  In this paper, we consider the sum $\alpha$-fair utility maximization problem
for joint downlink (DL) and uplink (UL) transmissions of a wireless powered
communication network (WPCN) via time and power allocation. In the DL, the
users with energy harvesting receiver architecture decode information and
harvest energy based on simultaneous wireless information and power transfer.
While in the UL, the users utilize the harvested energy for information
transmission, and harvest energy when other users transmit UL information. We
show that the general sum $\alpha$-fair utility maximization problem can be
transformed into an equivalent convex one. Tradeoffs between sum rate and user
fairness can be balanced via adjusting the value of $\alpha$. In particular,
for zero fairness, i.e., $\alpha=0$, the optimal allocated time for both DL and
UL is proportional to the overall available transmission power. Tradeoffs
between sum rate and user fairness are presented through simulations.
