Optimistic log-based rollback recovery protocols have been regarded as an attractive fault-tolerant solution in distributed systems based on message-passing paradigm due to low overhead in failure-free time. These protocols are based on a Piecewise Deterministic (PWD) Assumption model. They, however, assumed that all logged non-deterministic events in a consistent global recovery line must be determinately replayed in recovery time. In this paper, we give the impossibility of deterministic replaying of logged non-deterministic event in a consistent global recovery line as a Ω Line Problem, because of asynchronous properties of distributed systems: no bound on the relative speeds of processes, no bound on message transmission delays and no global time source. In addition, we propose a new optimistic log-based rollback recovery protocol, which guarantees the deterministic replaying of all logged non-deterministic events belonged in a consistent global recovery line and solves a Ω Line Problem in recovery time.