Energy and channel transmission management algorithm for resource harvesting body area networks

Abstract

In body area networks, sustainable energy supply and reliable data transmission are important to prolong the service cycle and guarantee the quality of service. In this article, we build a system model to capture the stochastic processes in body area networks, including energy harvesting process, spectrum pricing process, and data sampling process. In the system model, energy harvesting technology and cognitive radio technology are adopted to provide green energy and improve transmission environment for body area networks. Based on the proposed model, we formulate an optimization problem of system utility maximization. Since this problem is a multi-objective mixed-integer problem under multiple restrictions, we decompose the problem into several subproblems by Lyapunov optimization theory. Based on this, we design an efficient online energy and channel transmission management algorithm to solve these subproblems and achieve a close-to-optimal system utility without any priori knowledge. We analyze the optimality of proposed algorithm and derive the required battery capacity and the size of data buffer. Simulation results demonstrate the effectiveness of the proposed algorithm.

Keywords

Body area networks cognitive radio energy harvesting resource allocation Lyapunov optimization

Introduction

The phenomenon of aging population has became an inescapable issue that the government, medicine, and academic must face. On one hand, more and more old people need to come to the hospital for regular medical checkup, which places a heavy burden on the limited medical resource. On the other hand, many chronic diseases require real-time monitoring to prevent further deterioration of physical condition.¹ However, traditional medical system cannot provide such services for the elderly at home or in public place. Thus, we need a new type of medical monitoring system which is able to run without the limits of time and place. Body area networks (BANs) have shown significant promise in healthcare domain and can be used in many scenarios,² such as health monitoring, telemedicine, and fitness tracking. A typical BAN includes a personal device (PD) and several sensors which are low-power, wearable, and implantable. Those sensors collect and store physiological data, such as temperature, oxygen saturation, blood pressure, and electrocardiogram, and transmit data to the PD. Then, PD sends the gathered data to medical institutions for data analysis and disease surveillance.³ Therefore, BANs are promising medical system for improving healthcare and quality of life while reducing medical costs. However, BANs still face many challenges.

One of the major challenges in BANs is the sustainable power supply,⁴ which determines whether sensors can provide long-term continuous healthcare service for the user. Therefore, it is necessary to replace the battery or recharge it. However, the battery capacity in the sensor is limited by its size, which results a small battery capacity. So, it is not desirable to replace the battery or recharge it frequently for users. In addition, changing the battery is unrealistic for the medical application that embeds sensors inside human body. In this condition, energy harvesting (EH) technology is studied and developed,⁵ and already deployed into BANs, known as energy harvesting body area networks (EHBANs). EHBANs are capable of supplying energy for sensors continuously, which prolongs network lifetime and reduces maintenance cost in communication. The energy that EHBANs can harvest usually comes from three basic types,⁶ which are light energy, heat energy, and kinetic energy. Among these sources, the heat energy is gained from the difference in temperature between vitro and vivo; the kinetic energy is gained from various internal and external motions, such as walking, heartbeat, and blood flow.

The other major challenge in BANs is communication performance,⁷ which determines whether BANs can guarantee the quality, integrity, and timeliness of data transmission. BANs typically operate in the 2.4–2.5 GHz band,⁸ which belongs to unlicensed industrial, scientific, and medical (ISM) bands. However, with the remarkable growth of wireless service in recent years, the ISM bands have become so crowded, which leads to an intense competition for spectrum resource among various unlicensed wireless networks. Meanwhile, a survey of global spectrum utilization reveals that the actual utilization time of licensed wireless spectrum is only around 5%–10%.⁹ Therefore, in order to improve the efficiency of wireless resource usage, an emerging technology is developed, which is cognitive radio (CR). Through equipping BANs with CR technology,¹⁰ in addition to the use of ISM band, sensors can dynamically and opportunistically access the unused licensed bands for transmission, thereby enhancing the robustness, scalability, and utility of BANs. Such networks are referred as cognitive radio body area networks (CRBANs).

Although the problems of energy supply and channel transmission can be solved by EH technology and CR technology, there are still some new challenges to face.¹¹ First of all, the EH process is dynamic and stochastic, which makes a challenge to balance energy consumption and supply.¹² Second, the price of licensed spectrum is also dynamic, which makes providing stable, fast access to licensed spectrum challenging.¹³ Since the aim of EH technology and CR technology is harvesting resource, the BANs equipped with the two technologies above are referred to as resource harvesting body area networks (RHBANs). In RHBANs, the process of accessing licensed spectrum involves the process of energy consumption, and there is an inherent link between each other. Therefore, in these stochastic and dynamic processes, managing and allocating resource for RHBANs becomes challenging.

Motivated by the challenges above, we develop a framework based on Lyapunov optimization in a single-hop RHBANs consisting of a PD and a few medical sensors, by jointly considering three stochastic processes: EH, spectrum pricing, and data sampling. The sensors harvest energy by EH technology to sense biologic signals and transmit it to PD over licensed spectrum. We propose an efficient online energy and channel transmission management (ECTM) algorithm to achieve a close-to-optimal system utility,¹⁴ while ensuring the system sustainability. The main contributions of this article are summarized as follows:

We propose a stochastic formulation of the system utility optimization problem subject to the stability of RHBANs system, by characterizing the stochastic nature of the three processes, that is, EH process, spectrum pricing process, and data sampling process.

We develop a framework based on Lyapunov optimization to decompose the system optimization problem into three subproblems, that is, energy control optimization (ECO), sampling optimization (SO), and channel transmission optimization (CTO).

Based on the developed framework, we design an online ECTM algorithm which runs at each time slot and achieves closed-to-optimal time-average system utility. Moreover, the algorithm does not need any priori knowledge of system running.

We analyze the performance of the ECTM algorithm in terms of the required battery capacity, the bounds of data queue and debt queue, and the optimality of the ECTM algorithm. Specifically, we compute the required battery capacity to support the channel trading and data transmission; we compute the upper bound of data queue and debt queue to guarantee the stability of system; and we compute the optimality of ECTM algorithm to compare the gap between time-average system utility and optimal system utility.

Related works

Energy efficient resource allocation scheme has been widely designed for EHBANs to improve the performance of networks in the works.^15–20 Liu et al.¹⁵ designed an optimization framework to maximize the energy efficient and proposed a transmission rate allocation policy to minimize energy consumption, subject to constrains of quality-of-service (QoS) metrics. Seyedi and Sikdar¹⁶ addressed the problem of developing energy efficient transmission strategies for BANs and studied the trade-off between the energy consumption and pack error probability. He et al.,¹⁷ Liu et al.,¹⁸ and Ventura et al.¹⁹ all used the Markov method to build resource allocation model. He et al.¹⁷ modeled the EH process at each sensor as a discrete-time Markov chain and formulated and solved the steady-rate optimization problem to prolong the lifetime of sensors. Liu et al.¹⁸ proposed an optimal transmission power and time slots allocation strategy which can adaptively change the transmission power and transmission time to provide a sustainable service. Ventura et al.¹⁹ developed a Markov model for capturing the energy states of sensors and provided simplified analytical models for predicting the probability of a sensor running out of energy. Through prediction models, the network designer can set the requirements for the battery capacity of sensor nodes. The models^15–19 developed do not well reflect the stochastic features of EH process as they require sufficient statistical knowledge of harvestable energy. Huang and Neely²⁰ developed an online energy-limited scheduling algorithm, which achieved a close-to-optimal system utility and did not require any knowledge of EH process.

