as(23)τi∗=AiD+y∗−1,∀i∈K.Up to this point, we have found the optimal solution, i.e.,τ0∗, τi∗ and Ri∗ for ∀i∈K of the problem (6) in close form.


## 4.3. Centralized Solution Method

It is noted that the problem (6) is solved at the UE by using a centralized manner. Thus, the channel state information (CSI) of the MTCDs is required by the UE, and thereafter, the scheduled transmission time should be broadcasted to the MTCDs by the UE. These messages can be conveyed between the UE and the MTCDs through a dedicated feedback channel.


## 5. Simulation Results

To examine the performance of the proposed rate allocation method, a simulated MTCN is set up as shown in Figure3. The dimension of the room is 10 m × 10 m × 3 m. A human equipped with K = 4 MTCDs in/on the body is walking around in the room. A PB is deployed in the center of the ceiling (with coordinate (5 m, 5 m, 3 m)). In order to avoid creating powerful electromagnetic interference to the BS, both the WPT and WIT links adopt the 915 MHz ISM (industrial, scientific, and medical) band as the center frequency. It is noted that the 915 MHz is also used by some products, e.g., the Powercaster Transmitter TX91501 [26] and the Powerharvester 114 [27] for RF-based WPT and EH. According to [16], the transmission power of the PB is fixed at PPB=3W. The noise power at the UE receiver is set as σ2=10−15W [28]. For the ith MTCD, ∀i∈K, we assume that the energy harvesting efficiency for WPT is ξi=0.5, and the portion of harvested energy used for information transmission is αi=0.5. It is noted that the symmetry axiom of NBS [19] orders that the MTCDs as the game players should have the same minimal utility requirement R1min=R2min=⋯=RKmin. Thus, in accord with this axiom, we set R1min=R2min=⋯=RKmin=500Kb/s in the simulation. In practical applications, we can choose an MTCD with the maximal minimum-rate demand in the system and set its minimum-rate demand as the disagreement point of the game. In addition, we assume that the bandwidth of the network is 1 MHz, and the WPT and WIT links are with the same propagation model. In detail, the path loss exponent is 3.8, the body shadowing is modeled as a Gauss-distributed random variable with zero mean and variance 15 dB, and the small-scale fading is modeled as Rayleigh fading with unit mean. For easy reference, the assumed parameter values used in the simulation are summarized in Table 1.Figure 3
The simulated network structure.Table 1
Assumed system parameter values.
Parameter
Definition
Value
K
Number of the MTCDs
4
P
PB
Transmission power at the PB
3 W [26]
σ
2
Noise power at the UE receiver
10−15 W [28]
ξ
i
Energy harvesting efficiency
0.5
α
i
Portion of harvested energy used for information transmission
0.5
R
i
min
The minimum rate requirement of an MTCD
500 Kb/s
n
Path loss exponent
3.8 [16]
φ
Body shadowing loss margin
15 dB [16]At the beginning of the simulation, the human is in the southwestern corner of the room. The coordinates of the UE and the MTCDs are set as (1, 1, 1.2), (0.9, 0.9, 1.8), (1.1, 1.1, 1.1), (0.9, 1.1, 0.6), and (1.1, 0.9, 0.1), respectively. Then, the human moves along a straight line from the corner to the center of the room. The proposed rate allocation method is termed as theNBS-rate method in the following analysis, and the maximum-rate method as well as the common-rate method is also simulated for comparison purpose.In Figure4, we show the sum rate achieved by all the MTCDs using different rate allocation methods. In Figure 5, we compare the fairness performance of these methods by showing their fairness index [28]. It should be noted that the displayed results are obtained by averaging over 1000 randomly generated channel realization at each position. Let R=R1,…,RK be the data rates achieved by K MTCDs. The fairness index is defined as I=1/K⋅∑i=1KRi2/∑i=1KRi2, which tends to 1 when the K MTCDs have the same data rate and tends to 1/K when the data rates achieved by different MTCDs are severely unfair.Figure 4
Thesum rate of different methods.Figure 5
Thefairness index of different methods.From Figures4 and 5, we can observe that the common-rate method has the lowest sum rate but the best fairness index, as it contributes most of the available time resource to the MTCD with the worst channel. By contrast, the maximum-rate method achieves the highest sum rate but bears the notably unfair problem. The fairness index is almost below 0.3 and is with large undulating. It is because that the MTCD with the best channel always dominates the resource allocation in the maximum-rate method. Therefore, the other MTCDs have to starve. As for the proposed NBS-rate method, it can obtain a good tradeoff between the fairness and the sum rate performance. The fairness index I is always larger than 0.6, and the sum rate loss towards the maximum-rate method is only 1/2 of the common-rate method. Since the maximum-rate method achieves the best system efficiency and the sum rate performance gap between the proposed NBS-rate method and the maximum-rate method is much lower than that between the common-rate method and the maximum-rate method, we can conclude that the proposed NBS-rate method is more efficient than the common-rate method but does not break the per-user fairness of the system [29].Next, we examine the QoS performance of different rate allocation methods. When the human moves to the coordinate (x, y) = (3 m, 3 m), we show the data rate and transmission time allocated to each MTCD, respectively in Figures 6 and 7.Figure 6
Rate allocation by using different methods.Figure 7
Transmission time allocation by using different methods.From Figures6 and 7, we observe that the QoS requirements of the MTCDs can be fulfilled by both the NBS-rate and the common-rate methods. As for the maximum-rate method, the data rates of the 3rd and 4th MTCDs are only 10 Kb/s, which is much lower than the required 500 Kb/s. In addition, by using the common-rate method, the transmission rate is allocated to the MTCDs in a strictly fair manner regardless of their channel difference. In the premise of QoS assurance, the NBS-rate method can allocate the system resource to the MTCDs proportionally according to their channel conditions. Thus, the efficiency of resource utilization is greatly improved.In conclusion, the reason causing the unfairness problem and the low efficiency problem is the distinction of the channel conditions of the MTCDs. As for themaximum-rate method, most of the radio resource is allocated to the MTCDs with better channel condition. In contrast, the common-rate method allocates more radio resource to the MTCDs with worse channel condition, which reduces the system efficiency. In comparison to the maximum-rate method and the common-rate method, our proposed NBS-rate method can achieve a good tradeoff between the per-user fairness and system efficiency. It is because that the utility function is associated with the Pareto optimum for the NBS-rate method.


## 6. Conclusions

In this paper, we propose a cooperative game theoretic method to deal with the resource allocation problem in wireless-powered MTCNs. We first propose a frame structure over time-and-spectrum domain for an SWIPT network. Then, the utility function for a MTCD is defined based on the achievable data rate. The resource allocation problem in a hierarchical SWIPT network is formulated as a cooperative game. By solving the NBS of the proposed game problem, the optimal duty time for the WPT and WIT links are jointly found in close form. The simulation results show that the communication and energy resource are allocated to a group of MTCDs fairly and efficiently in the Pareto optimal sense.



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*Source: 3286356-2019-06-26.xml*