## 7 Conclusions  
For the exact hit ratio analysis of a cache of fixed size under independent (IRM) requests, the product form (1)-(2) provides the most general solution, which applies to FIFO, RANDOM and clock-based strategies, as well as to combinations of those methods. The product form solution can be computed by the scalable scheme of Equation (3) [30], but it needs an extended number representation, when there are many objects in the cache.  
The solution extends to multilevel/multisegment caches [20] with the same set of methods (FIFO, RANDOM, CpR) being applied on each level. However, the hit ratio results differ for those methods if data in a cache is varying in size. The analytic solution 7 for LRU caches even extends to data of different sizes in the form of Equation (13). However, the LRU solutions are not scalable for large caches.  
As an alternative, IRM hit ratio approximations have been proposed for LRU by [PERSON] [23] and [PERSON] et al. [24] as well as for FIFO by [PERSON] and [PERSON] [25]. Those approaches were proven to be asymptotically exact for large caches with data of unit size [26; 27; 28; 29]. Moreover, our quantitative analysis identifies the format of Equation (6) for request probabilities with maximum approximation error. The results of the quantitative evaluation lead to error bounds which are summarized as follows:  
* The maximum absolute deviations \(|\Delta h_{\text{Che}}|\) of [PERSON]'s approximation are decreasing with the cache size \(M\) from 8.25% for \(M=1\) down to less than 1% for \(M\geq 10\).
* The maximum absolute deviations \(|\Delta h_{\text{Fagin}}|\) of [PERSON]'s approximation are decreasing with the cache size \(M\) from 5.2% for \(M=2\) down to less than 1.3% for \(M\geq 10\).
* The maximum deviations of the FIFO approximation [25] are decreasing with the cache size \(M\) from 16.5% for \(M=1\) down to less than 3% for \(M\geq 10\).  
Extensions of those approximations for data of different sizes are provided [21; 28], which are still expected to yield asymptotic exact results for large caches. However, a fraction of unused cache space contributes to larger errors for small caches. Therefore, an exact analysis is recommended to avoid the largest approximation errors for small caches, as far as Markovian and other exact solutions are scalable.  
The presented framework of analysis and approximation results provides a basic tool set for a performance evaluation of content delivery in current CDNs and cloud architectures, as well as in ICN, CCN and NDN concepts towards a future Internet. The limited scope of exact solutions and insufficient accuracy of approximations for varying data sizes are major challenges for future research.  
**Author Contributions:** Formal analysis, G.H., K.N. and O.H.; software, F.H. and K.N.; methodology, G.H. and O.H.; writing\(-\)first draft, G.H.; writing\(-\)review and editing, all authors. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable, the study does not report any data.  
**Conflicts of Interest:** The authors declare no conflict of interest.