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This paper investigates the problem of revealing the tradeoff between energy efficiency (EE) and delay in device-to-device (D2D) communications underlaying cellular networks. Considering both stochastic traffic arrivals and time-varying channel conditions, we formulate it as a stochastic optimization problem, which optimizes EE subject to the average power, interference-control, and network stability constraints. With the help of fractional programming and the Lyapunov optimization technique, we develop an algorithm, referred to as the TRADEOFF, to solve the problem. To deal with the nonconvex and NP-hard power allocation subproblem in the TRADEOFF, we adopt the prismatic branch and bound algorithm to find its globally optimal solution, where only a linear programming needs to be solved in each iteration. Thus, the TRADEOFF serves as an important benchmark to evaluate performance of other heuristic …