Reducing the energy consumption in WSNs: A data scientific mechanism

Reducing the energy consumption in WSNs: A data scientific mechanism Gabriel Martins Dias∗, Boris Bellalta, Simon Oechsner Pompeu Fabra University, Barcelona, Spain

arXiv:1509.08778v1 [cs.NI] 29 Sep 2015

Abstract Radio transmissions are the operations with the highest energy consumption in wireless sensor nodes. Therefore, the most promising way to extend Wireless Sensor Network (WSN) lifetime is to reduce the number of packet transmissions, if this does not mean to lose the quality of the information that they can offer. In this work, we present a WSN model and design a mechanism for data reduction. Then, based on a data study, we show how effective can be its use to reduce the number of transmissions in WSNs and increase their lifetime. Simulation results show that the energy consumption can be reduced by almost 85% in the sensor nodes with the highest work load, and we detail the impact of predicting and aggregating the transmissions in the results. Keywords: wireless sensor networks, data science, predictions, data reduction

1. Introduction Wireless sensor nodes (sensor nodes, for brevity) are small computer devices with a radio antenna. They are often equipped with sensors that are capable of sensing one or more environmental parameters [1]. As an example, temperature and relative humidity sensors are some of the cheapest and smallest sensor chips available, and are commonly used in real applications. Sensor nodes are usually organized as a Wireless Sensor Network (WSN), in which a Gateway (GW) is responsible for the communication with the sensor nodes, i.e., to manage their operation and collect their measurements. In some cases, WSNs are internally divided into clusters, and each cluster has its own Cluster Head (CH) that is responsible for the communication between its sensor nodes and the GW [2]. Due to the sensor nodes’ portable size, the energy resources are constrained and usually limited to one battery each. Usually, these batteries cannot be recharged nor replaced during their operation, because it may be more costly than deploying new sensor nodes. A potential problem of having a sensor node running out of battery in a WSN is that the exact data they could produce would not to be available anymore. Such an impact can be better understood if we consider that most of the WSNs are data-oriented networks, i.e., ∗ Corresponding

author Email address: [email protected] (Gabriel Martins Dias)

Preprint submitted to Pervasive and Mobile Computing

September 30, 2015

the data that the sensor nodes can produce are their most valuable asset [3]. Moreover, when a sensor node has no energy to continue its work, the WSN connectivity may decrease, which happens in cases where a sensor node is responsible for providing a communication bridge between its neighbors that cannot reach each other. In the best case, having one sensor node less may still make the other sensor nodes consume more energy trying to find new routes towards the GW. In the last years, many studies focused on better exploiting the scarce resources of the sensor nodes and increase their lifetime while keeping a minimum level for the Quality of Information (QoI) that could be provided by the WSN. The data produced by sensor nodes can be transformed into different types of information, according to their application. For example, temperature measurements provided to the user may not have an error greater than 1o C, which forces the sensor nodes to transmit their measurements if the difference from the last measurement is greater than this threshold. Another case could be that some actions are taken if the number of insects observed in a trap exceeds a certain limit1 . In this work, we do not focus on the information that can be produced. Instead, we assume that each application contains its own mechanism to scale and evaluate it. This work presents a WSN model based on the sensor nodes’ connectivity and their maximum distance to the GW in number of hops. Based on this model, we design a mechanism to incorporate predictions into the WSN workflow and show how efficient it can be, depending on the accuracy of the predictions and the correlation between the measurements made by the sensor nodes. Based on the model described with the mechanism, it is possible to choose the prediction model with the best benefits for the WSN. In the next sections, we will evaluate the gains that can be obtained by reducing the number of transmissions and compare with the costs of choosing a certain prediction model, considering its expected (in)accuracy. Given that our mechanism is highly dependent on the correlation between the measurements made by different sensor nodes, we recommend its use in WSNs for environmental monitoring. Other WSN types, such as those used for object tracking and event detection, usually do not observe high correlations in their measurements. The rest of the paper is organized as follows: Section 2 shows the current state of the art and gives a better idea about the contributions of this work; Section 3 details how we model the data using Normal distributions, which is fundamental for understanding the assumptions we will make further; Section 4 describes the WSN model which will be used to develop the whole mechanism; Section 5 contains the mechanism that describes how to make predictions and better exploit the characteristics of the WSN components; Section 6 models the energy consumption for the mechanism; Section 7 presents the simulation results for a given use case; and Section 8 shows our conclusions and ideas for future work. 1 Entomatic

Project - http://entomatic.upf.edu/

2

2. Related Work There are several works that have applied prediction techniques based on the data retrieved by the sensor nodes to substitute the communication between sensor nodes with local computation. Current approaches mainly differ on where the extra computation takes place, which may vary between the GW, the CHs and the sensor nodes (including combinations of different component types). The mechanism proposed in [4] (called BBQ) is a centralized approach in which the CHs are responsible for predicting the measurements that the sensor nodes will make. It exploits the higher computational power and energy availability of the GW, besides exploring the spatio-temporal correlation between the measurements done by the sensor nodes. Since the general costs of retrieving data from many nodes can be extremely high, they incorporate statistical models of the data measured by the sensors in order to answer user queries. Thus, the CHs only acquire data from the sensor nodes when the predictions are not sufficiently accurate to answer the queries with an acceptable confidence. The sensor nodes that must transmit their measurements are those which consume less energy in the WSN (taking into account that the transmissions will be forwarded in cases where intermediate sensor nodes are needed) and will provide at least the minimum confidence level set by the user, according to the statistical models. In BBQ, the GW is responsible for choosing which sensor nodes are going to transmit which measurements. After receiving the selected measurements, it must predict the missing values based on its data models. Differently from our work, the authors did not take into account that the sensor nodes close to the GW usually provide connectivity to the others, which makes them having a higher impact in the WSN lifetime. Furthermore, the main difference of this mechanism to our solution is that, in BBQ, the sensor nodes are not able to take decisions and transmit their measurements in case the predictions made by the GW fail. Hence, changes in the environment cannot be noticed if they are not covered in the model, which implies on an eventual reduction in the QoI provided to the user. Other mechanisms give higher autonomy to the sensor nodes, which can compute their decisions locally, based on the available historical data, instead of transmitting every measurement to the CHs. For instance, dual prediction mechanisms run the prediction models both in the CHs and in the sensor nodes, based on the strong assumption that the energy necessary to execute an instruction is some orders of magnitude smaller than the energy required by the radio transmissions (which is shown in [5, 6, 7]). Given that, it is worth to increase the computing time in the sensor nodes to decrease the number of transmissions. Thus, sensor nodes make it by producing the same “a priori” knowledge as the CHs and compare the measurements with the predictions locally. If the predictions are not accurate, the actual measurements are transmitted to the CHs. The efficiency of such mechanisms depends not only on the quality of the predictions, but also on the number of transmissions required to distribute the prediction models and/or their parameters, on the power required to compute them, and on the bit error rate and the quality of the channel used. The approach

