Completion Time in Two-User Channels: An Information-Theoretic Perspective

Consider a multi-user channel, where each user has a large but non-replenishable bit pool to transmit. Completion time refers to the number of channel uses spent by each user to complete its transmission. In this paper, an information-theoretic formulation of completion time is based on the concept of constrained rates, which are defined over possibly different number of channel uses. Analogous to the capacity region, the completion time region characterizes all possible trade-offs among users' completion times. For a two-user 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 regions (or inner and outer bounds) are established for a two-user Gaussian broadcast channel and a two-user Gaussian interference channel. It is observed that although consisting of two convex subregions, the completion time region may not be convex in general. Finally, optimization problems of minimizing the weighted sum completion time for a Gaussian multi-access channel and a Gaussian broadcast channel are solved, demonstrating the utility of the completion time approach.