The application of CR technology in BANs has been investigated to improve the system’s performance.^21–24 Yu et al.²¹ proposed a network architecture for cognitive and cooperative communications in BANs and presented two cooperative transmission schemes for different applications to decrease the bit error rate and reduce energy consumption. Mahbub et al.²² investigated the interference temperature limit and outage probability in CRBANs, while focusing on the interference and power constraint issues. Moungla et al.²³ developed an analytical model using a continuous-time Markov chain for the interference attenuation in channel switching systems. The proposed approach insured the channel quality by reducing the packet loss rate and mitigating interferences. Syed and Yau²⁴ presented two architectures, two applications of CRBANs to address two critical concerns, which were the reduction of electro-magnetic interference to sensors and the enhancement of system performances.

In RHBANs, there is interrelation between energy management and channel transmission. However, for the improvement of system performance, it is not enough only to study one of them, just like the works above.^15–24 To fill this research gap, we propose a framework to capture the stochastic processes of EH and channel transmission and design a online ECTM algorithm to achieve an close-to-optimal system utility.

System model and problem formulation

We consider the single-hop RHBANs in which all the sensors with $n = {1, 2, \dots, N}$ directly communicate to the PD. The BANs are in a multichannel environment of licensed spectrum. We denote $K$ as the supportable number of frequency bands for PD, which means the PD can communicate over $K$ different frequency bands in a time slot. Sensors sense and transmit data over the time slots $t \in τ = {0, 1, 2, \dots}$ using licensed channels forming the set $m = {1, 2, \dots, M}$ . As shown in Figure 1, the channel agent (CA) is in charge of the trading of licensed channels between sensors and primary users (PUs) who have the privilege to access. And, we summarize the notations used in this article in Table 1.

Figure 1.

Resource harvesting body area networks architecture.

Table 1.

Key notations.

Symbol	Notation
$N$	The set of all sensors
$M$	The set of all channels
$K$	The supportable number of frequency bands for PD
$μ_{n} (t)$	The sampling rate of sensor $n$ in t^th
$β_{n}$	The budget of sensor $n$ for buying channel
$p_{n} (t)$	The channel price for sensor $n$ in t^th
$A (t)$	The channel allocation matrix
$D_{n} (t)$	The debt queue of sensor $n$ in t^th
$ξ_{n} (t)$	The data transfer rate of sensor $n$ in t^th
$θ_{n, m} (t)$	Channel capacity of sensor $n$ in t^th over m^th
$I_{n} (t)$	The data queue of sensor $n$ in t^th
$δ$	Stable energy harvesting
$η_{n} (t)$	Dynamical energy harvesting of sensor $n$ in t^th
$ρ_{n} (t)$	Dynamical energy supply rate of sensor $n$ in t^th
$H_{n} (t)$	Total energy harvesting of sensor $n$ in t^th
$C_{S}$	Sampling power
$C_{T}$	Transmission power
$C_{n} (t)$	Total energy consumption of sensor $n$ in t^th
$E_{n} (t)$	The energy queue of sensor $n$ in t^th
$Φ$	The storage capacity of battery
${\hat{Φ}}_{n} (t)$	The spare capacity of the sensor $n$ in t^th
$R (t)$	The system utility in one time slot
$V$	The weight of system utility
$ϵ$	The maximum of the $U' (μ_{n} (t))$
$μ_{\max}$	The maximum of $μ_{n} (t)$
$ζ$	The minimum of $p_{n} (t)$
$p_{\max}$	The maximum of $p_{n} (t)$
$ρ_{\max}$	The maximum of the $ρ_{n} (t)$
$θ_{\max}$	The maximum of $θ_{n, m} (t)$
$H_{\max}$	The maximum of $H_{n} (t)$
$C_{\max}$	The maximum of $C_{n} (t)$

Sampling rate and system utility

At every time slot, sensor device senses data from body at a sampling rate $μ_{n} (t)$ , which is bounded by the maximal sampling rate $μ_{\max}, \forall n \in N$ , that is

0 \leq μ_{n} (t) \leq μ_{\max}, \forall n \in N

(1)

Similar to the work by Huang et al.,²⁰ we define the utility function of sensor $n$ at time slot $t$ , that is, $U (μ_{n} (t)) = \log (1 + μ_{n} (t)), \forall n \in N$ , which is strictly concave and bounded by the upper bound of first order derivative $U' (μ_{n} (t))$ . The maximum of $U' (μ_{n} (t))$ is denoted by $ϵ$ and obtained at $μ_{n} (t) = 0$ . Based on this, we define the system utility as $R (t) = \sum_{n \in N} U (μ_{n} (t))$ in one time slot.

Channel trading and debt control model

At the beginning of each time slot $t$ , sensor $n$ makes a budget $β_{n}$ for buying channel and sends the budget to CA. Meanwhile, CA gives sensor $n$ an offer price $p_{n} (t)$ which comes form PUs. For PU activity on channel $m$ , there are two states: occupancy or idle. The state of PUs is assumed to vary randomly with independent and identical distribution (i.i.d),¹¹ which means the occupancy and idle durations are also independent. We assume that the spectrum pricing process varies randomly based only on the state duration across time slots. Moreover, a common control channel is assumed to exchange the channel price information among PUs, sensors, and CA and for delivering the deal decision of CA to BANs.²⁵ We define the deal decision state at slot $t$ , that is

{\begin{matrix} p_{n} (t) > 0, & CA sells sensorn one channel \\ p_{n} (t) = 0, & CA has no channel to sell sensor n \end{matrix}

The state $p_{n} (t) = 0$ does not mean that sensor $n$ get a free channel, but means no channel is allocated to sensor $n$ , which is same as the channel allocation state $\sum_{m = 1}^{M} a_{n, m} (t) = 0$ defined in the following section. Therefore, we use $\sum_{m = 1}^{M} a_{n, m} (t) = 0$ to denote sensor $n$ has not obtained any channel in slot $t$ instead of $p_{n} (t) = 0$ . Based on this, the lower and upper bounds of $p_{n} (t)$ are denoted as follows

ζ \leq p_{n} (t) \leq p_{\max}, \forall n \in N, t \in T

(2)

where $ζ$ is the minimum non-negative value of offer price, and $p_{\max}$ is the maximum of offer price.

