Enjoy the silence: Analysis of stochastic Petri nets with silent transitions

Abstract

Capturing stochastic behaviour in business and work processes is essential to quantitatively understand how nondeterminism is resolved when taking decisions within the process. This is of special interest in process mining, where event data tracking the actual execution of the process are related to process models, and can then provide insights on frequencies and probabilities. Variants of stochastic Petri nets provide a natural formal basis to represent stochastic behaviour and support different data-driven and model-driven analysis tasks in this spectrum. However, when capturing business processes, such nets inherently need a labelling that maps between transitions and activities. In many state of the art process mining techniques, this labelling is not 1-on-1, leading to unlabelled transitions and activities represented by multiple transitions. At the same time, they have to be analysed in a finite-trace semantics, matching the fact that each process execution consists of finitely many steps. These two aspects impede the direct application of existing techniques for stochastic Petri nets, calling for a novel characterisation that incorporates labels and silent transitions in a finite-trace semantics. In this article, we provide such a characterisation starting from generalised stochastic Petri nets and obtaining the framework of labelled stochastic processes (LSPs). On top of this framework, we introduce different key analysis tasks on the traces of LSPs and their probabilities. We show that all such analysis tasks can be solved analytically, in particular reducing them to a single method that combines automata-based techniques to single out the behaviour of interest within an LSP, with techniques based on absorbing Markov chains to reason on their probabilities. Finally, we demonstrate the significance of how our approach in the context of stochastic conformance checking, illustrating practical feasibility through a proof-of-concept implementation and its application to different datasets.