In Vehicular Fog Computing (VFC), the RSU-to-Vehicle (R2V) task offloading process is highly affected by undesirable yet sometimes inevitable events ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g.</i> , buffer exhaustion, task HoL blocking and deadline expiry), the occurrence of which will notably alter the RSU's performance in terms of crucial Quality-of-Service (QoS) metrics such as the average system response time, the blocking and deadline expiry probabilities. This paper proposes a novel R2V Deadline-Constrained Task Offload (R2V-DCTO) scheduling scheme with the objective of improving the RSU's performance in terms of the above-mentioned metrics; hence, filling an important literature gap. For this purpose, a stochastic modelling framework is established to capture the RSU's functional dynamics and assess its performance as it operates under R2V-DCTO. Extensive simulations are conducted to confirm the model's validity and accuracy and then compare R2V-DCTO's performance to that of the often adopted FIFO scheme. Results indicate that, on average, R2V-DCTO outperforms FIFO by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$33.4\%$</tex-math></inline-formula> in terms of the probability of deadline mismatch and by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$83.3\%$</tex-math></inline-formula> in terms of mean system response time; those being critical QoS performance metrics.