mathematics Article Using Markov Models to Characterize and Predict Process Target Compliance Sally McClean   Citation: McClean, S. Using Markov Models to Characterize and Predict Process Target Compliance. Mathematics 2021, 9, 1187. https:// doi.org/10.3390/math9111187 Academic Editors: Andreas C. Georgiou and Panagiotis-Christos Vassiliou Received: 7 May 2021 Accepted: 21 May 2021 Published: 24 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). School of Computing, Ulster University, Belfast BT37 0QB, Northern Ireland, UK; si.mcclean@ulster.ac.uk Abstract: Processes are everywhere, covering disparate fields such as business, industry, telecom- munications, and healthcare. They have previously been analyzed and modelled with the aim of improving understanding and efficiency as well as predicting future events and outcomes. In recent years, process mining has appeared with the aim of uncovering, observing, and improving processes, often based on data obtained from logs. This typically requires task identification, predicting future pathways, or identifying anomalies. We here concentrate on using Markov processes to assess com- pliance with completion targets or, inversely, we can determine appropriate targets for satisfactory performance. Previous work is extended to processes where there are a number of possible exit options, with potentially different target completion times. In particular, we look at distributions of the number of patients failing to meet targets, through time. The formulae are illustrated using data from a stroke patient unit, where there are multiple discharge destinations for patients, namely death, private nursing home, or the patient’s own home, where different discharge destinations may require disparate targets. Key performance indicators (KPIs) of this sort are commonplace in healthcare, business, and industrial processes. Markov models, or their extensions, have an important role to play in this work where the approach can be extended to include more expressive assumptions, with the aim of assessing compliance in complex scenarios. Keywords: process mining; process modelling; phase-type models; process target compliance 1. Introduction Processes are widespread, encompassing disparate areas such as business, production, telecommunications, and healthcare. They have previously been analyzed and modelled with the aim of improving understanding and efficiency as well as predicting future events and outcomes. With the burgeoning capability of IT systems to collect, process, store, and exchange data, and the upsurge of suitable technologies for Big Data, recently, process mining has appeared, providing a bridge between data mining and process modelling [1]. Process mining provides an opportunity and framework for service design and improve- ment, as well as a scientific rationale for decision-making. In general, we consider processes comprising several tasks each with start and end times and associated durations. A process instance completes these tasks according to the logic and rules prevailing in the real-world setting. The process data features mainly consist of data such as duration, customer id, etc., and are held in log files. Hence, such log files provide an automated time-stamped record of tasks performed during the execution of a given process. Consequently, process mining may include discovering the tasks and trajectories that comprise the process, predicting trajectories, or identifying anomalies. Such activities can employ traditional methods for data mining such as classification, clustering, regression, as- sociation rules, sequence mining, or deep learning. However, model-based approaches can also provide opportunities for incorporating structural process knowledge into the analysis, thereby facilitating improved understanding and prediction. As such, process mining can be employed in diverse areas, such as manufacturing [2], telecommunications [3], financial processing, and healthcare [4]. Mathematics 2021, 9, 1187. https://doi.org/10.3390/math9111187 https://www.mdpi.com/journal/mathematics