In recent years, the energy harvesting(EH) technology is leading to significant interests in systems powered by harvested ambient energy. In contrast to traditional systems powered by battery, using energy from nature not only reduces the carbon emission but also sustains device equipped with a rechargeable battery with an infinite lifetime. However, the intermittent and stochastic nature of the harvested energy put forward a fundamental challenge when applying EH technique. Hence, in this paper, we propose an adaptive policy for adjusting data sampling rate to maximize the effective information in wireless sensor application.

The application ofEH technology in wireless sensor network has attracted substantial research attention, especially to seek novel EH scheduling policies for improving the system performance. To mitigate efficiency loss of workload, Ref. [1] proposed an architecture to achieve maximum energy efficiency tracking for the overall sensor node. In Ref. [2], the authors studied the properties of the conditional expected rewards and further presented a sub-optimal threshold-based transmission scheduling policy for EH sensors nodes through maximizing the conditional expected rewards with respect to data storage and energy storage, respectively. By using virtual queues and techniques from Lyapunov optimization, Ref. [3] formulated a long-term time-averaged joint scheduling and sensing allocation problem in wireless sensor networks with finite energy and data buffers. Based on realistic energy and network models, the authors of Ref. [4] incorporated CPU-intensive edge operations with sensing and networking to jointly optimize EH wireless sensor network performance. Further, a MIMO network control system with an EH sensor was considered in Ref. [5], and MIMO precoding was designed at the sensor so as to stabilize the unstable dynamic plant. The work in Ref. [6] proposed a two-stage water filling policy to achieve throughput maximization in EH and power grid coexisting wireless communication systems. In Ref. [7], the power-delay tradeoff is formulated as a stochastic optimization problem, and solved by Lyapunov optimization technique. The scenario that one user harvests energy from an energy access point to power its information transmission to a data access point is investigated in Ref. [8]. It should be noted that the works mentioned above focus on the energy efficiency optimization. But currently the EH technology offers only a small amount of energy storage and is capable of harvesting only a trivial amount of energy. Therefore, new technique for managing the energy associated with sensor node is required.

In this paper, we focus on the estimation of energy deficiency probability (EDP) for characterizing the degree of matching between the energy demand and the harvested energy. We employ thelarge deviation theory(LDT) to formulate a model for estimating the EDP, which is applied to assist the sensor in promptly controlling the data sampling rate. The proposed method relies on online measurements instead of any prior statistical knowledge of the harvested energy and the channel state. The main contributions of this paper are summarized as follows:

1) We transform the requirement of information efficiency into the energy deficiency which is kept lower than a desired level. Accordingly, the problem of adjusting data sampling rate overEH aided wireless sensor system is formulated as maximizing the sampling rate subject to a constraint imposed on the EDP in order to guarantee high information efficiency.

2) An EDP estimation model based onLDT is proposed by monitoring the energy-buffer fullness and its variations in the rechargeable battery. By applying LDT, this model accurately characterizes the probability of the rare events of energy deficiency, which assist the sensor in controlling the data sampling rate.

3) We conduct numerical simulations for demonstrating the effectiveness of the proposeddata sampling rate adjustment algorithm. Q-learning, a greedy algorithm is implemented as our benchmark algorithms. The simulation results show that the proposed method achieves an improved performance.

The rest of this paper is organized as follows. Section1 describes the system model and formulates the data sampling rate adjustment problem. In Section 2, we propose the EDP estimation method based on the LDT. Section 3 derives the online measurement based adaptive data sampling rate adjustment algorithm based on the EDP estimation model. Our simulation results and performance analysis are presented in Section 4. Finally, the conclusions are offered in Section 5.

1 System model

We consider a wireless sensor equipped with an EH device and a rechargeable battery having a limited capacity, as depicted in Fig. 1. The device can collect natural renewable energy from the ambience and store in the battery. By utilizing the observations of the energy-buffer fullness and its variations in the rechargeable battery, the prediction calculator estimates the EDP for instructing the decision-maker to generate the data sampler commands. The data sampler executes the data sampling and forms a data stream to control panel.

