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Drive-thru Internet systems are multiple-access wireless networks in which users in moving vehicles can connect to a roadside access point (AP) to obtain Internet connectivity for some period of time as the vehicles pass through the AP's coverage range. In order to evaluate the type of communication services and the quality-of-service that these systems can provide, in this paper, we investigate the data communication performance of a vehicle in Drive-thru Internet systems. In particular, we derive analytical models with tractable solutions to characterize the average and the distribution of the number of bytes downloaded by a vehicle by the end of its sojourn through an AP's coverage range, in the presence of other vehicles contending for the same AP's resources. Our models are able to quantify the impact of road traffic density, vehicle speed, service penetration rate, AP's transmission range and the corresponding bit rate, on the amount of data downloaded by an individual vehicle. In terms of analysis technique, we map the study of our vehicular data downloading process into the transient analysis of a series of Markov reward processes. Our use of Markov reward model is novel in the sense that we only select from the corresponding Markov chain, a subset of relevant sample paths that matches the required behavior of our vehicular flow model. We also validate our proposed analytical models through extensive simulations, driven by empirical vehicular traffic traces. We believe our work offers a unique analytical framework based on which the interplay between vehicular traffic parameters and a vehicle's data communication performance in a Drive-thru Internet system can be studied and optimized in a systematic, quantitative manner.