Workflow security has become increasingly important and challenging in today's open service world. While much research has been conducted on various security issues of workflow systems, the workflow satisfiability problem, which asks whether a set of users together can complete a workflow, is recently identified as an important research problem that needs more investigation. In this paper, we study the computational complexity of the problem along two directions: one is by considering either one path or all paths of a workflow, and the other is by considering the possible patterns in a workflow. We further investigate the problem for access control policies with various constraints and workflows with different task constraints. We have shown that the general workflow satisfiability analysis problem is intractable. This complexity result motivates us to consider restrictions on workflow control-flow patterns and access control policies, and to identify tractable cases of practical interest.