  In this paper, we study a mobile edge computing (MEC) system in which the
mobile device is assisted by a base station (BS) and a relay node. The mobile
device has sequential tasks to complete, whereas the relay assists the mobile
device on both task offloading and task computation. In specific, two cases are
investigated, which are 1) the relay has no tasks to complete itself, and 2)
the relay has tasks to complete itself. Our target is to minimize the total
energy consumption of the mobile device and the relay through optimizing the
transmit duration in task offloading, CPU frequency in task computing along
with the task index to offload in the sequential tasks. In the first case, we
decompose the mixed-integer non-convex problem into two levels. In the lower
level problem, thanks to the convexity, Karush-Kuhn-Tucker (KKT) conditions are
utilized to simplify the problem, which is then solved with bisection search.
In the upper level problem, to find solution of the task index to offload,
rather than utilizing traversal method, we develop a monotonic condition to
simplify the searching process. In the second case, in order to guarantee the
successful computation of the mobile device and relay, the uploading
transmission is classified into three schemes. Within each scheme, the
non-convex problem is decomposed. In the lower level problem, semi-closed
solution is found by KKT conditions. In the upper level problem, traversal
method is applied to find the optimal offloading index.
