Energy performance of Internet of Things (IoT) networks for pipeline monitoring

Energy performance of Internet of Things (IoT) networks for pipeline monitoring Godlove Suila Kuaban, Tadeusz Czach´ orski, Erol Gelenbe, Piotr Pecka Institute of Theoretical and Applied Informatics, Polish Academy of Sciences (IITiS PAN), Gliwice, Poland, [email protected] Valery Nkemeni Laboratory of Electrical Engineering and Computing, Faculty of Engineering and Technology, University of Buea, Cameroon nkemeni.v[email protected] Piotr Czekalski Faculty of Automatic Control, Electronics, and Informatics, Silesian University of Technology Gliwice, Poland piotr[email protected] Abstract—Pipelines are the most convenient ways to transport fluids (e.g., water, oil, and gas). However, leakage of fluids into the environment results in resource wastage (primarily water, which is becoming a scarce resource) and environmental pollution (in the case of leakage of toxic fluids like oil and gas). Emerging technologies like the Internet of Things (IoT), Wireless Sensor Networks (WSNs), Artificial Intelligence (AI), distributed computing, and cloud computing enable continuous monitoring of pipelines to detect leakages and corrosion on the pipeline. The main challenge with using battery-powered sensor nodes to monitor pipelines is the energy constraint, necessitating frequent battery replacement. Thus, there is a need to develop energysaving mechanisms to prolong the lifetime of these sensor nodes. In this paper, we use the diffusion approximation modelling framework in which the data from the experimental testbed are used to model the dynamics of the battery’s energy content and to estimate the mean and variance of the device’s lifetime. The novelty in the proposed diffusion model of the battery of an IoT node is the introduction of multiple energy thresholds that split the energy state-space of the battery into multiple energysaving regimes. As the battery discharges, the node gradually transitions into energy-saving regimes by reconfiguring some of its parameters to reduce energy consumption (sometimes at the cost of trading off some performance metrics). We investigate the impact of energy-saving regimes or the number of thresholds on the node’s lifetime. Index Terms—Green Internet of Things (IoT), energy performance, diffusion models, adaptive sensing, distributed computing, duty cycling. I. INTRODUCTION Water pipeline leakages pose significant challenges for water utility companies. The loss of water through leaks is widely acknowledged as a costly issue, [1]–[3]. The increasing water demand and decreasing water resources due to global warming and climate change pose a significant challenge worldwide [1], [4]. Water scarcity is pervasive, affecting both developing and developed countries [1]. In light of this scarcity, it is imperative This paper was partially supported by Reactive Too project that has received funding from the European Union’s Horizon 2020 Research, Innovation and Staff Exchange Programme under the Marie Skłodowska-Curie Action (Grant Agreement No871163) and the international project co-financed by the program of the Minister of Science and Higher Education entitled ”PMW” in the years 2021 - 2025; contract no. 5169/H2020/2020/2. to minimise water losses from leaks by promptly detecting and localising leakages in real-time. Additionally, leaks pose a risk of contaminating treated drinking water, potentially leading to the outbreak of diseases [5]. Therefore, swift identification of leaks in the Water Distribution Network (WDN) is crucial for safeguarding treated water from contamination. One effective method for detecting and locating leaks in water pipelines is through WSN-based Water Pipeline Monitoring (WWPM) systems. These systems comprise multiple sensor nodes strategically placed along the pipeline, which periodically gather data on leak signals emitted by the pipe. Subsequently, the collected signals undergo processing to ascertain the presence or absence of a leak in the pipeline. Since Water Distribution Networks (WDNs) typically follow linear structures, implementing a WWPM solution as a linear WSN is highly feasible and advantageous. Pipeline monitoring schemes are influenced by various factors, including communication mechanisms, evaluation methods, power management, monitoring types, sensor connectivity, sensing coverage, sensing methods, and types of sensors [6]. The primary challenge in leak detection within WWPM systems using low-cost sensors lies in the potential inaccuracies of leak signals. This inaccuracy can arise from the sensors’ low sensitivity and environmental noise, leading to false alarms in the leak detection system. Consequently, reliably identifying genuine leak signals amidst noise poses a fundamental challenge for any leak detection system [7], [8]. Existing WWPM solutions encounter significant challenges, including high cost, high energy consumption, and nonrealtime leak detection, primarily due to their centralised nature and reliance on intrusive sensors such as pressure and