## 5 Analytical Modeling
### Modeling the Backpropagation Dynamics of ACK  
As mentioned above, when a new coded packet was received by the destination node, an ACK \(q\) was fed back to the source by epidemic routing. ACK \(q\) would overwrite the existing ACK, which had a larger index value \(q^{\prime}\). This means that only one ACK message could be stored in the cache of node. Since ACKs in the cache of nodes would not be cleared at the end of each cycle, the backpropagation state of ACKs in current cycle would be affected by the past cycles. Therefore, we investigated the backpropagation of ACKs from a global perspective. In this section, the CDF of the \(D_{Y_{q}}\), which is the delivery delay of ACK \(q\) from the destination to the source, was calculated. Hence, the probability that the source node had received ACK \(q\) by time \(t\), denoted as \(P_{Y_{q}}(t)\), was obtained.  
Let \(Y_{q}(t)\) represent the number of relay nodes that store ACK \(q\) at time \(t\), and \(y_{q}(t)\) represent the number of ACK \(q\) that reaches the source node at time \(t\). \(\left\{y_{q}(t),t\geq 0\right\}\) can be viewed as a non-homogeneous Poisson process with parameter \(\lambda\cdot Y_{q}(t)\). It is observed that \(Y_{q}(T)\), the number of ACK \(q\) reaches the source node in the interval \([0,\) T\(]\), conditioned on the natural filtration \(\mathcal{F}_{Y_{q}}\), is a random variable that obeys the Poisson distribution with parameter \(\lambda\cdot\int_{0}^{T}Y_{q}(t)dt\). Then, the probability that the delivery delay \(D_{Y}\) of ACK \(q\) is greater than \(T\), conditioned on the natural filtration \(\mathcal{F}_{Y_{q}}\), is given by:  
\[P\Big{(}D_{Y_{q}}>\text{T}|\mathcal{F}_{Y_{q}}\Big{)}=P\Big{(}y_{q}(\text{T})= 0\Big{|}\mathcal{F}_{Y_{q}}\Big{)}=e^{-\lambda\int_{0}^{T}Y_{q}(t)dt} \tag{13}\]  
Take the expectation from both sides of Equation (13):  
\[P(D_{Y_{q}}>T)=E\Big{(}e^{-\lambda\int_{0}^{T}Y_{q}(t)dt}\Big{)} \tag{14}\]  
According to literature [18] by [PERSON] et al., Equation (14) is bounded for the number of all relay nodes \(N.\) Using the dominated convergence theorem, we obtain:  
\[P(D_{Y_{q}}>T)=\lim_{N\to\infty}E\Bigg{(}e^{-\lambda\int_{0}^{T}Y_{q}(t)dt} \Bigg{)}=e^{-\lambda\int_{0}^{T}Y_{q}(t)dt} \tag{15}\]  
Therefore, for large \(N\), the CDF of delay \(D_{Y_{q}}\) of ACK \(q\) (\(q=0,1,2,\ldots,r-1\)), \(P_{Y_{q}}(t)\), can be approximated by the solution of the following ordinary differential Equation (16):  
\[\frac{dP_{Y_{q}}(t)}{dt}=\lambda\cdot Y_{q}(t)\cdot\Big{(}1-P_{Y_{q}}(t)\Big{)} \quad(0<t<+\infty\quad) \tag{16}\]  
with the initial condition \(P_{Y_{q}}(0)=0\). Where \(Y_{q}(t)\) can be obtained by solving the following ordinary differential equation:  
\[\frac{dP_{Y_{q}}(t)}{dt}=\lambda\cdot Y_{q}(t)\cdot\Bigg{(}N-\sum_{j=1}^{r}Y_ {j}(t)\Bigg{)}+\lambda\cdot S_{q,r}(t)\cdot\Big{(}N-Y_{q}(t)\Big{)}+\lambda \cdot Y_{q}(t)\cdot\sum_{m>q}Y_{m}(t)-\lambda\cdot Y_{q}(t)\cdot\sum_{m<q}Y_{ m}(t) \tag{17}\]  
with the initial condition: \(Y_{q}(0)=0\), \(q=0,1,2,\ldots,r-1\), where \(S_{q,..,}(t)\) represents the probability that the destination node has collected \(r\)-\(q\)-coded packets at time \(t\) within the current cycle. It can be obtained from the \(F_{i}(t)\) which was calculated in the Equations (5) and (6):  
\[S_{q,r}(t)=F_{r-q}(t)-F_{r-q+1}(t) \tag{18}\]  
To understand Equation (18), the first item on the right of the equal sign indicates that the ACK \(q\) is copied by intermediate carriers to nodes without any ACK stored in unit time; the second item indicatesthe diffusion of ACK \(q\) by the destination node, and the \(\lambda\cdot Y_{q}(t)\cdot\sum_{m>q}Y_{m}(t)\cdot\lambda\cdot Y_{q}(t)\cdot\sum_{m <q}Y_{m}(t)\) indicates the replacement of the existing ACKs with the new ACKs.