## 3 System Design
### System Model
#### 3.3.2 Optimal Che Approximation  
The Che approximation only considers a simple situation of a single cache node while it ignores the correlation of requests between adjacent nodes. The key to applying Che approximation to a general network node \(i\) is to analyze all possible events that happened in the duration of \(\left[t-T_{M}^{i},t\right]\) for NDO \(c\) in node \(i\) at time \(t\)[23]. The correlation of requests between adjacent nodes differs under different cache strategies that affect possible events that happened in a different way. We optimized the Che approximation by analyzing LCP, LCE, and LCD one by one, as these are the most widely used on-path caching strategies in ICN. Suppose the cache node \(j\) belongs to set J and is adjacent to node \(i\). This implies that \(j\) is located behind \(i\) in a content transmission path so that \(j\) forwards the miss request flow of \(c\) to \(i\). Our inference was based on a hypothesis: \(T_{M}^{i}>T_{M}^{j}\), otherwise \(P_{hit}^{i}=0\) because the request of \(c\) will hit the cache node \(j\). Therefore, the average arrival rate of requests for \(c\) at cache node \(i\) is equal to:  
\[\overline{\lambda}_{c}(i)=\ \sum_{j\in J}\overline{\lambda}_{c}(j)\Big{(}1-P_{hit }^{j}(c)\Big{)} \tag{3}\]  
We first analyzed the optimal Che approximation in LCP, because LCE is only a special case of LCP when \(p=1\). Leave-Copy-Probability  
As for LCP, each node along a transmission path caches content with a fixed probability \(p\), which can be adjusted according to the cache condition in network. Suppose that a request of \(c\) hits node \(i\) at time \(t\), which means that NDO \(c\) is stored in \(i\) rather than \(j\). Hence, the previous request of \(c\) arrives \(i\) at time duration \(\left[t-T_{M}^{i},t\right]\), and this may bring about two possibilities: One is that node \(i\) has a hit, and the other is that node \(j\) caches \(c\) with probability \(p\) when the request missed in \(i\). We estimated time average probability \(P_{in}^{i}(c)\) by the standard Che approximation of request rate \(\overline{\lambda}_{c}(i)\) without considering the variation of \(T_{M}^{i}\) because the \(T_{M}^{i}\) for single node is hard to change when the content catalog \(C\) is large enough. We obtained:  
\[P_{in}^{i}(c)\approx\left[P_{hit}^{i}(c)+p\Big{(}1-P_{hit}^{i}(c)\Big{)} \right]\Big{(}1\ -e^{-\overline{\lambda}_{c}(i)T_{M}^{i}}\Big{)} \tag{4}\]  
\(P_{hit}^{i}(c)\) cannot be simply estimate like \(P_{in}^{i}(c)\) because it is changeable and is influenced by the correlation between nodes. Additionally, to ensure that there is no copy of \(c\) cached in node \(j\) at time \(t\), there are 3 possible events that could have happened in the duration of \(\left[t-T_{M}^{i},t\right]\). The last request of \(c\) either arrives \(i\) from given \(j\) at the time duration of \(\left[t-T_{M}^{i},t-T_{M}^{j}\right]\), it arrives at \(\left[t-T_{M}^{j},t\right]\) while failing to trigger cache action for \(c\) with probability \(1-p\), or it arrives \(i\) from any node \(k(k\in J)\) different from \(j\) at \(\left[t-T_{M}^{i},t-T_{M}^{j}\right]\). The latter two possible events can be regarded as standard Poisson flow, but the first is not because it depends on the cache state of NDO \(c\) at node \(j\).