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presented in this paper is a dual prediction mechanism in which the GW is responsible for predicting large sequences of measurements and transmit their expectations to the sensor nodes. After a measurement, the sensor nodes are able to compare the measured values with the ones provided by the model and transmit them only when the prediction has failed. In [8], the authors showed that the combination of accurate predictions with an aggregation scheme could lead to excellent results in terms of energy savings. Using least mean squares, they reduced the number of transmissions to up to 92% without significantly reducing the accuracy of the data. Their work was relevant to set up the potentiality of the mechanism, but they did not focus on proving the importance of each component. In this work, we rely on the features of normal distributions to explain the impact of each “gear” of this engine in the final results: the aggregation scheme, the correlation between the measurements, the use of dual predictions and their accuracy. The work presented in [9] was the first of this type to introduce statistical methods to choose which prediction model better fits to a certain environment. It considers the percentage of transmitted measurements (r) and the user desired level of accuracy (α). Later, in [10], the authors designed an extended model for the Prediction Cost (PC), which is more generic and also considers the computational costs of executing each algorithm in the sensor nodes with respect to their memory footprint (Ec):

PC = [ αf (e) + (1 − α)r ] Ec

(1)

where e is the measure of the predictions’ accuracy (e.g., mean squared error, root-mean-square deviation or symmetric mean absolute percentage error) and f (e) is the accuracy according to the measure chosen. The approach presented in [11] calculates whether it is worth to make predictions in the sensor nodes or not, based on the relation between the predictions’ accuracy, the correlation between measurements and the error tolerated by the user. It takes the parameters observed at a given moment as input in order to adapt the behavior of the nodes, which can vary among:(i) being in the sleep mode (and making no measurements); (ii) making measurements and transmitting every measurement done; and (iii) making measurements, transmitting them to the CH whenever the prediction differs by more than an accepted value, and updating the prediction model parameters when necessary. Our mechanism also incorporates statistical methods to decide whether it is better or not to use predictions, based on the predictions’ accuracy and the correlation between measurements. The difference is that we model a multi-hop WSN in order to centralize the decisions into an unique GW, instead of having several CHs that can directly communicate with each of their children.

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3. Data modeling A Normal distribution is characterized by its probability density function whose pattern is often encountered in several types of observations. In [4], for example, the authors showed that environmental readings - such as temperature, light and humidity - done by outdoor WSNs can be approximated to normal distributions, if properly managed. In case of having several normal distributions, a Multivariate Normal (MVN) distribution can be defined based on their correlation, i.e., the relationship between each pair of distributions. For years, MVN distributions have been the focus of several studies that discovered their properties and made them an established main topic, which makes it a potential candidate for describing a system’s behavior as a model using mathematical equations. In this work, we will assume that a WSN is composed by a set of sensor nodes S and each sensor node i (such that i ∈ S) is responsible for measuring a certain parameter from the environment, such that the set of observations follows a Normal distribution with mean µi and variance σi2 . By convention, this is represented as Yi = N (µi , σi2 ). The GW is able to compute new predictions y¯i (for example, y¯i = µi ) and disseminate each prediction to the respective sensor node. We define the accepted threshold εi , i.e., the prediction is told to be correct if the real observation (yi ) is in the interval [¯ yi − εi , y¯i + εi ]. Assuming that the data is normally distributed, the chances of observing a new value inside the accepted interval can be calculated by normalizing the value of εi , i.e., rewriting it in terms of the variance σi2 . The normalized value of εi is represented by zi as

zi =

εi − y¯i σi

(2)

Thus, in this case, the accuracy of the predictions (αi ) can be calculated based on the cumulative distribution function of the normal distribution (Φ). That is, the percentage of observations that will fall outside the accepted interval is represented by the two-tailed Z-test (i.e., 2Φ(−|zi |)), and αi is αi = 1 − 2Φ(−|zi |).

(3)

By substituting the Equation 2 into Equation 3, we can observe that   εi − y¯i , αi = 1 − 2Φ − σi

(4)

which shows that the accuracy of the predictions depends on the accepted threshold, on the mean and on the variance of the data. We will assume unbiased predictions and random errors in further developments.

