Emerging wireless technologies are envisioned to support a variety of applications that require simultaneously maintaining low latency and high reliability. Non-orthogonal multiple access techniques constitute one candidate for grant-free transmission alleviating the signaling requirements for uplink transmissions. In open-loop transmissions over fading channels, in which the transmitters do not have access to the channel state information, the existing approaches are prone to facing frequent outage events. Such outage events lead to repeated re-transmissions of the duplicate information packets, penalizing the latency. This paper proposes a multi-access broadcast approach in which each user splits its information stream into several information layers, each adapted to one possible channel state. This approach facilitates preventing outage events and improves the overall transmission latency. Based on the proposed approach, the average queuing delay of each user is analyzed for different arrival processes at each transmitter. First, for deterministic arrivals, closed-form lower and upper bounds on the average delay are characterized analytically. Secondly, for Poisson arrivals, a closed-form expression for the average delay is delineated using the Pollaczek-Khinchin formula. Based on the established bounds, the proposed approach achieves less average delay than single-layer outage approaches. Under optimal power allocation among the encoded layers, numerical evaluations demonstrate that the proposed approach significantly minimizes average sum delays compared to traditional outage approaches, especially under high arrival rates.