Power Is Knowledge: Distributed and Throughput Optimal Power Control in Wireless Networks

Consider N devices that transmit packets for T time slots, where device n uses transmission power <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{n}\left ({{t}}\right)$ </tex-math></inline-formula> at time slot t. Independently at each time slot, a packet arrives at device n with probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda _{n}$ </tex-math></inline-formula>. The probability of successfully transmitting a packet <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu _{n}\left ({{\boldsymbol {P}}}\right)$ </tex-math></inline-formula> is a function of the transmission powers of all devices <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {P}$ </tex-math></inline-formula> and the channel gains <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\left \{{{ g_{m,n}}}\right \} $ </tex-math></inline-formula> between them. This function is unknown to the devices that only observe binary reward <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r_{n}\left ({{\boldsymbol {P}}}\right)$ </tex-math></inline-formula> of whether the transmission was successful (ACK/NACK). All packets of device n that were not successfully transmitted yet at time slot t wait in a queue <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q_{n}\left ({{t}}\right)$ </tex-math></inline-formula>. The centralized max-weight scheduling (MWS) can stabilize the queues for any feasible <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\lambda }$ </tex-math></inline-formula> (i.e., throughput optimality). However, MWS for power control is intractable even as a centralized algorithm, let alone in a distributed network. We design a distributed yet asymptotically throughput optimal power control for the wireless interference channel, which has long been recognized as a major challenge. Our main observation is that the interference <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$I_{n}\left ({{t}}\right)=\sum g_{m,n}^{2}P_{m}\left ({{t}}\right)$ </tex-math></inline-formula> can be leveraged to evaluate the weighted throughput if we add a short pilot signal with power <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P_{m}\propto Q_{m}\left ({{t}}\right)r_{m}\left ({{\boldsymbol {P}}}\right)$ </tex-math></inline-formula> after transmitting the data. Our algorithm requires no explicit communication between the devices and learns to approximate MWS, overcoming its intractable optimization and the unknown throughput functions. We prove that, for large T, our algorithm can achieve any feasible <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\boldsymbol {\lambda }$ </tex-math></inline-formula>. Numerical experiments show that our algorithm outperforms the state-of-the-art distributed power control, exhibiting better performance than our theoretical bounds.