flow sensors. These sensors are not only costly but also need to be improved during installation and consume substantial amounts of energy. In recent years, WWPM systems utilising vibration sensors have gained popularity [8]–[10]. Vibration sensors offer a promising alternative for monitoring water pipelines. They leverage the relationship between water pipeline pressure fluctuations and surface vibrations, wherein a transient change in pressure during a leak event leads to increased The 20th International Wireless Communications & Mobile Computing Conference 1490 2024 International Wireless Communications and Mobile Computing (IWCMC) | 979-8-3503-6126-1/24/$31.00 ©2024 IEEE | DOI: 10.1109/IWCMC61514.2024.10592530 979-8-3503-6126-1/24/$31.00 ©2024 IEEE Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply. pipe surface acceleration at corresponding locations along the pipe length [11]. This relationship is nonlinear but monotonic, allowing vibration sensors to detect and locate leaks [11], [12]. Various types of vibration sensors, including accelerometers, piezoelectric transducers, and force-sensitive resistors, can be employed. Unlike intrusive sensors, vibration sensors are easy to install, require lower maintenance and operational costs, and consume less energy. These attributes make them a more costeffective and energy-efficient solution for WWPM systems, addressing some of the key challenges associated with traditional monitoring approaches. In many developing countries, water distribution networks predominantly consist of plastic pipes. However, studies have demonstrated that leak signals (vibrations) do not propagate far in plastic pipes [13], [14]. Therefore, achieving reliable leak detection necessitates placing sensors very close to each other to attain higher spatial resolution [15]. High-accuracy accelerometers affixed to the outer surface of the pipe can effectively detect sudden increases in pipe surface acceleration resulting from pipeline leaks. Deploying such accelerometers requires reducing intersensor distances, which can significantly increase overall costs. It renders them impractical for installation in developing countries. Hence, low-cost MEMS accelerometers emerge as a feasible and economically viable solution for deployment in these regions. Nonetheless, the use of low-cost accelerometers presents challenges, notably in reliably detecting leaks amid random environmental noise due to the sensors’ low accuracy [8], [9], [16], [17]. Another challenge lies in achieving realtime detection while preserving the lifetime of the WWPM system. Motivated by the need for real-time monitoring, affordability, low maintenance cost, energy efficiency and sustainability in WWPM systems, we propose some Green IoT (G-IoT) strategies. These strategies integrate distributed computing through a distributed Kalman filter, duty cycling facilitated by interrupt-enabled sleep/wakeup functionality, and adaptive sensing achieved by leveraging low-power, low-accuracy sensors and high-power, high-accuracy sensors. Another energysaving strategy is using energy-saving thresholds at which the IoT nodes are forced to enter energy-saving regimes at the cost of sacrificing some of their functionalities to prolong their lifetime. The overarching objective is to introduce a sustainable solution for water pipeline monitoring that is costeffective, energy-efficient, real-time, and reliable. The energy consumption measurements from the testbed designed to conduct experiments to investigate the impact of the energy-saving strategies on the device’s lifetime, exhibit some degree of variability. Thus, the time-dependent energy content of the battery is modelled as a stochastic process. One approach is to discretise the energy stored in the battery into energy units or energy packets and then apply queueing theorybased models in the analysis as in [18]–[20]. A Markov fluid queue model of the battery of an IoT device with adaptive sensing was proposed in [21]. Also, Markov fluid queue models of the battery of an IoT device with energy-saving threshold or energy-saving regimes was proposed in [21]–[23]. A. Main contribution of the paper Most of the studies that model the influence of energysaving thresholds on the dynamics of the energy content of energy storage systems (ESSs) and the density of the time required to deplete all the stored energy are based on Markov models. The limitation of the Markov-based models is the assumption that delivering energy to the ESS (from renewable energy sources) follows a Poison process and that the energy consumption process is exponentially distributed. These assumptions are not valid for some IoT applications. Although most existing studies consider energy harvesting, our study considers IoT nodes powered only by energy storage systems (e.g., batteries or supercapacitors). In this paper, we use the diffusion approximation modelling framework in which the data from the experimental testbed are used to model the dynamics of the battery’s energy content and to estimate the mean and