## 2 An ODE-Based Model for OORS  
For the sake of clarity in presenting the model, we provide a brief overview of the OORS scheme, which was initially proposed in [7]. While we summarise the protocol here to facilitate understanding of the model development, the full details and pseudo-code are available in [7]. We recall that the goal of this study is not to propose a novel offloading scheme, but to show how ODE-based models can be applied to an offloading scheme that adopts time-limited caching and forwarding to minimise memory and network resource usage during the offloading process. Let us consider a set of \(N+1\) users (the terms user, node, and device are used as synonyms) moving in a specific geographical region. At random times, users request content \(c\) from a central data server (CDS). Each content \(c\) has a hard delivery delay constraint \(T_{c}\) (or _content timeout_). A user \(n\) that has requested a content item can retrieve it through an opportunistic D2D communication when encountering a user that has a copy of it in their local cache. On the other hand, \(n\) receives the requested content directly from the CDS through a long-range cellular communication when the content timeout expires. A simple caching policy is adopted. In particular, a copy of content \(c\) is removed from a node cache when a _sharing timeout_\(T_{s}\) expires. Figure 1 exemplifies the time evolution of the \begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Literature Work** & **Seed Selection** & **Parameter Selection** & **Cache** & **Evaluation** & **Mobility** \\ \hline This work & No & Utility maximisation problem & Unlimited & Simulation (custom and ODE-based model & Synthetic (vehicular) and real-world (pedestrian, IEEE INFOCOM’06 [25]) \\ [PERSON] et al. [4] & Yes & Reinforcement learning & Unlimited & Simulation (custom) and ODE-based model & Synthetic (HCMM [26]) \\ [PERSON] et al. [6] & Yes & Control loop design & Unlimited & Simulation (custom and SUMO [27]) & Real-world (vehicular) large-scale dataset from the city of Bologna, no longer available) \\ [PERSON] et al. [7] & No & Fixed & Unlimited & Simulation (ns-3 with the LENA module for LTE) & Synthetic (vehicular) \\ [PERSON] et al. [14] & Yes (scheme from literature) & Submodular function maximization problem & Limited & Simulation (ONE [28]) & Synthetic (random waypoint) and real-world (MAP-driven DITN model, pedestrian by Haggle Project [29] and taxi GPS trace of Shanghai, no longer available) \\ [PERSON] et al. [15] & Yes & Heuristic (node influence evaluation model) Heuristic (community-based source selection) & Unlimited & Simulation (custom) & Synthetic (vehicular and pedestrian) \\ [PERSON] et al. [17] & Yes & Greedy, heuristic, random prototype & Unlimited & Simulation (custom) & Real-world (pedestrian, NUS student trace [30]) \\ [PERSON] et al. [20] & No & Utility maximisation problem & Limited & Simulation (custom) & Real-world (pedestrian, CanuMobSim [32]) \\ [PERSON] et al. [21] & Yes & Greedy Algorithm Predictive model and submodular optimization problem & Limited & Simulation (ns-2) & Synthetic (pedestrian, Own in ONE [28]) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Taxonomy and comparison of mobile data offloading schemes. content dissemination process from the point of view of a content requester \(n\). To improve readability, Table 2 shows the main mathematical notations used throughout the paper.