4. WSN Model Langendoen and Meier [12] presented a ring model for WSN topologies to describe a multi-hop network based on the average number of neighbors (C) of a sensor node and on the number of hops from the GW to 5

Figure 1: WSN model based on the density of the sensor nodes and their coverage.

the furthest nodes (D). The distance from the GW also defines in which ring a sensor node is placed. That is, since transmissions made by a component (either the GW or a sensor node) can reach neighbors that are up to 1 unit of length from it, to reach the GW from ring d, a sensor node has to make a d-hop transmission. Therefore, assuming an uniform node density on the plane and defining it as C + 1 nodes per the unit disk, the first ring will contain C nodes, and subsequently the number of nodes Nd in ring d can be calculated based on the surface area of the annulus2 :

Nd =

  0,

if d = 0

 Cd2 − C(d − 1)2 = (2d − 1)C,

otherwise

(5)

The number of nodes in the WSN is equal to CD2 and, given that the first ring has C sensor nodes, it is expected C branches with D2 sensor nodes each. In this work, each branch will be referenced as a sub-tree. As Langendoen and Meier also described, the expected number of direct children of a node in ring d does not depend on the value of C and is

Id =

  0,

if d = D (6)

  2d+1 , otherwise 2d−1 Figure 1 shows an example of a WSN based on this model with D = 3 and C = 5. In this work, we use a similar structure to derive energy consumption models based on the nodes’ positions. First, we define the set of children nodes of a node i as Hi . Its expected size does not depend on C, but it can be calculated based on which ring the sensor node is located. Therefore, we define the expected size of Hi as Kd , where d is the ring in which the sensor node i is. The value of Kd is the expected number of direct children times 2 The

region bounded by two concentric circles.

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the expected number of their children plus one (representing themselves):

Kd =

  0,

if d = D (7)

 Id (Kd+1 + 1), otherwise Since the expected number of sensor nodes is CD2 and the first ring is expected to have C nodes, the expected number of children of the nodes in the first ring (i.e., K1 ) is always D2 − 1. The sensor nodes in the first ring have the highest work load and are usually the first to run out of battery, determining the end of the communications in the WSN. This is commonly observed in real-world irregular topologies and often neglected by other models (as we could observe in [4]), due to its complexity. The main advantage of our model is its simplicity to identify and describe the operation of such bottlenecks in a WSN. 4.1. Energy consumption for node-to-GW transmissions As we mentioned in Section 2, some works have already shown that the energy necessary to execute an algorithm is several orders of magnitude smaller than the energy required to make a radio transmission (either transmitting or receiving). Thus, we assume that the number of transmissions and receptions made by the sensor nodes are the major concern about energy consumption in WSNs and focus on reducing them without diminishing the QoI provided by the WSN. To achieve that, we assume that the GW has neither energy nor memory constraints, i.e., its energy sources and storage capacity are unlimited or, at least, significantly larger than in the sensor nodes. Hence, it is able to run compute-intensive algorithms without compromising the WSN lifetime. In monitoring WSNs, sensor nodes transmit their data in a pre-defined interval that can vary from few seconds to hours. These transmissions, which we call node-to-GW, do not necessarily happen right after a measurement, because the sensor nodes can aggregate more data received from other sensor nodes or from past measurements. In the simplest approach, measurements are transmitted right after their creation, and we use them as the baseline in our comparisons. If no communication is made, a minimum energy (EMIN ) is consumed per time unit. Such a value may vary according to the hardware, the chosen MAC protocol and other configurations, but it contains the average energy required for the minimum operation of the sensor nodes. Besides that, we define as ETX and ERX the extra energy spent in a sensor node to respectively make a transmission and a reception. The total energy consumed in a WSN is proportional to the sum of the amount of extra energy used to transmit and receive the data from the sensors. It is modeled by the following equation:

EWSN = S WSN ETX + RWSN ERX + CD2 EMIN ,

7

(8)

where S WSN is the number of transmissions and RWSN the number of receptions that occur in the WSN during the considered period. Their values are proportional to the frequency of measurements (f ) and the observed time period (T ), and can be calculated based on the size of the network (number of rings) and its node density. Given that sensor nodes must forward the information from their children towards the GW, the number of transmissions during a period of 1/f seconds in a sensor node in ring d is |Kd | + 1 and the number of receptions is |Kd |. Thus, in T seconds, the total number of transmissions and receptions of a sensor node in ring d are: S d = (|Kd | + 1) f T

(9)

Rd = |Kd | f T

(10)

Ed = S d ETX + Rd ERX + EMIN

(11)

and its energy consumption is equal to

Based on (7), we can affirm that K1 > Kd , if d > 1. Applying this inequality to (11), we mathematically show that E1 > Ed , if d > 1, which makes us expect that they run out of battery earlier than the others, limiting the WSN lifetime. Therefore, we focus on modeling their energy consumption of the sensor nodes in the first ring in the following. 4.2. Energy consumption for GW-to-node transmissions GW-to-node transmissions are those initiated by the GW, for example, in order to change the configuration or update the software of the sensor nodes. Assuming one unicast transmission per packet, the sensor nodes in the first ring will receive and forward every GW-to-node transmission to their children. In case of transmitting a packet to every sensor node in the WSN, the number of transmissions made by the GW to a subtree is D2 , i.e., the number of nodes in each subtree. Therefore, the extra energy required by the sensor nodes in the first ring to disseminate the transmissions from the GW through the WSN can be represented as

EDISS = RDISS ERX + S DISS ETX

(12)

= D2 ERX + (D2 − 1) ETX If packets to the same subtree are aggregated or if the GW uses broadcast (or multicast) transmissions, the sensor nodes in the first ring will receive only one packet that will be split before being retransmitted to the direct children in the second ring. In such cases, the energy required for the dissemination is

EDISS = ERX + I1 ETX , 8

(13)

Parameter

Description

f

Number of measurements between two predictions

T

Time interval during which the mechanism will be applied without interventions

C

Expected number of neighbors of each sensor node

D

Expected number of rings/hops of the WSN

ρ

Correlation between the measurements in a certain sub-tree

α

Expected accuracy of the predictions made by the GW

Table 1: Parameters taken into account to calculate the number of transmissions and receptions using the model.

where, from (6), the expected value of I1 is 3, if there is more than one ring in the WSN.