We assume that this wireless sensor operates in a time-slotted fashion which means the time is divided into slots of equal duration and normalized to unity. At the beginning of each time slot, the decision-maker makes the data sampling rate command to control the data sampler based on the battery level. Let R_n be the data sampling rate made by decision-maker at time slot n. R_n∈{R₀, R₁, …, R_max} characterize the sensor operation level. Higher R_n, implies more precise information will transmit to control panel. Correspondingly, the size of data that should be delivered is given by:

$ {D_n} = f\left( {{R_n}} \right) $

(1)

Where f(·) indicates the relationship between the data sampling rate and data size. The wireless channel is assumed to be constant over a slot duration, but it changes at the slot boundaries. Let N₀ be the white Gaussian noise and W the wireless band width, according to Shannon's capacity formula, the amount of transmission power in the th time slot is

$ {P_n} = \frac{W}{{{N_0}{H_n}}}\left( {{2^{\frac{{{D_n}}}{{dW}} - 1}}} \right) $

(2)

where H_n is the channel state and d is the duration of a slot.

Since the channel state H_n and the amount D_n of the data are available for transmission during the time slot n, the amount of energy E_n^t required to transmit D_n bits is calculated as:

$ E_n^{\rm{t}} = P\left( {{H_n}, {D_n}} \right)d $

(3)

Let E_n^h∈ε={0, 1, …, e_max} denote the amount of energy harvested in the nth time slot. The process E_n^h is assumed to be an i.i.d random variable. Let B_n represent the energy stored in the battery at the beginning of the nth time slot, and the capacity of the battery is B_max. The energy E_n^h harvested in the nth slot, is available until the beginning of the (n+1)th slot. Hence, the dynamics of the energy in the battery are characterized by

$ {B_{n + 1}} = {B_n} - E_n^{\rm{t}} + E_n^{\rm{h}}, n = \{ 1, 2, \cdots \} $

(4)

For sensor operation, the data transmission interruptions substantially reduce the precision of information. Hence, it is desired that the battery always holds sufficient energy for transmitting data of the forthcoming time slots in order to avoid transmission interruptions. Motivated by this, we define a low threshold of B_min as an indicator of the battery being partially depleted. In order to characterize the energy deficiency, we define the EDP as

$ {p_{{\rm{ED}}}} = P\left( {B < {B_{\min }}} \right) $

(5)

where B is a random variable representing the energy-buffer fullness. Higher EDP, p_ED, implies a lower energy fullness in the near future, hence energy deficiency is more likely to occur.

In order to achieve a more accurate data service, the EDP should be kept low. Hence, the data transmission problem in wireless sensor system is interpreted as that of choosing the maximumdata sampling rate for maximizing the information precision subject to a given EDP, which can be formulated as

$ \begin{array}{*{20}{c}} {\mathop {\max }\limits_{_{{R_n} \in \{ 1, 2, \cdots , N\} }} {R_n}}\\ {{\rm{ s}}{\rm{.t}}{\rm{. }}P\left( {B < {B_{\min }}} \right){p_{{\rm{SIR}}}}} \end{array} $

(6)

where p_SIR is a given threshold value of sufferable interrupt rate. In this optimization problem, we transform the transmission interruptions into a probability constraint that the EDP is kept below a certain tolerable level. Different values of p_SIR correspond to different SIR levels.

2 Estimation model of energy deficiency probability

We assume that the index of the current time slot is n, and the current battery level is B_n, while R_n represents the data sampling rate during the time slot. We predict the EDP at the (n+N)th (N > 0) slot under the condition of supporting the data sampling rate R_n in the future N time slots. Let P_ES^n+N denote the EDP at the (n+N)th slot, which is defined as

$ P_{{\rm{ES}}}^{n + N} = P\left( {{B_{n + N}} < {B_{\min }}} \right) $

(7)

where N is the length of the prediction interval.

2.1 Reinterpretation of energy deficiency probability

For a given time slot k, the reduction of the energy available for the transmitter is characterized by

$ \Delta {E_k} = E_k^{\rm{t}} - E_k^{\rm{h}} $

(8)

where ΔE_k∈{-e_max^h, …, 0, 1, …, e^t_max}. Let π_i=P(ΔE_k=i) enote the probability that the energy reduction of the battery is ΔE_k=i. Since E^t_k is the energy consumed in the kth slot and E_k^h is the energy collected in the kth slot, the difference ΔE_k characterizes the instantaneous energy mismatch between the energy required for keeping R_n data sampling rate and the harvested energy.

