The energy-harvesting sensor network is a new network architecture to further prolong the lifetime of sensor networks and enhance the quality of IoT services. Due to the inherent problems of energy-harvesting sensor networks, it is really hard to collect fresh and useful sensory data. To solve the above problems, we investigate the data collection scheme in edge-assisted energy-harvesting sensor networks and try to collect fresh and useful sensory data from such networks. Enlightened by the concept of the age of information, we define a new metric, the age of useful information (AoUI), to measure the usefulness and freshness of the sensory data. Furthermore, we define the Minimizing the Maximum Age of Useful Information problem (Min-AoUI) to construct a sensory data collection method to minimize the AoUI of the sensory data. We prove that the Min-AoUI problem is NP-Hard, and approximation algorithms are proposed to solve this problem. The time complexity and the approximation ratio of this algorithm are analyzed. The performance of the algorithm is also verified by extensive experimental results.
1 Introduction
The Internet of Things (IoT) [12, 25] is a key technique that can connect the physical world and the digital world. The information about the physical world can be gathered by IoT devices (UAVs [54, 55], sensors, RFIDs, etc.) and uploaded to a remote cloud to be further analyzed. The analysis results can help users to make decisions in the physical world. Due to the limited battery capacity of IoT devices, a key challenging issue of the IoT application is how to monitor the physical world for a long period and return the analysis results in real time.
In traditional IoT architectures, such as wireless sensory networks (WSN), the network lifetime is limited due to the limitation of battery capacity, and it is not feasible for a long time of monitoring. The energy-harvesting sensor network (EHN) [36, 37], also known as battery-free sensor network[3, 34], is a new technique that can improve the performance of IoT applications by prolonging the network lifetime. An energy-harvesting sensor network consists of energy-harvesting nodes (EH nodes) that can harvest energy from unlimited ambient energy, such as solar energy, wind energy, RF signal energy, and so on. For example, in Reference [13], the authors have designed a wearable smart bracelet that can harvest solar energy from the environment, and the amount of harvested power is sufficient to support one blood oxygenation measurement per minute. The authors in Reference [18] have designed a system that can generate energy from human knee joints and power the on-body WSN. Thus, compared to WSN, the lifetime of an EHN is unlimited in terms of energy. Figure 1 illustrates two types of energy-harvesting nodes.
Fig. 1.
Fig. 1. The energy-harvesting nodes [38] powered by RF-signal (a) and solar energy (b).
To transmit the analysis results in real-time, the gathered sensory data should be both useful and fresh. The “useful” means the sensory data should contain enough information and the remote cloud can obtain useful knowledge from the sensory data. The “fresh” means the sensory data uploaded to the cloud is the most recent sensory data. However, it is really hard to maintain the data usefulness and freshness because of the inherent limitation of the EHN. First, the ambient energy is weak and distributes unevenly, and an EH node cannot work at any time [39] due to the energy limitation. When an EH node wants to send its sensory data, it may have to wait a long time until its neighbor has harvested enough energy to receive. Thus, it will take a much longer time to collect the sensory data from all EH nodes. From another perspective, it is very difficult for a multi-hop energy-harvesting sensor network to collect high-quality sensory data in real-time due to the packet loss of wireless communications [51].
Thanks to the growth of the number of edge servers (Base Stations, APs...) [10, 35, 56], we can reduce the data transmission distance and data transmission latency by deploying an edge-assisted EHN (E-EHN) [26, 27, 50]. Since the density of edge servers is increasing, we can firmly assume that for each EH node there is at least one edge server in its transmission radius. Based on the edge computing technique, we can use edge servers as sinks to collect sensory data from EH nodes. Thus, the sensory data of an EH node can be directly sent to an edge server with only one hop. Figure 2 illustrates an E-EHN example.
Fig. 2.
Fig. 2. An edge-assisted energy-harvesting sensor network.
In this article, we aim to propose a method to collect useful and fresh sensory data from E-EHNs. A new concept, the Age of Information [16, 17, 21, 44, 45, 46, 47], is usually adopted as a metric to measure the freshness of periodically generated sensory data. There are many works focusing on optimizing the AoI of sensor networks. However, due to the following two reasons, these works cannot be adopted in the E-EHNs directly.
First, these works have not considered the data usefulness. Existing works mainly concentrate on how to minimize the AoI of the sensory data generated by the individual sensor node. However, we have figured out that the useful sensory data should be the data generated by a set of nodes and contain enough information, and the data generated by individual sensor nodes may not be useful enough.
Second, these works have not considered the data transmission collisions between EH nodes. When an EH node i is transmitting data to another node j, the radio sent out by i may cause signal interference to other EH nodes that intend to receive data from other EH nodes.
Based on these two reasons, we need to define a new metric to measure the usefulness and freshness of sensory data and design a transmission collision-free data collection method to optimize the metric. Our contributions are summarized as follows:
(1)
We defined a new metric, the Age of Useful Information (AoUI), to measure the usefulness and freshness of sensory data.
(2)
We defined the Minimizing the maximum AoUI (Min-AoUI) problem to construct a collision-free data collection method and optimize the freshness and usefulness of the sensory data. We have proved that the Min-AoUI problem is NP-Hard.
(3)
We considered the Min-AoUI problem in E-EHNs with single and multiple edge servers, respectively. We have also proposed two approximation algorithms to solve these problems and analyzed the approximation ratio of these algorithms.
(4)
Extensive experiments were carried out to examine the performance of the proposed algorithms, and the experimental results show that these algorithms are effective and efficient.
The rest of the article is organized as follows: Section 2 surveys the related works. Section 3 defines the Min-AoUI problem and proves the NP-Hardness of the Min-AoUI. Section 4 and Section 5 propose approximated algorithms to solve the Min-AoUI in E-EHNs with single and multiple edge servers, respectively. Furthermore, Section 4 and Section 5 also provide theoretical analysis of these algorithms. Section 6 shows the simulation results, and Section 7 concludes the article.
2 Related Works
2.1 Energy-harvesting Sensor Networks
The authors in References [32, 33, 41] first designed a type of energy-harvesting node that is a modified RFID tag and can be recharged by RF-signal power. The authors in References [1, 29] have designed the energy-harvesting node powered by solar power and wind power. However, none of these works has considered how to construct energy-harvesting sensor networks with these nodes. In Reference [36], we have first considered the energy-harvesting sensor network consisting of RF-powered energy-harvesting nodes, and in Reference [37], we have investigated the network composed of energy-harvesting nodes powered by general ambient energy resources. More details about the concept of the energy-harvesting sensor network can be found in Reference [40].
2.2 Data Collection in Energy-harvesting Sensor Networks
Data collection is important in traditional IoT applications, such as wireless sensor networks [11, 15, 19, 23, 24, 31, 42, 53, 57]. Compared with data collection (or data aggregation) in traditional wireless sensor networks, collecting sensory data in energy-harvesting sensor networks is more challenging, and few works are focusing on these areas. The authors in Reference [7] have analyzed the difficulties of data aggregation in energy-harvesting sensor networks and defined the “energy collision” in the processing of data aggregation. The authors in Reference [5] have proposed a distributed algorithm for data aggregation in energy-harvesting sensor networks to reduce the data aggregation delay. In References [8, 20, 22], the authors have considered the data-gathering strategy in energy-harvesting sensor networks with mobile sinks, such as drones. These works have remarkable contributions. However, they only consider the data aggregation problems. Some information of the sensory data may be omitted through the data aggregation process. Thus, to maintain more information on the sensory data, we consider the raw data collection method in this article.
2.3 Age of Information
The Age of Information (AoI) [43] is a new metric to measure the freshness of sensory data. The authors in References [16, 17] have proposed the concept to the AoI. The AoI of a target i is the time that elapsed since the last received update of i was generated. From the edge server’s perspective, if the AoI of a target is high, then it means the sensory data of such a target is not fresh. Thus, plenty of works focus on how to minimize the peak AoI or average AoI in an IoT system [2, 9, 21, 44, 45, 46, 47]. However, these works only aim to optimize the AoI of specific targets. The authors in References [4, 6] have first considered minimizing the AoI in the energy-harvesting sensor network. However, they have not considered the data usefulness.
