## 2 An ODE-Based Model for OORS
### Optimal Selection of the \(T_{s}\) Parameter  
In this subsection, we present a utility-aware technique to determine the optimal value of the \(T_{s}\) parameter, leveraging the analysis of the stationary regime. First, we model the total utility \(\mathbf{U}(T_{s})\) of the content dissemination process within the stationary regime given the \(T_{s}\) parameter. In the following, we assume that \(\mathbf{U}(T_{s})\) increases if either the content latency decreases or the probability of receiving content from D2D communications increases, as this saves resources of the cellular network. Conversely, \(\mathbf{U}(T_{s})\) decreases if the number of content copies in the nodes' caches increases, as the content dissemination process consumes nodes' resources. More formally, let \(U^{O}(T_{s})\) denote the utility of receiving content in the stationary regime (i.e., when \(t\in[t_{a},t_{b}]\)) from D2D communications, which can be expressed as  
\[U^{O}(T_{s})=p_{o}(t)=1-(1-\epsilon_{s})^{\omega_{0}+\frac{T_{c}}{\theta}}\, \tag{11}\]  
and let the utility \(U^{D}(T_{s})\) represent the reduction in content latency, which is given by  
\[U^{D}(T_{s})=1-\frac{\overline{W}(t)}{T_{c}}=1-\frac{\theta(1-\epsilon_{s})^{ \omega_{0}}[1-(1-\epsilon_{s})^{\frac{T_{c}}{\theta}}]}{\epsilon_{s}T_{c}}. \tag{12}\]  
Finally, let \(U^{C}(T_{s})\) denote the normalised cost of using nodes' cache memory, which is given by  
\[U^{C}(T_{s})=1-\epsilon_{s}. \tag{13}\]  
Expression (13) yields a low utility value if many content copies are stored in the nodes' caches. Finally, we can express the total utility as the weighted sum of the individual utilities, given by  
\[\mathbf{U}(T_{s})=\alpha_{1}U^{O}(T_{s})+\alpha_{2}U^{D}(T_{s})+\alpha_{3}U^{ C}(T_{s})\, \tag{14}\]  
with \(\alpha_{1}+\alpha_{2}+\alpha_{3}=1\). Then, the utility maximisation problem for the content dissemination process can be formulated as  
\[\mathbf{P}\ : \max_{T_{s}} \mathbf{U}(T_{s})\] (15) s.t. \[0<T_{s}\leq\frac{N+1}{\lambda_{0}}. \tag{16}\]  
The optimisation problem \(\mathbf{P}\) is a nonlinear programming problem (NLP) with a single decision variable that can be solved using simple numerical optimisation techniques. However, in some special cases, it may be possible to find the optimal solution analytically by computing the derivative of the objective function. For instance, when \(\alpha_{2}=0\), the optimisation problem \(\mathbf{P}\) simplifies to finding the maximum of the function \(\mathbf{U}(T_{s})=\alpha_{1}[1-(1-\epsilon_{s})^{\overline{k}_{T}}]+\alpha_{3 }(1-\epsilon_{s})\), with \(\overline{k}_{T}=\omega_{0}+T_{c}/\theta\). Setting the derivative of \(\mathbf{U}(T_{s})\) equal to zero, after some algebraic manipulations, we obtain that the optimal \(T_{s}\) value, denoted by \(T_{s}^{*}\), is given by  
\[T_{s}^{*}=\frac{N+1}{\lambda_{0}}\left[1-\left(\frac{\alpha_{3}}{\alpha_{1}} \frac{1}{\overline{k}_{T}}\right)^{\frac{1}{T_{s}-1}}\right]. \tag{17}\]  
To satisfy the constraint in (16), the parameters in (17) must meet the following conditions: \(k_{T}>1\), \(1/(k_{T}+1)<\alpha_{1}<1\), and \(0<\alpha_{3}<1/(1+\frac{1}{k_{T}})\). It is worth noting that, in practical scenarios, \(k_{T}\) is typically much greater than one, allowing for a wide range of choices for weight values. We conclude this section by discussing practical considerations on how configuration rule (17) can be implemented. First, we highlight that the Access and Mobility Management Function, or AMF, is a control plane function in the 5G core network that handles connection and management mobility tasks. Thus, the CDS can leverage AMF capabilities to know the number \(N\) of mobile devices located in a specific region and are subscribed to a content topic.