## 4 Models and Algorithm Design
### Algorithm with Multiple Uncertainties
#### 4.5.4 Drop-Off Packet Representation  
The term \"drop-off packet,' also known as a \"lost packet\", is commonly used in communication scenarios. It can be modeled using the Bernoulli process, denoted as \(X\sim Ber(p)\). The Bernoulli process is a discrete-time stochastic process involving binary random variables, taking either 0 or 1 values. Specifically, it is characterized by \(P(X=1)=p\) and \(P(X=0)=(1-p)\). An alternative model can be Binomial distribution. Binomial distribution means the instance frequency of a random event in \(N\) independent trials. The probability can be formulated as \(X\sim Bi(n,p)\). Parameter \(n\) represents the number of attempts, and parameter \(p\) denotes the probability. Binomial distribution thus describes that the event will occur \(x\) times out of a total of \(n\) attempts, with a probability of \(p\). \[P[x=X]=\Big{(}\binom{n}{x}\Big{)}p^{x}(1-p)p^{n-x}\]  
The mean is represented by \(E\), and the variance is marked as \(D\). They can be expressed as:  
\[E[X]=np,\quad D[X]=np(1-p).\]  
The Binomial distribution thus works with the result for multiple trials for a given event. In contrast, the Bernoulli process is used if the result is needed only once. This means that Bernoulli is designed for the outcome of a single trial for the given event. If the traffic delay approaches infinity or some invalid value, this state can be labelled as a Drop-off packet. \[\text{drop-off packet}:\tau_{i}(k)\to\infty\]  
If the traffic delay \(\tau_{i}(k)\) has a constant value lower than the limit given in (35), then the algorithm converges, as was shown in [65]. However, the traffic delay can take on values higher than this limit, which makes it possible to add the representation and treatment of these events. In case that the traffic delay \(\tau_{i}(k)\) rises above limit value \(\tau_{\max}+1\), then it can be marked as a Drop-off packet. \[\tau_{i}(k)= \begin{cases}\tau_{i}(k)&\text{if}\quad\tau_{i}(k)<\tau_{\max}+1 \\ \\ \infty&\text{if}\quad\tau_{i}(k)\geq\tau_{\max}+1\end{cases} \]From the Equation (37)-(39) is clear that the traffic delay only affects the calculation of the Equations for \(\lambda\) (37) and \(y\) (39). \[\lambda_{i}(k)= \begin{cases}\lambda_{i}(k)&\text{if}\quad\tau_{i}(k)\
eq\infty\\ \\ \lambda_{i}(k-1)&\text{if}\quad\tau_{i}(k)=\infty\end{cases}  \tag{36}\]  
The same rule shown in (36) applies to \(y_{i}(k)\), substituting \(\lambda\) by \(y\). This straightforward adjustment effectively addressed oscillations and instability caused by significant traffic delays. It also prevented potential issues, such as fatal collapse, if negative indices were involved in the discrete algorithm. With the provided insights, these elements can be integrated into the existing algorithm expressed by Equations (22)-(24). The modified algorithm version will take the following form:  
\[\lambda_{i}(k+1) =\sum_{j=1}^{N}w_{ij}\lambda_{j}(k-\tau_{ij})- \begin{cases}\gamma _{i}(k)\frac{\
abla\Phi_{i}\left(\lambda_{i_{j}}(k)\right)+\
u_{i}(k)}{y_{ii}( k)},&\text{if}\,i\in O_{c},\\ \\ 0,&\text{if}\,i\in O_{s,w},\\ \\ \gamma_{i}(k)\frac{\
abla\Phi_{i}\left(\lambda_{i_{j}}(k)\right)+\
u_{i}(k)}{ y_{ii}(k)},&\text{if}\,i\in O_{b},\end{cases}  \tag{37}\] \[P_{i}(k+1) = \begin{cases}\varphi_{c_{i}}\big{(}\
abla C_{c_{i}}^{-1}( \lambda_{c_{i}}(k+1)\cdot\mu_{c_{i}}^{-1}(k))\big{)},&\text{if}\,i\in O_{c}, \\ \\ \varphi_{s,w_{ii}}\big{(}P_{s,w_{i}}\cdot\mu_{s,w_{i}}(k)\big{)},&\text{if}\, i\in O_{s,w},\\ \\ \varphi_{b_{i}}\big{(}\big{(}E_{b_{i}}(t)\big{)}\cdot\mu_{b_{i}}(k)\big{)},& \text{if}\,i\in O_{b},\end{cases} \] (38) \[y_{i}(k+1) =\sum_{j=1}^{N}w_{ij}y_{j}(k-\tau_{ji}). \tag{39}\]  
**Theorem 2** will be presented in the next step, which describes that the algorithm represented by Equations (37)-(39) is robust against all the inaccuracies described above and still converges to the optimal solution. **Theorem 2**. : _Assumptions 1-3 and Definitions 1 and 2 still hold again.