variance of the device’s lifetime. Similar diffusion models have been applied in [24]–[28] to model the battery of IoT devices, sometimes in the presence of energy harvesting. Here, we extend the model in [28] to incorporate the impact of imposing energy-saving thresholds on the lifetime of a battery-powered IoT device. We model the changes in the battery’s energy content by a diffusion process that uses the statistical parameters of the power measurements from the testbed to determine the density of the time required to deplete all the energy stored in the battery. We develop a diffusion-based model for a batterypowered IoT node configured to switch to more energy-saving regimes when defined energy-saving thresholds are reached. We investigate the impact of energy-saving regimes or the number of thresholds and their magnitude on the node’s lifetime. II. RELATED WORKS We present an overview of stochastic models for evaluating the performance of energy storage systems with thresholdbased energy-saving policies. One approach to model energy storage systems using the energy packet concept introduced by Gelenbe in [18], [19], [29]. The authors in [] applied the energy packet concept to develop Markov-based models for selfpowered green IoT with state-dependent energy consumption. When the battery’s energy content reaches a defined threshold, the node switches to an energy-saving regime by reducing energy consumption. The limitation of this model is that the energy consumption process was assumed to be exponentially distributed, which is not always the case. Since energy is a continuous variable, fluid queue models have been proposed where changes in the energy content or charge level of the battery are modelled as a fluid process. The authors in [22] modelled the dynamic changes in the charge content of the battery of an LTE-A user equipment using a fluid queue model in which the changes in the charge level of the battery at time tare treated as the changes in the level The 20th International Wireless Communications & Mobile Computing Conference 1491 Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply. of fluid in a reservoir. They defined a threshold below which the charge controller gradually decreases the current drawn from the battery by executing the Discontinuous Reception (DRX) mechanism, in which most of the User Equipment (UE) components are turned off for a prolonged period. They derived the cumulative distribution function of the lifetime of the UE (the time required to decrease the charge level of the battery to zero). The authors in [23] used a similar modelling approach: the device enters an energy-saving regime by reducing its power consumption when a defined charge level threshold is reached. The authors in [30] proposed an adaptive threshold energy management algorithm for IoT nodes used in pipeline monitoring. In their approach, two energy thresholds are defined to enable the IoT nodes to switch between three energy-saving regimes: the fully functional, the semi-functional, and the nonfunction functional regime. They demonstrated the advantage of using multiple thresholds to reduce the energy consumption of IoT nodes but did not present the modelling of the battery with multiple thresholds. The authors in [21] proposed a Markv fluid queue model evaluating the energy thresholdbased energy management policy. In their proposed analysis, any number of thresholds can be used to define multiple energy consumption regimes. The nodes can switch between these energy regimes depending on their energy content or charge levels. The numerical results they presented were obtained considering a single energy threshold and adaptive sensing. In their energy management policy, when the energy threshold is reached, the high power consumption sensor is turned off, and the low power sensor is activated. Existing studies on the performance modelling of the influence of the energy threshold on energy storage systems (e.g., batteries or supercapacitors) considered energy harvesting, but IoT and wireless sensor nodes are sometimes deployed with energy harvesting sources. The authors in [28] proposed a diffusion-based energy depletion model for the battery of an IoT node without energy harvesting sources. Their energy management policy considered that when the energy threshold is reached, the node’s battery should be replaced or recharged. The advantage of the diffusion-based model is that it allows the use of measured energy data to evaluate the impact of the threshold-based energy management policy and considers fluctuations in energy consumption. III. EXPERIMENTAL TESTBED SETUP AND MEASUREMENTS The experiments were conducted using a laboratory testbed constructed from a high-pressure plastic pipe measuring 12 m in length and with a diameter of 25 mm. To simulate leak events, a control valve was installed at the midpoint of the pipe. The monitoring system comprised two sensor nodes positioned 2 m apart on both sides of the valve location. A visual representation of the experimental setup is depicted in