We define the channel allocation matrix $A (t) = {a_{n, m} (t)}_{N \times M}$ , which denotes the channel $m$ whether to be allocated to sensor $n$ at time slot $t$ , that is

a_{n, m} (t) = {\begin{matrix} 1, & if channel m is allocated to sensors n \\ 0, & else \end{matrix}

To avoid the channel collision, we define each sensor can only use one channel and each channel can only be allocated to one sensor at most in each time slot. In addition, since the PD communicates over $K$ different frequencies, the number of channels obtained by sensors cannot exceed $K$ , that is

{\begin{matrix} \begin{matrix} \sum_{m = 1}^{M} a_{n, m} (t) \leq 1, & \forall n \in N \\ \sum_{n = 1}^{N} a_{n, m} (t) \leq 1, & \forall m \in M \\ \sum_{n = 1}^{N} \sum_{m = 1}^{M} a_{n, m} (t) \leq K \end{matrix} \end{matrix}

(3)

To evaluate the debt of sensors about the budget and expense, for all $n \in {1, 2, \dots, N}$ , we define $D_{n} (t)$ as the debt queue of sensor $n$ in time slot $t$ , and a vector $D (t) = (D_{1} (t), \dots, D_{N} (t))$ , which denotes the debt queue of all sensors. The debt queue varies with an input process and an service process, that is, the expenses $\sum_{m = 1}^{M} a_{n, m} (t) p_{n} (t)$ and the budget $β_{n}$ . Define the debt queue as follows

D_{n} (t + 1) = [D_{n} (t) - β_{n}, 0]^{+} + \sum_{m = 1}^{M} a_{n, m} (t) p_{n} (t)

(4)

The debt queue is stable only if time-average input rate $lim_{t \to \infty} (1 / t) \sum_{τ = 0}^{t - 1} a_{n, m} (τ) p_{n} (τ)$ less than the time-average service rate $lim_{t \to \infty} (1 / t) \sum_{τ = 0}^{t - 1} β_{n} (τ)$ , that is

lim_{T \to \infty} \frac{1}{T} \sum_{t = 0}^{T - 1} \sum_{n = 1}^{N} E [D_{n} (t)] \leq \infty

(5)

Data transmission and data queue dynamics model

The channel capacity is denoted by $θ_{n, m} (t) = \log (1 + C_{T} f_{n, m} (t) / (l_{n}^{4} S))$ , where $C_{T}$ denotes transmission power, $l_{n}$ denotes the the distance between sensor $n$ and PD, $S$ denotes the noise power, and $f_{n, m} (t)$ denotes channel fading coefficients.²⁶

The channel capacity $θ_{n, m} (t)$ determines data transfer rate $ξ_{n} (t)$ of sensor $n$ in time slot $t$ over channel $m$ . In view of the influence of body tissue and changing environment to channel (e.g. body motion, angle, and distance between sensor and PD), we assume that $θ_{n, m} (t)$ randomly varies over time slots in an i.i.d fashion and is bounded by $θ_{n, m} (t) \leq θ_{\max}, \forall n \in N, m \in M$ . Thus, the data transfer rate $ξ_{n} (t)$ is bounded by

ξ_{n} (t) \leq θ_{n, m} (t), \forall n \in N

(6)

We build the dynamic data queue $I_{n} (t)$ of sensor $n$ by input process which is sampling rate $μ_{n} (t)$ and service process which denoted by $a_{n, m} (t) ξ_{n} (t)$ , as shown below

I_{n} (t + 1) = I_{n} (t) - \sum_{m = 1}^{M} a_{n, m} (t) ξ_{n} (t) + μ_{n} (t)

(7)

To ensure the single-serve queuing system is stable, data queue should meet the constraint that the time-average occupancy of sensor is finite, that is

lim_{T \to \infty} \frac{1}{T} \sum_{t = 0}^{T - 1} \sum_{n = 1}^{N} E [I_{n} (t)] \leq \infty

(8)

Besides, the amount of data transmitted by the $n^{th}$ sensor must not exceed the length of data queue $I_{n} (t)$ in the t^th time slot, that is

0 \leq ξ_{n} (t) \leq I_{n} (t), \forall n \in N

(9)

EH model

We consider that the energy can be harvested from two aspects. One is stable energy source, like heat energy from the body. The other is dynamic energy source, like movement or light energy from the surrounding environment.

We define the stable energy harvested by the $n^{th}$ sensor in each time slot as $δ_{n}$ . Since the body temperature change very little under normal circumstances, we assume all sensors have been equipped with the same EH devices, which means the stable energy harvested by any sensor is same, that is

δ = δ_{1} = δ_{2} = . . . = δ_{n}, \forall n \in N

(10)

However, we define the dynamic energy harvested by the $n^{th}$ sensor in time slot $t$ as $η_{n} (t)$ . Since dynamic energy supply rate $ρ_{n} (t)$ determines the amount of energy which can be harvested by sensor $n$ in time slot $t$ , the dynamic energy $η_{n} (t)$ is bounded by $ρ_{n} (t)$ , that is

0 \leq η_{n} (t) \leq ρ_{n} (t) \forall n \in N

(11)

The dynamic energy supply rate $ρ_{n} (t)$ has a upper bound which is denoted by $ρ_{n} \leq ρ_{\max}, \forall n \in N$ . Thus, the totally harvested energy $H_{n} (t)$ by the $n^{th}$ sensor in the t^th time slot is

H_{n} (t) = δ_{n} + η_{n} (t), \forall n \in N

Since the stably harvested energy is a fixed value $δ$ and the dynamically harvested energy $η_{n} (t)$ is bounded by $ρ_{\max}$ , the totally harvested energy is bounded by $H_{n} (t) \leq δ + ρ_{\max}$ . And, we use $H_{\max} = δ + ρ_{\max}$ as the upper bound of totally harvested energy in each time slot.

Energy consumption and energy queue dynamics model

In RHBANs, the factors that affect energy consumption come from two aspects: data sampling and data transmission. For data sampling, senor senses health data from body area with sampling rate $μ_{n} (t)$ and sampling power $C_{S}$ which is assumed to be a constant value. So, we can get the sampling energy consumption with $C_{S} μ_{n} (t)$ . For data transmission, sensor sends data to PD with constant power $C_{T}, \forall n \in N$ , on the assumption that the $n^{th}$ sensor can access to channel $m$ . So, the total energy consumption $C_{n} (t)$ of the $n^{th}$ sensor in the t^th time slot is

C_{n} (t) = C_{S} μ_{n} (t) + \sum_{m = 1}^{M} a_{n, m} (t) C_{T}, \forall n \in N

Since the sampling rate $μ_{n} (t)$ is bounded by $μ_{\max}$ and $\sum_{m = 1}^{M} a_{n, m} (t) \leq 1$ , the total energy consumption is bounded by $C_{n} (t) \leq C_{S} μ_{\max} + C_{T}$ . And, we use $C_{\max} = C_{S} μ_{\max} + C_{T}$ as the upper bound of energy consumption in certain time slot.

