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In this paper, we study random access in a drive-thru scenario, where roadside access points (APs) are installed on a highway to provide temporary Internet access for vehicles. We consider vehicle-to-roadside (V2R) communications for a vehicle that aims to upload a file when it is within the APs' coverage ranges, where both the channel contention level and transmission data rate vary over time. The vehicle will pay a fixed amount each time it tries to access the APs, and will incur a penalty if it cannot finish the file uploading when leaving the APs. First, we consider the problem of finding the optimal transmission policy with a single AP and random vehicular traffic arrivals. We formulate it as a finite-horizon sequential decision problem, solve it using dynamic programming (DP), and design a general dynamic optimal random access (DORA) algorithm. We derive the conditions under which the optimal transmission policy has a threshold structure, and propose a monotone DORA algorithm with a lower computational complexity for this special case. Next, we consider the problem of finding the optimal transmission policy with multiple APs and deterministic vehicular traffic arrivals thanks to perfect traffic estimation. We again obtain the optimal transmission policy using DP and propose a joint DORA algorithm. Simulation results based on a realistic vehicular traffic model show that our proposed algorithms achieve the minimal total cost and the highest upload ratio as compared with some other heuristic schemes. In particular, we show that the joint DORA scheme achieves an upload ratio 130% and 207% better than the heuristic schemes at low and high traffic densities, respectively.