Figure 1. Each sensor node comprises several components, including an ESP32 MCU serving as the processing unit, an nRF24L01+ transceiver for communication, LSM9DS1(high-power sensor) Sensor Node USB A USB Power Meter Fig. 1. The architecture of the experimental setup and ADXL344 (low-power sensor) accelerometers as sensors, and a 3.7 V 2000 mAh Li-Po rechargeable battery as the power source. A USB power meter is employed To monitor the power consumption of the sensor nodes during experimentation. This power meter consists of an INA226 module interfaced with an STM nucleo-32 F303k8 MCU. The power measurements are visualized on a display and simultaneously recorded on the SD card module throughout experimentation, see Figs. 2. To realize a low-cost, energy-efficient solution with realtime monitoring, a hybrid energy conservation technique encompassing distributed computing, adaptive sensing, and duty cycling is proposed. This hybrid approach ensures effective energy conservation across multiple facets of the sensor node’s operation, promoting sustainability and prolonged battery life while maintaining real-time monitoring capabilities. Duty cycling is implemented through an interrupt-driven sleep/wakeup mechanism to reduce the energy consumption of the sensor node and prolong its lifespan. Duty cycling aims to deactivate node components, including the MCU, radio, and sensors, thereby minimizing energy consumption. The ESP32 MCU, utilized as the node’s processing unit, exhibits high power consumption when operating in active mode (modem sleep mode), consuming current ranging from 20 mA to 68 mA. Continuous operation in active mode significantly diminishes the node’s lifespan. Hence, it is imperative to minimize the duration during which the MCU remains in active mode by transitioning it to deep sleep mode, characterized by current consumption of 150 µA when inactive. However, in deep sleep mode, the node cannot detect events. To facilitate realtime leak detection, the node must switch from deep sleep mode to active mode upon detecting a leak event. It can be achieved by continuously monitoring the pipeline using the low-power, low-accuracy accelerometer and triggering an The 20th International Wireless Communications & Mobile Computing Conference 1492 Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply. interrupt to activate the other components of the sensor node when a leak event is detected. The operation of the sensor nodes implementing this hybrid technique is detailed as follows: 1) Upon startup, the main core of the ESP32 enters deep sleep mode while the nRF24L01+ transceiver and LSM9DS1 accelerometer are in power-down mode. Meanwhile, the ESP32 Ultra Low Power (ULP) coprocessor and ADXL344 accelerometer are active. 2) The ADXL344 accelerometer continuously monitors the pipeline for activity. An activity indicative of a leak event is detected when the measured acceleration exceeds the predefined threshold value (1.01 g) stored in the activity register of the ADXL344 accelerometer. Upon activity detection, an external interrupt is triggered to awaken the other components of the sensor node. 3) If the ADXL344 accelerometer detects no activity, the ESP32 remains in deep sleep mode with the ULP coprocessor active, while the nRF24L01+ transceiver and LSM9DS1 accelerometer remain in power-down mode. Item Upon activity detection, the ADXL344 triggers a wake-up interrupt for the ESP32, LSM9DS1, and nRF24L01+. The LSM9DS1 wakes up to collect more precise measurements, which are then processed by the ESP32 main core using the Distributed Kalman Filter (DKF) algorithm. The nRF24L01+ transceiver facilitates communication of the sensor node’s local estimates to its direct neighbours to achieve distributed data fusion. 4) Following the fusion of local estimates from neighbouring nodes, the computed estimate is compared with the baseline value for leak detection. If the final estimate exceeds the baseline value by a predefined threshold, a leak alarm is triggered, and the node returns to sleep. Otherwise, no leak alarm is triggered, and the node returns to sleep. To investigate the impact of each energy conservation technique on overall energy reduction, we conducted four sets of experiments, as depicted in Table I. More technical details about implementing the testbed and the measurement procedure can be found in [31]. IV. ENERGY CONSUMPTION MODEL FOR THE IOTNODES The average power consumption of the device can be estimated by considering the time spent in the various modes of operation. The IoT nodes in the considered experiments can be in any of the following modes at any given time: P1: Sensing, distributed computing, and idle radio listening. P2: Radio communication (connection setup, data transfer, and connection tear-down). P3: CPU idling and radio listening. P4: Sleep mode (if the device is configured to go to sleep mode and wake up later). The average power consumption of an IoT device that is Fig. 2. A snapshot of the power profile of the IoT node with configured duty cycling and adaptive sensing. always