Based on the EH $H_{n} (t)$ and the energy consumption $C_{n} (t)$ in time slot $t$ , we can build the energy queue of the sensor $n$ by

E_{n} (t + 1) = E_{n} (t) - C_{n} (t) + H_{n} (t)

(12)

in which we use $E_{n} (t)$ to denote the energy queue length of sensor n. And, there are two kinds of constraints about $E_{n} (t)$ . For one thing, the total energy consumption of sensor $n$ cannot exceed the energy in $E_{n} (t)$ , that is

C_{n} (t) \leq E_{n} (t), \forall n \in N

(13)

For another, we assume the storage capacity of battery $Φ$ is a constant value and same to all sensors. So, the sum energy of $E_{n} (t)$ and $H_{n} (t)$ cannot exceed the $Φ$ in time slot $t$ , that is

E_{n} (t) + H_{n} (t) \leq Φ, \forall n \in N

(14)

Optimization problem formulation

Based on the models discussed above, we formulate the time-average system utility optimization problem, which can be written as

\bar{R} = li m_{T \to \infty} \frac{1}{T} \sum_{t = 0}^{T} - E [R (t)]

(15)

For the sake of convenience, we use $μ (t), ξ (t), η (t), a (t))$ to stand for the vectors of sampling rate $μ_{n} (t)$ , data transmission rate $ξ_{n} (t)$ , dynamically harvested energy $η_{n} (t)$ , the element $a_{n, m} (t)$ of matrix $A (t)$ in time slot $t$ , respectively. In addition, we use $Θ (t) \overset{Δ}{=} (μ (t), ξ (t), η (t), a (t))$ to express the collection of variables in time slot t.

At this point, we can get the maximum system utility (MSU) by optimizing $Θ (t)$ under the constraints of the formulae (1)–(14), that is

\begin{matrix} (MSU) max_{Θ (t)} \bar{R} \\ s . t . equations (1) - (14) \end{matrix}

Because the MSU is a multi-objective optimization problem under multiple restrictions, is also a mixed-integer problem, for instance, $a_{n, m} (t)$ is an integer variable and $μ_{n} (t)$ is a continuous variable. It is not easy to solve MSU directly. Therefore, we decompose MSU into several single-objective subproblems by Lyapunov optimization theory in the next section.

Proposed framework

The framework proposed in this article is based on Lyapunov optimization. We decompose the utility maximization problem MSU into three definite subproblems by four steps: define Lyapunov function $L (t)$ , define conditional Lyapunov drift $Δ (t)$ , define improved drift function $Δ_{V} (t)$ , and derive the upper bound of $Δ_{V} (t)$ .

Lyapunov optimization

Based on the three queue formulas (4), (7), and (12), we define the Lyapunov function $L (t)$ , which is equal to the sum of squares of the queue lengths of debt, data and energy, that is

L (t) = \frac{1}{2} \sum_{n = 1}^{N} [D_{n}^{2} (t) + I_{n}^{2} (t) + (- {\hat{Φ}}_{n} (t))^{2}]

(16)

In $L (t)$ , ${\hat{Φ}}_{n} (t)$ means the remaining capacity of the n^th sensor battery, that is, ${\hat{Φ}}_{n} (t) = Φ - E_{n} (t)$ . For easy description, we use $K (t) \overset{Δ}{=} (D (t), I (t), E (t))$ to denote the system state in time slot $t$ . This represents the occupancy of debt queue, data queue, and energy queue. In addition, we define the conditional Lyapunov drift $Δ (t)$ , which equals to the difference of $L (t)$ in a time slot conditional upon $K (t)$ , that is

Δ (t) = E [L (t + 1) - L (t) | K (t)]

(17)

By minimizing $Δ (t)$ in each time slot, we can stabilize the debt queue, data queue, and energy queue and push $D_{n} (t)$ , $I_{n} (t)$ , and ${\hat{Φ}}_{n} (t)$ to zero. In other words, we can ensure the mean of debt queue and the mean of data queue are both finite values throughout the whole time slots, and the remaining capacity can get close to battery capacity. Thus, constraints (5), (8), and (13) can be satisfied.

It is obvious that minimizing $Δ (t)$ make no difference to maximize system utility. So, we define improved drift function $Δ_{V} (t)$ , in which we insert weighted system utility into Lyapunov drift, as in the following

Δ_{V} (t) \overset{Δ}{=} E [Δ (t) - V R (t) | K (t)]

(18)

where $V$ is the non-negative weight of system utility. By now, by minimizing $Δ_{V} (t)$ , we can realize minimizing $Δ (t)$ and maximizing $V R (t)$ at one time. Besides, we can make a trade-off between $Δ (t)$ and $R (t)$ by adjusting $V$ .

Since $Δ_{V} (t)$ is a quadratic function, which is not easy to find minimum value. In view of this, we derive the upper bound of $Δ_{V} (t)$ and minimize the upper bound instead of $Δ_{V} (t)$ .

Lemma 1

The upper bound of $Δ_{V} (t)$ is as follows

Δ_{V} (t) \leq E [F_{V} (t) | K (t)] + B

(19)

where the $F_{V} (t)$ is given in equation (25). And, the $B$ is a constant whose value is as follows

\begin{matrix} B = \frac{N}{2} [θ_{\max}^{2} + μ_{\max}^{2} + H_{\max}^{2} + C_{\max}^{2}] \\ + \frac{1}{2} [\sum_{m = 1}^{M} {(β_{n})}^{2} + p_{\max} M] \end{matrix}

(20)

Proof

We can get $Δ_{V} (t)$ by substituting equations (16) and (17) into equation (18). We derive the upper bound of $Δ_{V} (t)$ in two steps: first, derive the upper bounds of various parts on the right-hand side of equation (21), respectively, as shown in equations (22)–(24); then, substitute equations (22)–(24) into equation (21) and rearrange equation (21). The derivation processes are shown as follows

\begin{array}{l} Δ_{V} (t) = \frac{1}{2} [(D_{n} {(t + 1))}^{2} - {(D_{n} (t))}^{2}] \\ + \frac{1}{2} [(I_{n} {(t + 1))}^{2} - {(I_{n} (t))}^{2}] \\ + \frac{1}{2} [(- {\hat{Φ}}_{n} {(t + 1))}^{2} - {(- {\hat{Φ}}_{n} (t))}^{2}] - V ℛ (t) \end{array}

(21)

\begin{matrix} \frac{1}{2} [(D_{n} (t + 1))^{2} - (D_{n} (t))^{2}] \\ \leq \frac{1}{2} {{(β_{m})}^{2} + [\sum_{n = 1}^{N} a_{n, m} (t) p_{n} (t)]^{2} + 2 D_{n} (t) [\sum_{n = 1}^{N} a_{n, m} (t) p_{n} (t) - β_{n}]} \\ \leq \frac{1}{2} [(β_{n})^{2} + (p_{\max})^{2}] + D_{n} (t) [\sum_{n = 1}^{N} a_{n, m} (t) p_{n} (t) - β_{n}] \end{matrix}

(22)

\begin{matrix} \frac{1}{2} [(I_{n} {(t + 1))}^{2} - {(I_{n} (t))}^{2}] \\ \leq \frac{1}{2} {[\sum_{m = 1}^{M} a_{n, m} (t) ξ_{n} (t {)]}^{2} + {[μ_{n} (t)]}^{2} + 2 I_{n} (t) [μ_{n} (t) - \sum_{m = 1}^{M} a_{n, m} (t) ξ_{n} (t)]} \\ \leq \frac{1}{2} [(θ_{\max})^{2} + {(μ_{\max})}^{2}] + I_{n} (t) [μ_{n} (t) - \sum_{m = 1}^{M} a_{n, m} (t) ξ_{n} (t)] \end{matrix}

