  In this paper, downlink delivery of popular content is optimized with the
assistance of wireless cache nodes. Specifically, the requests of one file is
modeled as a Poisson point process with finite lifetime, and two downlink
transmission modes are considered: (1) the base station multicasts file
segments to the requesting users and selected cache nodes; (2) the base station
proactively multicasts file segments to the selected cache nodes without
requests from users. Hence the cache nodes with decoded files can help to
offload the traffic upon the next file request via other air interfaces, e.g.
WiFi. Without proactive caching placement, we formulate the downlink traffic
offloading as a Markov decision process with random number of stages, and
propose a revised Bellman's equation to obtain the optimal control policy. In
order to address the prohibitively huge state space, we also introduce a
low-complexity sub-optimal solution based on linear approximation of the value
functions, where the gap between the approximated value functions and the real
ones is bounded analytically. The approximated value functions can be
calculated from analytical expressions given the spatial distribution of
requesting users. Moreover, we propose a learning-based algorithm to evaluate
the approximated value functions for unknown distribution of requesting users.
Finally, a proactive caching placement algorithm is introduced to exploit the
temporal diversity of shadowing effect. It is shown by simulation that the
proposed low-complexity algorithm based on approximated value functions can
significantly reduce the resource consumption at the base station, and the
proactive caching placement can further improve the performance.