Due to the limitations of the literature above, we concentrate on both the age and the usefulness of sensory data.
3 Problem Description
We formulate the Minimizing the Maximum AoUI problem in this section, and the main notations adopted in this section are illustrated in Table 1.
The set of all useful datasets from 0 to t and the end of monitoring
\(A_{\mathbf {D}}(t)\)
The AoUI at time slot t
Table 1. Summary of Main Notations
3.1 Network Model
As illustrated in Figure 2, an edge-assisted energy-harvesting sensor network (E-EHN), \(\mathbf {N}(V,S,\mathbf {O}, \mathbf {T}, \mathcal {C}(M))\) , contains a set of energy-harvesting nodes (EH nodes) \(V=\lbrace 1,2,\ldots ,n\rbrace\) , a set of edge servers \(S=\lbrace n+1,n+2,\ldots ,n+m\rbrace\) , a set of targets \(\mathbf {O}=\lbrace o_1,o_2,\ldots ,o_H\rbrace\) , a monitoring duration \(\mathbf {T}\) , and a remote cloud \(\mathcal {C}(M)\) with a data analyzing engine (neural networks or data-mining engines) M. The monitoring duration \(\mathbf {T}=\lbrace 1,2,\ldots ,K\rbrace\) is slotted into K time slots. During the monitoring duration \(\mathbf {T}\) , the sensory data of targets in \(\mathbf {O}\) will be sensed and gathered by EH nodes and edge servers and then be offloaded to the remote cloud \(\mathcal {C}\) . In the cloud, sensory data is fed into M to be further analyzed. For example, if the network is deployed in a forest, then the EH nodes will transmit the sensory data of the targets, such as temperature and humidity, to the cloud, and the analyzing engine in the cloud will determine if there is a potential fire hazard.
Let \(r_s\) and \(r_c\) be the sensing and communication radius of each EH node. A target \(o \in \mathbf {O}\) can be monitored by node i if the distance between o and i is less than \(r_s\) , and EH nodes can communicate with each other if their distance is within \(r_c\) . Thus, each EH node can collect data from the targets and transmit the collected data to edge servers. Without loss of generality, we assume that the interference radius of each EH node is equal to the communication radius.
3.2 Energy Model
Each EH node equips a supercapacitor with a capacity B to store energy. Based on the energy-harvesting technique, an energy-harvesting node i can harvest \(\mu _i(t)\) energy at time slot t. Thus, the energy kept by a node i at the beginning of time t, \(B_i(t)\) , can be calculated by
where \(\varepsilon _i(t-1)\) is the energy consumed by i during time slot \(t-1\) . Based on the properties of energy-harvesting nodes, a node i is available at time slot t if and only if its energy \(B_i(t)\) is no less than a threshold \(B_f\) . Let \(V_t= \lbrace i | B_i(t) \ge B_f \wedge i \in V \rbrace\) be the set of available EH nodes at time t. Then, each EH node in \(V_t\) can sense and transmit sensory data to edge servers at time t. Since the energy consumed by sensing is far less than that consumed by transmitting or receiving, we omit the sensing energy. Assume that each EH node expends \(\varepsilon ^B\) energy to maintain basic functionality and uses an additional \(\varepsilon ^T\) energy to transmit a single data packet. If an EH node i has transmitted \(n_i(t)\) packets during time slot t, then we have
Let \(D_i(t)\) be the data stored in EH node i at the beginning of the time slot t and if \(t=0\) , then \(D_i(t)=\emptyset\) . Denote \(d_j^t\) by the sensory data generated by \(o_j\) at time slot t. Let \(O_i = \lbrace o_j | dis(o_j, i) \le r_s\rbrace\) be the set of targets within EH node i’s sensing range. An available EH node i can gather a set of sensory data from targets in \(O_i\) at time slot t and update its memory as \(D_i(t)=D_i(t-1) \cup d(O_i,t),\) where \(d(O_i,t) = \lbrace d_j^t | o_j \in O_i\rbrace\) is the data generated by targets in \(O_i\) . We use a four-tuple \([i,j,\bar{D}_i(t),t]\) to represent that EH node i will transmit dataset \(\bar{D}_i(t)\) to edge server j at time slot t. We say that the four-tuple is a transmission. During the monitoring duration \(\mathbf {T}\) , a Data Collection \(\mathbf {D}\) is adopted to collect the sensory data. The Data Collection is defined as follows:
Definition 1 (Data Collection).
A Data Collection \(\mathbf {D} = \lbrace [i,j,\bar{D}_i(t),t] | i \in V \wedge j \in S \wedge \bar{D}_i(t) \wedge t \in \mathbf {T} \rbrace\) satisfies the following requirements:
where \(Ne(i)=\lbrace j | dis(i,j) \le r_c \wedge j \in S \rbrace\) is the set of one-hop neighbors of an EH node i.
The Energy Collision-free was first defined in Reference [7]. It implies that the EH node should have enough energy to transmit data packages. The Data Collision-free means each EH node can only transmit the data that it has stored. The Time Collision-free [7] requires that two transmissions can be scheduled at the same time if and only if their receivers are not in the interference range of other senders.
3.4 Problem Definition
In a real-time system—the E-EHN, for example—one major objective is to maintain the whole picture of all targets. From the cloud perspective, it has the following two demands:
(1) Usefulness: If the cloud can gather enough sensory data from which the analyzing engine deployed in the cloud can absorb precise and useful information.
(2) Freshness: If these sensory data are fresh enough to represent the current situations of the monitored targets.
Unfortunately, obtaining these two demands is really hard in the E-EHNs. At a time slot, only the available EH nodes can work and they can only monitor part of the targets. Even worse, an available EH node may not be able to transmit or receive data due to the Energy Collision and the Time Collision. Therefore, maintaining the usefulness and the freshness of the collected sensory data remains a problem in E-EHNs.
To further investigate this problem, we define the data usefulness and the data freshness in Section 3.4.1 and Section 3.4.2, respectively. In Section 3.4.3, we give a formal definition of the Min-AoUI problem, which aims to output a data collection schedule to keep the data usefulness and improve the data freshness as well.
3.4.1 Data Usefulness.
In a time slot t, we call the sensory data generated by all targets a snapshot of t. We have the following definition:
Definition 2 (Snapshot).
Given the set of targets \(\mathbf {O}=\lbrace o_1,o_2,\ldots ,\) \(o_H\rbrace\) and a time slot t, the snapshot of t is denoted by \(S_N(t) = \lbrace d_i^{t} | o_i \in \mathbf {O}\rbrace\) .
If the remote cloud \(\mathcal {C}\) has received \(S_N(t)\) , then it will feed \(S_N(t)\) into the analyzing engine M and obtain an analyzing result \(M(S_N(t))\) . Obviously, \(S_N(t)\) contains the information of all targets at time t and the analyzing result \(M(S_N(t))\) is precise. However, as discussed in the above section, collecting \(S_N(t)\) is really hard in the E-EHNs. Therefore, we are wondering if we can collect the sensory data from a small set of targets and keep the data useful as well. Considering the spatial relationship between sensory data, we noticed that for each pair of targets \(o_i,o_j\) , if the distance between them is within \(\delta\) , i.e., \(dis(o_i,o_j) \le \delta\) , then the sensory data \(d_i^t,d_j^t\) generated by \(o_i\) and \(o_j\) at the same time slot t are highly similar to each other. Here, \(\delta\) is a factor related to the monitored region. If the distance grows, then the similarity between the sensory data decreases. Thus, instead of feeding the analyzing engine M with the complete snapshot of t, using a subset of \(S_N(t)\) , \(\bar{S}_N(t) \in S_N(t)\) , can obtain similar analyzing results. The utility of \(\bar{S}_N(t)\) is defined as follows:
Definition 3 (Useful Dataset).