(23)

\begin{matrix} \frac{1}{2} [(- {\hat{Φ}}_{n} (t + {1))}^{2} - {(- {\hat{Φ}}_{n} (t))}^{2}] \\ \leq \frac{1}{2} {[C_{n} (t {)]}^{2} + {[H_{n} (t)]}^{2} - 2 C_{n} (t) H_{n} (t) + 2 \hat{Φ} [C_{n} (t) - H_{n} (t)]} \\ \leq \frac{1}{2} [(C_{\max})^{2} + {(H_{\max})}^{2}] + \hat{Φ} [C_{n} (t) - H_{n} (t)] \end{matrix}

(24)

\begin{array}{l} F_{V} (t) = \underset{ECO}{\underset{︸}{\sum_{n = 1}^{N} [- {\hat{Φ}}_{n} (t) η_{n} (t)]}} \\ + \underset{SO}{\underset{︸}{\sum_{n = 1}^{N} {μ_{n} (t) [I_{n} (t) + C_{S} {\hat{Φ}}_{n} (t)] - V U (μ_{n} (t))}}} \\ + \underset{CTO}{\underset{︸}{\sum_{n = 1}^{N} \sum_{m = 1}^{M} a_{n, m} (t) [D_{n} (t) p (t) - I_{n} (t) ξ_{n} (t) + C_{T} {\hat{Φ}}_{n} (t)]}} \end{array}

(25)

By squaring both sides of equation (4), we can get equation (22). Note that $p_{n} (t) \leq p_{\max}$ in equation (2) and $a_{n, m} (t) = {0, 1}$ , we have $[\sum_{n = 1}^{N} a_{n, m} (t) p_{n} (t)]^{2} \leq (p_{\max})^{2}$ , as shown in equation (22). Similarly, we can get equation (23) from equation (7) and get equation (24) from equation (12). Note that $ξ_{n} (t) \leq θ_{n, m} (t)$ in equation (6) and $θ_{n, m} (t) \leq θ_{\max}$ , we have $[\sum_{m = 1}^{M} a_{n, m} (t) ξ_{n} (t)]^{2} \leq (θ_{\max})^{2}$ , as shown in equation (23).

In the process of rearranging equations (22)–(24), we combine the terms who have the same variable to one collection. Finally, we get the $F_{V} (t)$ with three parts, as shown in equation (25).

So, we can minimize the $F_{V} (t)$ by minimizing the three parts. In particular, the three parts, respectively, correspond to three problems, which are ECO, SO, and CTO. In the following, we optimize the above problems separately.

ECO problem

For energy control problem, it is the optimization goal that minimizing the first term of equation (25) under the constraints (11) and (14), as follows

\begin{matrix} (ECO) min_{η_{n} (t)} - {\hat{Φ}}_{n} (t) η_{n} (t) \\ s . t . {\begin{matrix} 0 \leq η_{n} (t) \leq ρ_{n} (t) \\ E_{n} (t) + δ + η_{n} (t) \leq Φ \end{matrix} \end{matrix}

Thus, we can get the optimal solution by maximizing dynamic energy $η_{n} (t)$ . In other words, when $Φ_{n} (t) > δ$ , the RHBANs should harvest dynamic energy as much as possible. So, we have the optimal solution

η_{n}^{*} (t) = {\begin{matrix} ρ_{n} (t), & if ρ_{n} (t) < {\hat{Φ}}_{n} (t) \\ Φ - δ - E_{n} (t), & if δ < {\hat{Φ}}_{n} (t) \leq ρ_{n} (t) \\ 0, & else \end{matrix}

SO problem

For SO problem, it is the optimization goal that minimizing the second term of equation (25) under the constraint (1), as follows

\begin{matrix} (SO) min_{μ_{n} (t)} μ_{n} (t) [I_{n} (t) + C_{S} {\hat{Φ}}_{n} (t)] - VU (μ_{n} (t)) \\ s . t . 0 \leq μ_{n} (t) \leq μ_{\max} \end{matrix}

Since the system utility $U (μ_{n} (t))$ is concave function, so the SO is a convex problem. Thus, we can get the optimal solution based on the convex optimal theory, as follows

μ_{n}^{*} (t) = {\begin{matrix} min ({U'}^{- 1} (y), μ_{\max}), & if {U'}^{- 1} (y) > 0 \\ 0, & else \end{matrix}

(26)

where $y = (1 / V) (I_{n} (t) + C_{S} {\hat{Φ}}_{n} (t))$ .

CTO problem

For channel transmission problem, it is the optimization goal that minimizing the third term of equation (25) under the constraints (3), (6), and (9), as follows

\begin{matrix} (CTO) \\ min_{a_{n, m} (t) ξ_{n} (t)} a_{n, m} (t) [D_{n} (t) p (t) - I_{n} (t) ξ_{n} (t) + C_{T} {\hat{Φ}}_{n} (t)] \\ s . t . (3), (6), (9) \end{matrix}

By optimizing CTO problem, CA sells channels to the sensors who have low debt, more data in buffer, and more remaining capacity in battery. It is often difficult to optimize the integer variable $a_{n, m} (t)$ and continuous variable $ξ_{n} (t)$ all at once. Therefore, we convert the mixed-integer problem CTO into integer programming problem, that is, channel optimization (CO) problem by removing continuous variable $ξ_{n} (t)$ . We assume that $a_{n, m}^{*} (t)$ and $ξ_{n}^{*} (t)$ are the optimal solution for the problem CTO. And, removing the continuous variable $ξ_{n} (t)$ in three steps: first, we show that $\sum_{m \in M} a_{n, m}^{*} (t) = 1$ is the optimal solution; second, we prove that $ξ_{n}^{*} (t) = θ_{n, m} (t)$ is the optimal solution under condition (27); last, we prove that $I_{n} (t) > θ_{\max}$ is the necessary condition of equation (28). Details are as follows.

First, there is no sense in optimizing the problem CTO when $\sum_{m \in M} a_{n, m}^{*} (t) = 0$ for sensor $n$ , that is, no channel is assigned to the sensor $n$ . It is because no channels, no transmission; no transmission, no optimization. Consequently, the problem CTO gets the optimal solution if a channel $m$ is assigned to sensor $n$ , that is

\sum_{m \in M} a_{n, m}^{*} (t) = 1, \forall n \in N

(27)

Second, by maximizing data transfer rate $ξ_{n} (t)$ under condition (27), we can minimize the third term of equation (25), that is, optimize the problem CTO. In other words, the sensor $n$ should transmit data as much as possible if it gets one channel. By constraint (6), it can be known that the data transfer rate $ξ_{n} (t)$ is subject to the maximum channel capacity $θ_{n, m} (t)$ . Therefore, problem CTO can get the optimal solution on condition that sensors transmit data at full channel capacity, that is

ξ_{n}^{*} (t) = θ_{n, m} (t)

(28)

Finally, data transfer rate $ξ_{n} (t)$ is bounded by the constraint (9). So, sensor $n$ will not be able to transmit data at full capacity if the length of data queue is less than the maximum channel capacity, that is, $I_{n} (t) < θ_{\max}$ , which also means $ξ_{n} (t) < θ_{\max}$ . Therefore, equation (28) can work if and only if sensor $n$ has enough data to transmit, that is, $I_{n} (t) \geq θ_{\max}$ . So, in order to ensure that sensors have enough data to transmit, we define optimal data queue ${\hat{I}}_{n} (t)$ to replace $I_{n} (t)$ in CTO, as follows

