## 4 Proposed Framework of IWO Algorithm
### Mathematical Model of IWOA
#### 4.2.2 Exploitation Phase  
To efficiently solve the application mapping problem, the IWOA uses the bubble net behavior of whales to achieve an optimum mapping solution within a faster convergence time.  
**1--Shrinking Encircling Mechanism: Exploitation for Mapping.** In a 2D space, Figure 2 depicts the potential positions starting from \((X)\) towards \((X^{*})\) that can be obtained by \(0\leq A\leq 1\). The shrinking encircling mechanism is modeled in a 2D NoC architecture such that \(X\) represents a mapping solution with a certain communication cost. In contrast, \(X^{*}\) is a better solution with a better communication cost than the earlier one. The mapping denoted as \(X\) tries to achieve a better solution, represented by \(X^{\star}\), with a better communication cost. This mechanism runs for a certain number of iterations until final mapping is achieved. This mechanism to achieve optimal mapping uses Equation (26) by altering a value. In the equation, \(A\) represents a random value with a range of \(-a\) to \(a\), where \(a\) gradually diminishes from 2 to 0 over a specified number of iterations.  
**2--Spiral Strategies: Elevating NOC Application Mapping.** This approach first approximates the deviation between the value of the objective function for the current mapping solution \(X\) and the optimal value that needs to be achieved. The spiral update position to achieve the optimal mapping is derived using the following equation:  
\[\vec{X}(t+1)=\vec{D}^{{}^{\prime}}(t)\cdot e^{bl}\cdot cos(2\pi l)+\vec{X}^{* }(t) \tag{28}\]  
where \(\vec{D}^{{}^{\prime}}(t)\) is provided by  
\[\vec{D}^{{}^{\prime}}(t)=|\vec{X}^{*}(t)-\vec{X}(t)|. \tag{29}\]  
The equation shows the value of the \(i\)th mapping from the best mapping solution obtained thus far, where \(b\) determines the logarithmic spiral's shape while \(l\) is a random number. As the mapping is achieved using the shrinking circle and the spiral-shaped mechanism simultaneously, it is presumed that there is an equal 50% probability of selecting either the shrinking circular mechanism or the spiral model to update the mapping methods. The mathematical model takes the following form:  
Figure 1: Task allocation and fitness updating process in IWOA for NoC platform.  
between \(1\) and \(-1\) in order to compel the search agents to distance themselves from the reference mapping solution and explore more of the overall search space. Compared to the exploitation phase, adjusting the placement of a search agent (mapping solution) does not take place during exploration based on the best solution. Instead, we choose a search agent (a random mapping solution) at random and perform updates based on it. This method, along with \(\mid A\mid>1\), emphasizes exploration and enables the IVOA to perform a global search. Its mathematical model is presented as follows:  
\[\vec{D}=|\vec{C}\cdot X^{\rightarrow}_{rand}-\vec{X}|, \tag{31}\]  
\[X(t+1)=X^{\rightarrow}_{rand}-\vec{A}-\vec{D}, \tag{32}\]  
where \(X\) rand is a mapping solution picked at random from the current population. Several mapping solutions surrounding a specific solution with \(A>1\) are created.