Given a snapshot of time slot t, \(S_N(t)\) , and a constant \(\alpha \le 1,\) then a set of sensory data \(\bar{S}_N(t)\) is useful if and only if \(\bar{S}_N(t) \subseteq S_N(t)\) and for each \(d_i^t \in S_N(t) / \bar{S}_N(t)\) there is a \(d_j^t \in \bar{S}_N(t)\) and \(dis(o_i,o_j) \le \frac{\delta }{\alpha }\) .
According to Definition 3, if \(\alpha = 1\) , then the analyzing engine M can precisely recover \(S_N(t)\) from \(\bar{S}_N(t)\) . Otherwise, M has to approximately recover \(S_N(t)\) . Thus, during the monitoring duration, the analyzing engine M needs to give a feasible \(\alpha\) , and the E-EHN will gather useful data at each time slot. We have discussed the utility of \(\bar{S}_N(t)\) and the selection of \(\alpha\) in Section 6.3. Without loss of generality, we assume that at time slot \(t=0\) , the freshest useful data is \(\bar{S}_N(0)=\emptyset\) .
Example. We use a fraction of data in dataset Krakow Air Pollution [30] as an example to further explain the data usefulness. The dataset [30] contains air-quality data from 2017 in Krakow, Poland. For convenience, we only consider the PM 2.5 of six spots (also known as targets in this article) in Krakow at 00:00, May 1st, 2017. The specific locations of these six spots (A, B, C,..., F) are shown in Figure 3(a). The table in Figure 3(b) illustrates the locations and the values of PM 2.5 of these spots. In this example, we assume that \(\delta = 0.8 km\) . Suppose the analyzing engine M aims to calculate the average PM 2.5 in this area and sets \(\alpha = 0.8\) . Then, the snapshot at \(t=00:00\) is \(S_N(t) = \lbrace 48, 48, 44, 45, 45, 43\rbrace\) and the average PM 2.5 in this area is \(M(S_N(t)) = 45.5\) . Constructing a useful dataset only requires collecting the sensory data from B, D, and F, and the useful dataset is \(\bar{S}_N(t) = \lbrace 48, 45, 43\rbrace\) . The average PM 2.5 calculated through \(\bar{S}_N(t)\) is \(M(\bar{S}_N(t)) = 45.3,\) which is highly similar to the real average PM 2.5 calculated through \(S_N(t)\) . Obviously, the useful dataset is not unique and we can also collect data from A, C, and E through which the average PM 2.5 is 45.6. This example implies that we can obtain similar analysis results by collecting the sensory data from a small set of targets.
Fig. 3.
Fig. 3. (a) The map that contains six targets and (b) the locations and the PM 2.5 of these six targets.
Given a data collection \(\mathbf {D}\) , the data obtained by the cloud at time slot t is
where \(\mathbf {S}(\mathbf {D}, t)\) is the set of all useful datasets from time slot 0 to t, and \(\varDelta (t)\) is the set of data that does not belong to any useful dataset. Based on the above equations, the cloud can hold multiple useful datasets during the monitoring duration.
3.4.2 Age of Useful Information.
Although the useful datasets contain the sketch of all targets at different time slots, it is also important to measure the freshness of such datasets. We have noticed that a cloud could have a series of useful datasets at a certain time slot. However, we only concentrate on the freshest useful dataset, since it is the most representative one. For each pair of useful datasets \(\bar{S}_N(t)\) and \(\bar{S}^{\prime }_N(t^{\prime })\) , if \(t \lt t^{\prime }\) , then \(\bar{S}^{\prime }_N(t^{\prime })\) is fresher than \(\bar{S}_N(t)\) . We define the AoUI to measure the freshness of the freshest useful dataset.
Definition 4 (Age of Useful Information (AoUI)).
Given a data collection \(\mathbf {D}\) , the data obtained by the cloud at \(t^{\prime }\) , \(\Pi (\mathbf {D}, t^{\prime })\) and the set of all useful datasets in \(\Pi (\mathbf {D}, t^{\prime })\) , \(\mathbf {S}(\mathbf {D},t^{\prime })\) . Let \(\bar{S}_N(t) \in \mathbf {S}(\mathbf {D},t^{\prime })\) be the freshest useful dataset in \(\mathbf {S}(\mathbf {D},t^{\prime })\) , i.e., \(t = \max \lbrace t | \bar{S}_N(t) \in \mathbf {S}(\mathbf {D},t^{\prime }) \rbrace\) . Based on \(\mathbf {D}\) , the AoUI at time slot \(t^{\prime }\) is calculated as
At the next time slot \(t^{\prime \prime }=t^{\prime }+1\) , if the cloud has not received any new useful dataset at \(t^{\prime \prime }\) , then the AoUI gets one time slot older
Example. Consider the example in Section 3.4.1 again. Figure 4 illustrates the time when different sensory data have arrived in the cloud based on a data collection \(\mathbf {D}\) , and Figure 5 records the value of AoUI at different time slots. At time slot \(t_3\) , the cloud has received three pieces of data \(\lbrace d_A^{t_1}, d_C^{t_1}, d_E^{t_1}\rbrace\) , and based on Definition 3, they comprise a useful dataset \(\bar{S}_N(t_1)\) . Therefore, due to the data transmission delay, the overall sensory data generated at \(t_1\) , a.k.a. the snapshot \(S_N(t_1)\) , can be approximately maintained by the cloud at \(t_3\) . Based on the definition of AoUI, \(A_{\mathbf {D}}(t_3) = 3-1=2\) . At time \(t_3\) , no fresher useful dataset has arrived. Thus, \(A_{\mathbf {D}}(t_4) = 2+1=3\) . Similarly, at time \(t_6\) , the cloud has received a fresher useful dataset \(\bar{S}_N(t_4)=\lbrace d_A^{t_4},d_C^{t_4},d_E^{t_4}\rbrace\) and \(A_{\mathbf {D}}(t_6) = 6-4=2\) .
Fig. 4.
Fig. 4. The arrival time of different data.
Fig. 5.
Fig. 5. The age of useful information. At time slot 3, the useful dataset at time slot 1 has arrived, and \(A_{\mathbf {D}}(t_3)=3-1=2\) . At time slot 4, \(A_{\mathbf {D}}(t_4)=2+1=3\) .
3.4.3 The Min-AoUI Problem.
From the cloud perspective, the AoUI is a key measurement of the freshness of useful datasets. A high AoUI is really harmful to a real-time monitor system. Based on Definition 4, the AoUI is related to the data collection strategy. To improve the monitoring performance of the E-EHN, an intuitive way is to construct a data collection schedule to minimize the average AoUI during the monitoring duration. However, minimizing the average AoUI cannot always avoid the existence of high AoUI. Therefore, we want to construct a data collection to minimize the maximum AoUI during the monitoring duration, and we define the Minimizing the maximum Age of Useful Information(Min-AoUI) problem.
Min-AoUI Problem
Input:
(1) an E-EHN \(\mathbf {N}(V,S,\mathbf {O},\mathbf {T},\mathcal {C}(M))\) ,
(2) the sensing and communication radius \(r_s\) , \(r_c\) ,
(3) the energy harvested by EH node i at time slot t, \(\mu _i(t)\) ,
(4) the energy consumed by transmitting one data packet, \(\varepsilon ^T\) , and the energy used to maintain the basic functionality, \(\varepsilon ^B\) ,
(5) the energy capacity of EH nodes B and the threshold \(B_f\) ,
(6) two constants given by users, \(\alpha\) and \(\delta\) .
Outputs: a data collection \(\mathbf {D}\) , such that
where \(\tilde{\mathbf {D}}\) is the universal set of data collections.
Theorem 1.
The Min-AoUI problem is NP-Hard.
The proof of this theorem can be found in the appendix.
4 Minimize AoUI in S-E-EHNs
We begin by exploring a scenario where only one edge server is present in the E-EHN. In the following section, the approach used in this simpler context will be extended to handle E-EHNs with multiple edge servers. This simpler single-server scenario is indicative of smart home applications where the monitored area is relatively small, making a single edge server sufficient for the task. We use S-E-EHN (“S” represents single) to denote the E-EHN with a single edge server. In this section, we will solve the Min-AoUI problem in an S-E-EHN (Min-AoUI-S). In the following, we first analyze the upper bound and lower bound of the AoUI in the S-E-EHN and then we propose an approximation algorithm to solve the Min-AoUI-S problem.