5. Proposed mechanism In this section, we adopt the model presented before in a mechanism for reducing the number of unnecessary transmissions that exploits two characteristics inherent to the WSNs:(i) the proximity of the sensor nodes to the sources of the data; and (ii) the higher computational power and the larger energy resources of the GW to make the predictions. Hence, the proposed mechanism handles the sensor nodes’ hardware limitations and avoid unnecessary computation that might reduce their lifetime, compromise the overall WSN lifetime or the reduce the QoI. In this mechanism, the responsibilities of the GW are mainly divided into:(i) calculating whether making predictions will help the WSN to save energy or not, based on the parameters explained in Section 4 (and illustrated in Table 1); (ii) predicting the sensor nodes’ measurements for a large interval; and (iii) inform these values and the time between consecutive measurements to the sensor nodes, in case of adopting the predictions. In order to reduce the number of transmissions from the sensor nodes to the GW, the predictions are compared with the measurements in the sensor nodes and the measurements are transmitted to the GW only in case of inaccuracy. Figure 2 illustrates step-by-step how the mechanism applies the predictions and the details are explained in the following. 5.1. Assumptions and limitations In this work we assume that the QoI can be scaled as “acceptable” if the values at the GW do not differ by more than a certain threshold. Since the sensor nodes can compare the predictions with the real measurements locally (without making any transmission), no transmission will be required if the prediction is accurate and does not differ by more than an accepted threshold from the measured value. Figure 2e shows both cases: the green points are inside the accepted threshold and, even though they are not exactly 9

the same as the predicted value, the sensor node does not have to make any transmission. On the contrary, the red points fall outside the accepted threshold and must be reported by the sensor node in order to inform inaccuracy of the predictions. Even though the sensor nodes have limited memory size and reduced computing power, we assume that they can store few extra variables which will be used to compute simple predictions based on simple machine instructions, such as additions and multiplications. Additionally, the mechanism considers WSNs that may combine several data types, for example, in case that the sensor nodes are able to measure temperature and relative humidity, both data types could be used for saving their energy through predictions. Hence, it does not require any information about the statistics of the data which is going to be retrieved by the sensor nodes, but it must be possible to have a learning phase before beginning to make predictions. Such an assumption makes it possible to use advanced prediction methods, like Artificial Neural Networks (ANNs), that require larger amounts of data to find stable models, due to their high complexity and the large number of parameters to estimate [13]. Nonetheless, although we assume that the GW is able to compute the accuracy of the predictions and take decisions based on their values, the mechanism does not contain any method to improve them. Finally, after making the predictions, the GW must summarize and transmit the predicted values to the sensor nodes. The efficiency of this mechanism depends on how many transmissions are required for this task, which is proportional to the number of hops between the sensor nodes and the GW. 5.2. Algorithm in the Gateway The algorithm defined by this mechanism has a different behavior in GW and sensor nodes. As explained before, we assume that the GW has more computing power, which is exploited by using more complex prediction models. After collecting some data from the sensor nodes, the GW starts a cycle of procedures that includes predicting the measurements of the next time interval, transmitting the predictions (or a short summary) to the sensor nodes and collecting any correction transmitted by the sensor nodes. The decision about adopting the mechanism is a responsibility of the GW and must be taken based on the energy consumption calculated using the model that we will further explain. 5.2.1. Prediction of the next time interval A prediction method is a class of algorithms used to compute predictions using a set of parameters and the historical data as input [14]. Each set of parameters defines a prediction model, which is a deterministic function that depends only on the historical data to compute its output. In other words, it is possible to create different prediction models using the same prediction method. In our case, the prediction model will be used to predict which values will be measured by the sensor nodes in the next period. The number of measurements and the interval between predictions may vary from case to case. For example, in 2a, we

10

Temperature (ºC)

Temperature (ºC)

27

22

17

1 day

27

22

17

1 day

surements that will be observed by a sensor node

we used the points in which the slope signal got

during the next interval.

inverted. They are transmitted to the sensor node.

Temperature (ºC)

(b) Summary of the predictions. In this example,

Temperature (ºC)

(a) The GW is responsible for predicting the mea-

27

22

17

1 day

27

22

17

1 day

(d) The sensor node can compare its measurements

GW are able to compute the same predictions. In

with the predictions locally. If a measurement falls

this example, we interpolate the values to predict

outside the accepted threshold, it is transmitted to

the other measurements.

the GW.

Temperature (ºC)

(c) Based on the summary, the sensor node and the

27

22

17

1 day

(e) Only the measurements wrongly predicted (in red) are transmitted to the GW. The green points are not transmitted, because they are inside the accepted threshold. Here, the accuracy is around 0.8.

Figure 2: Step by step of the algorithm.

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assume that the sensor nodes make one measurement per minute and the GW predict all measurements of a day, which means that 1440 measurements are predicted at a time. The prediction method used by the GW can be chosen after considering the asymmetric resource availability in the WSNs. In other words, the extra storage capacity and computing power inherent to the GW can be exploited in this decision. For instance, ANNs are examples of prediction methods that are able to provide a high accuracy, but do not fit to the sensor nodes constraints, because they require a computation intensive training phase over a large amount of data. Simpler methods, such as an average inside an hourinterval through a set of days, may also require a large amount of memory and would not provide a high accuracy if computed in the sensor nodes. 5.2.2. Transmitting the predictions Accurately predicting all the measurements beforehand may be a lightweight task for the GW. However, transmitting them to the sensor nodes consumes an extra amount of energy that may turn the mechanism into an inefficient option. One option to summarize the predictions and reduce their length is to summarize the values, which can be done by transmitting only the differences between consecutive predictions or using simple time series methods, such as autoregressive-moving average models [15]. Figure 2b shows a summary of the predictions using the points in which the slope defined by two consecutive measurements gets its signal inverted. The decision about adopting a prediction model to summarize the future measurements depend on the expected amount of energy that can be saved. That is, the GW must calculate whether adopting prediction models is really energy-efficient given the extra transmissions to distribute their parameters and the expected number of wrong predictions. In case of having a wrong prediction and measuring a different value, the sensor nodes will transmit their values to the GW in order to keep the QoI of the WSN. In the end, each sensor node must receive a message informing (i) the sampling interval, i.e., the time between two measurements; plus (ii) the prediction model parameters, if a model is expected to provide substantial energy savings, or otherwise a notification about the absence of prediction models. 5.2.3. Reporting measurements to the user Once the GW has transmitted the parameters to the sensor nodes, it is able to provide information about their measurements to the user (or to an application attached to it). In order to keep the consistency, the GW must compute the same values as the sensor nodes, as illustrated in Figure 2c. That is, if the GW uses prediction models to summarize the predicted measurements for the sensor nodes, it must use the same models to compute the measurements to be informed to the user. Measurements eventually received from the sensor nodes must take the place of the predictions once they fix the predictions’ inaccuracy. Furthermore, when the GW receives a measurement (informing that the prediction was wrong), it incre-

