  We consider cross-layer design of delay optimal transmission strategies for
energy harvesting transmitters where the data and energy arrival processes are
stochastic. Using Markov decision theory, we show that the value function is
weakly increasing in the queue state and weakly decreasing in the battery
state. It is natural to expect that the delay optimal policy should be weakly
increasing in the queue and battery states. We show via counterexamples that
this is not the case. In fact, we show that for some sample scenarios the delay
optimal policy may perform 5-13% better than the best monotone policy.
