  With the proliferation of latency-critical applications, fog-radio network
(FRAN) has been envisioned as a paradigm shift enabling distributed deployment
of cloud-clone facilities at the network edge. In this paper, we consider
proactive caching for a one-user one-access point (AP) fog computing system
over a finite time horizon, in which consecutive tasks of the same type of
application are temporarily correlated. Under the assumption of predicable
length of the task-input bits, we formulate a long-term weighted-sum energy
minimization problem with three-slot correlation to jointly optimize
computation offloading policies and caching decisions subject to stringent
per-slot deadline constraints. The formulated problem is hard to solve due to
the mixed-integer non-convexity. To tackle this challenge, first, we assume
that task-related information are perfectly known {\em a priori}, and provide
offline solution leveraging the technique of semi-definite relaxation (SDR),
thereby serving as theoretical upper bound. Next, based on the offline
solution, we propose a sliding-window based online algorithm under arbitrarily
distributed prediction error. Finally, the advantage of computation caching as
well the proposed algorithm is verified by numerical examples by comparison
with several benchmarks.
