This paper addresses the resource allocation problem in a multiuser Mobile-Edge Computation Offloading (MECO) system, focusing on time-division multiple access (TDMA) and orthogonal frequency-division multiple access (OFDMA). The primary goal is to minimize the weighted sum of mobile energy consumption while ensuring computation latency constraints. For TDMA, the optimal resource allocation is formulated as a convex optimization problem, and an offloading priority function is derived to determine the optimal policy structure. This policy is threshold-based, with users performing complete or minimum offloading based on their priority. For finite cloud capacity, a sub-optimal algorithm is proposed to reduce computational complexity. For OFDMA, a non-convex mixed-integer problem is formulated, and a low-complexity algorithm is proposed by transforming it into a TDMA problem. The average offloading priority function is defined, and the resource allocation is derived, showing close-to-optimal performance in simulations. The paper also discusses special cases and extensions, including the impact of finite cloud capacity and non-negligible computing time.This paper addresses the resource allocation problem in a multiuser Mobile-Edge Computation Offloading (MECO) system, focusing on time-division multiple access (TDMA) and orthogonal frequency-division multiple access (OFDMA). The primary goal is to minimize the weighted sum of mobile energy consumption while ensuring computation latency constraints. For TDMA, the optimal resource allocation is formulated as a convex optimization problem, and an offloading priority function is derived to determine the optimal policy structure. This policy is threshold-based, with users performing complete or minimum offloading based on their priority. For finite cloud capacity, a sub-optimal algorithm is proposed to reduce computational complexity. For OFDMA, a non-convex mixed-integer problem is formulated, and a low-complexity algorithm is proposed by transforming it into a TDMA problem. The average offloading priority function is defined, and the resource allocation is derived, showing close-to-optimal performance in simulations. The paper also discusses special cases and extensions, including the impact of finite cloud capacity and non-negligible computing time.