When analysing business processes through process mining, cases are typically considered in isolation. A case is seen as one process execution, whose control-flow is only influenced by case-specific features. However, this isolated view on cases does not consider inter-case effects, which likely occur when cases compete for the availability of a fixed number of shared resources. In this paper, we study the stochastic implications of (1) transition life cycles and (2) resources. That is, we extend stochastic Petri nets with the notion of a transition life cycle to capture the non-atomicity of resources executing transitions. This then allows us to describe the competition between cases for resources stochastically and analyse how scarcity of resources influences the behaviour of real-life processes. We provide an according automated discovery method, and evaluate our technique on multiple real-life event logs by showing its applicability and feasibility, and demonstrating that new types of insights can be obtained for real-life processes.