## 1 Introduction
### Contributions of the Paper  
Motivated by the observations outlined in the _Introduction_ and the analysis of prior work discussed in the _Related Work_ section, this paper addresses the challenge of modelling delayed D2D-based offloading under time constraints. Specifically, we focus on how _hard delivery delay constraints_ and _time-constrained caching memory_ impact the performance of such systems. The main contributions of this study can be summarised as follows:  
* We leverage the proposed ODE model to derive explicit expressions for functions that capture the temporal evolution of the system state. Two main novelties are associated with our model. First, it enables the analysis of constant delivery delays and time-limited caching with _constant and asynchronous timeouts_. Second, we rely on a _simple discretisation technique to model content latency_, making our model more practical for real-world applications. Additionally, we analyse the _stationary regime_ of the proposed ODE model and derive a necessary condition for its existence.
* We formulate a _utility-aware maximisation problem to optimally tune the parameters of the content dissemination process_. Furthermore, we leverage the properties of the stationary regime to derive a closed-form solution for this optimisation problem.
* We demonstrate the ability of our approximated ODE-based framework to capture the complexity of the offloading process by deriving the _closed-form expressions_ of the protocol parameters as a function of node density and content request speed for an illustrative use case: a one-dimensional linear vehicular scenario.
* We conduct an extensive simulation study to validate the model's accuracy using both a _synthetic_ mobility trace and IEEE INFOCOM'06 [25], a _real-world_ mobility dataset. We analyse aggregate performance metrics (e.g., content latency and offloading efficiency) alongside the time evolution of the system state. Additionally, we provide empirical evidence supporting the existence of the stationary regime.
* _We compare OORS--optimally tuned with respect to protocol parameters--with two state-of-the-art data offloading schemes_ that ensure guaranteed delays but rely on unlimited caching, namely, Push-and-track (PAT) [6] and SNSNI, a seed node selection algorithm based on node influence [15]. Our results show that OORS achieves a similar offloading efficiency to both benchmarks while reducing the number of stored content copies by at least 50%. Additionally, selecting an optimal sharing timeout results in offloading performance that is nearly unaffected by the rate of content requests.  
Finally, Table 1 shows a qualitative comparison between our study and the related works that we have discussed in Section 1.1.1. In particular, we use the following categories as classification criteria: \((1)\) the use of an initial subset of nodes as seeds; \((2)\) the approach for selecting the parameters of the offloading algorithm; \((3)\) the size of the node cache; \((4)\) the evaluation methodology; and \((5)\) the type of mobility dataset used for the evaluation. As shown in the table, our work is the only one that provides a mathematical model for the analysis of the offloading efficiency that accounts for both content delivery guarantees and time limitations for storing content copies in local device caches.  
The remainder of the paper is organised as follows. Section 2 describes our ODE model and applies it to a one-dimensional vehicular scenario for illustrative purposes.  
In Section 3, simulation results confirm the model predictions. Section 4 compares the performance of OORS, optimally tuned using the solution of the optimisation problem formulated in Section 2, with PAT and SNSNI. Finally, Section 5 concludes the paper by discussing future work.