One of the most appealing challenges, especially in mobile robotics, is the real-time inferencing of the vision data offloaded from a mobile robot for its enhanced navigation performance. Leveraging the benefits of an on-premise and an integrated fog-cloud computing platform for processing vision data is a viable solution. This paper addresses the mentioned challenge via a data-driven control approach for the development of an autonomic fog-cloud computing platform suitable for processing the offloaded vision data within a prescribed time bound called the service time. The approach comprises developing a data-driven linear parameter varying (LPV) framework for modeling and design of a model predictive controller (MPC) for the fog and the cloud platforms independently. A heuristic algorithm performs the distribution of the mobile robot vision data (MRVD) between the fog and the cloud platforms. The LPV model for the fog and the cloud platforms are developed by characterizing the MRVD as the variable frame rate and resolution of the objects (under active consideration in a given frame) acquired during navigation. We validate the developed theory for a mobile robot while navigating in the application environment such as the warehouse. The experimental results presented for object detection under service time bounds show the efficacy of the proposed approach.