Unslotted ALOHA protocol has been adopted as a channel access mechanism in commercial low-power wide-area networks (LPWANs), such as Sigfox and long-range (LoRa) alliance. This work examines the throughput and random access (RA) delay distribution of unslotted ALOHA systems by considering exponential random backoff (ERB) or uniform random backoff (URB) algorithm. We further characterize the operating region of the systems as unsaturated stable, bistable, and saturated regions in terms of the new packet arrival and retransmission rates. To run the system stably with the maximum throughput, we propose a Bayesian online backoff algorithm that estimates the number of backlogged devices. Its performance is compared with other algorithms, such as particle filter (PF)-based algorithm, binary exponential backoff (BEB) algorithm, and the algorithm of exploiting exact backlog size information. Through extensive simulations, it is demonstrated that the performance of the proposed algorithm is very close to the upper bound and robust to time-varying traffic condition.