  This paper introduces an analytical framework to investigate optimal design
choices for the placement of virtual controllers along the cloud-to-things
continuum. The main application scenarios include low-latency cyber-physical
systems in which real-time control actions are required in response to the
changes in states of an IoT node. In such cases, deploying controller software
on a cloud server is often not tolerable due to delay from the network edge to
the cloud. Hence, it is desirable to trade reliability with latency by moving
controller logic closer to the network edge. Modeling the IoT node as a
dynamical system that evolves linearly in time with quadratic penalty for state
deviations, recursive expressions for the optimum control policy and the
resulting minimum cost value are obtained by taking virtual fog controller
reliability and response time latency into account. Our results indicate that
latency is more critical than reliability in provisoning virtualized control
services over fog endpoints, as it determines the swiftness of the fog control
system as well as the timeliness of state measurements. Based on a realistic
drone trajectory tracking model, an extensive simulation study is also
performed to illustrate the influence of reliability and latency on the control
of autonomous vehicles over fog.
