  In a two-user channel, completion time refers to the number of channel uses
spent by each user to transmit a bit pool with some given size. In this paper,
the information-theoretic formulation of completion time is based on the
concept of constrained rates, where users are allowed to employ different
numbers of channel uses for transmission as opposed to the equal channel use of
the standard information-theoretic formulation. Analogous to the capacity
region, the completion time region characterizes all possible trade-offs among
users' completion times. For a multi-access channel, it is shown that the
completion time region is achieved by operating the channel in two independent
phases: a multi-access phase when both users are transmitting, and a
point-to-point phase when one user has finished and the other is still
transmitting. Using a similar two-phase approach, the completion time region
(or inner and outer bounds) is established for a Gaussian broadcast channel and
a Gaussian interference channel. It is observed that although consisting of two
convex subregions, the completion time region may not be convex in general.
Finally an optimization problem of minimizing the weighted sum completion time
for a Gaussian multi-access channel and a Gaussian broadcast channel is solved,
demonstrating the utility of the completion time approach.
