## 3 Results
### Response Time of Slave and Master Modules
#### 3.4.3 Mathematical Model with Virtual Wires and Multiprocessing  
By adding this improvement to the software, the communication speed between the modules increases, thus reducing transfer times. Modification of the mathematical model with the I2C virtual wire system:  
* For the slave module: The equation for the enhanced response time of the slave module (_T_improved_slave_) will now consider the additional improvement due to the I2C virtual wire system:  
\[T\_improved\_slave(C)=ae^{-bC}f(C) \tag{6}\]  
where '\(f(C)\)' is a function representing the improvement provided by the I2C virtual wire system as a function of the workload (\(C\)). * For the master module: The equation for the enhanced response time of the master module (_T_improved_master_) will also be adjusted to account for the additional improvement from the I2C virtual wire system:  
\[T\_improved\_master(C)=ce^{-dC}g(C) \tag{7}\]  
where '\(g(C)\)' is a function representing the improvement provided by the I2C virtual wire system as a function of the workload (\(C\)). * Improvement function for the slave module: '\(f(C)\)': For the enhanced efficiency of the slave module. As the workload increases, the enhanced efficiency also increases, but with a lower growth rate. \[f(C)=1-e^{-k_{1}C} \tag{8}\]  
where 'k1' is a constant that controls the improvement rate as a function of workload. As '\(C\)' increases, the efficiency will approach 100%, but will never fully reach it. * Improvement function for the master module '\(g(C)\)': Where '\(k_{2}\)' is a constant that controls the initial improvement rate. As '\(C\)' increases, the improvement slows down, but it never reaches 100% efficiency. \[g(C)=1-\frac{1}{k_{2}C+1} \tag{9}\]  
As observed in Figure 12, there is a difference in the successful response of the master and slave modules to workloads (data downloads) between using virtual wires, multiprocessing separately, and combining these two technologies. When the master and slave modules use only virtual wires with an average workload of 60 \(C\) at 1 Mbps, they take approximately 12 and 24 s, respectively, to respond successfully. However, the configuration that uses both wires and multiprocessing takes 5 and 6 s, respectively, reducing the time by more than half for the master modules and fourfold for the slave modules. This results in both the slave and master modules taking the same amount of time, thus avoiding delays in action processing and saturation in action transmission. The master modules will not have to wait as long to receive a response from the slave modules, enabling homogeneous communication among all modules. This is crucial in this field, considering that the software is subject to constant changes, and seamless communication without delays is necessary to resolve errors in the shortest possible time. Probability of Repair and Error Resolution of the Self-Adaptable Software for Pre-Programmed Internet Tasks  
In the mathematical model, we consider three key aspects:  
1. The probability of failure of each module (A, B, and C), where A represents the master module, B represents the slave module, and C represents the repair module. 2. The probability that the system will repair the code in case of a fatal failure is 80%, according to the tests we conducted, with an 80% success rate. 3. The probability that the code will execute correctly after a repair. We denote the following probabilities:  
P(A): Probability of a failure occurring in module A.  
P(B): Probability of a failure occurring in module B.  
P(C): Probability of a failure occurring in module C.  
P(R): Probability of the system repairing the code in case of a fatal failure. P(TF): Probability of software fault tolerance (probability of the software functioning correctly). P(RF): Probability of recovery after a failure.