## 2 An ODE-Based Model for OORS
For the sake of model development, we introduce the following notation: (i) \(\mathcal{M}(t)\) (we omit the time dependency \(t\) when there is no ambiguity), defined as the set of users that, at time \(t\), have not yet requested content \(c\); (ii) \(\mathcal{S}(t)\), defined as the set of nodes that have requested content \(c\) but have not yet received it; (iii) \(\mathcal{I}(t)\), defined as the set of nodes that have already received content \(c\) and are storing a copy of it in their local cache for sharing; and (iv) \(\mathcal{R}(t)\), defined as the set of nodes that have received content \(c\) and have discarded the cached copy because the sharing timeout \(T_{s}\) has exp  
Figure 1: Time evolution of the content dissemination process from the point of view of a content requester \(n\). (**a**) Content received before \(T_{c}\) expires; (**b**) content received from the CDS. \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Definition** \\ \hline \(N\) & Number of nodes \\ \(T_{c}\) & Content timeout \\ \(T_{s}\) & Sharing timeout \\ \(\mathcal{M}(t)\) & Set of nodes that, at time \(t\), have not yet requested content \(c\) \\ \(\mathcal{S}(t)\) & Set of nodes that, at time \(t\), have requested but not yet received content \(c\) \\ \(\mathcal{I}(t)\) & Set of nodes that, at time \(t\), store a copy of content \(c\) \\ \(\mathcal{R}(t)\) & Set of nodes that, at time \(t\), have received the requested content but no longer \\  & store a copy of it \\ \(\lambda(t)\) & Global generation rate of content requests at time \(t\) \\ \(\mu(t)\) & Normalised rate, at time \(t\), at which users in \(\mathcal{S}\) receive content \\ \(p_{o}(t)\) & Probability of receiving the requested content via D2D communications at time \(t\) \\ \(p_{c}(t)\) & Probability of receiving the requested content directly from the CDS at time \(t\) \\ \(\Omega_{o}(t)\) & Total number of content items delivered via opportunistic contacts up to time \(t\) \\ \(\Omega_{c}(t)\) & Total number of content items delivered using cellular network up to time \(t\) \\ \(\gamma(t)\) & Per-user transition rate from \(\mathcal{I}\) to \(\mathcal{R}\) \\ \(\theta\) & Average inter-contact time between a node and any other node \\ \(W_{c}(t)\) & Per-user content latency at time \(t\) \\ \(\epsilon(t)\) & Offloading efficiency at the end of time interval \([0,t]\) \\ \(l_{lx}\) & Fixed transmission range of the nodes \\ \(\rho\) & Node density as observed by the content requester \\ \(V_{i}\) & Speed of node \(n_{i}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: List of notation. that \(\mathcal{M}(t)+\mathcal{S}(t)+\mathcal{I}(t)+\mathcal{R}(t)=N+1\). We highlight that we use the same notation to indicate both a set and its cardinality when there is no ambiguity. Let \(\lambda(t)\) be the global generation rate of content requests, i.e., the transition rate of users from set \(\mathcal{M}\) to set \(\mathcal{S}\), as depicted in Figure 2, representing a Markov chain model of content dissemination. Let us denote by \(\mu(t)\) the normalised rate at which users in set \(\mathcal{S}\) receive content, i.e., the per-user transition rate to set \(\mathcal{I}\). As shown in Figure 1, two categories of events can trigger this state transition, namely, an opportunistic encounter with a node \(m\in\mathcal{I}\), or a direct data transfer from the cellular network. Formally, let us denote by \(p_{o}(t)\) and \(p_{c}(t)=[1-p_{o}(t)]\) the probabilities of these two events, respectively. Then, it holds that the total number \(\Omega_{o}(t)\) of content items delivered through opportunistic contacts in the time interval \([0,t]\) is given by  
\[\Omega_{o}(t)=\int_{0}^{t}p_{o}(\tau)\mu(\tau)\mathcal{S}(\tau)d\tau\;. \tag{1}\]  
Following the same line of reasoning as in (1), we derive the total number \(\Omega_{c}(t)\) of content items delivered via the cellular network in the time interval \([0,t]\).