# Two-Layer Network Caching for Different Service Requirements
## 3 Model
### Caching Problems
#### 3.3.3 Case 3: Guaranteed Download Time
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Case** & **Configuration Parameters** & **Main Performance Evaluation Results** \\ \hline \hline Case 1: & fixed item size, known content popularity & Algorithm1 outperforms LFU and approaches theoretical limit (2). Known popularity makes the effect of layer-1 queues appreciable \\ Case 2: & fixed item size, popularity estimation with \(N\) = 2000 & Algorithm1 outperforms LFU, but it is below the theoretical limit (2) \\ Case 2: & fixed item size, popularity estimation with \(N\) = 10,000 & Algorithm1 slightly outperforms LFU, the distance from theoretical limit (2) is not so significant thanks to the longer \(N\) \\ Case 3: & known content popularity and variable content size & Algorithm2 outperforms theoretical limit (2) thanks to the usage of density of popularity, LFU is not considered. Known popularity makes the effect of layer-1 queues appreciable \\ Case 4: & variable content size and popularity estimation with \(N\) = 2000 & Algorithm2 approaches theoretical limit (2), LFU is not considered. Estimated popularity makes hit ratio to decrease and less appreciable the effect of layer-1 queues \\ Case 4: & variable content size and popularity estimation with \(N\) = 10,000 & Algorithm2 nearly overlaps to theoretical limit (2) thanks to long \(N\), LFU is not considered. Long estimation window \(N\) makes less remarkable the performance penalty \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of performance evaluation results. obtained hit ratio values.The numerical analysis included also a comparison with the well-known LFU eviction strategy, adapted to the analyzed systems. Results show very good performance, under the assumption of either known or unknown content popularity. The future research objective consists of optimal caching for mobile users and variable connection with layer-2 caches. Conceptualization, G.R. and M.F. ; methodology, G.R. ; validation G.R. and M.F. ; writing--original draft preparation, G.R. ; writing--review and editing, M.F. ; funding acquisition, M.F. All authors have read and agreed to the published version of the manuscript. This work was performed in the framework of the EU projects 5G-EVE and 5G-CARMEN under grant agreements Nos. 815074 and 825012, respectively. The views expressed are those of the authors and do not necessarily represent the projects. The Commission is not responsible for any use that may be made of the information it contains. Not applicable. The authors declare no conflict of interest. ## Appendix A

A matroid is an algebraic organization of the elements of a finite set that generalizes the concept of linear independence from linear algebra. Formally, a matroid is a pair \(M_{1}(X,I_{1})\) where X is a finite _ground set_ of elements and \(I_{1}\) is a family of subsets, called _independent sets_, defined by using the elements of X. These subsets have the following properties. The empty set is an independent set. Hereditary property: A subset of an independent set is an independent set. Exchange property: If A and B are two independent sets and \(|\) A\(|\)\(>\)\(|\)B\(|\), then \(3x\)\(\in\)A\(\smallsetminus\)B \(:\)\(\triangleright\)\(\{x\}\)\(\in\)\(I_{i}\). For example, given a set of integers \(\beta_{i}\), \(M_{1}(X,I_{1}):I_{1}=\{I:I\subseteq X,I=\cup_{i}I_{i},|I_{i}|\leq\beta_{i} \forall i\}\), is a particular matroid called _uniform_ matroid. Essentially, for a uniform matroid a set \(I_{i}\) is independent if and only if it contains at most \(\beta_{i}\) elements. The intersection of a matroid with a uniform matroid defined on the same ground set generates a uniform matroid. We check the exchange property. Let \(M_{1}(X,I_{1})\) be a uniform matroid and \(M_{2}(X,I_{2})\) a generic matroid. Let the independents \(S_{1},S_{2}\in I_{1}\cap I_{2},|S_{1}|<|S_{2}|\); By definition of uniform matroid \(\exists\)\(\beta:|S_{2}|\leq\beta\). Since \(|M_{2}|\) is a matroid, \(3^{\circ}_{\xi}\in S_{2}\smallsetminus S_{1}:S_{1}\cup\{\xi\}\in I_{2}\). But \(|S_{1}\cup\{\xi\}|\leq S_{2}\leq\beta_{i}\).