{\hat{I}}_{n} (t) = [I_{n} (t) - θ_{\max}]^{+}

(29)

After above analysis, we can replace $ξ_{n} (t)$ and $I_{n} (t)$ with $θ_{\max}$ and ${\hat{I}}_{n} (t)$ in CTO and relax constraints (6) and (9). Thus, the problem CTO is converted to problem CO as follows

\begin{matrix} (CO) min_{a_{n, m} (t)} a_{n, m} (t) [D_{n} (t) p (t) - {\hat{I}}_{n} (t) θ_{n, m} (t) + C_{T} Φ_{n} (t)] \\ s . t . equation (3) \end{matrix}

CO problem is a matching problem, which can be solved by the Hopcroft–Karp algorithm. The complexity of Hopcroft–Karp algorithm is $O (N M \sqrt{K})$ ,²⁷ which increases linearly with the number of sensors $N$ .

Algorithm and performance analysis

ECTM algorithm

In this subsection, we design the ECTM algorithm in Algorithm 1 to optimize the three subproblems mentioned above, that is, ECO, SO, and CTO. In Algorithm 1, we can get the optimal dynamically harvested energy $η^{*} (t)$ , sampling rate $μ^{*} (t)$ , data transmission rate $ξ^{*} (t)$ , and channel allocation $a_{n, m}^{*} (t)$ at each time slot by updating the debt queue $D (t)$ , data queue $I (t)$ , and energy queue $E (t)$ .

Since ECO and SO problems can be solved at the each sensor only through, respectively, local information, their complexity will not increase as the number of sensors $N$ increases. For CO problem, its complexity increases linearly with the number of sensors $N$ . So, Algorithm 1 can be applied to relatively large-scale RHBANs.

Algorithm 1.

Energy and channel transmission management algorithm.

Data:

D (t)

I (t)

E (t)

p_{n} (t)

H_{n} (t)

\forall n \in N

θ_{n, m} (t)

\forall n \in N, m \in ℳ

Result:

η^{*} (t)

μ^{*} (t)

A^{*} (t)

ξ^{*} (t)

D (t + 1)

I (t + 1)

E (t + 1)

/*Solve ECO problem
1 for each

n \in N

do
2 if

ρ_{n} (t) < {\hat{Φ}}_{n} (t)

then
3

η_{n}^{*} (t) = ρ_{n} (t)

4 else if

δ < {\hat{Φ}}_{n} (t) \leq ρ_{n} (t)

then
5

η_{n}^{*} (t) = Φ - δ - E_{n} (t)

;
6 else
7

η_{n}^{*} (t) = 0

/*Solve SO problem
8 Compute

U'^{- 1} (y)

as (26);
9 for each

n \in N

do
10 if

U'^{- 1} (y) > 0

then
11

μ_{n}^{*} (t) = min ({U'}^{- 1} (y), μ_{\max})

;
12 else
13

μ_{n}^{*} (t) = 0

/*Solve CTO problem
14 Solve CO problem and set

a_{n, m}^{*} (t)

;
15 for each

n \in N

do
16 if

\sum_{m \in ℳ} a_{n, m}^{*} (t) = = 1

then
17

ξ_{n}^{*} (t) = θ_{n, m} (t)

;
18 else
19

ξ_{n}^{*} (t) = 0

/*Queues Updating
20 for each

N \in N

do
21 Compute

D_{n} (t + 1)

based on equation (4);
22 Compute

I_{n} (t + 1)

based on equation (7);
23 Compute

E_{n} (t + 1)

based on equation (12);

Required battery capacity

To ensure sensors have enough energy to work, we derive the required battery capacity in the case that the available energy is less than the maximum energy consumption. In other words, sensors will not collect or transmit data in this case. We show the required battery capacity $Φ$ in Theorem 1.

Theorem 1

For $\forall n \in N$ , the required battery capacity $Φ$ as follows

Φ = \max (\frac{V ϵ}{C_{S}} + C_{\max}, \frac{I_{\max} θ_{\max}}{C_{T}} + C_{\max})

(30)

if and only if $E_{n} (t) \leq C_{\max}$ , that is, the length of energy queue is less than the maximum energy consumption, sensors will not collect data or get one channel, that is, $μ_{n} (t) = 0$ and $\sum_{m \in M} a_{n, m} (t) = 0$ .

Proof

We prove Theorem 1 in two ways.

First, we derive $Φ$ in the case that sensor $n$ does not collect any data, that is, $μ_{n} (t) = 0$ , if the available energy is less than the maximum energy consumption. Since system utility $U (μ_{n} (t))$ is concave function, so $U'^{- 1} (μ_{n} (t))$ and $μ_{n} (t)$ are inversely proportional. Therefore, based on equation (26), we can get $μ_{n} (t) = 0$ , if

y \geq ϵ

(31)

Considering that ${\hat{Φ}}_{n} (t) = Φ - E_{n} (t)$ , we rearrange equation (31) to $Φ \geq (V ϵ - I_{n} (t)) / (C_{S}) + E_{n} (t)$ . To ensure sensor $n$ cannot collect any data when $E_{n} (t) \leq C_{\max}$ , we minimize $I_{n} (t)$ and maximize $E_{n} (t)$ . Then, we have

Φ \geq \frac{V ϵ}{C_{S}} + C_{\max}

Second, we derive $Φ$ in the case that sensor $n$ does not get any channel, that is, $\sum_{m \in M} a_{n, m} (t) = 0$ , if $E_{n} (t) \leq C_{\max}$ . To problem (CO), sensor $n$ has no channel if $D_{n} (t) p (t) - {\hat{I}}_{n} (t) θ_{n, m} (t) + C_{T} {\hat{Φ}}_{n} (t) \geq 0$ , that is

Φ \geq \frac{{\hat{I}}_{n} (t) θ_{n, m} (t) - D_{n} (t) p_{n} (t)}{C_{T}} + E_{n} (t)

(32)

To ensure sensor $n$ cannot get any channel when $E_{n} (t) \leq C_{\max}$ , we maximize ${\hat{I}}_{n} (t)$ , $θ_{n, m} (t)$ , and $E_{n} (t)$ and minimize $D_{n} (t) p_{n} (t)$ in equation (32). Then, we have

Φ \geq \frac{I_{\max} θ_{\max}}{C_{T}} + C_{\max}

Thus, Theorem 1 is proved.

Bounded data queue

In Theorem 2, the upper bound $I_{\max}$ of data queue is derived. Thus, we can ensure the stability of data queue $I_{n} (t)$ , that is, guarantee the data queue has a finite time-average occupancy.

Theorem 2

The upper bound of data queue as follows

I_{\max} = V ϵ + μ_{\max}, \forall n \in N

where $V$ is the non-negative weight of system utility. Thus, we have

0 \leq I_{n} (t) \leq I_{\max}, \forall n \in N

(33)

Proof

Theorem 2 is proved by induction. First, at $t = 0$ , equation (33) holds. Then, we assume it holds in time slot $t$ and prove it holds in time slot $t + 1$ :

If sensor $n$ does not collect data in time slot $t + 1$ , we can get $I_{n} (t + 1) \leq I_{n} (t) \leq I_{\max}$ .

