It is well known that the average transmit power can be traded off by average delay. This paper strives to study the relation between the maximal energy efficiency (EE) and the delay bound with a given violation probability for wireless systems serving random arrivals. We show that if the minimal average transmit and circuit powers consumed at a base station linearly increase with the required average service rate, i.e., the power-rate relation is linear, then a non-tradeoff region will appear in the EE-delay relation. By taking multi-input-multi-output system as an example, we show that the power-rate relation will be linear if transmit power and bandwidth are jointly allocated, and bandwidth constraint is inactive to support a required delay bound. The impacts of bandwidth constraint on the power-rate and EE-delay relations are then analyzed. To study fundamental EE-delay relation, aqueue length-dependent two-state policy is optimized. By further considering a compound Poisson arrival process in large number of transmit antennas asymptotics, we find the boundary of the tradeoff and non-tradeoff regions,and provide a lower bound of the Pareto optimal EE-delay relation in the tradeoff region, all with closed-form expressions. Our results show that the non-tradeoff region increases with the maximal bandwidth and the number of transmit antennas.