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ments a local counter that can be used to assess the accuracy of the predictions and to decide whether to recalculate and retransmit the prediction model to the respective sensor node. This procedure is repeated until the time interval is over and the GW starts a new cycle with new predictions (5.2.1). 5.3. Algorithm in the sensor nodes At the beginning, all actions are taken by the GW. That is, the sensor nodes do not have to synchronize with their neighbors nor pre-compute any kind of prediction model. Thus, the first step taken by every sensor node is to wait for instructions from the GW, which may vary between:(i) an announcement that they must transmit every measurement in the next time interval and the sampling interval; or (ii) a message containing a sampling interval and the values predicted by the GW, which can be summarized as a simpler prediction model and will be used to compute the same values as the GW and compare to the real measurements locally. Each reception can be acknowledged in order to inform the GW that it is working properly and that an eventual absence of transmissions is due to the high accuracy of the predictions. If a sensor node receives a message with the prediction model parameters, it must compare its measurements with the predictions computed using such parameters. If a prediction fails (e.g., it differs from the actual value by more than a given threshold), the sensor node must transmit the real measurement to the GW in order to keep the QoI of the WSN, as illustrated in Figure 2e.

6. Energy consumption of the proposed mechanism As explained before, the GW is responsible for deciding whether to adopt the scheme with predictions or not. Assuming that the WSN lifetime increases whenever the energy consumption in the first ring reduces, we use our model to calculate the energy savings and suggest its use to decide whether to adopt predictions or not. In this Section, we explain the energy consumption based on different possible approaches, one with no data aggregation in the intermediate nodes and another in which sensor nodes are able to aggregate transmissions and reduce the traffic load in the WSN. 6.1. Energy consumption with no data aggregation Assume that αi is the accuracy of the predictions in sensor node i, i.e., αi is the probability that a measurement of i matches to the prediction and does not have to be transmitted to the GW, and αic = 1−αi . Notice that the procedure to estimate αi may depend on the metrics chosen to assess the QoI, the types of measurements and the system requirements. Since the sensor nodes do not transmit their own measurements P P when the predictions are correct, a sensor node i expects to make αic + j∈Hi αjc transmissions and j∈Hi αic receptions between any two measurements (i.e., every 1/f seconds).

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Finally, the number of transmissions during the observed time (f T ) is represented by S 0i and R0i as 



S 0i = αic +

X

αjc  f T ,

(14)

j∈Hi

 R0i = 

 X

αic  f T

(15)

j∈Hi

and its energy consumption is Ei0 = S 0i ETX + R0i ERX + EDISS + EMIN

(16)

A low accuracy in the predictions used in the sensor nodes that are far from the GW has higher impact in the WSN’s energy consumption than a low accuracy in the predictions from sensor nodes in the first rings. However, the WSN lifetime still depends on the energy consumption in the sensor nodes in the first ring. Therefore, let us define a minimum desired accuracy (αmin ) according to the size of the network and its number of rings. This value can be used to define the maximum number of transmissions (S 0i,max ) and receptions (R0i,max ) in a sensor node i in ring d:

S 0i,max = (1 − αmin ) +

X

(1 − αmin )

j∈Hi

(17)

= (1 + Kd ) (1 − αmin ) and

R0i,max =

X

(1 − αmin )

j∈Hi

(18)

= Kd (1 − αmin ) As explained in Section 4, Kd = |Hi |, for a sensor node i in ring d. 0 Finally, the proposed architecture may be able to extend the WSN lifetime if Ei,1 < Ei,1 . After some

mathematical development (shown in Appendix A), we arrive at the following expression for the minimum accuracy of the predictions:

αmin > 1 −

ETX (f T D2 − D2 + 1) + ERX (f T D2 − D2 − f T ) [ETX D2 + ERX (D2 − 1)] f T

(19)

In conclusion, the mechanism requires a minimum accuracy to ensure the extension of the WSN lifetime. As explained before, the values ETX and ERX may vary according to the hardware, the MAC protocol and the operational system, but usually they are not changeable after the deployment. Hence, the minimum accuracy 14

Y2

Y1 Figure 3: Values of Y1 and Y2 are correlated (ρ = 0.7), and each line represents a different density of points.

