<p>Random delays between the occurrence of accident events and the corresponding reporting times of insurance claims is a standard feature of insurance data. The time lag between the reporting and the processing of a claim depends on whether the claim can be processed without delay as it arrives or whether it remains unprocessed for some time because of temporarily insufficient processing capacity that is shared between all incoming claims. We aim to explain and analyze the nature of processing delays and build-up of backlogs. Development patterns for incoming reported claims that form the basis for claims reserving may be distorted by backlogs when transformed into processed (or paid) claims. In a first step, we show how to infer hidden development patterns from processed claims data. In a second step, we discuss how backlogs impact claims costs, and we show how to select processing capacity optimally in order to minimize claims costs, taking delay-adjusted costs and fixed costs for claims settlement capacity into account. Theoretical results are combined with a large-scale numerical study that demonstrates practical usefulness of our proposal.</p>