The integration of mixed-critical tasks into a platform is an increasingly important trend in the design of real-time systems due to its efficient resource usage. With a growing variety of activation patterns considered in real-time systems, some of them capture arbitrary activation patterns. As a consequence, the existing scheduling approaches in mixed-criticality systems (MCs), which assume the sporadic tasks with implicit deadlines, have sometimes become inapplicable or are ineffective. In this paper, we extend the sporadically activated task model to the arbitrarily activated task model in MCs with the preemptive fixed-task-priority schedule. By using the event arrival curve to model task activations, we present the necessary and sufficient schedulability tests that are based on the well-established results from Real-Time Calculus. We propose to use the busy-window analysis to do the sufficient test because it has been shown to be tighter than the sufficient test of using Real-Time Calculus. According to our experimental results, for sporadic task sets, our proposed test can achieve the same performance as the state-of-the-art schedulability test. However, compared with the previous schedulability analysis of preemptive fixed-task-priority, our approaches can handle more general tasks with blocking, jitter, and arbitrary deadlines.