is a lower bound that depends only on the network layout (i.e., the number of rings D), the frequency of the measurements (f ) and the time between two predictions (T ). If the accuracy of the predictions does not reach this limit, the GW has two options, either it sets new values for f and T , or it turns this mechanism off. 6.2. Energy consumption with data aggregation Apart from the prediction mechanism explained above, we expect that in most cases the sensor nodes are able to aggregate the data received from their children and transmit only after making its own measurement. Thus, we model an aggregation scheme that can be adopted to enhance the gains provided by the mechanism. First, we introduce a scenario with only two sensor nodes in order to clarify the normalization of the data and its application. Later, we extend the model to a more complex scenario with D rings. 6.2.1. WSN with two sensor nodes Let us consider a section of the WSN with the GW and a sensor node i with a single child j. We assume that their measurements follow the Normal distributions respectively represented by Yi = N (µi , σi2 ) and Yj = N (µj , σj2 ). An illustration of the multivariate normal distribution containing Yi and Yj is shown in Figure 3. Due to the WSN topology, the transmissions from sensor node j can reach the GW only through the sensor node i. Thus, every 1/f seconds, i may transmit a message to the GW if its prediction has failed or if it had happened to j. Assuming that the predictions are not biased, we may also approximate them to normal distributions and label an outcome as incorrect whenever a measurement falls outside the interval 15

defined by the accepted threshold εj . In such case, the probability that the sensor node j will have a packet to transmit (including its own) after 1/f seconds is 1 − αj . Thus, after a period of f T seconds, the expected number of transmissions is S 00j = (1 − αj )f T,

(20)

which is the same number of receptions in i (R00i ). The energy consumption of the sensor node j can be calculated as the energy to transmit with no additional energy required for reception: Ej = S 00j ETX + EMIN

(21)

Similarly, sensor node i will make a transmission if the prediction about its measurement fails (i.e., it falls out of the accepted threshold εi ) or if the prediction in sensor node j had failed and the real measurement has been received. In other words, there will be an transmission if at least one of the two predictions fail. We assume that he correlation between Yi and Yj is defined by the Pearson correlation coefficient and represented by ρi,j . Therefore, to model the probability of having at least one wrong prediction, we must calculate the correlation matrix (Σ). It is defined as 

σi2

Σ= ρi,j σi σj

ρi,j σi σj σj2

 

(22)

Finally, given the lower limits

li = y¯i − εi and lj = y¯j − εj ,

(23)

ui = y¯i + εi and uj = y¯j + εj ,

(24)

the upper limits

and the correlation matrix (Σ), it is possible to calculate the following multivariate normal probability: Z ui Z uj 1 t −1 1 p F (yi , yj ) = e(− 2 θ Σ θ) dθ (25) 2 |Σ|(2π) li lj The value of F (yi , yj ) represents the probability that both predictions (in i and j) are correct and can be illustrated by the density inside the crosshatched rectangle in Figure 4. Thus, the probability that at least one prediction fails can be calculated as 1 − F (yi , yj ), and the value of S i as 1 − F (yi , yj ). Finally, the energy consumption in sensor node i is modeled as Ei = S 00i ETX + R00i ERX + EDISS + EMIN ,

(26)

which determines the WSN lifetime. Note that the number of transmissions made by the sensor node i (S 00i ) is the number of packets received by the GW and the number of receptions in the sensor node j (R00j ) is equal to zero. 16

y1 − n 1

y1 + n 1

y1

Y2

y2 + n 2 y2 y2 − n 2

Y1 Figure 4: The hashed rectangle in the center illustrates the points in which both predictions are correct. 6.2.2. Large WSNs Now we extend the example above to larger sections. The correlation matrix (Σ) of several data distributions can be calculated as 

σa2

   ρb,a σb σa Σ= ..   .  ρz,a σz σa



ρa,b σa σb

···

ρa,z σa σz

σb2 .. .

··· .. .

ρz,b σz σb

···

  ρb,z σb σz  , ..   .  σz2

(27)

Therefore, similarly to the two-dimensional model, the expected number of transmissions made by the sensor node i (represented by S 00i ) depends on the probability of all predictions being correct (including its own), which can be calculated as S 00i = 1 − F (i, a, b, . . . , z),

(28)

where {a, b, . . . , z ∈ Hi }, and the function F is the multivariate normal probability function integrated from the lower accepted limits to the upper accepted limits over the k = Kd distributions:

F (a, b, . . . , z) = p

1 |Σ|(2π)k

Z

ua

Z

ub

Z

uz

··· la

lb

e(− 2 θ 1

t

Σ−1 θ )

dθ,

(29)

lz

which can be efficiently calculated with the use of Monte Carlo methods for higher dimensions [16]. The number of receptions at the node i (R00i ) is slightly different from the previous example, since now the sensor node may have several children in the next ring and their transmissions happen independently. Let 17

us define Hi0 as the set of direct children of i, which expected size is Id , as explained before. The expected number of receptions can be calculated as R00i =

X

S 00j ,

(30)

j∈Hi0

and its energy consumption is defined by Ei00 = (S 00i ETX + R00i ERX )f T + EDISS + EMIN

(31)

Even though the function F has no closed formula, it is possible to set a lower bound based on a case when there is absolutely no correlation between the values measured by i and its children. When the correlation is equal to zero, the expected number of transmissions and receptions in sensor node i are the maximum possible: S 00i,max = 1 − α1+Kd

(32)

R00i,max = Id (1 − αKd /Id )

(33)

We claim that

Ei00 ≤ [ (1 − α1+Kd ) ETX + Id (1 − αKd /Id ) ERX + EMIN ]f T

+ EDISS ,

(34)

which means that the approach that aggregates the data is able to save more energy than the previous one. Comparing (34) with (16), we have that for any α ∈ [0, 1] and Kd ≥ 0, it can be shown based on the proof detailed in Appendix B that 1 − α1+Kd ≤ (1 + Kd ) (1 − α) and, hence, S 00i,max ≤ S 0i,max . Moreover, R00i,max ≤ R0i,max and Id (1 − αKd /Id ) ≤ Kd (1 − α), which can be similarly proved to be true, since Kd /Id ≥ 1 when Kd > 0 and α ∈ [0, 1]. In case of being in the last ring, since there are no children (Kd = Id = 0), no reception is made. For now, we assume that the energy consumption to transmit and receive aggregated packets are the same as before (without aggregation). In a real implementation, this could be done by adopting simple aggregation functions, such as the maximum, minimum and the average of the measurements. In the next Section, for the simulations, we incorporate a different packet length and show that the mechanism is still more energy efficient than the other approaches.

