Abstract: In this paper, packet scheduling with average transmit power constraint over Gaussian channels is considered with the objective to minimize the maximum delay constraint. It is proved that the optimal average transmit power is decreased with the maximum delay constraint for non-time-varying schedulers with maximum delay constraint. Then the equivalence between the optimal scheduler for average transmit power and for maximum delay constraint is proved according to the decreasing property. Then, the form of non-time-varying optimal scheduler based on this equivalence is given. The schedulers parameter, which also reflects the performance, is determined under some certain arrival process, with an example of Poisson arrival. A practical design for this scheduler is presented and simulations show that this design can realize the optimal scheduler.