Recently, in order to extend the computation capability of smart mobile devices (SMDs) and reduce the task execution delay, mobile edge computing (MEC) has attracted considerable attention. In this paper, a stochastic optimization problem is formulated to maximize the system utility and ensure the queue stability, which subjects to the power, subcarrier, SMDs, and MEC server computation resource constraints by jointly optimizing congestion control and resource allocation. With the help of the Lyapunov optimization method, the primal problem is transformed into five subproblems including the system utility maximization subproblem, SMD congestion control subproblem, SMD computation resource allocation subproblem, joint power and subcarrier allocation subproblem, and MEC server scheduling subproblem. Since the first three subproblems are all single variable problems, the solutions can be obtained directly. The joint power and subcarrier allocation subproblem can be efficiently solved by utilizing alternating and time‐sharing methods. For the MEC server scheduling subproblem, an efficient algorithm is proposed to solve it. By solving the five subproblems at each slot, we propose a delay‐aware task congestion control and resource allocation (DTCCRA) algorithm to solve the primal problem. Theoretical analysis shows that the proposed DTCCRA algorithm can achieve the system utility and execution delay trade‐off. Compared with the intelligent heuristic (IH) algorithm, when the control parameter V increases from 10⁶ to 10⁷, the total backlogs are decreased by 5.03% and the system utility is increased by 3.9% on average for the extensive performance by using the proposed DTCCRA algorithm.