7. Simulations The model presented before can be used to draw conclusions about the energy consumption in a WSN based on the number of transmissions and receptions made during a time interval, which depends on the 18

Nr. of transmissions

4e+05 3e+05 2e+05 1e+05 0e+00 1

2

3

4

5

D

6

7

8

9

10

Nr. of transmissions

(a) The aggregation reduces the number of transmissions from quadratic to linear order.

60000

No prediction, no aggregation

40000

α = 0.5, no aggregation α = 0.7, no aggregation α = 0.9, no aggregation

20000

α = 0.95, no aggregation Aggregation only

0 1

2

D

3

4

(b) When number of rings is small (D ≤ 4), the use of predictions can lead to less transmissions than the aggregation scheme.

Figure 5: The impact of the network size in the number of transmissions.

parameters shown in Table 1. In this Section we show how the number of transmissions vary according to the scenario and corroborate our expectations with some simulation results.

19

7.1. Model application In Figure 5a, we show how different values of D can impact the number of transmissions in the first ring if the other parameters are not changed. In these simulations, sensor nodes make one measurement per minute and the GW predicts their measurements once a day during three days (T = 3). Therefore, 4320 measurements are expected, which 1440 (= f ) happen between each prediction. Finally, GW-to-node transmissions were aggregated. We assume that the connectivity between the sensor nodes does not vary when the size of the WSN changes, following the proposed model. The results show that the number of transmissions does not depend on the density of sensor nodes (C), but on the number of rings. Furthermore, the number of transmissions increases exponentially (as well as the number of sensor nodes) when new rings are added, if the aggregation is not adopted. On the other hand, the number of transmissions is constant and independent of the number of rings, if the transmissions are aggregated. Let us categorize the tests into two sets:(i) with (some) prediction and no aggregation; and (ii) with aggregation, but no prediction. When the predictions are highly accurate and the number of rings is small (D ≤ 4), the aggregations are not enough to significantly reduce the number of transmissions, if compared with the scenario with high accuracy (0.9), as highlighted in Figure 5b. However, for larger WSNs, the use of aggregation reduces the number of transmissions to a level that cannot be achieved even in case of the highest accuracy considered (0.95), as shown in Figure 5a. Similar results were observed by the authors of [8], but they did not realize that the predictions had less impact in the final savings and concluded that such optimal achievements happened due to the high accuracy of the predictions. However, as we can observe here, the aggregation scheme plays a major role in the results. To detail the power of the aggregation scheme, we considered a WSN with 5 rings in which the aggregation scheme produces nearly the same number of transmissions observed in the scenario with the most accurate predictions. In Figure 6 we highlight the gains obtained by adopting both schemes. First, we observe that the number of transmissions can be reduced to 15% of its maximum in the best scenario: the predictions are highly accurate and the measurements in the sub-tree are highly correlated. Second, there are no significant gains when the predictions are less accurate (around 0.5) nor when the predictions are more accurate (around 0.7) and the correlations are less than 0.7. Finally, increasing the accuracy from 0.5 to 0.9 with an average correlation (0.5) reduces around 30% of the transmissions, while increasing the correlation from 0.5 to 0.9 with an accuracy of 0.5 reduces only in 6.5% the number of transmissions, which illustrates that the impact of making accurate predictions is much higher than having a high correlation between the measurements. As we can notice, the energy consumption of this mechanism is highly attached to the correlation of the measurements made by the nodes in a sub-tree and the accuracy of their predictions. Assuming normally

20

Data: n = number of nodes, α = accuracy, ρ = correlation Result: P (n, α, ρ) = probability that no transmission happens 1 2 3

if n = 0 then return P ← 1 else

4

q ←| Φ−1

5

Q ← {q, q, . . . , q}1×n

6

Y ← {Y1 , Y2 , . . . , Yn }   1 ρ ··· ρ     ρ 1 · · · ρ   Σ ← . . . . . ...    .. ..  

7

ρ 8 9

1−α 2

ρ




···

|

1

n×n

return P ← Φ (Y, Σ, Q) end Algorithm 1: Algorithm to calculate the probability that no transmission will be made.

distributed values, the expected number of transmissions and receptions rely on cumulative density functions of multivariate normal distributions to calculate the probability that no transmission is made in a group of n nodes measuring data with correlation ρ, as illustrated in Algorithm 1. Based on these values and considering that the WSN lifetime is defined by the energy of the nodes in the first ring, we model the number of transmissions in a sensor node i in ring 1: S 00i = ( ( (1 − P (K1 , α, ρ) ) f ) + I1 ) T

(35)

R00i = (1 − P (K1 , α, ρ) ) I1 f T

(36)

and the number of receptions as

They can be used to compare the energy consumption of the mechanism with the other cases, when predictions or aggregations are adopted. The energy consumption in the sensor nodes in the first ring is addressed as the bottleneck of the WSN lifetime and it was the one considered in the plots. Thus, in order to show the potential of this mechanism, we simulated 3 days of a WSN with the same topology described in Section 4, with D = 5 and C = 3, which sums up to 75 sensor nodes plus the GW. The efficiency of the aggregation scheme explained in Section 6.2 depends on the correlation between the measurements in a sub-tree. In order to show their differences, we simulated different correlations among their measurements: 0.1, 0.2, . . . , 0.9 and 0.95. 21

% of transmissions

100

75

Aggregation only α = 0.5, plus aggregation

50

α = 0.7, plus aggregation α = 0.9, plus aggregation α = 0.95, plus aggregation

25

0 0.25

0.50

ρ

0.75

Figure 6: The effectiveness of the aggregations depend on the correlation between the measurements in a sub-tree.