If sensor $n$ collects data with sampling rate $μ_{n}^{*} (t)$ , we can get $VU' (μ_{n}^{*} (t)) = I_{n} (t) + C_{S} {\hat{Φ}}_{n} (t)$ by equation (26). Since $0 \leq C_{S} {\hat{Φ}}_{n} (t)$ and $U' (μ_{n}^{*} (t)) \leq ϵ$ , we have $I_{n} (t) \leq VU' (μ_{n}^{*} (t)) \leq V ϵ$ . At last, since $μ_{n}^{*} (t) \leq μ_{\max}$ , we have $I_{n} (t + 1) \leq I_{n} (t) + μ_{\max} \leq V ϵ + μ_{\max}$ .

Thus, equation (33) holds at time slot $t + 1$ . Theorem 2 is proved.

From the Theorem 2, we can see the bound of data queue increases linearly with the weight of system utility. So, a large V means sensors need a longer data buffer to realize a bigger system utility.

Bounded debt queue

In Theorem 3, we derive the upper bound of debt queue to ensure the the length of debt queue is limited, that is, guarantee its stability.

Theorem 3

For $\forall n \in N$ , the upper bound of debt queue as follows

D_{\max} = \frac{I_{\max} θ_{\max}}{ζ} + p_{\max}

we have

0 \leq D_{n} (t) \leq D_{\max}, \forall n \in N

(34)

Proof

We use induction to prove Theorem 3. First, at $t = 0$ , the debt queue $D_{n} (t)$ is initialized as an empty queue, equation (34) holds. Then, we prove equation (34) holds in time slot $t + 1$ , if it holds in time slot $t$ .

For CO problem

If $D_{n} (t) p (t) - {\hat{I}}_{n} (t) θ_{n, m} (t) + C_{T} {\hat{Φ}}_{n} (t) > 0$ , we have $a_{n, m} (t) = 0$ , that is, sensor $n$ does not obtain any channel. Therefore, we have $D_{n} (t + 1) \leq D_{n} (t) \leq D_{\max}$ .

If $D_{n} (t) p (t) - {\hat{I}}_{n} (t) θ_{n, m} (t) + C_{T} {\hat{Φ}}_{n} (t) \leq 0$ , we have $a_{n, m} (t) = 1$ , that is, sensor $n$ obtains one channel. Since $C_{T} {\hat{Φ}}_{n} (t) \geq 0$ , we can get $D_{n} (t) p (t) \leq {\hat{I}}_{n} (t) θ_{n, m} (t)$ . Then, since ${\hat{I}}_{n} (t) \leq I_{n} (t) \leq I_{\max}$ , $θ_{n, m} (t) \leq θ_{\max}$ , and $p (t) \geq ζ$ , we have $D_{n} (t) \leq (I_{\max} θ_{\max} / ζ)$ . Therefore, the debt queue $D_{n} (t + 1) \leq (I_{\max} θ_{\max} / ζ) + p_{\max}$ in time slot $t + 1$ .

Summarily, we have $D_{n} (t + 1) \leq D_{\max}$ . Theorem 3 is proved.

From Theorem 3, although sensors may continue to be in debt, there will be a cap on their debt levels, that is, the debt constraint (5) can be satisfied.

Optimality of algorithm

In Theorem 4, we analyze the optimality of ECTM algorithm and compare the performance between time-average system utility and optimal system utility.

Theorem 4

We assume that the optimal system utility is $\bar{R}$ and time-average system utility is $R^{*}$ . We have

\bar{R} \geq R^{*} - \frac{\hat{B}}{V}

(35)

where $\hat{B} = B + NM θ_{\max}^{2}$ .

Proof

We prove Theorem 4 by comparing Lyapunov drift of ECTM algorithm with that of another random algorithm, which denoted by $Ξ$ . And, we use superscript $Ξ$ to tag variables generated by algorithm $Ξ$ . Since EH process change in i.i.d, according to Theorem 4.5 by Neely,²⁸ we have

{\begin{matrix} E [\sum_{n \in N} U (μ_{n}^{Ξ} (t))] & \leq R^{*} + γ \\ | E [\sum_{n \in N} (\sum_{m = 1}^{M} a_{n, m}^{Ξ} (t) p_{n} (t) - β_{n})] | & \leq σ_{1} γ \\ | E [\sum_{n \in N} (μ_{n}^{Ξ} (t) - \sum_{m = 1}^{M} a_{n, m}^{Ξ} (t) ξ_{n}^{Ξ} (t))] | & \leq σ_{2} γ \\ | E [\sum_{n \in N} (H_{n}^{Ξ} (t) - C_{n}^{Ξ} (t))] | & \leq σ_{3} γ \end{matrix}

(36)

where $γ$ is arbitrarily small positive number, and $σ_{1}$ , $σ_{2}$ , and $σ_{3}$ are constant scalars.

In each slot time, we minimize the four parts of Lyapunov drift ${\hat{F}}_{V} (t)$ in equation (37) by ECTM algorithm

\begin{matrix} {\hat{F}}_{V} (t) = \sum_{n = 1}^{N} D_{n} (t) (\sum_{m = 1}^{M} a_{n, m}^{Ξ} (t) p_{n} (t) - β_{n}) \\ - \sum_{n = 1}^{N} Φ_{n} (t) (H_{n}^{Ξ} (t) - C_{n}^{Ξ} (t)) \\ + \sum_{n = 1}^{N} (I_{n} (t) μ_{n} (t) - VU (μ_{n} (t))) \\ - \sum_{n = 1}^{N} \sum_{m = 1}^{M} a_{n, m}^{Ξ} (t) ξ_{n}^{Ξ} (t) {\hat{I}}_{n} (t) \end{matrix}

(37)

According to Theorem 2 by Huang et al.,²⁰ equation (37) can be proved. Note that $Δ (t) - V E [R (t)] \leq \hat{B} + E [{\hat{F}}_{V} (t) | K (t)]$ , where $\hat{B} = B + NM θ_{\max}^{2}$ , we can get

\begin{matrix} Δ (t) - V E [R (t)] \leq \hat{B} + E [{\hat{F}}_{V}^{UA} (t) | K (t)] \\ \leq \hat{B} + E [{\hat{F}}_{V}^{Ξ} (t)] \\ \leq \hat{B} + (σ_{1} + σ_{2} + σ_{3} + 1) γ - V R^{*} \end{matrix}

(38)

where the superscript of ${\hat{F}}_{V} (t)$ indicates which algorithm it comes from. By setting $γ$ to 0, we can have

Δ (t) - V E [R (t)] \leq \hat{B} - V R^{*}

(39)

Take expectations on both sides, we have

\frac{L (T - 1) - L (0)}{T} - \frac{1}{T} \sum_{t = 0}^{T - 1} E [R (t)] \leq \hat{B} - V R^{*}

(40)

Since $L (t)$ is finite, by $T \to \infty$ , we can get $\bar{R} \geq R^{*} - (\hat{B} / V)$ . Thus, Theorem 4 is proved.

