Information systems have been widely adopted to support service processes in various domains, e.g., in the telecommunication, finance, and health sectors. Information recorded by systems during the operation of these processes provides an angle for operational process analysis, commonly referred to as process mining. In this work, we establish a queueing perspective in process mining to address the online delay prediction problem, which refers to the time that the execution of an activity for a running instance of a service process is delayed due to queueing effects. We present predictors that treat queues as first-class citizens and either enhance existing regression-based techniques for process mining or are directly grounded in queueing theory. In particular, our predictors target multi-class service processes, in which requests are classified by a type that influences their processing. Further, we introduce queue mining techniques that derive the predictors from event logs recorded by an information system during process execution. Our evaluation based on large real-world datasets, from the telecommunications and financial sectors, shows that our techniques yield accurate online predictions of case delay and drastically improve over predictors neglecting the queueing perspective.