7.2. Energy consumption We calculated the extra energy necessary to make transmissions and receptions, based on TelosB nodes [17] simulated using a TDMA-based MAC protocol in OMNET++ [18]. In the MAC protocol adopted, each sensor node has a reserved slot to transmit. Therefore, we do not expect collisions during the transmissions and there is no overhearing. We point out that other MAC protocols may obtain different results, due to concurrent transmissions, although similar gains could be expected. The final results of our simulations were compared with our model and are shown in Figure 7. In our model, the number of children is used to define how many distributions will be used, which means that decimal values cannot be used. Thus, we rounded all of them up to the next integer, which resulted on an upper bound for the number of transmissions in the simulations. We highlight that the aggregation scheme may require larger packets to accommodate all the information in a transmission, which would mean higher values of ETX and ERX in comparison with the standard transmissions. In our simulations, we used packets with 8 times the payload of the normal packets. In Figure 7, we can see that simply adopting the aggregation scheme (without making predictions) reduces the extra consumption to 60% of the total, yet larger packets are used. The greatest gains, nonetheless, are obtained after adopting the predictions and the aggregation scheme: they can save up to nearly 92% of the energy consumed by the transmissions. As explained before, the accuracy of the predictions are more significant and have higher impact than the correlation between the measurements in a sub-tree. Hence, a very low correlation (0.1) with highly accurate predictions (0.95) gives better results than a high correlation

22

Extra energy consumption (W . s)

20

No prediction, no aggregation Aggregation only α = 0.5, plus aggregation

15

α = 0.7, plus aggregation α = 0.9, plus aggregation

10

α = 0.95, plus aggregation type model

5

simulation 0.25

0.50

ρ

0.75

Figure 7: The model provides reliable results when compared with the simulations.

(0.9) with an average accuracy (0.5). Regardless of the values shown in the plot, the amount of energy saved depends on the hardware of the sensor nodes, their operational system and the chosen MAC protocol, besides other configurations. Nonetheless, it is mainly driven by the relation between the minimum energy necessary to keep them making measurements and the amount of battery required for transmitting and receiving a packet.

8. Conclusion and Future Work In this work, we presented two main contributions to the field: a mathematical framework to calculate the energy consumption in a WSN, and a mechanism that exploits the characteristics of the WSN components in order to adopt predictions and improve their lifetime. Differently from most of the works about this topic, we used this model to show that, in terms of energy savings, the benefits of an aggregation scheme are better than using only predictions, and that combining both leads to the highest savings. Then, we showed that the accuracy of the predictions impacts more than the correlation between the measurements made by the sensor nodes. Considering that this mechanism has been designed to attend different types of WSNs, there are some challenges to set up the best parameters for each use case. From our experiments, we expect that the critical points to adopt and adjust this mechanism may vary between:(i) finding the precise correlation of the measurements; (ii) approximating the measurements to normal distributions, since it may require some data analysis in advance; (iii) having restrictions about changing the operation of some sensor nodes, because they may be fundamental for the WSN connectivity and its QoI; and (iv) calculating the energy necessary 23

for each step (i.e., transmission, reception, etc.), since details in the software and hardware implementations may influence such values. On the other hand, the mechanism provides means to support the adoption of predictions to reduce the number of transmissions in a WSN and extend its lifetime. The decisions are based on statistical proofs, such as the central limit theorem, which supports the normalization of the data measured by the sensor nodes. From the results, we observed that the energy consumed by making transmissions and receptions in the “key” sensor nodes (i.e., those in the first ring) can be reduced by nearly 85%, as shown in Figure 7. This result is achieved when accurate predictions are made by the GW and the intermediate transmissions are aggregated in the nodes. We plan to include this mechanism in a larger system that is able to improve the accuracy of the predictions made by the GW after consulting external sources of information. Furthermore, a more complete system will be able to choose the best mechanism to evaluate the QoI provided by a WSN. Finally, all these features make it be possible to run self-managed systems that adapt the WSNs operation according to their surroundings (and not only based on the observed environment) in order to achieve the best results, i.e., the highest QoIs and the lowest energy consumption possible.

Acknowledgment This work has been partially supported by the Spanish Government through the project TEC2012-32354 (Plan Nacional I+D), by the Catalan Government through the project SGR-2014-1173 and by the European Union through the project FP7-SME-2013-605073-ENTOMATIC.

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[4] A. Deshpande, C. Guestrin, S. R. Madden, J. M. Hellerstein, W. Hong, Model-driven data acquisition in sensor networks, in: Thirtieth international conference on Very large data bases, 2004, pp. 588–599. URL http://dl.acm.org/citation.cfm?id=1316741 [5] M. Li, D. Ganesan, P. Shenoy, Presto:

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Appendix A. Minimum accuracy Let us suppose that a sensor node i is in ring 1. The minimum accuracy (αmin ) necessary to make the mechanism save energy must satisfy the following equation:

S 0i ETX + R0i ERX + EDISS < S i ETX + Ri ERX c c ((Kd + 1) αmin ETX + (Kd αmin )ERX )f T + EDISS < ((Ki,d + 1)ETX + Kd ERX )f T c αmin (D2 ETX + (D2 − 1)ERX )f T < (D2 ETX + (D2 − 1)ERX )f T − EDISS c αmin (D2 ETX + (D2 − 1)ERX )f T < f T D2 ETX + f T (D2 − 1)ERX − (D2 (ETX + ERX ) − ETX ) c αmin [D2 ETX + (D2 − 1)ERX ]f T < ETX (f T D2 − D2 + 1) + ERX (f T D2 − D2 − f T )

ETX (f T D2 − D2 + 1) + ERX (f T D2 − D2 − f T ) [D2 ETX + (D2 − 1)ERX ]f T ETX (f T D2 − D2 + 1) + ERX (f T D2 − D2 − f T ) >1− [ETX D2 + ERX (D2 − 1)] f T

c αmin
−1, i 6= 0 is a real number and j ≥ 2 an integer value. Substituting the values of α and x in Equation B.1 respectively by i + 1 and j, the claim is proved.

27