## 3 System Model
### Model Analysis  
To simplify the matrices, we simplify some formulas. The concrete representation of some symbols in the below matrices is as follows:  
\[\left\{ \begin{array}{ll}\zeta=\bar{\alpha}_{11}\bar{\alpha}_{12}\bar{\alpha }_{2},&\delta=\bar{\alpha}_{11}\bar{\alpha}_{12}\alpha_{2}\\ \eta=\bar{\alpha}_{11}\bar{\alpha}_{12}\bar{\alpha}_{2},&\theta=\bar{\alpha}_ {11}\bar{\alpha}_{12}\alpha_{2}\end{array} \right. \tag{2}\]  
The system's total number of user packets varies from 0 to \(k+1\), where \(k\) (\(k>1\)) is the cache capacity. The total state transition matrix \(\mathbf{H}\) is shown in Equation (3), and its submatrix \(\mathbf{H}_{a,b}\) is shown in Equations (4)-(15), where \(\mathbf{H}_{a,b}\) represents the one-step state transition matrix of the total user packets from \(a\) to \(b\). \[\mathbf{H}=\left( \begin{array}{cccccccc}\mathbf{H}_{0,0}&\mathbf{H}_{0,1}&\mathbf{H}_{0,2}&&&& \\ \mathbf{H}_{1,0}&\mathbf{H}_{1,1}&\mathbf{H}_{1,2}&\mathbf{H}_{1,3}&&\\ &\mathbf{H}_{2,1}&\mathbf{H}_{2,2}&\mathbf{H}_{2,3}&\mathbf{H}_{2,4}&&\\ &&\ddots&\ddots&\ddots&\\ &&&\mathbf{H}_{k,k-1}&\mathbf{H}_{k,k}&\mathbf{H}_{k,k+1}\\ &&&&\mathbf{H}_{k+1,k}&\mathbf{H}_{k+1,k+1}\end{array} \right) \tag{3}\]  
1. \(\mathbf{H}_{0,0}\), \(\mathbf{H}_{0,1}\), \(\mathbf{H}_{0,2}\), \(\mathbf{H}_{1,0}\) are represented as Equations (4)-(7). \[\mathbf{H}_{0,0}=\left( \begin{array}{cc}\bar{f}\zeta&f\zeta\\ r\zeta&\bar{r}\zeta\end{array} \right) \tag{4}\]  
\[\mathbf{H}_{0,1}=\left( \begin{array}{cccccccc}\bar{f}\delta&\bar{f}\eta&\bar{f} \alpha_{11}\bar{\alpha}_{2}&f\delta&f\eta&f\alpha_{11}\bar{\alpha}_{2}\\ r\delta&r\eta&\alpha_{11}\bar{\alpha}_{2}&\bar{r}\delta&\bar{\eta}\eta&\pi \alpha_{11}\bar{\alpha}_{2}\end{array} \right) \tag{5}\]  
\[\mathbf{H}_{0,2}=\left( \begin{array}{cccccccc}0&\bar{f}\theta&\bar{f}\alpha_{11} \alpha_{2}&0&f\theta&f\alpha_{11}\alpha_{2}\\ 0&r\theta&r\iota_{11}\alpha_{2}&0&\bar{r}\theta&\bar{r}\alpha_{11}\alpha_{2} \end{array} \right) \tag{6}\]  
\[\mathbf{H}_{1,0}=\left( \begin{array}{cccccccc}\beta_{2}\zeta\zeta&\beta_{2} \zeta\\ \beta_{12}\bar{f}\zeta&\beta_{12}\zeta\\ \beta_{11}\bar{f}\zeta&\beta_{11}\bar{f}\zeta\\ 0&0&0\\ e_{12}r\zeta&e_{12}r\zeta\\ e_{11}r\zeta&e_{11}r\zeta\end{array} \right) \tag{7}\]  
2. When \(2\leq c\leq k\), \(\mathbf{H}_{c,c-1}\) is represented as Equation (8). \[\mathbf{H}_{c,c-1}=\left( \begin{array}{cccccccc}\beta_{2}\zeta\zeta&0&0&\beta_{ 2}f\zeta&0&0\\ \beta_{12}\bar{f}\zeta&0&0&\beta_{12}f\zeta&0&0\\ \beta_{11}\bar{f}\zeta&0&0&\beta_{11}f\zeta&0&0\\ 0&0&0&0&0&0\\ e_{12}r\zeta&0&0&e_{12}r\zeta&0&0\\ e_{11}r\zeta&0&0&e_{11}r\zeta&0&0\end{array} \right) \tag{8}\]  
3. When \(1\leq c\leq k-1\), \(\mathbf{H}_{c,c}\), \(\mathbf{H}_{c,c+1}\) and \(\mathbf{H}_{c,c+2}\) are represented as Equations (9)-(11). \[\mathbf{H}_{c,c}=\left( \begin{array}{cccccccc}\bar{f}(\delta\beta_{2}+\zeta\bar{ \beta}_{2})&\beta_{2}\bar{f}\eta&\beta_{2}\bar{f}\alpha_{11}\bar{\alpha}_{2}&f( \delta\beta_{2}+\zeta\bar{\beta}_{2})&\beta_{2}f\eta&\beta_{2}f\alpha_{11}\bar{ \alpha}_{2}\\ \beta_{12}\bar{f}\delta&\bar{f}(\eta\beta_{12}+\bar{\alpha}_{2}\bar{\beta}_{ 12})&\beta_{12}\bar{f}\alpha_{11}\bar{\alpha}_{2}&\beta_{12}f\delta&f(\eta \beta_{12}+\bar{\alpha}_{11}\bar{\alpha}_{2}\bar{\beta}_{12})&f\alpha_{11}\bar {\alpha}_{2}\\ \beta_{11}\bar{f}\delta&\beta_{11}\bar{f}\eta&\bar{f}\bar{\alpha}_{2}(\alpha_{1 1}\beta_{11}+\bar{\beta}_{11})&\beta_{11}f\delta&\beta_{11}f\eta&f\bar{\alpha }_{2}(\alpha_{11}\beta_{11}+\bar{\beta}_{11})\\ r\zeta&0&0&\bar{r}\zeta&0&0\\ e_{12}r\delta&r(\eta e_{12}+\bar{\alpha}_{2}\bar{e}_{12})&e_{12}r\kappa_{11} \bar{\alpha}_{2}&e_{12}r\delta&r(\eta e_{12}+\bar{\alpha}_{11}\bar{\alpha}_{2 }\bar{e}_{12})&r\kappa_{11}\bar{\alpha}_{2}\\ e_{11}r\delta&e_{11}r\eta&r\bar{\alpha}_{2}(\alpha_{11}e_{11}+\bar{e}_{11})&e _{11}r\delta&e_{11}r\eta&r\bar{\alpha}_{2}(\alpha_{11}e_{11}+\bar{e}_{11}) \end{array} \right) \tag{9}\]  
\[\mathbf{H}_{c,c+1}=\left(