This paper explores strategies for achieving higher long-term energy efficiency while preserving better network stability in a wireless-powered mobile edge computing environment. Using the Lyapunov drift-plus-penalty method, we transform the long-term stochastic EE maximization problem into deterministic subproblems at each time slot, thus simplifying the overall problem solving process. In each slot’s convex subproblem, we employ the Lagrange multiplier and interior-point methods to jointly optimize CPU frequency, as well as data and energy transmission power, achieving globally optimal resource allocation. A pivotal factor in the Lyapunov drift-plus-penalty method is the parameter V, which traditionally implies a O(1/V) trend for energy efficiency and a O(V) trend for queue length. Contrary to this conventional view, we observe that queue length may initially decrease and then increase as V grows. This non-monotonic behavior implies an interval in which increasing V can simultaneously yield better energy efficiency and improved stability. To exploit this insight, we propose an algorithm that quantifies system performance under varying V values. Specifically, we introduce the achievement scalarizing function (ASF) to systematically evaluate how V impacts the overall system. This provides a theoretical and methodological foundation for choosing the optimal V, filling a gap in the existing literature regarding practical tuning strategies. Experimental results show that the ASF-based method simultaneously accounts for multiple goals, steering the selection toward a solution with an optimal balance.