  High energy efficiency and low latency have always been the significant goals
pursued by the designer of wireless networks. One efficient way to achieve
these goals is cross-layer scheduling based on the system states in different
layers, such as queuing state and channel state. However, most existing works
in cross-layer design focus on the scheduling based on the discrete channel
state. Little attention is paid to considering the continuous channel state
that is closer to the practical scenario. Therefore, in this paper, we study
the optimal cross-layer scheduling policy on data transmission in a single
communication link with continuous state-space of channel. The aim of
scheduling is to minimize the average power for data transmission when the
average delay is constrained. More specifically, the optimal cross-layer
scheduling problem was formulated as a variational problem. Based on the
variational problem, we show the optimality of the deterministic scheduling
policy through a constructive proof. The optimality of the deterministic policy
allows us to compress the searching space from the probabilistic policies to
the deterministic policies.