active (not configured to switch to sleep mode) is given by Pnode =Psen ·tsen +Pcomp ·tcomp +Pconn ·tconn(1) +Ptx ·ttx +Pdiscon ·tdiscon +Pidle ·tidle where, Psen,Pcomp,Ptx,Pconn,Pdiscon,Pidle are the power consumed by the node when sensing, performing distributed computing, setting-up connection, data transmission, tearing down the connection, in the CPU idling and radio listening modes respectively, and tsen,tcomp,tconn,ttx,ttx,tdiscon, and tidle are the times spent in those modes. If duty cycling is configured, then the power consumption of the node is Pnode =D·Pact + (1 −D)·Psleep (2) where, D=tact tact +tsleep is the duty cycle ratio, tact =tsen +tcomp +tconn +ttx + tdiscon +tidle is the duration of the active mode and Pact =Psen ·tsen +Pcomp ·tcomp +Pconn ·tconn tact +Ptx ·ttx +Pdiscon ·tdiscon +Pidle ·tidle tact is the active mode power consumption. Also, tsleep is the duration of the sleep mode and Psleep is the power consumed by the node during the sleep mode. The mean power can be computed from a power profile obtained from experimental measurements. Given a power profile of the IoT node Pnode(t), the mean power consumption with a measurement interval [0, T]is µ=1 TZT 0 Pnode(τ)dτ. (3) The 20th International Wireless Communications & Mobile Computing Conference 1493 Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply. The device stays in sleep mode until a leakage event is triggered, and the device wakes up to measure the vibration with high accuracy, using a high-power accelerometer sensor. The device then performs distributed computing to obtain a more accurate estimate, which is then compared with the leakage threshold to determine if a leakage occurred. This way, the sleep times tsleep are independent and identically distributed random variables. We may consider a completely random energy consumption process where the energy drawn from the battery per time unit is scattered independently and uniformly in the sense of a Poison process [32]. In this case, the energy consumption process becomes Enode =tact ·PactN(1/ti) t, where Nµ tdenotes a standard Poisson process on the half line with constant intensity µ. That is, energy is drawn from the battery in small jumps of energy tact ·Pact which occur interspaced by independent and exponentially distributed waiting times with expected value ti=tact +tsleep. The mean number of energy packets drawn from the energy storage system per time unit in the time is µ=tact ·Pact ti (4) V. ENERGY MODEL OF BATTERY OF THE NODE We model the battery discharging process by a diffusion process process {X(t), t ≥0}its the value x∈[0, B] corresponds to the energy still in the battery.Denote by Bthe battery’s maximum volume; x=Bmeans that the battery is fully charged; x= 0 means the battery is empty. The density function (pdf) of the diffusion process f(x, t;x0)dx =P[x≤X(t)< x +dx |X(0) = x0] is defined by the diffusion equation, which, in the case of the unbounded interval, is ∂f(x, t;x0) ∂t =α 2 ∂2f(x, t;x0) ∂x2−β∂f(x, t;x0) ∂x .(5) Parameter βrepresents mean changes of the process and αits variation; x0is the initial value of the process (in our case, x0=B). If we assume that the time to consume a unit of energy has 1/µ and the variation σ2 Bthan β=−µand α=µ3σ2 B=µC2 Bwhere C2 Bis the squared coefficient of variation of the distribution of this time, see, e.g. [33]. To model the depletion time, we use the notion of the first passage time, i.e. the time it takes for the diffusion process to travel a certain distance. If the process starts at a certain x=x0and ends at x= 0, then the density of the first passage time is obtained by placing at zero an absorbing barrier; the process ends when it reaches the barrier, The density function of this first passage time is obtained by placing at the destination point an absorbing barrier: when the process finishes when it reaches this barrier. It corresponds to the boundary condition limx→0f(x, t;x0) = 0.Thus, the Probability Density Function (PDF) of the diffusion process that starts at x=B(the device is deployed with full battery energy capacity, B) at time t= 0 and ends at x= 0 at time tis given by, see [34], f(x, t;B) = eβ α(x−B)−β2 2αt √2παt e−(x−B)2 2αt −e−(x+B)2 2αt .(6) and represents the current content of the battery. A. Lifetime model of the node The node’s lifetime is modelled as the first passage time of the diffusion process that starts at x=B(the device is deployed with full battery energy capacity, B) at time t= 0 and ends at x= 0 at time t. The PDF of the first passage time process T=inf{t > 0 : X(t) = 0}from x=Bto x= 0 is, [34] γB→0(t) = B √2παt3e−(B+βt)2 2αt ,(7) with the Laplace transform ¯γB→0(s) = e−x0β+√β2+2αs α.(8) Using γB→0(t)we compute the mean lifetime of the IoT device L=−B βand its variance σ2 T=−Bα β3. This way, γB,0(t)presents a model for the density of the battery lifetime distribution if the diffusion parameters do not change. In our numerical examples, the battery capacity B= 18500 mWh (i.e., Q= 5000 mAh and v = 3.7 V, and DoD = 1). The measured in our experiments values of µand σ2 Bfor various system operation modes are given in the table I. They define the diffusion parameters αand β. A few distributions of the density of the depletion time for chosen parameters are presented in Fig. 3. 