According Theorem 4, we can see the performance gap between $\bar{R}$ and $R^{*}$ is within $O (1 / V)$ . And, we can also see the bigger the $V$ , the bigger the system utility $\bar{R}$ .

Simulation results

In this section, we evaluate the performance of ECTM algorithm by MATLAB simulation. We consider the RHBANs composed of $N = 8$ sensors and one PD. And the PD has $K = 3$ transceivers to communicate. The RHBANs run on $M = 5$ licensed channels.

We assume that the budget $β_{n}$ for each sensor $n$ is same in each slot, and $β = 3$ . The minimum channel price is set to $ζ = 1$ , and the maximum channel price is set to $p_{\max} = 7$ . The stable energy supply $δ = 0.1$ , and the upper bound of the dynamic energy supply $ρ_{\max} = 0.5$ . The sampling power of sensor $C_{S} = 2 \times 10^{- 2}$ , and the transmission power of sensor $C_{T} = 0.2$ . The upper bound of sampling rate $μ_{\max} = 10$ .

For the relevant parameters of channel capacity $θ_{n, m} (t)$ , they are set as follows: $S = 10^{- 5}$ , $f_{n, m} (t)$ is uniformly distributed between (0.9, 1.1), and i.i.d across time slots.²⁶ And, the maximum channel capacity $θ_{\max} = 5$ .

We assume that the battery capacity is full in time slot $t = 0$ , whereas debt queue and data queue are empty at $t = 0$ .

Network utility and queue length

The system utility is plotted in Figure 2 under different weights $V$ ranging from 100 to 1000. The system utility for each $V$ is taken by the average of system utility values generated from 10,000 time slots in one experiment. Table 2 shows the 95% confidence interval of the system utility corresponding to each $V$ , whose size is within an acceptable range. From Figure 2, we can see that the system utility grows with the weight $V$ , but its growth rate is slowing. And, after $V > 700$ , the value of system utility tends to be gentle. This confirms what we mentioned in equation (35), which is a concave function of $V$ .

Figure 2.

System utility versus the value of $V$ ranging from 100 to 1000.

Table 2.

The 95% confidence interval of the system utility.

Weight, $V$	System utility	Confidence interval
V = 100	3.11	(2.736, 3.464)
V = 200	3.73	(3.340, 4.087)
V = 300	4.12	(3.729, 4.471)
V = 400	4.38	(3.984, 4.751)
V = 500	4.52	(4.090, 4.947)
V = 600	4.61	(4.223, 5.008)
V = 700	4.68	(4.328, 5.063)
V = 800	4.71	(4.337, 5.082)
V = 900	4.73	(4.338, 5.127)
V = 1000	4.74	(4.352, 5.124)

Figure 3 shows the relationship between $V$ and time-average length of the three queues, which are debt queue, data queue, and energy queue. Similar to Figure 2, the time-average queue length for each $V$ is taken by the average from 10,000 time slots in one experiment. We can see that their time-average length increase linearly with the increase in the $V$ , which are consistent with the conclusions of equation (30) in Theorem 1, equation (33) in Theorem 2, and equation (34) in Theorem 3. This is also verified in Figures 4 –6, that is, the larger $V$ will produce a larger queue length.

Figure 3.

Time-average queue length versus $V$ .

Figure 4.

Debt queue dynamics.

Figure 5.

Data queue dynamics.

Figure 6.

Energy queue dynamics.

Besides, comparing Figures 2 and 3, we can get that while increasing $V$ can increase system utility, but at the same time, the system needs more budget, larger battery capacity, and data cache to keep it running.

Queue dynamics

Figure 4 shows the debt queue dynamics with different values of V over 10,000 time slots. The lengths of debt queues increase rapidly with the increase in the time $t$ in the beginning. It is because the system begins to receive and send data after $t = 0$ , which leads to a rapid increase in the demand for channel. After this, the lengths of debt queues converge quickly and fluctuate around a time-average value. This is due to the ECTM algorithm plays a regulatory role in system by giving the low-debt sensor priority to get the channel. In addition, it can be seen that the queue length increases with increasing $V$ , which similar to Figure 3.

Figure 5 shows the data queue dynamics with different values of V over 10,000 time slots. Similar to the debt queue dynamics shown in Figure 4, the lengths of data queues increase and converge quickly after $t = 0$ and fluctuate around a time-average value after the convergence. It is because that the length of data queue is $0$ and the battery is full at $t = 0$ . So, at $t = 1$ , sensor begin to collect data, resulting in a rapid increase in data queue. Meanwhile, sensor sends data via channel to ensure the stability of data queue.

Figure 6 shows the energy queue dynamics with different values of V over 10,000 time slots. We can see that the time-average lengths of energy queues fluctuate around a time-average value in the whole time slots. This is regulated by two aspects. On one hand, sensor consumes energy in the data collection and transmission process; on the other hand, the sensor itself is constantly gaining energy.

Impact of parameter variations

In this section, we evaluate the impacts of system parameters on the system utility. Similar to Figure 2, the system utility for each $V$ and each $β$ is taken by the average from 10,000 time slots in one experiment. First, we evaluate the impact of budget $β$ on network utility. Figure 7 shows that the system utility increases quickly with the budget $β$ increases from 1 to 3. It is because sensors have a greater chance to get channel resource. However, the growth rate of system utility decays with the budget $β$ increases from 4 to 7. This is because, at this stage, the system utility is not limited by budget, but energy supply.

Figure 7.

Impact of $β$ on system utility.

Figure 8 shows the impact of the maximum dynamic energy supply $ρ_{\max}$ on the system utility. As we can see, system utility increases with the increase in the $ρ_{\max}$ because the system has enough energy to transmit data. However, similar to Figure 7, the growth rate of system utility is decreasing. This indicates system utility is also limited by channel resource.

Figure 8.

Impact of $ρ_{\max}$ on system utility.

From Figures 7 and 8, we can see that, for improving system utility, it is not enough to simply increase one of the system parameters.

Conclusion

In this article, we have designed a framework for optimizing system utility in RHBANs. In this framework, three stochastic process are included, which are channel trading, EH, and data transmission. Moreover, in order to solve the system utility optimization problem, the framework decomposes it into three subproblems: ECO, SO, and channel trading and data transmission optimization. Then, the ECTM algorithm is designed to solve the three subproblems. Besides, we have derived the bounds on debt, data, and energy queues, as well as the gap between optimal system utility and approximately optimal solution. At last, simulations verify that the ECTM algorithm can stabilize system status and improve system utility. The outcomes of this work can be used to guide the design of ECTM algorithm in practical RHBANs.

For the future work, we plan to investigate channel auction and energy conversion efficiency. In addition, the channel interference will be considered.

Footnotes

Handling Editor: Katsuya Suto

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported, in part, by the Major Program of National Natural Science Foundation of China (No. 71633006), the National Natural Science Foundation of China (Nos 61672540 and 61379057), and the Fundamental Research Funds for the Central Universities of Central South University (No. 2017zzts203). Also, this work was supported partially by “Mobile Health” Ministry of Education—China Mobile Joint Laboratory.

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