The total reduction of the energy in the rechargeable battery during the period spanning from the nth slot to the (n+N)th slot is characterized as

$ \Delta {E^{n + N}} = \sum\limits_{i = 1}^N \Delta {E_{n + i}} $

(9)

Then, the energy-buffer fullness B_n+N in the (n+N)th time slot is given as

$ {B_{n + N}} = {B_n} - \Delta {E^{n + N}} $

(10)

According to Eq.(7), the EDP at the (n+N)th slot is rewritten as

$ P_{{\rm{ES}}}^{n + N} = P\left( {{B_n} - \Delta {E^{n + N}} < {B_{\min }}} \right) $

(11)

We define the maximum tolerable average energy reduction during each of the forthcoming N slots as

$ {a_{{\rm{ES}}}} = \frac{{{B_n} - {B_{\min }}}}{N} $

(12)

and the expected value of the average reduction of the energy fullness in the battery in each slot during the future n slots as

$ {m_{{\rm{ES}}}} = E\left[ {\frac{{\sum\limits_{i = 1}^N \Delta {E_{n + i}}}}{N}} \right] $

(13)

where E[·] denotes the expectation operator. Furthermore, m_ES≥a_ES implies while keeping the current data sampling rate, after the forthcoming N slots, transmission interruption will definitely occur.

Let us rewrite Eq.(11) as

$ \begin{array}{*{20}{c}} {P_{{\rm{ES}}}^{n + N} = P\left( {{B_n} - \Delta {E^{n + N}} < {B_{\min }}} \right) = }\\ {P\left( {\frac{{\Delta {E^{n + N}}}}{N} > \frac{{{B_n} - {B_{\min }}}}{N}} \right) = }\\ {P\left( {\frac{{\sum\limits_{i = 1}^N \Delta {E_{n + i}}}}{N} > {a_{{\rm{ES}}}}} \right)} \end{array} $

(14)

The term $\frac{{\sum\limits_{i = 1}^N \Delta {E_{n + 1}}}}{N}$ in Eq.(14) represents the average variation of the energy in the battery in each slot, which is determined by the energy collection and consumption, while a_ES is the tolerable average energy reduction in each of the forthcoming N slots, which is determined by the energy fullness of the battery in the nth slot.

2.2 Large deviation theory based EDP estimation

Since our objective is to provide an interruption-free data transmission, the occurrences of the transmission interruption are expected to be rare events. According to Ref. [9], the LDT can be employed to estimate the probability of rare or tail events. Hence, Cramér's theorem applied in the context of the LDT constitutes an appropriate method of estimating the EDP in Eq.(7).

According to Cramér's theorem^[10], the sequence ΔE_i(i=1, 2, …) obeys the LDT which is rewritten as

$ \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\log P\left( {\frac{1}{N}\sum\limits_{i = 1}^N {{R_i}} > a} \right) = - I(a) $

(15)

where the function I(a) is referred to as the "rate function". Therefore, if a_ES > m_ES, we have

$ \mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\log P\left( {\frac{1}{N}\sum\limits_{i = 1}^N \Delta {E_{n + i}} > {a_{{\rm{ES}}}}} \right) = - I\left( {{a_{{\rm{ES}}}}} \right) $

(16)

where

$ I\left( {{a_{{\rm{ES}}}}} \right) = \mathop {\sup }\limits_{\theta > 0} \left\{ {{a_{ES}}\theta - \log M(\theta )} \right\} $

(17)

and

$ \log M(\theta ) = \log \left\{ {\sum\limits_{i = - e_{\max }^{\rm{h}}}^{\theta _{\max }^{\rm{t}}} {{\pi _i}} \exp [i\theta ]} \right\} $

(18)

The rate function I(a_ES) is usually referred as the Fenchel-Legendre transform or convex conjugate^[9]. Note that both logM(θ) and the rate function I(a_ES) are convex^[9].

Since Eq.(16) is logarithmically asymptotic, for a sufficiently large N, the EDP of the (n+N)th slot can be approximated by

$ P_{{\rm{ES}}}^{n + N} \approx \exp \left[ { - NI\left( {{a_{{\rm{ES}}}}} \right)} \right] $

(19)

Owning to the rapid exponential decay of the EDP estimate with N, a moderate value of N can be used to acquire an accurate EDP estimate.

2.3 Online estimation of energy deficiency probability

In order to estimate the EDP according to Eq.(19), a_ES, m_ES, π_i are required. It is straightforward to calculate a_ES according to Eq.(12). However, because of the absence of any prior knowledge about the probability distribution of ΔE_k, it remains an open challenge to derive analytical expressions for m_ES and π_i. Consequently, we have to utilize historical observations for estimating these parameters by invoking a sliding window-based method.

Let us denote the observed sequence of the energy fullness decrements in the battery by ΔE₁, ΔE₂, ΔE₃, …. The sliding window covers the N_s most recent entries in this sequence, which is slid over this sequence. For the nth window, the observation vector is denoted by W_n=[ΔE_n, ΔE_n-1, …, ΔE_n-Ns+1].