4.1 Upper Bound and Lower Bound of AoUI
An S-E-EHN is illustrated in Figure 6. Due to the Time Collision, only one available EH node can send sensory data to the edge server in a time slot. Let \(s_0\) be the edge server in the E-EHN and \(\mathbf {D}\) be a feasible data collection. For convenience, we use \(A(\cdot)\) instead of \(A_\mathbf {D}(\cdot)\) in the following paragraphs: Suppose that based on \(\mathbf {D}\) , the cloud has received a set of useful sensory data \(\mathbf {S}(\mathbf {D})\) at the end of the monitoring duration. For a \(\bar{S}_N(t) \in \mathbf {S}(\mathbf {D})\) , let \(s(t) \in \mathbf {T}\) be the time slots when the network starts collecting sensory data in \(\bar{S}_N(t)\) and \(e(t) \in \mathbf {T}\) be the time slots when \(\bar{S}_N(t)\) has been received by the edge server. Then, we have the definition of Efficient Data Collection.
Fig. 6.
Fig. 6. At each time slot, only one EH node can transmit to the edge server.
Definition 5 (Efficient Data Collection (EDC)).
Given an E-EHN \(\mathbf {N}(V,S,\mathbf {O}, \mathbf {T}, \mathcal {C}(M))\) , a feasible data collection \(\mathbf {D}\) and the set of useful datasets \(\mathbf {S}(\mathbf {D})\) , then \(\mathbf {D}\) is an Efficient Data Collection if it satisfies the following conditions:
(1)
For each \(\bar{S}_N(t) \in \mathbf {S}(\mathbf {D})\) , there is \(s(t)=t\) .
(2)
For each pair of \(t,t^{\prime }\) and \(\bar{S}_N(t) \in \mathbf {S}(\mathbf {D})\) , \(\bar{S}_N(t^{\prime }) \in \mathbf {S}(\mathbf {D})\) , if \(t \gt t^{\prime }\) , then \(e(t) \gt e(t^{\prime })\) .
The first condition implies that the EDC should always collect the freshest data. The second condition means that the cloud should always receive the freshest sensory data. Obviously, the optimal data collection is an EDC.
For each two useful datasets \(\bar{S}_N(t^{\prime })\) , \(\bar{S}_N(t^{\prime \prime })\) and \(t^{\prime }\lt t^{\prime \prime }\) , they are adjacent if \(\forall \bar{S}_N(t) \in \mathbf {S}(\mathbf {D})\) , \(t \notin [t^{\prime },t^{\prime \prime }]\) . Then, for an EDC, we have the following theorem:
Theorem 2.
Given an EDC \(\mathbf {D}\) and the set of useful datasets \(\mathbf {S}(\mathbf {D})\) , then for each pair of adjacent useful datasets \(\bar{S}_N(t^{\prime })\) , \(\bar{S}_N(t^{\prime \prime })\) , we have
As shown in Figure 5, for each \(\bar{S}_N(t) \in \mathbf {S}(\mathbf {D})\) , \(A(e(t)-1)\) is the local peak value. We also have the following theorem:
Theorem 3.
Given a time duration \(\mathbf {T}=\lbrace 1,2,\ldots ,K\rbrace\) , an EDC \(\mathbf {D}\) , a set of useful dataset \(\mathbf {S}(\mathbf {D})\) and \(|\mathbf {S}(\mathbf {D})| = m\) , then
(1) \(A(K)=\max \lbrace A(t)|t \in \mathbf {T} \rbrace\) and \(K \notin \lbrace t | \bar{S}_N(t) \in \mathbf {S}(\mathbf {D})\rbrace\) .
(2) \(A(t^{\prime }) = \max ~\lbrace A(t) | t \in \mathbf {T} \rbrace\) and \(t^{\prime }\lt K\) .
In Case 1, according to Theorem 2 and Definition 4, \(A(K) = K - s(t)\) where \(t = \max \lbrace t | \bar{S}_N(t) \in \mathbf {S}(\mathbf {D})\rbrace\) . Since transmitting each sensory data costs at least a time slot, we have
Given an arbitrary EDC \(\mathbf {D}\) , the upper bound of AoUI during the monitoring duration \(\mathbf {T}\) is \(|\mathbf {T}| - m\) , where \(m = | \mathbf {S}(\mathbf {D}) |\) and \(\mathbf {S}(\mathbf {D})\) is the set of useful datasets.
To analyze the lower bound of AoUI, we define the Representative Set.
Definition 6 (Representative Set).
Given an E-EHN \(\mathbf {N}(V,S,\) \(\mathbf {O}, \mathbf {T}, \mathcal {C}(M))\) , a representative set at time slot t, \(R(t)\) , is a set of EH nodes that satisfies that \(\lbrace d_j^t | \exists i \in R(t), dis(i,o_j) \le r_s\rbrace\) is a useful dataset.
In a time slot, there exist many representative sets, and the minimum representative set at time slot t is \(R_m(t) = \arg \min \lbrace |R(t)| | R(t)~is~a~representative~set~at~t\rbrace\) . The minimum representative set during \(\mathbf {T}\) is \(\bar{R}_m = \arg \min \lbrace |R_m(t)| | \bar{S}_N(t) \in \mathbf {S}(\mathbf {D}^{opt}) \wedge t\in \mathbf {T}\rbrace\) where \(\mathbf {D}^{opt}\) is the optimal data collection. We have the following theorem:
Theorem 4.
Given an S-E-EHN \(\mathbf {N}(V,S,\mathbf {O}, \mathbf {T}, \mathcal {C}(M))\) and an optimal data collection \(\mathbf {D}^{opt}\) , then the maximum AoUI is at least \(|\bar{R}_m|\) .
Proof.
Suppose that the maximum AoUI caused by the optimal data collection is less than \(|\bar{R}_m(t)|\) , then there must exist a time slot \(t^{\prime }\) that satisfies \(\bar{S}_N(t^{\prime }) \in \mathbf {S}(\mathbf {D})\) and
Then, we can construct a representative set \(R(t^{\prime }) = \lbrace i | i \in V\ and\ i\ has\ transmitted\ data\ to edge\ server\ during\ [s(t^{\prime }),e(t^{\prime })]\rbrace\) . At each time slot, there is only one EH node that can communicate with the edge server. Thus, we have
So, \(R(t^{\prime })\) is the minimum representative set, a contradiction. Thus, the theorem is proved. □
The size of the minimum representative set is only based on the distance between targets and EH nodes. Thus, the minimum representative set \(\bar{R}_m\) is an inherent property of an E-EHN.
4.2 Approximated Algorithm for Min-AoUI-S
Theorem 3 implies that maximizing the number of useful datasets can indirectly minimize the value of AoUI. Thus, in this algorithm, we want to maximize the size of \(\mathbf {S}(\mathbf {D})\) and solve the Min-AoUI-S problem approximately.
One major challenge is that the actual energy-harvesting rate of each EH node at each time slot is hard to predict, and thus it is really hard to determine whether an EH node is available at a specific time slot. Fortunately, based on the historical statistic, we can easily obtain the minimum energy-harvesting rate, \(\mu _{min}\) , at each time slot, where \(\mu _{min}=\min \lbrace \mu _i(t) | i \in V \wedge t\in \mathbf {T} \rbrace\) . We can use \(\mu _{min}\) to approximately estimate the minimum energy kept by each EH node.
At each time slot, there is at most one EH node that can transmit sensory data. Furthermore, the EH node can only transmit a limit number of sensory data due to the energy limitation. Thus, to maximize the size of \(\mathbf {S}(\mathbf {D})\) , we should minimize the transmission delay of each useful dataset and an EH node should transmit the most “representative” sensory data. For sensory data \(d_j^t\) , its representativeness is defined as follows:
Definition 7 (Sensory data Representativeness).
