# Energy-Optimized Content Refreshing of Age-of-Information-Aware Edge Caches in IoT Systems
## 3 System Overview and Model
### Model of the Power Consumption  
Devices consume energy when performing three main tasks [30]: (i) data sampling, e.g., sensing from the environment, for example, the temperature, the humidity, the pressure, the fluid flow, etc.; (ii) data processing, performed after sampling and involving operations like storage, denoising, etc.; (iii) data communication, which includes all necessary networking tasks like packet transmissions and receptions, protocol overheads due to control traffic, etc.  
We model the energy consumption as depending on two components: (i) sampling energy consumption, due to the sensing operation and the processing of the sampled data; (ii) communication energy consumption, due to the transmission of the updates. The computational energy cost can be considered negligible, as it becomes significant only in some specific cases involving complex mathematical operations or very long sleep times. Denote \(c_{T}\) as the energy consumption for transmitting an update message and denote \(c_{S}\) as the energy consumption for generating a new sample. The energy consumption of the device depends on the frequency of these two operations, i.e., on the transmission frequency and on the sampling frequency.  
In our system model, the transmissions frequency, that we denote as \(f_{T}\), is the poll frequency, i.e., it is the inverse of the average time between two requests that trigger a cache refresh: \(f_{T}(w,s)=1/E\{T\}\). The sampling frequency, that we denote as \(f_{S}\), is instead the inverse of the sampling period: \(f_{S}(w,s)=1/s\). So, the energy consumption per time unit \(c\) on the device is  
\[c=c_{T}f_{T}+c_{S}f_{S} \tag{8}\]  
Given the wide diversity of the IoT devices, \(c_{T}\) and \(c_{S}\) can range from very small values to very large values, relative to each other. Without losing generality, we normalize \(c\) with respect to the sum of \(c_{T}\) and \(c_{S}\), i.e., with respect to the sum of the energy cost of one transmission operation and the energy cost of one sampling operation:  
\[\frac{c}{c_{T}+c_{s}}=\frac{c_{T}}{c_{T}+c_{s}}f_{T}+\frac{c_{S}}{c_{T}+c_{s} }f_{S} \tag{9}\]  
Denote \(c/(c_{T}+c_{S})\) as \(c_{\overline{\beta}}\), and \(c_{T}/(c_{T}+c_{S})\) as \(\overline{\beta}\), it is:  
\[c_{\overline{\beta}}=\overline{\beta}f_{\mathrm{T}}+\big{(}1-\overline{\beta} \big{)}f_{\mathrm{S}} \tag{10}\]  
with \(\overline{\beta}\in[0,1]\).  
This means that \(c_{\overline{\beta}}\) takes into account the relationship between the energy consumption of a transmission operation and a sampling operation, but it does not depend on their absolute values. Indeed, the parameter \(\overline{\beta}\) indicates the energy cost of a transmission operation with respect to the sum of the energy costs of a transmission operation and a sampling operation. The value of \(\overline{\beta}\) depends on the type of device, e.g., for a device where the energy cost of a sampling operation is negligible with respect to the energy cost of a transmission operation \(\overline{\beta}\) tends to one; on the contrary, for a device where the energy cost of a transmission operation is negligible with respect to the energy cost of a sampling operation \(\overline{\beta}\) tends to zero. In [3] [PERSON] et al. compute the operational energy costs in wireless sensor networks focusing on energy consumption during a single sampling period. They consider several commercial sensors, and they present a comparison of their sensing and communication energy costs. Comparisons are normalized with respect to communication energy. In Table 1 we report the results for six exemplary sensors for which we compute the corresponding value of \(\overline{\beta}\) starting from the given values of \(c_{S}/c_{T}\). Clearly, for sensors where the cost of a sensing operation is much higher than the cost of a transmission, the value of \(\overline{\beta}\) is close to zero; instead, for sensors where the cost of transmission is much higher than the cost of a sampling operation, the value of \(\overline{\beta}\) is close to one.