Hybrid automatic repeat request (HARQ) is a key enabler for ultrareliable communications. This paper optimizes transmit power for the initial transmission and the subsequent retransmissions of HARQ with either incremental redundancy or Chase combining, aiming to minimize the expected energy consumption given the target outage probability and the target latency. The main challenge is due to the fact that the outage probability is a complicated function of the power variables which are nested in successive convolutions. The existing works mostly use a classic upper bound to approximate the outage probability by assuming unbounded transmit power, then convert the original problem to a geometric programming (GP) problem. In contrast, we propose a novel and much tighter upper bound by taking the practical power limit into consideration. The new bound and the resulting new GP method are further extended to a broader group of channel models with various fading, multiple antennas, and multiple receivers. As shown in simulations, the GP method based on the new bound significantly outperforms the existing strategies that either fix transmit power or optimize power by the classic bounding technique.Index Terms-Hybrid automatic repeat request (HARQ), incremental redundancy, Chase combining, ultrareliable communications, power control, energy consumption, new upper bound on outage probability, geometric programming (GP).
I. INTRODUCTIONU LTRARELIABLE communications refer to transmitting data with a target outage probability lower than 10 −5 , as compared to the traditional cellular systems typically with an outage probability around 10 −2 [2]. This stringent performance standard is driven by a multitude of evolving applications of the Internet of Things (IoT), e.g., ultra highdefinition (UHD) video [3]. This paper seeks an energyefficient implementation of the hybrid automatic repeat request Manuscript accepted in IEEE Transactions on Wireless Communications.