# Energy-Optimized Content Refreshing of Age-of-Information-Aware Edge Caches in IoT Systems
## 3 System Overview and Model
### Model of the Cache-Management Scheme  
Here, for the sake of completeness, we briefly report the model of the cache management scheme we proposed in [10], which is the basis for the optimization method proposed in this work.  
We model the cache-management scheme as a \(2w\)-states Discrete-Time Markov Chain (DTMC) \(\left\{X_{k}\right\}_{k\in\mathbb{N}}\), where transitions occur at time instants \(t_{k}\). We assume that the generation of server requests follows a Poisson distribution with an aggregate rate \(\lambda\). We also assume that the cache parameter \(W\) is a multiple of the device sampling period, i.e., \(W=w\)s with \(w\geq 1\). The discrete-time Markov chain is a simple yet effective model that allows us to derive in closed forms the following system KPIs: the average AoI of the items and their distribution at the steady-state, and the number of transmissions per unit of time needed to update the cache. This model makes only a few assumptions on the underlying system, i.e., we only assume that the edge-based proxy experiences small delays as compared to AoI requirements, and that the link between the proxy and the server is almost deterministic. Finally, the probability distribution function of the AoI can be used for the model-driven optimization of the cache.  
**States:** We denote \(X_{k}\) as the state of the DTMC at time \(t_{k}\). \(X_{k}\) is defined by two components: (i) the first specifies if in the previous interval \((t_{k-1},t_{k})\) at least one request has arrived; (ii) the second specifies the number of remaining intervals during which the cached item is still considered fresh. Hence, the states of the DTMC are the following (see Figure 2):  
1. \(X_{k}=(0,w)\): no request arrived during the interval \((t_{k-1},t_{k})\) and there is not a fresh item in the cache. So, if a request arrives in the interval \((t_{k},t_{k+1})\), the proxy fetches the latest update from the device, which will be cached and will expire in \(w\) intervals.
2. \(X_{k}=(0,w-j)\), \(1<j\leq w\): no request arrived in the interval \((t_{k-1},t_{k})\), and the cached item will expire in \(w-j\) intervals.
3. \(X_{k}=(1^{+},w-j)\), \(1\leq j\leq w\): at least one request arrived during the interval \((t_{k-1},t_{k})\), and the cached item will expire in \(w-j\) intervals.  
Figure 2: Discrete Time Markov Chain.  
**Transition probabilities:** The model needs only to keep track if any request has arrived or not during the current interval, so the transition probabilities are the probability of having no requests in a period of length \(s\), called \(p_{0}\), and the probability of having at least one request in a period of length \(s\), called \(p_{1^{+}}\), as shown in Figure 2. Since the generation of server requests follows a Poisson distribution, it is \(p_{0}=e^{-\lambda s}\) and \(p_{1^{+}}=1-e^{-\lambda s}\).  
The DTMC is irreducible and positive recurrent, so it is possible to compute the steady-state probabilities, denoted as \(\pi\)[10]. Based on this system model we can derive in closed forms the network cost, the average AoI of the items, and the probability distribution function of the AoI at the steady-state.