0 5000 10000 15000 20000 t in hours (h) 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 γB → 0( t ) µ =2 . 48 mW, C 2 B =0 . 139070 µ =0 . 99 mW, C 2 B =0 . 022647 Fig. 3. Comaparison of the density of the lifetime of the node (depletion time), γB→0(t)of the device for µ= 2.48,C2 B= 0.139070 (experiment 3) and µ= 0.99,C2 B= 0.022647 (experiment 4) Note that the results depend on two parameters µand C2 B; if we apply C2 B= 1 (corresponding to the exponentially distributed time to consume the energy unit, the curves of fig. (3) will change to those in Fig. (4). The mean time to depletion is represented in Fig. 5 as a function of µand B. The 20th International Wireless Communications & Mobile Computing Conference 1494 Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply. TABLE I THE IMPACT OF THE VARIOUS ENERGY-SAVING STRATEGIES ON THE POWER CONSUMPTION AND LIFETIME OF THE IOTNODE Experiment Configured energy saving mechanism µ (mW )ρ2 BC2 BL(h)σ2 T Experiment 1Distributed Computing only 125.42 1281.80 0.081486 147.50 0.000764 Experiment 2Distributed Computing + Adaptive Sensing 123.76 1233.39 0.080526 149.48 0.000786 Experiment 3Distributed Computing + Duty Cycling 2.48 0.85533 0.139070 7459.68 168.6749 Experiment 4Distributed Computing + Adaptive Sensing + Duty Cycling 0.99 0.022197 0.022647 18686.87 431.8012 0 5000 10000 15000 20000 25000 30000 35000 t in hours (h) 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 γB → 0( t ) µ =2 . 48 mW µ =0 . 99 mW Fig. 4. Comaparison of the density of the lifetime of the node (depletion time), γB→0(t)of the device for µ= 2.48,C2 B= 1 and µ= 0.99,C2 B= 1 012345 Mean power consumption, µ in mW 0 50000 100000 150000 200000 Device lifetime, L in hours (h) B =7400 mWh B =11100 mWh B =14800 mWh B =18500 mWh Fig. 5. Mean lifetime of the node for various Band µ,C2 B= 1 B. Modelling the impact of energy-saving threshold In addition to adaptive sensing and duty cycling, energysaving thresholds have been adopted to further increase the energy efficiency of IoT nodes (e.g., [21]–[23], [30]). The node may switch between the various energy consumption regimes over time, e.g. after reaching a certain depletion threshold, it passes to the more energy-efficient operating regime. In the case of a single barrier K, we have two regimes, the normal regime and the energy-saving regime, and the density of the time required to deplete all the energy stored in the battery is γB→0(t) = γ1 B→K(t)∗γ2 K→0(t),(9) where * is the operator of the convolution and with the mean and variance being the sum of means and variances of corresponding two phases i.e., L=L1+L2and σ2 T=σ2 T1+σ2 T1. The Laplace transform of (9) is ¯γB→0(s) = ¯γ1 B→K(s)·¯γ2 K→0(s).(10) The density functions of the process (i.e. of the energy content) f1(x, t;B),f2(x, t;B)n both intervals are based on the solution f(x, t;B)given by Eq. (6) with necessary changes: for the first function the starting point is x=B, and the absorbing barrier is at x=K; the second process starts at x=Kwith the intensity γB→K(t), hence f2(x, t;K) = f(x, t;K)∗γB→K(t). Energy-saving regime Normal or fully functional regime Fig. 6. Battery model with a single energy-saving threshold Fig. 3 compares the densities depletion time distribution where after a defined threshold K= 0.6B, the node switches from the energy consumption regime with µ1= 2.48 mW and C2 B1= 0.139070 to a more energy-efficient regime with µ2= 0.99 mW and C2 B2= 0.0022647. The authors in [30] proposed the use of two thresholds (e.g., K1and K2,K1> K2) with three intervals or energysaving regimes in which the nodes can switch between fully functional, semi-functional, and non-functional regimes to optimise their energy consumption. In this case, the density of the time required to deplete all the energy stored in the battery is ¯γB→0(s) = ¯γ1 B→K1(s)·¯γ2 K1→K2(s)·¯γ3 K2→0(s).(11) We consider that the node is configured to switch between four energy consumption regimes (e.g., distributed computing, The 20th International Wireless Communications & Mobile Computing Conference 1495 Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply. 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 t in hours (h) 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 γB → 0( t ) µ 1=2 . 48 mW µ 1=2 . 48 mW, µ 2=0 . 99 mW Fig. 7. The impact of the energy-saving threshold on the density of the lifetime of the node (depletion time), γB→0(t)of the device for µ= 2.48, C2 B= 0.139070 (experiment 3) and µ= 0.99,C2 B= 0.022647 (experiment 4) distributed computing + adaptive sensing, distributed computing + duty cycling, distributed computing + adaptive sensing + duty cycling) defined by three thresholds (e.g., K1,K2, and K3,K1> K2> K3). In this case, the density of the time required to deplete all the energy stored in the battery is ¯γB→0(s) = ¯γ1 B→K1(s)·¯γ2 K1→K2(s)·¯γ3 K2→K3(s)·¯γ4 K3→0(s) (12) 0 5000 10000 15000 20000 25000 30000 t in hours (h) 0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 γB → 0( t ) µ 1=2 . 