Vehicular fog computing is attractive for sharing computing resources and data for safety and infortainment of self-driving cars. Recently, the V2X communication technology using mm-Wave frequency spectrum accelerates such future mobile computing with large bandwidth and beam-forming using a directional antenna. Although the beam-forming technique requires a complicate procedure for beam alignment, it can reduce mutual interference by spatial diversity. From the beam-forming scheduling, the vehicular fog can improve network performance, which is limited by data locations. Beams toward a vehicle for the same content should be scheduled in the time domain. Instead, we propose to replicate the content to multiple vehicles nearby to diversify beam directions. However, it is a challenge for vehicles to cache the content because the content caching costs not only limited local storage, but data transmission for other vehicles. For this, we adopt evolutionary game theory in which vehicles learn an evolutionarily stable strategy (ESS) from repeated games and maximize social utility. In this paper, we contribute to modeling a road segmentation for the mm-Wave V2X communication in order to derive connectivity probability with distributed content caches for the vehicular fog, and centralized and distributed algorithms for the evolutionary content cache game. From experiments, we confirm that content cache can improve V2X connectivity and the proposed evolution algorithm leads vehicles to choose the ESS for the content cache in the vehicular fog.
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