  We consider a node where packets of fixed size are generated at arbitrary
intervals. The node is required to maintain the peak age of information (AoI)
at the monitor below a threshold by transmitting potentially a subset of the
generated packets. At any time, depending on packet availability and current
AoI, the node can choose the packet to transmit, and its transmission speed. We
consider a power function (rate of energy consumption) that is increasing and
convex in transmission speed, and the objective is to minimize the energy
consumption under the peak AoI constraint at all times. For this problem, we
propose a (customized) greedy policy, and analyze its competitive ratio (CR) by
comparing it against an optimal offline policy by deriving some structural
results. We show that for polynomial power functions, the CR upper bound for
the greedy policy is independent of the system parameters, such as the peak
AoI, packet size, time horizon, or the number of packets generated. Also, we
derive a lower bound on the competitive ratio of any causal policy, and show
that for exponential power functions (e.g., Shannon rate function), the
competitive ratio of any causal policy grows exponentially with increase in the
ratio of packet size to peak AoI.
