# -- Select and Attempt task -- ```
## 4 Enhancing Performance by Avoiding Unwarranted Task Executions
### A Semi-Formal Analysis of the CA Algorithm to Identify and Avoid Unwarranted Task Executions  
To identify potential improvements to the performance of the CA algorithm, we conducted the following semi-formal analysis. While this analysis incorporates formal elements such as assumptions, definitions, propositions, and a proof by induction, it does not constitute a strictly formal analysis in the traditional mathematical or computer science sense. Instead, it represents a blend of formal reasoning and empirical evaluation for the following reasons:  
1. Although induction is used to support Proposition 1, many conclusions rely more on empirical observation than on rigorous formal logic;
2. The conclusions are drawn from specific experimental conditions (e.g., Threshold = 0.5, a = \(\beta\) = 0.1), whereas a purely formal analysis would typically strive to derive results that hold regardless of particular parameter values;
3. Several conclusions are derived from example scenarios rather than universally valid logical proofs. In our testbed, each consumer requesting the service at a certain performance level waits for a period equal to WT for the service to be provided by a provider in its operational range (nearby provider). If the requested \(\text{task}=(\text{service\_ID},\text{performance\_level})\), where \(\text{performance\_level}\in\{\text{PERFECT},\text{GOOD},\text{OK},\text{BAD}, \text{WORST}\}\) is not performed, either because there are no nearby providers or none of the nearby providers are willing to perform the task, then the consumer requests the service at the next lower performance level, assuming that any consumer can manage with a lower-quality service. **Assumption 1**. : _WT is large enough so that if there is a capable provider in the operational range of the consumer, then the task will be successfully executed._  
**Assumption 2**. : _The reason why a nearby provider may not be willing to perform a task is only because the weight of the relevant connection is less than the threshold rule. We assume that there are no other reasons, i.e., the providers are not selfish or malicious. In other words, we assume that the providers are always honest and comply with the implemented protocols, having no intention to harm the trustors._  
Since every rational consumer aims to maximize its profit, the sequence of tasks that it may request until a provider is found to provide the service is as follows: \(task_{1}\), \(task_{2}\), \(task_{3}\), \(task_{4}\), and \(task_{5}\). In Table 3, the requirements of each task are specified. **Definition 1**. : _Given that \(C_{j}\) is a consumer requesting the successful execution of \(task_{i}\), \(i\in\{1,\cdots,5\}\), \(P_{k}\) is a provider in the operational range of \(C_{j}\) receiving a request from \(C_{j}\) for the execution of \(task_{i}\), and \(P_{k}\) does not yet have a connection for \(C_{j}\) and \(task_{i}\), we define \(P_{k}\) as having learned its incapubility to perform \(task_{i}\) (meaning that it cannot always perform \(task_{i}\) successfully) when it initializes the weight of a new connection for \(P_{k}\) and \(task_{i}\) to a value smaller than the threshold value, which will result in not executing \(task_{i}\) for \(C_{j}\)._  
Since only bad and intermittent providers have negative (causing damage to the consumers) performances, we analyze how bad providers learn their own capabilities in each of the five tasks, aiming to identify ways to reduce task executions with negative performances and thus improve the performance of the CA algorithm. We consider a system of four consumers, \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\), and only one provider: the bad provider \(BP_{1}\). Each consumer has in its operational range the provider \(BP_{1}\). The provider has no knowledge of its capabilities in performing tasks, meaning that it has not formed any connections yet. Phase A: Provider \(BP_{1}\) learns its incapability in performing \(task_{1}\).