Given a pair of sensory data \(d_j^t\) and \(d_i^t\) , we say that \(d_j^t\) can be represented by \(d_i^t\) if \(dis(o_i,o_j) \le \frac{\delta }{\alpha }\) and vice versa. The representativeness of \(d_j^t\) is \(Re(d_j^t) = \lbrace d_i^t | dis(o_i,o_j) \le \frac{\delta }{\alpha } \wedge o_j \in \mathbf {O} \rbrace\) .
According to Definition 3, if \(d_j^t\) can be represented by \(d_i^t\) , then \(d_j^t\) and \(d_i^t\) are similar to each other. The representativeness of a sensory data \(d_j^t\) illustrates how much sensory data that are similar to \(d_j^t\) and can be represented by \(d_j^t\) .
Based on this strategy, we propose the Greedy Useful Data Sets Collection Algorithm in S-E-EHN(GRUSS-S), which is illustrated in Algorithm 1. The GRUSS-S algorithm scans the time slots in \(\mathbf {T}\) and collects useful datasets of specific time slots greedily. Let \(t_s\) be the timestamp of the freshest useful dataset and \(t_c\) be the current time slot. The GRUSS-S algorithm has the following steps:
Step 1 (Lines 1–3). Initialize \(\mathbf {D}=\emptyset\) , \(t_s=1,\) and \(t_c=1\) . The scan begins at the first time slot. Because each EH node can work at the beginning, the algorithm first collects the useful sensory data in the first time slot.
From Step 2–Step 4, the algorithm tries to construct transmissions to collect the useful dataset of a specific time slot \(t_s\) . The following steps will be processed in iteration until all time slots have been scanned, i.e., \(t_c \le |\mathbf {T}|\) .
Initially (Line 5), the useful dataset of \(t_s\) is initialized as an empty set, \(\bar{S}_N(t_s) = \emptyset\) .
Step 2 (Lines 6–10). If there exist EH nodes that can work at time slot \(t_c\) , then calculate the weight of each EH node according to the following two substeps:
(1)
If an EH node \(i \in V\) has enough energy to transmit at least one packet of sensory data, then it will calculate the weight of each sensory data in \(\lbrace d_j^{t_s} | dis(o_j, i)\le r_s \wedge o_j \in \mathbf {O}\rbrace\) .
Obviously, \(|\bigcup \nolimits _{d_j^{t_s} \in \bar{S}_N(t_s)} Re(d_j^{t_s})|\) is the set of sensory data that has been represented by the data in \(\bar{S}_N(t_s)\) . According to Definition 3, if all sensory data in \(t_s\) have been represented, then \(\bar{S}_N(t_s)\) is a useful dataset. The weight of a sensory data \(d_j^t\) illustrates how much new sensory data it can represent. The weight of an EH node i, \(W_i^j\) , means if i transmits the sensory data \(d_j^t\) , then it will represent \(w_i(d_j^{t_s})\) new sensory data.
If there is no EH nodes that can transmit at time slot \(t_c\) , then update \(t_c\) as \(t_c++\) (Line 20).
Step 3 (Lines 12–16). Construct a transmission to collect sensory data in \(t_s\) . The EH node with the highest weight, \(W_{\bar{i}}^{\bar{j}}\) , will be selected to transmit sensory data \(d_{\bar{j}}^{t_s}\) . The transmission is constructed based on the following two cases:
(1)
If \(\bar{i}\) has already transmitted other sensory data with timestamp \(t_s\) and after transmitting these data \(\bar{i}\) has enough energy to transmit \(d_{\bar{j}}^{t_s}\) , i.e., \(\exists [\bar{i},s_0, d_j^{t_s}, t^{\prime }] \in \mathbf {D}\) and \(B_{\bar{i}}(t^{\prime }+1) - B_f - \mu _{min} \ge \varepsilon ^T + \varepsilon ^B\) , then we can schedule \(\bar{i}\) to transmit at time slot \(t^{\prime }\) , i.e., \([\bar{i},s_0, d_{\bar{j}}^{t_s}, t^{\prime }]\) .
(2)
If \(\bar{i}\) has not transmitted any data with timestamp \(t_s\) , then we can schedule \(\bar{i}\) to transmit at current time slot \(t_c\) .
Step 4 (Lines 17–18). Update the energy of each EH node based on the new data collection schedule. Furthermore, update the \(t_c\) as \(t_c++\) if an EH node has transmitted in \(t_c\) .
The above three steps process in iteration until \(\bar{S}_N(t_s)\) is a useful dataset.
Step 5 (Lines 21–23). Update \(t_c\) and \(t_s\) when a useful dataset has been collected. To effectively use the energy kept in each EH node, we update \(t_c\) as the first time slot if there exists an EH node that can transmit at least a packet of data. According to Definition 5, we set \(t_s = t_c\) .
Obviously, the GRUSS-S algorithm can construct a feasible EDC \(\mathbf {D,}\) and the performance of the GRUSS-S algorithm is analyzed in the following section.
4.3 Performance Analyzing
4.3.1 Time Complexity.
From Line 4 to Line 20, there are \(|\mathbf {T}|\) iterations at most. In Step 2, calculating the weight of each EH node costs \(O(|V||N_{max}|)\) time, where \(N_{max}\) is the maximum number of targets monitored by each EH node. In Step 3, constructing a transmission needs \(O(1)\) time. In Step 4, updating the energy of each EH node costs \(O(|V|)\) time. Therefore, the time complexity of GRUSS-S algorithm is \(O(|\mathbf {T}||V|N_{max})\) .
4.3.2 Approximation Ratio.
Lemma 1.
Let \(\mathbf {D}\) be the data collection generated by the GRUSS-S algorithm and \(\Pi (\mathbf {D}, t)\) be the data obtained by the cloud at time slot t. If there exists a set of EH nodes \(V_t\) such that for each \(i\in V_t\) , \(B_i(t)-B_f \ge \varepsilon ^T + \varepsilon ^B\) , then based on GRUSS-S, there must exist \(i^{\prime } \in V_t\) , which will transmit a sensory data d to the edge server and \(d \notin \Pi (\mathbf {D}, t)\) .
Proof.
According to Step 4 of the GRUSS-S algorithm, if there exist EH nodes that can work at time t, then the node with the highest weight will be selected to transmit sensory data. Obviously, in a time slot t, if t is the most recent time, then the sensory data of t have not been received by the cloud. Thus, the theorem is proved. □
Lemma 2.
Let \(\Pi (\mathbf {D})\) be the data received by the cloud during \(\mathbf {T}\) . We have \(|\Pi (\mathbf {D})| \ge \tfrac{\mu _{min}|\mathbf {T}|}{\varepsilon ^T + \varepsilon ^B}\) .
Proof.
Obviously, there must exist at least one EH node that can transmit at least one packet of sensory data for every \(\tfrac{\varepsilon ^T + \varepsilon ^B}{\mu _{min}}\) time slots. Based on Lemma 1, the cloud can collect at least \(\tfrac{\mu _{min}|\mathbf {T}|}{\varepsilon ^T + \varepsilon ^B}\) sensory data. □
Let \(A^g\) be the AoUI obtained by the GRUSS-S algorithm. We have the following two theorems:
Obviously, for an arbitrary useful dataset \(\bar{S}_N(t)\) , \(|\bar{S}_N(t)|\le |\mathbf {O}|\) . According to Lemma 1 and 2, the cloud can collect at least \(\tfrac{\mu _{min}|\mathbf {T}|}{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)}\) useful datasets. Then, based on Theorem 3 and Corollary 1, we have
Each EN node could be able to work at least once in every \(\tfrac{\varepsilon ^T + \varepsilon ^B}{\mu _{min}}\) time slots. Therefore, based on Step 2 of the GRUSS-S algorithm, there exists at least one EH node that can transmit a packet of data in every \(\tfrac{\varepsilon ^T + \varepsilon ^B}{\mu _{min}}\) time slots. Because \(|\bar{S}_N(t)|\le |\mathbf {O}|\) for every arbitrary \(t \in \mathbf {T}\) , we have
Thus, we have \(A^g \le \tfrac{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)}{\mu _{min}}\) . □
Based on Theorems 4, 5, and 6, we have the following two corollaries:
Corollary 2.
