  An energy-limited source trying to transmit multiple packets to a destination
with possibly different sizes is considered. With limited energy, the source
cannot potentially transmit all bits of all packets. In addition, there is a
delay cost associated with each packet. Thus, the source has to choose, how
many bits to transmit for each packet, and the order in which to transmit these
bits, to minimize the cost of distortion (introduced by transmitting lower
number of bits) and queueing plus transmission delay, across all packets.
Assuming an exponential metric for distortion loss and linear delay cost, we
show that the optimization problem is jointly convex. Hence, the problem can be
exactly solved using convex solvers, however, because of the complicated
expression derived from the KKT conditions, no closed form solution can be
found even with the simplest cost function choice made in the paper, also the
optimal order in which packets should be transmitted needs to be found via
brute force. To facilitate a more structured solution, a discretized version of
the problem is also considered, where time and energy are divided in discrete
amounts. In any time slot (fixed length), bits belonging to any one packet can
be transmitted, while any discrete number of energy quanta can be used in any
slot corresponding to any one packet, such that the total energy constraint is
satisfied. The discretized problem is a special case of a multi-partitioning
problem, where each packet's utility is super-modular and the proposed greedy
solution is shown to incur cost that is at most $2$-times of the optimal cost.
