## 2 An ODE-Based Model for OORS
### Modelling Transition Rates  
By definition, the transition rate \(\mu(t)\) is computed as \(1/E[W_{c}(t)]\), where \(W_{c}(t)\) is a stochastic continuous variable modelling the per-user delay of receiving the requested content at time \(t\) (as explained later, we can omit the dependency of \(W_{c}(t)\) on the time \(t\) because it will be accounted for in the function \(\mathcal{I}(t)\) used to model \(W_{c}\)). It is straightforward to note that \(W_{c}\in[0,T_{c}]\). In our system, the exact probability distribution of \(W_{c}\) is impossible to derive because it depends not only on the distribution of contact events between users, but also on the time constraints of the offloading process, namely, the sharing and content timeouts. Without loss of generality, we assume that each pair of users has the same average contact rate, which is fixed over time and equal to \(1/\theta\), where \(\theta\) is the average inter-contact time (note that we do not require the inter-contact time to be exponentially distributed). We highlight that, unlike [9], we do not use a pairwise inter-contact time but the inter-any-contact time, which is defined as the time elapsed between two successive contacts of a node with any other node in the network [33]. For the sake of brevity, hereafter we simply refer to inter-contact time instead of inter-any-contact time. Then, we apply a _discretisation method_ to approximate \(W_{c}\). Specifically, we _approximate_\(W_{c}\) using a discrete random variable \(\widehat{W}_{c}\) taking values in the set \(\{k\theta|k=0,1,\dots,k_{c}\}\cup\{T_{c}\}\), where \(k_{c}=\lfloor T_{c}/\theta\rfloor\). The rationale behind this approximation is the simplifying assumption that nodes _only_ meet at multiples of the inter-contact time \(\theta\). In other words, we replace the stochastic mobility pattern with a deterministic one of equal rate. To clarify the practical implications of this discretisation technique used for modelling the content latency, Figure 3 shows an illustrative example of a probability density function of the discretised content latency. As depicted in the figure, this probability density function collapses into a probability mass function that gives us the probabilities of the possible \(k_{c}+2\) values of \(\widehat{W}_{c}\). The following lemma derives the probability mass function of \(\widehat{W}_{c}\), under the assumption that \(\mathcal{I}(t)\) is a slowly varying function and can be considered constant in the time interval between the content request and the content delivery (i.e., \(W_{c}\)). In other words, in the following lemma, \(\mathcal{I}(t)\) is considered as a constant independent of \(k\). **Lemma 1**. : _Under the above assumptions of deterministic inter-contact times and \(\mathcal{I}(t)\approx\mathcal{I}\), it holds that_  
\[p_{W}(k,\mathcal{I}) =\Pr\{\widehat{W}_{c}=k\theta|\mathcal{I}\} \tag{3}\] \[= \begin{cases}1-\left(1-\frac{\mathcal{I}}{N}\right)^{\omega_{0 }}&k=0\\ \frac{\mathcal{I}}{N}\left(1-\frac{\mathcal{I}}{N}\right)^{\omega_{0}+k-1}&1 \leq k\leq k_{c}\end{cases} \]  
Figure 3: Illustrative example of the probability density function for the discretised content latency. The probability values in the figure are obtained for \(N=20\), \(\mathcal{I}=4\), \(\omega_{0}=1.5\), \(\theta=1\) s, and \(T_{c}=7.5\) s.  
\[p_{W}(k_{c}+1,\mathcal{I})=\Pr\{\widehat{W}_{c}=T_{c}\}=\left(1-\frac{\mathcal{I}} {N}\right)^{\omega_{0}+k_{c}}, \tag{4}\]  
_where \(\omega_{0}\) is the number of neighbouring nodes of the content requester at the time of the content request. Note that in (3) and (4), \(\mathcal{I}\) represents an unknown quantity._  
Proof. : Under deterministic inter-contact times, a node retrieves content from another node it encounters according to a _Bernoulli_ process. Furthermore, \(\mathcal{I}/N\) gives the average probability that a content requester meets a node that has a stored copy of that content.