The AoUI obtained by the GRUSS-S algorithm is less than \(\min \lbrace |\mathbf {T}|(1-\tfrac{\mu _{min}}{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)})+1,\tfrac{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)}{\mu _{min}}\rbrace\) .
Corollary 3.
The approximation ratio of the GRUSS-S algorithm is \(\tfrac{\rho }{|\bar{R}_m|}\) , where \(\rho = \min \lbrace |\mathbf {T}|(1-\tfrac{\mu _{min}}{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)})+1,\tfrac{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)}{\mu _{min}}\rbrace\) and \(\bar{R}_m\) is the minimum representative set.
Estimate the minimum representative set. One should notice that the minimum representative set is an inherent property of an E-EHN. However, how to estimate the minimum representative set is a difficult problem. For each EH node i, let \(O_i = \lbrace o_j | dis(i,o_j) \le r_s\rbrace\) be the set of targets that are monitored by i and \(\bar{O}_i = \lbrace o_j | dis(o_j, o_l)\le \tfrac{\delta }{\alpha } \wedge o_l \in O_i\rbrace\) . Obviously, for each \(o_j \in \bar{O}_i\) , its sensory data can be represented by the sensory data of targets in \(\bar{O}_i\) . Thus, a representative set is a set cover of the targets set. For a representative set \(R(t)\) , we have \(\mathbf {O} = \bigcup \nolimits _{i \in R(t)} \bar{O}_i\) , and the minimum representative set \(\bar{R}_m\) must be smaller than the minimum set cover. The minimum set cover can be estimated by many existing works[49] and then we can estimate the minimum representative set.
5 Minimize AoUI in M-E-EHNs
5.1 Upper Bound and Lower Bound of AoUI
We use M-E-EHN to represent the E-EHN with multiple edge servers. In this section, we will solve the Min-AoUI problem in the M-E-EHNs (Min-AoUI-M). In a time slot, multiple EH nodes can transmit in an M-E-EHN. Obviously, Theorem 3 still holds in the M-E-EHNs. Thus, the upper bound of the AoUI in the M-E-EHNs is still related to the number of useful datasets the cloud has received. However, the lower bound of the AoUI can be modified.
Theorem 7.
Given an M-E-EHN \(\mathbf {N}(V,S,\mathbf {O},\) \(\mathbf {T}, \mathcal {C}(M))\) , and an optimal data collection \(\mathbf {D}^{opt}\) , then the maximum AoUI is at least \(\tfrac{|\bar{R}_m|}{|S|}\) .
Proof.
In the M-E-EHNs, there are at most \(|S|\) EH nodes that can transmit at a time slot. Therefore, similar to the proof of Theorem 4, the maximum AoUI is at least \(\tfrac{|\bar{R}_m|}{|S|}\) . □
5.2 Approximated Algorithm for Min-AoUI-M
In this section, we want to extend the GRUSS-S algorithm to minimize the AoUI in the M-E-EHNs. However, multiple edge servers may result in multiple collisions in data transmission. Let \(Ne(s)=\lbrace i | i \in V \wedge dis(i,s)\le r_c\rbrace\) be the set of EH nodes that can transmit sensory data to edge server s. For each pair of edge servers s and \(s^{\prime }\) , if there exists \(i^{\prime } \in Ne(s) \cap Ne(s^{\prime })\) , then according to the time collision, when \(i^{\prime }\) is transmitting to s (or \(s^{\prime }\) ), other EH nodes in \(Ne(s) \cup Ne(s^{\prime })\) should keep silent.
An extreme scenario is, if \(i^{\prime } \in \bigcap \nolimits _{s \in S} Ne(s)\) , then when \(i^{\prime }\) is transmitting, all other EH nodes cannot transmit. For example, in Figure 7, when EH node i is transmitting, all other EH nodes cannot transmit. From another perspective, when an EH node \(i^{\prime }\) is transmitting, none of its neighbors can transmit either. Thus, the interference set is defined as follows:
Fig. 7.
Fig. 7. The transmission collision in an M-E-EHN.
Definition 8 (Interference Set).
Given an E-EHN \(\mathbf {N}(V,S,\mathbf {O},\) \(\mathbf {T},\mathcal {C}(M))\) and an EH node i that will transmit at time slot t, then the Interference Set of i at t, \(I(i,t)\) , satisfies that
(1) for each \(j \in I(i,t)\) , \(j \in V\) and \(B_i(t)-B_f \ge \varepsilon ^T + \varepsilon ^B\) ;
(2) for each \(j \in I(i,t)\) , \(j \in Ne(i)\) or \(j \in \lbrace j | j \in Ne(s) \wedge s \in S \wedge i \in Ne(s)\rbrace\) .
The first condition implies that each EH node in the interference set \(I(i,t)\) should be able to work at time slot t. The second condition means that if i will transmit at time slot t, then the EH nodes in \(I(i,t)\) cannot transmit at t. The interference set records the EH nodes that are interfered by the transmission of i at time slot t. Therefore, when scheduling an EH node to transmit, besides the representativeness of the transmitted sensory data, we should also consider the interference set.
The Greedy Useful Data Sets Collection Algorithm in M-E-EHN(GRUSS-M) is illustrated in Algorithm 2. The GRUSS-M algorithm is similar to the GRUSS-S algorithm.
The major difference between GRUSS-M and GRUSS-S is the weight calculation part (Lines 7–11). In a current time slot \(t_c\) , if there exists an EH node i that does not belong to any interference set and i can transmit, i.e., \(B_i(t_c) - B_f \ge \varepsilon ^T + \varepsilon ^B\) , then the weight of i will be calculated. Otherwise, update \(t_c\) as \(t_c++\) (Line 20). Different from GRUSS-S, the weight of i is
where \(|I(i,t_c)|\) is the size of i’s interference set.
5.3 Performance Analyzing
In the GRUSS-M algorithm, the calculation of the interference set costs \(O(|V|)\) time. Based on the time complexity analyzed in Section 4.3.1, the time complexity of the GRUSS-M algorithm is \(O(|\mathbf {T}||V|N_{max})\) .
Obviously, Theorem 5,6 still hold in the GRUSS-M. Thus, we have the following corollary:
Corollary 4.
The approximation ratio of the GRUSS-M algorithm is \(\tfrac{\rho }{|\bar{R}_m|}\) , where \(\rho = \min \lbrace |\mathbf {T}|(1-\tfrac{\mu _{min}}{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)})+1,\tfrac{|\mathbf {O}|(\varepsilon ^T + \varepsilon ^B)}{\mu _{min}}\rbrace\) and \(\bar{R}_m\) is the minimum representative set.
6 Experiment
6.1 Simulation Settings
In this section, we evaluate the performance of the GRUSS-S and GRUSS-M algorithms by simulations. We deploy an S-E-EHN, \(\mathbf {N}_S\) , in a \(20m \times 20m\) square, and an M-E-EHN, \(\mathbf {N}_M\) , in a \(60m \times 60m\) square, to evaluate the performance of GRUSS-S and GRUSS-M separately. The \(20m \times 20m\) square consists of 10 targets, and it simulates an office or a lobby. The \(60m \times 60m\) square simulates a plaza or a small park and has 40 targets. We assume that, for each pair of targets, if their distance is within \(\delta = 5m\) , then at each time slot, they collect the same sensory data. There is one edge server in \(\mathbf {N}_S\) and nine edge servers in \(\mathbf {N}_M\) . Both \(\mathbf {N}_S\) and \(\mathbf {N}_M\) consist of homogeneous EH nodes. We simulate the parameters of EH nodes as follows [4, 48, 52]: the energy capacity is \(B=10mJ\) , the threshold is \(B_f= 5mJ\) , the sensing and communication radius is \(r_c=r_s=10m\) , and sending one data packet will cost 3mJ energy. To simulate the ambient energy, we adopt the solar radiation data in Reference [14]. Based on Reference [14], the ambient energy harvested by each EH node varies from \(0.5mJ\) to \(1.5mJ\) .
