Process mining reveals how processes in organisations are actually performed and pinpoints deviations from the desired process execution. Process delay is one type of deviation that can be detected. Specific activities may take longer than expected or the waiting times between activities may deviate from service agreements. However, the quantification of processing or waiting times is often only the starting point in identifying the underlying root causes for process delay. One such root cause are adverse incidents in the environment of the process such as malfunctioning of supporting systems or unavailability of resources. Data about these external factors is often neither included in the event log nor recorded precisely enough to be directly linkable to a specific set of process instances. This paper presents a method for estimating process delay caused by incidents for which only the approximate occurrence time is known. We link incidents that are recorded in an incident log to process delay and calculate the effect of incidents on process delay using a Markov chain Monte Carlo sampling (MCMC) approach. Our proposed method was evaluated in a project conducted with the infrastructure manager of the Norwegian railway system. We applied it to a large event log of more than 120 million events capturing block-level movements of trains in the railway network and estimated the impact on process delay of about 50 000 infrastructure-related incidents. This showed that the method is useful for providing decision support and insights on the effects of maintenance. Since then the method has become part of the standard toolbox of the infrastructure manager.