The parameter m_ES can be approximated by the sample mean of the observation vector W_n, yielding

$ {{\hat m}_{{\rm{ES}}}} = \frac{{\sum\limits_{i = n - {N_s} + 1}^n \Delta {E_i}}}{{{N_s}}} $

(20)

For the parameter π_i, i∈{-e^h_max, …, 0, 1, …, e^T_max}, we propose the following estimation technique.

Let N_i denote the number of ΔE_k=i events happening within the sliding window, which is counted by

$ {N_i} = \sum\limits_{k = n - {N_s} + 1}^n {{I_{\left\{ i \right\}}}} \left( {\Delta {E_k}} \right) $

(21)

where

$ {I_{\left\{ i \right\}}}\left( {\Delta {E_k}} \right) = \left\{ {\begin{array}{*{20}{l}} {1, }&{{\rm{ if }}\Delta {E_k} = i}\\ {0, }&{{\rm{ otherwise }}} \end{array}} \right. $

(22)

is an indicator function. Then, the frequency of ΔE_k=i can be estimated as

$ {{\hat q}_i}(n) = \frac{{{N_i}}}{{{N_s}}} $

(23)

Having too small an N_s may result in an excessive estimation error of $\hat q$_i(n), while too large a value may reduce the sensitivity of the estimation model to the variance. Hence, N_s should be set to a moderate value according to the timescale.

Although $\hat d$_i(n) may be used for estimating in Eq.(18), it may result in an undesired fluctuation of the EDP. Hence, we invoke an exponential smoothening of the estimated value, which is formulated as

$ {{\hat \pi }_i}(n) = \rho {{\hat \pi }_{i - 1}} + (1 - \rho ){{\hat q}_i}(n) $

(24)

where the forgetting parameter ρ∈[0, 1] controls the weighting of historical estimates. This smoothening method is especially useful in a non-stationary environment. If ρ approaches 1, the current estimate $\hat \pi $_i(n) largely depends on the most recent past estimate $\hat \pi $_i(n-1), while if ρ=0, the past estimates are neglected, and $\hat \pi $_i(n) entirely depends on the current estimate $\hat q$_i(n. Specifically, ρ∈[0.7, 0.9] is recommended in Gardner's report^[11]. By invoking the estimation of π_i in Eq.(18), we can now calculate logM(θ) and finally estimate the EDP according to Eq.(17) and Eq.(19).

3 Online measurement-based algorithm for data sampling rate adjustment

In this section, we will discuss the details of our online measurement-based algorithm conceived for data sampling rate adjustment, which invokes the online EDP estimation model of Section2 for finding the highest data sampling rate under our SIR constraint. The problem(6) that maximizes the data sampling rate in the th slot can be solved according to:

$ \left. {\begin{array}{*{20}{l}} {P\left( {{B_n} - E_n^{\rm{t}}\left( {{R_n}} \right) + E_n^{\rm{h}} < {B_{\min }}} \right){p_{{\rm{SIR}}}}}\\ {P\left( {{B_n} - E_n^{\rm{t}}\left( {{R_n} + 1} \right) + E_n^{\rm{h}} < {B_{\min }}} \right){p_{{\rm{SIF}}}}} \end{array}} \right\} $

(25)

where E_n(R_n) denotes the energy allocated for transmitting the data with R_n data sampling rate. Since the current data sampling rate is R_n, the sequence E^t_n(R_n+1) is not observable. Hence, the EDP in the second inequality of Eq.(25) cannot be directly calculated according to the steps of Section 2.3. Here, we consider a heuristic iteration policy for solving Eq.(25). The basic idea is to constrain the data sampling rate adjustments to a single level, yielding R_n∈{R_n-1, R_n, R_n+1}.

For the case of $\hat m$_ES≥a_ES, the average reduction of the energy is higher than the tolerable reduction per slot in the future N time slots. This implies that the battery energy would run out after N slots, if the R_n data sampling rate is executed. Correspondingly, the transmission of the data stream would be interrupted because of energy exhaustion. Therefore, in order to prevent future information deficiency, the sensor should reduce the transmission energy consumption by reducing data sampling rate, which can be expressed a

$ {R_{n + 1}} = \max \left( {{R_n} - 1, {R_0}} \right) $

(26)

By contrast, for the case of $\hat m$_ES < a_ES, the average reduction of the energy fullness in the forthcoming N slots is below the tolerable average energy reduction per slot. However, this does not necessarily imply that no energy deficiency will happen in the future N time slots, because $\hat m$_ES represents the average reduction per slot, which cannot exactly characterize the specific energy reduction in a single slot. Hence, energy deficiency may still occur. Since $\hat m$_ES < a_ES, energy deficiency remains a rare event. Hence, the LDT is immediately applicable to characterize the EDP.