48 mW, µ 2=0 . 99 mW µ 1=123 . 76 mW, µ 2=2 . 48 mW, µ 3=0 . 99 mW µ 1=125 . 42 mW, µ 2=123 . 76 mW, µ 3=2 . 48 mW, µ 4=0 . 99 mW Fig. 8. The influence of the number of energy-saving thresholds on the density of the lifetime of the node (depletion time), γB→0(t)of the device Fig. 8 compares the depletion time densities for exemplary scenarios with one, two or three thresholds. The variances of the distributions are large, and the maximum values of the functions are small. In this scale, the computing errors introduced by the numerical inversion of the Laplace transforms in Eqs. (10) - (12) are visible. The inversion was done using the Stehfest algorithm [35], and its accuracy should be improved. VI. CONCLUSION This study highlights the critical importance of energy efficiency in addressing the operational challenges faced by battery-powered sensor nodes in IoT applications, particularly in remote or inaccessible environments such as pipeline monitoring systems. By implementing a hybrid approach incorporating distributed computing, duty cycling, and hierarchical sensing, we have demonstrated significant advancements in energy reduction, with a remarkable 99% improvement observed. Our findings underscore the pivotal role of duty cycling as the most influential factor in enhancing energy efficiency. In this paper, we have proposed a practical modelling framework for a battery-powered IoT node. Data from experimental testbed measurements are used to model the battery’s energy content dynamics and estimate the mean and variance of the node’s lifetime. We have developed a diffusion-based model for a battery-powered IoT node configured to switch to more energy-saving regimes when defined thresholds are reached. We also investigated the impact of energy-saving regimes or the number of thresholds of the node’s lifetime. Although duty cycling was observed as the most influential energy-saving strategy, we saw a discrepancy between the expected power consumption of the node in sleep mode and the measured value, which may be attributable to energy leakage during sleep mode. We will extend the proposed diffusion model to incorporate the influence of energy leakage. One way to extend the lifetime of the IoT nodes is to use energy harvesters to harvest energy from ambient or external sources. We intend to conduct experiments using energy harvesters (e.g., solar and flow) to power the nodes. REFERENCES [1] A. Iyeswariya, R. Shamila, M. JayaLakshm, K. Maharajan, and V. Sivakumar, “A study on water leakage detection in buried plastic pipes using wireless sensor networks,” Int. J. Sci. Eng. Res, vol. 3, no. 1, 2012. [2] O. I. Okeya, “Detection and localisation of pipe bursts in a district metered area using an online hydraulic model,” 2018. [3] K. B. Adedeji, Y. Hamam, B. T. Abe, and A. M. Abu-Mahfouz, “Towards achieving a reliable leakage detection and localization algorithm for application in water piping networks: An overview,” IEEE Access, vol. 5, pp. 20 272–20 285, 2017. [4] S. M. F. Islam and Z. Karim, “World’s demand for food and water: The consequences of climate change,” Desalination-challenges and opportunities, pp. 1–27, 2019. [5] B. Adelodun, F. O. Ajibade, J. O. Ighalo, G. Odey, R. G. Ibrahim, K. Y. Kareem, H. O. Bakare, A. O. Tiamiyu, T. F. Ajibade, T. S. Abdulkadir et al., “Assessment of socioeconomic inequality based on virus-contaminated water usage in developing countries: a review,” Environmental Research, vol. 192, p. 110309, 2021. [6] M. Z. Abbas, K. Abu Baker, M. Ayaz, H. Mohamed, M. Tariq, A. Ahmed, and M. Faheem, “Key factors involved in pipeline monitoring techniques using robots and wsns: Comprehensive survey,” Journal of Pipeline Systems Engineering and Practice, vol. 9, no. 2, p. 04018001, 2018. [7] M. Henrie, P. Carpenter, and R. E. Nicholas, Pipeline leak detection handbook. Gulf Professional Publishing, 2016. [8] S. Yazdekhasti, K. R. Piratla, J. Sorber, S. Atamturktur, A. Khan, and H. Shukla, “Sustainability analysis of a leakage-monitoring technique for water pipeline networks,” Journal of Pipeline Systems Engineering and Practice, vol. 11, no. 1, p. 04019052, 2020. [9] S. El-Zahab, E. M. Abdelkader, and T. Zayed, “An accelerometer-based leak detection system,” Mechanical Systems and Signal Processing, vol. 108, pp. 276–291, 2018. The 20th International Wireless Communications & Mobile Computing Conference 1496 Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply. [10] S. Tariq, Z. Hu, and T. Zayed, “Micro-electromechanical systemsbased technologies for leak detection and localization in water supply networks: A bibliometric and systematic review,” Journal of Cleaner Production, vol. 289, p. 125751, 2021. [11] G. Nwalozie and A. Azubogu, “Design and implementation of pipeline monitoring system using acceleration-based wireless sensor network,” Int. J. Eng. Sci, vol. 3, no. 9, pp. 49–58, 