On the tradeoff of average delay and average power for fading point-to-point links with monotone policies

We consider a fading point-to-point link with packets arriving randomly at rate $\lambda$ per slot to the transmitter queue. We assume that the transmitter can control the number of packets served in a slot by varying the transmit power for the slot. We restrict to transmitter scheduling policies that are monotone and stationary, i.e., the number of packets served is a non-decreasing function of the queue length at the beginning of the slot for every slot fade state. For such policies, we obtain asymptotic lower bounds for the minimum average delay of the packets, when average transmitter power is a small positive quantity $V$ more than the minimum average power required for transmitter queue stability. We show that the minimum average delay grows either to a finite value or as $\Omega\brap{\log(1/V)}$ or $\Omega\brap{1/V}$ when $V \downarrow 0$, for certain sets of values of $\lambda$. These sets are determined by the distribution of fading gain, the maximum number of packets which can be transmitted in a slot, and the transmit power function of the fading gain and the number of packets transmitted that is assumed. We identify a case where the above behaviour of the tradeoff differs from that obtained from a previously considered approximate model, in which the random queue length process is assumed to evolve on the non-negative real line, and the transmit power function is strictly convex. We also consider a fading point-to-point link, where the transmitter, in addition to controlling the number of packets served, can also control the number of packets admitted in every slot. Our approach, which uses bounds on the stationary probability distribution of the queue length, also leads to an intuitive explanation of the asymptotic behaviour of average delay in the regime where $V \downarrow 0$.