In the following, we first evaluate the impact of the network parameters on the AoUI obtained by GRUSS-S and GRUSS-M: The parameters of the networks include the length of the monitoring duration, the value of \(\alpha\) , the energy-harvesting rate, the number of EH nodes, the data transmission energy, and the initial energy of each EH node. Then, we compare the GRUSS-based algorithm with the state-of-the-art method [4] under a real-world dataset.
6.2 The Impact of Different Parameters
6.2.1 The Impact of the Length of \(\mathbf {T}\) .
Based on Theorem 5, the length of the monitoring duration, \(|\mathbf {T}|\) , is related to the upper bound of the AoUI. To examine the impact of \(|\mathbf {T}|\) , we vary the length of \(\mathbf {T}\) in \(\mathbf {N}_S\) from 20 time slots to 100 time slots and vary the length of \(\mathbf {T}\) in \(\mathbf {N}_M\) from 50 time slots to 400 time slots. There are 5 EH nodes in \(\mathbf {N}_S\) , 50 EH nodes in \(\mathbf {N}_M\) , and in each E-EHN \(\alpha = 0.5\) and \(\mu _{min}=0.5mJ\) . We process the GRUSS-S and GRUSS-M algorithms on \(\mathbf {N}_S\) and \(\mathbf {N}_M\) hundreds of times and record the average, the minimal, and the maximal of the obtained AoUI. We also calculate the upper bound and lower bound AoUI under different monitoring durations. The results are shown in Figure 8(a) and Figure 8(b).
Fig. 8.
Fig. 8. The impact of \(|\mathbf {T}|\) .
We can see that the AoUI lower bound does not have a remarkable change, since the lower bound of AoUI is majorly related to the network topology. Furthermore, we can also notice that the AoUI obtained by the GRUSS-based algorithm is very close to the lower bound. The AoUI obtained by the GRUSS-S algorithm is 1.47 times larger than the lower bound, and the AoUI obtained by the GRUSS-M algorithm is 1.49 times larger than the lower bound.
6.2.2 The Impact of \(\mu _{min}\) .
The value of \(\mu _{min}\) implies the minimum energy-harvesting ability of each EHN. To investigate the influence of \(\mu _{min}\) , we vary the value of \(\mu _{min}\) from \(0.5mJ\) to \(1.5mJ\) . Furthermore, there are 5 EH nodes in \(\mathbf {N}_S\) and 50 EH nodes in \(\mathbf {N}_M\) , and the monitoring duration contains 50 time slots. The GRUSS-S and GRUSS-M algorithms are processed in these networks under different values of \(\mu _{min}\) . We also record the maximal, average, and minimal AoUI obtained by these algorithms. The results are illustrated in Figure 9(a) and Figure 9(a).
Fig. 9.
Fig. 9. The impact of \(\mu _{min}\) .
According to Corollary 2, the upper bound decreases when the value of \(\mu _{min}\) grows. Furthermore, the AoUI obtained by the GRUSS-based algorithms also decreases when the value of \(\mu _{min}\) grows. When \(\mu _{min}\) grows from 0.5 to 1.5, the values of AoUI obtained by GRUSS-S and GRUSS-M decrease 43.1% and 47.6%, respectively. We have also noticed that, when \(\mu _{min}\) is small, the AoUI obtained by the GRUSS-based algorithms has a high deviation. When \(\mu _{min} = 0.5mJ\) , the maximal AoUI returned by the GRUSS-M algorithm is 97.
6.2.3 The Impact of \(\alpha\) .
\(\alpha\) is a key measurement that implies the spatial relationship between sensory data. When the value of \(\alpha\) is low, the spatial relationship between sensory data is high. To measure the impact of the \(\alpha\) , we adjust the value of \(\alpha\) from 0.3 to 0.9. We also deploy 5 EH nodes and 50 EH nodes in \(\mathbf {N}_S\) and \(\mathbf {N}_M\) , and the length of monitoring duration \(|\mathbf {T}|=200\) , \(\mu _{min}=0.5mJ\) . We process GRUSS-S and GRUSS-M algorithms on these networks and illustrate the results and the upper bound and lower bound of AoUI in Figure 10(a) and Figure 10(b).
Fig. 10.
Fig. 10. The impact of \(\alpha\) .
As shown in Figure 10(a) and Figure 10(b), when the value of \(\alpha\) grows, a.k.a. the spatial relationship between sensory data decrease, the values of AoUI obtained by the GRUSS-based algorithms and the lower bound of AoUI increases. We also notice that the AoUI obtained by GRUSS-S and GRUSS-M is close to the lower bound. The values of AoUI returned by GRUSS-S and GRUSS-M are 1.6 and 1.32 times larger than the lower bound, respectively.
6.2.4 The Impact of the Number of Nodes.
To evaluate the impact of the number of nodes on the value of AoUI, we vary the number of EH nodes in \(\mathbf {N}_S\) from 3 to 9 and the number of EH nodes in \(\mathbf {N}_M\) from 20 to 100. In this group of simulations, \(\mu _{min}=1.5mJ\) and \(\alpha =0.5\) . The simulation results are shown in Figure 11(a) and Figure 11(b). The AoUI decreases when the number of EH nodes increases. As for the GRUSS-S algorithm, when the number of EH nodes grows from 3 to 9, the AoUI drops about 24%. When the number of EH nodes rises from 20 to 100, the AoUI obtained by the GRUSS-M algorithm declines by about 31.4%. It is easy to see that the value of AoUI does not have significant changes when the size of the network dramatically grows. The reason is that when the size of the network grows, the interference between EH nodes also increases.
Fig. 11.
Fig. 11. The impact of \(|V|\) .
6.2.5 The Impact of \(\varepsilon ^T\) .
The energy consumption for transmitting one data packet is represented by \(\varepsilon ^T\) , which plays a significant role in influencing the data collection process within the E-EHN. To assess the impact of \(\varepsilon ^T\) on the AoUI, we vary its value within the range of 1mJ to 5mJ. The simulation results are shown in Figure 12(a) and Figure 12(b). As \(\varepsilon ^T\) increases, there is a corresponding rise in AoUI. Notably, the AoUI resulting from the GRUSS-based algorithm exhibits a slow growth rate. When \(1 \le \varepsilon ^T \le 3\) , the algorithms grounded on GRUSS approach optimality, with the AoUI they yield closely aligning with the lower bound. Compared to the upper bound, the AoUI of the GRUSS-based algorithms is almost 10 times smaller than the lower bound.
Fig. 12.
Fig. 12. The impact of \(\varepsilon ^T\) .
6.2.6 The Impact of the Initial Energy.
In the aforementioned simulations, each EH node starts with a full charge. Nonetheless, the initial energy can influence the AoUI. To assess this effect, we adjust the starting energy of each EH node, ranging from 8mJ to 10mJ. Figure 13(a) and Figure 13(b) illustrate the simulation results. For GRUSS-S, the AoUI remains largely stable despite the increase in initial energy, and for GRUSS-M, the AoUI is reduced 57% when the initial energy grows from 8mJ to 10mJ. Significantly, the performance of GRUSS-M exhibits greater sensitivity to initial energy fluctuations compared to the GRUSS-S algorithm. There is a pronounced variance in the AoUI produced by GRUSS-M when the initial energy falls below 9mJ. Given the increased number of EH nodes and targets, it is logical for GRUSS-M to exhibit heightened sensitivity to variations in initial energy.
Fig. 13.
Fig. 13. The impact of initial energy.
6.2.7 Discussion.
We assessed how various system parameters affect the performance of GRUSS-based algorithms. Our findings indicate that these algorithms frequently surpass the upper bound and closely align with the lower bound in terms of performance. We observed that the AoUI of GRUSS-based algorithms is significantly influenced by the values of \(\mu _{min}\) , \(\alpha\) , and \(\varepsilon ^T\) , as these parameters affect execution rounds and data representativeness. Given that all EH nodes undergo recharging and generally adhere to a consistent working cycle, factors such as initial energy, the number of EH nodes, and monitoring duration length do not profoundly impact algorithmic performance.