According to the estimation model of Section 2, the EDP in the (n+N)th slot can be approximated as

$ \hat P_{{\rm{ES}}}^{n + N} = \exp \left[ { - NI\left( {{a_{{\rm{ES}}}}} \right)} \right] $

(27)

Thus, the EDP $\hat P$_ES^n+N of the (n+N)th slot can be estimated by applying the online measurement-based method of Section 2.3 using a moderate prediction interval of N.

Based on the above discussions, the proposedonline measurement-based adaptive data sampling (OM-ADS) is summarized in Algorithm 1.

Algorithm 1  OM-ADS

Data: the historical observations: B_i, E_i^t, and E_i^h

(i={1, 2, …, n}); the data sampling rate in the nth slot: R_n;

the desired SIR level: p_SIR; the SIR threshold P_T; a given constant K_T; a temporary counter K.

While true do

  Calculate $\hat m$_ES and a_ES according to Eq. (20) and Eq. (12).

if $\hat m$_ES≥a_ES then

  j_n+1←j_n-1

else

Calculate $\hat P$_ES^n+N according to the steps in Section

2.3.

if $\hat P$ _ES^n+N≥P_SIR then

  j_n+1←j_n-1

  else

    if $\hat P$_ES^n+N≤R^_t then

      K←K+1

      if K≥K_^t then

        j_n+1←j_n+1

        K←0

      end

    end

  end

 end

end

4 Experimental results

In this section, we carried out a series of experiments for evaluating the performance of the proposed methods. In order to quantify the attainable performance improvements, we also implemented a heuristic greedy method, an online Q-learning method^[12] as benchmark algorithms.

We consider a nondispersive Rayleigh fading channel with bandwidth w=2 MHz and noise spectral density of N₀=4×10^-9 W/Hz. The amount of the energy harvested is assumed to follow the Poisson distribution. We also consider different data sampling rates for different data transfers. In order to characterize the capability of adaptive data sampling rates, we set the harvest energy, varying expectations from 0.2 to 1.6 with the step size of 0.2. The amount of the energy harvested is assumed to follow the Poisson distribution.

For the proposed OM-ADS, the parameters were set as follows: the length of sliding window was N_s=110, the size of prediction interval was N=50, the forgetting parameter was ρ=0.7, the desired EDP level was P_SIR=0.001 and the low threshold value was P_T=0.1×P_SIR. For the Q-learning method, we follow to set the learning rate and the discount factor to 1 and 0.8, respectively.

Fig. 2(a) characterizes different data sampling rates versus the average energy harvest rate, while Fig. 2(b) characterizes the transmission interruption rates that reflect the integrity of data collection versus the average energy harvest rate. It can be observed that although the greedy method performs better than the other methods in terms of data sampling rates, it suffers from a higher transmission interruption rate, especially in situation of low energy harvest rate. This is because the greedy method maximizes the data sampling rate according to the current available energy in the battery, it frequently suffers from energy scarcity, which lead to the highest transmission interruption rate, thus reducing the integrity of the sampled data. This implies that excessive data collection may result in energy insufficient in the future. Similarly, as the Q-learning method merely aims for maximizing the system's throughput by adjusting the data sampling rate according to the system's state, but without considering energy storage. This implies that excessive data collection may result in energy insufficient in the future. The proposed OM-ADS method achieves a slightly worse data sampling rate, but it gains a much lower transmission interruption rate. The reason for this trend is that the OM-ADS benefits from evaluating the energy shortage probability of the forthcoming slot in each time slot for prudently adjusting data sampling rate for keeping the likelihood of energy shortage under a certain tolerance level.

We also can see that as the capture energy increases, thedata sampling rate of each algorithm increases and the probability of transmission interruption is reduced, and it meets our expectations.

5 Conclusions

In this paper, we discussed the SIR-guaranteed adaptive data sampling rate adjustment in wireless sensor network. The SIR constraint is interpreted as keeping the energy deficiency probability below a given threshold, in order to reduce the rate of transmission interruptions that substantially reduce the information efficiency. Large deviation theory was invoked for estimating the EDP via online measurements. A heuristic method, OM-ADS, was proposed based on the EDP estimation to maximize the information efficiency associated with the EDP constraint. The experimental results demonstrated that the proposed adaptive data sampling rate adjustment methods is capable of effectively improving the information efficiency while keeping the energy available for the future.