# Dynamic and Energy Efficient Cache Scheduling Framework for IoMT over ICN
## 3 Materials and Methods
### System Model  
The network system model comprises of N routers, i.e., set \(r_{n}\) is the number of each router, where \(n=1,2,3,\ldots,N\) as shown in Figure 1. The \(r\) router is the nearest to the content consumer, while router \(r_{N}\) is the nearest to the content generator. The term \(c\) is used to represent the size of the cache on the router \(r_{k}\). The Poisson distribution is used to follow the average number of requests of content service with the request rate 1 for class \(k\) using the Zipf-Mandelbrot distribution with type factor \(\lambda_{k}\) for class k. Network contents are partitioned into three different classes, i.e., class \(a\), \(b\), and \(c\), based on characteristics as discussed in [39]. The content classification depends on the jitter, delay, content utilization, and the pattern of request rate. Class \(a\) requires low jitter and delay integrated with repeatedly requested contents. Class \(b\) is user-generated content with less jitter and a medium required delay. This class includes the content that has been generated and distributed by volunteer subscribers, like sites, web-based social media content, etc. Class \(c\) comprises video-on-demand content which is not the target of this research study. This research follows Poisson distribution to make several requests for content service classes. Due to content popularity, the Zipf-Mandelbrot distribution is used for request modeling, which comprises a flattened factor [40; 41; 42]. The content on top rankings has more chance of being requested as compared with lower rankings content.  
Figure 1: Workflow of popular content over the network. (**a**) Has no caching content (**b**) Both routers have the same content but the content of \(R1\) is older than the content of \(R2\). (**c**) After some time, the content of \(R2\) expires. (**d**) Client needs less popular content, which is cached at \(R2\). The age of the content is determined by the distance from the server or source node.  
The cache receives request rate \(\beta\), which is contingent upon the condition \(\beta\geq 0\), denoting the request intensity. Let \(P(i)\) be the probability of content requests; \(N\) is the total number of requests, \(q\) is the vector, and \(\alpha\) is the exponential factor for content popularity. In this context, we can express the probability of accessing content \(i\) using Equation (1).  
\[P(i)=\frac{\left(i+q\right)^{-\alpha}}{\sum_{l=1}^{N}\left(i+q\right)^{-\alpha}} \tag{1}\]  
Each content may receive updates to replace the previous data item. It is assumed that the content \(n\) receives updates as per the Poisson distribution request model bounded with the refresh rate of \(\lambda_{n}\geq 0\). As all the content of the data is subject to updates, then the same content item that resides in the local cache can be considered as an older version of the content. To measure the freshness of the content, the age \(\Delta_{n}\) with respect to time t can be bounded as \(\Delta_{n}(t)\in\{0,1,2,\ldots n\}\), where \(n\) represents the number of updates. The cost of the fetching content is subject to the condition, i.e., \(c_{f}\geq 0\), in case of operations. For performance, the cost metric can be integrated with time as \(c_{a}=\Delta_{n}(t)\) subject to \(c_{a}\geq 0\), and linear growth rate.