  Mobile edge computing and fog computing are promising techniques providing
computation service closer to users to achieve lower latency. In this work, we
study the optimal offloading strategy in the three-tier federated computation
offloading system. We first present queueing models and closed-form solutions
for computing the service delay distribution and the probability of the delay
of a task exceeding a given threshold. We then propose an optimal offloading
probability algorithm based on the sub-gradient method. Our numerical results
show that our simulation results match very well with that of our closed-form
solutions, and our sub-gradient-based search algorithm can find the optimal
offloading probabilities. Specifically, for the given system parameters, our
algorithm yields the optimal QoS violating probability of 0.188 with offloading
probabilities of 0.675 and 0.37 from Fog to edge and from edge to cloud,
respectively.
