  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 cooperative node. The
mobile device has sequential tasks to complete, whereas the cooperative node
assists the mobile device on both task offloading and task computation. In
specific, two cases are investigated, which are 1) the cooperative node has no
tasks to complete itself, and 2) the cooperative node has tasks to complete
itself. Our target is to minimize the total energy consumption of the mobile
device and the cooperative node 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 cooperative node, 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 Lagrangian dual method. In the upper level problem,
traversal method is applied to find the optimal offloading index.