6.3 Comparison with the SOTA
In this set of simulations, we evaluate the performance of the GRUSS-based algorithm in comparison with the state-of-the-art (SOTA) algorithm, MAoIG, presented in Reference [4]. Although the authors of Reference [4] have addressed AoI optimization in EHNs, MAoIG primarily focuses on minimizing the average AoI across all targets. We further model a real-world scenario—PM2.5 monitoring—leveraging the Krakow Air Pollution dataset [30]. The objective, at every time slot, is for the EHN to relay the average PM2.5 concentration across the entire monitoring region. As depicted in Figure 14(a), the monitoring infrastructure comprises seven EH nodes \(V=\lbrace 1, 2,\ldots ,7\rbrace\) and a single edge server. Seven targets \(\mathbf {O}=\lbrace o_1,o_2,\ldots ,o_7\rbrace\) overlapped with these seven EH nodes. This monitoring spans a total of 30 hours, containing 180 time slots \(\mathbf {T}=\lbrace 1,2,\ldots ,180\rbrace\) , with each time slot lasting 10 minutes. Each EH node is represented by a LoRa node, augmented with an energy-harvesting component. With a communication radius of 1km, each EH node expends 2mJ to transmit a data packet. The energy-harvesting rate is set to a minimum threshold of \(\mu _{min} = 1.5mJ\) and the similarity threshold \(\delta = 300m\) . Recall the network model in Section 3.1, the data analyzing engine M in this application is an average function
where \(\bar{S}_N(t)\) is the useful dataset, and \(d_i^t\) is the data generated by \(o_i\) at time slot t. We use the root mean squared error (RMSE) to evaluate the usefulness of the data received by the edge server. Let \(\tilde{P_{2.5}^t}\) be the groundtruth of the average PM 2.5 at time slot t. Then, the data utility during the monitoring duration is calculated by
Fig. 14. Comparison between GRUSS-based algorithm and MAoIG under a real-world dataset.
We vary the value of \(\alpha\) from 0.2 to 1 and assess the AoUI and utility yielded by both GRUSS-S and MAoIG. Their performances are depicted in Figure 14(b). Furthermore, Figure 15 shows the average PM 2.5 levels as reported by GRUSS-S, MAoIG, and the actual ground truth for values of \(\alpha =0.4\) and \(\alpha =0.9\) . We can notice that the AoUI and utility achieved by GRUSS-S are both better than that of MAoIG in all cases. The reason is that the MAoIG aims to minimize the average AoI of all targets and does not take the AoUI into account. It costs MAoIG more time slots to obtain a useful dataset. Notably, the utility and AoUI are negatively correlated when \(\alpha \ge 0.4\) . The utility of GRUSS-S increases dramatically when \(\alpha = 0.4\) , and then it decreases when \(\alpha\) grows from 0.5 to 1.0. Intuitively, a higher value of \(\alpha\) may result in an accurate dataset. However, it will also increase the AoUI, which in the end decreases the data utility. Thus, choosing a proper value of \(\alpha\) that can benefit both AoUI and utility is important for a real-world application.
Fig. 15.
Fig. 15. The average PM 2.5 returned by different algorithms under different values of \(\alpha\) .
7 Conclusion
In this article, we investigated the data collection problem in edge-assisted energy-harvesting sensory networks. This problem aims to construct a data collection method to minimize the maximum age of useful information and maintain the freshness and usefulness of the sensory data. We proved that this problem is NP-Hard. Two approximated algorithms are proposed to solve this problem in S-E-EHNs and M-E-EHNs, respectively. We analyzed the approximation ratio of these algorithms. We also verified the performance of these algorithms by conducting experiments. The experimental results show that these algorithms are effective and efficient.
Appendix
Theorem 1.
The Min-AoUI problem is NP-Hard.
Proof.
To prove the NP-Hardness of the Min-AoUI problem, we first prove that the following sub-Min-AoUI problem is NP-Hard:
sub-Min-AoUI Problem
Input:
(1) an E-EHN \(\mathbf {N}(V,S,\mathbf {O},\mathbf {T},\mathcal {C}(M))\) ,
(2) a set of energy-harvesting nodes V,
(3) an edge server \(s_0\) ,
(4) a set of targets \(\mathbf {O}\) , \(|\mathbf {O}|=|V|\) and for each pair of \(o_i \in \mathbf {O}\) and \(i\in V\) , \(o_i\) and i share the same location,
(5) a monitoring duration \(\mathbf {T}=\lbrace t_1,t_2,\ldots \rbrace\) which contains infinite time slots,
(6) the sensing and communicating radius \(r_s\) , \(r_c\) , and \(r_s=r_c\) ,
(7) a remote cloud \(\mathcal {C}(M)\) with an analyzing engine M,
(8) the ambient energy harvested by each EH node i at each time slot t, \(\mu _i(t)=+\infty\) ,
(9) the energy consumption of transmitting one data packet, \(\varepsilon ^T\) , and the energy used to maintain the basic functionality, \(\varepsilon ^B = 0\) ,
(10) the energy capacity of each EH node B and the threshold \(B_f\) , and \(B-B_f=\varepsilon ^T\) ,
(11) two constants given by users \(\alpha =1\) and \(\delta =r_s=r_c\) .
Outputs: a data collection schedule \(\mathbf {D}\) , such that the maximum AoUI during the monitoring duration \(\mathbf {T}\) is minimized.
We reduce the Minimum Dominating Set in Unit Disk Graph (MDS-UD) [28] problem to the sub-Min-AoUI problem. Given an instance of the MDS-UD problem, \([G(V^{\prime },E^{\prime }),r]\) , where r is the radius of the unit disk, the reduction has the following steps:
Step 1. Construct a virtual edge server \(s_0\) and a virtual cloud \(\mathcal {C}(M)\) ;
Step 2. Construct a set of virtual EH nodes V and for each EH node \(i \in V\) there is a corresponding \(v_i \in V^{\prime }\) , \(r_s=r_c=r\) and the distance between \(s_0\) and each EH node i is within r;
Step 3. Construct a set of virtual targets \(\mathbf {O}\) , and for each \(o_i\in \mathbf {O}\) , there is a corresponding \(i\in V\) and each pair of \(o_i\) and i share the same location.
Based on the above three steps, we can construct an instance of the sub-Min-AoUI problem. According to the sub-Min-AoUI problem and the time collision (Definition 1), at each time slot, only one EH node can transmit sensory data to the edge server. Figure 6 shows an example. Furthermore, since \(B-B_f = \varepsilon ^T\) , each EH node can only transmit one data pack at each time slot.
Let \(V_{opt}\) be the optimal solution of the MDS-UD problem, such that for each \(v\in V\) , there is a \(v^{\prime } \in V_{opt}\) and \(dis(v,v^{\prime }) \le r\) and the size of \(V_{opt}\) is minimized. Then, there is a corresponding solution of the sub-Min-AoUI problem,
where \(t_m\) is the mth time slot, \(v_{i_k} \in V_{opt}\) \((1\le k \le m)\) and \(|V_{opt}|=m\) . Obviously, the maximum AoUI is \(A(t_m)=A(t_{2m})=\cdots =m-1\) . Suppose there is an optimal solution of the sub-Min-AoUI problem,
with \(m^{\prime } \le m\) and \(A(t_{m^{\prime }}) \le A(t_m)\) . Then, there is a corresponding solution of the MDS-UD problem, \(V_{opt}^{\prime }=\lbrace v_{i_1^{\prime }},v_{i_2^{\prime }},\ldots ,v_{i_m^{\prime }}\rbrace\) , and \(|V_{opt}^{\prime }| \le |V_{opt}|\) . Because \(V_{opt}\) is the optimal solution, we have \(|V_{opt}^{\prime }| = |V_{opt}|\) and \(m^{\prime }=m\) . Thus, \(\mathbf {D}\) is the optimal solution of the sub-Min-AoUI problem.
Similarly, given an optimal solution to the sub-Min-AoUI problem, we can also construct an optimal solution to the MDS-UD problem.
Therefore, the sub-Min-AoUI problem is NP-Hard, and the Min-AoUI problem is NP-Hard, too. □
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Tuo Shi, 