2014. [12] M. I. M. Ismail, R. A. Dziyauddin, N. A. A. Salleh, F. MuhammadSukki, N. A. Bani, M. A. M. Izhar, and L. A. Latiff, “A review of vibration detection methods using accelerometer sensors for water pipeline leakage,” IEEE access, vol. 7, pp. 51 965–51 981, 2019. [13] K. Marmarokopos, D. Doukakis, G. Frantziskonis, and M. Avlonitis, “Leak detection in plastic water supply pipes with a high signal-tonoise ratio accelerometer,” Measurement and Control, vol. 51, no. 1-2, pp. 27–37, 2018. [14] S. Hamilton and B. Charalambous, Leak detection: technology and implementation. IWA Publishing, 2013. [15] M. J. Brennan, D. N. Chapman, P. F. Joseph, N. Metje, J. M. Muggleton, and E. Rustighi, Achieving Zero Leakage by 2050: Leak Detection and Location Methods-Acoustic Leak Detection. UK Water Industry Research Limited, 2017. [16] U. Baroudi, A. A. Al-Roubaiey, and A. Devendiran, “Pipeline leak detection systems and data fusion: A survey,” IEEE Access, vol. 7, pp. 97 426–97 439, 2019. [17] M. I. M. Ismail, R. A. Dziyauddin, N. A. M. Salleh, R. Ahmad, M. H. Azmi, and H. M. Kaidi, “Analysis and procedures for water pipeline leakage using three-axis accelerometer sensors: Adxl335 and mma7361,” IEEE Access, vol. 6, pp. 71 249–71 261, 2018. [18] E. Gelenbe, “Energy packet networks: Ict based energy allocation and storage,” in International Conference on Green Communications and Networking. Springer, 2011, pp. 186–195. [19] ——, “Energy packet networks: smart electricity storage to meet surges in demand,” in SIMUTOOLS’12 Proceedings of the 5th International ICST Conference on Simulation Tools and Techniques, no. ISBN 9781450315104, 2012, pp. 1–7. [20] G. S. Kuaban, T. Czach´ orski, E. Gelenbe, S. Sharma, and P. Czekalski, “A markov model for a self-powered green iot device with statedependent energy consumption,” in 2023 4th International Conference on Communications, Information, Electronic and Energy Systems (CIEES), 2023, pp. 1–7. [21] C. Tunc and N. Akar, “Markov fluid queue model of an energy harvesting IoT device with adaptive sensing,” Performance Evaluation, vol. 111, pp. 1–16, 2017. [22] A. Gautam and S. Dharmaraja, “An analytical model driven by fluid queue for battery life time of a user equipment in LTE-A networks,” Physical Communication, vol. 30, pp. 213–219, 2018. [23] G. L. Jones, P. G. Harrison, U. Harder, and T. Field, “Fluid queue models of battery life,” in Proceeding of the 2011 IEEE 19th Annual International Symposium on Modelling, Analysis, and Simulation of Computer and Telecommunication Systems. IEEE, 2011, pp. 278–285. [24] O. H. Abdelrahman and E. Gelenbe, “A diffusion model for energy harvesting sensor nodes,” in 2016 IEEE 24th International Symposium on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (MASCOTS). IEEE, 2016, pp. 154–158. [25] T. Czach´ orski, E. Gelenbe, and G. S. Kuaban, “Modelling energy changes in the energy harvesting battery of an iot device,” in Proceedings of the 2022 30th International Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS). Nice, France: IEEE, 2022, pp. 81–88. [26] T. Czach´ orski, E. Gelenbe, G. S. Kuaban, and D. Marek, “Energy optimization for an unmanned aerial vehicle (e.g drone) during its mission,” in Security in Computer and Information Sciences. EuroCybersec 2021. Communications in Computer and Information Science, vol. 1596. Springer, 2022, pp. 61–75. [27] E. Gelenbe and K. Y. Murat, “Battery attacks on sensors: Wireless nodes with battery attacks,” in EuroCybersec 2018: International symposium on computer and information sciences, cybersecurity workshop. Springer Cham, 2018. [28] G. S. Kuaban, E. Gelenbe, T. Czach´ orski, P. Czekalski, and J. K. Tangka, “Modelling of the energy depletion process and battery depletion attacks for battery-powered internet of things (iot) devices,” Sensors, vol. 23, no. 13, 2023. [29] E. Gelenbe, “A sensor node with energy harvesting,” ACM SIGMETRICS Performance Evaluation Review, vol. 42, no. 2, pp. 37–39, 2014. [30] L. Miao, Z.-M. Huo, Y. Rong, H.-W. Mu, and Z.-X. Sun, “Iot adaptive threshold energy management algorithm based on energy harvesting,” Ad Hoc Networks, vol. 149, p. 103241, 2023. [31] V. Nkemeni, F. Mieyeville, and P. Tsafack, “Energy consumption reduction in wireless sensor network-based water pipeline monitoring systems via energy conservation techniques,” Future Internet, vol. 15, no. 12, p. 402, 2023. [32] I. Kaj and V. Konan´ e, “Modeling battery cells under discharge using kinetic and stochastic battery models,” Applied Mathematical Modelling, vol. 40, no. 17, pp. 7901–7915, 2016. [33] E. Gelenbe, “On approximate computer system models,” Journal of the ACM (JACM), vol. 22, no. 2, pp. 261–269, 1975. [34] R. P. Cox and H. D. Miller, The Theory of Stochastic Processes. London, UK: Chapman and Hall, 1965. [35] H. Stehfest, “Numeric inversion of laplace transform,” Communication of ACM, vol. 13, no. 1, pp. 47–49, 1970. The 20th International Wireless Communications & Mobile Computing Conference 1497 Authorized licensed use limited to: Politechnika Slaska. Downloaded on September 11,2025 at 10:28:30 UTC from IEEE Xplore. Restrictions apply.