State-Dependent Estimation of Delay Distributions in Fork-Join Networks

Problem definition: Delay announcements have become an essential tool in service system operations: They influence customer behavior and network efficiency. Most current delay announcement methods are designed for relatively simple environments with a single service station or stations in tandem. However, complex service systems, such as healthcare systems, often have fork-join (FJ) structures. Such systems usually suffer from long delays as a result of both resource scarcity and process synchronization, even when queues are fairly short. These systems may thus require more accurate delay estimation techniques than currently available. Methodology/results: We analyze a network comprising a single-server queue followed by a two-station FJ structure using a recursive construction of the Laplace–Stieltjes transform of the joint delay distribution, conditioning on customers’ movements in the network. Delay estimations are made at the time of arrival to the first station. Using data from an emergency department, we examine the accuracy and the robustness of the proposed approach, explore different model structures, and draw insights regarding the conditions under which the FJ structure should be explicitly modeled. We provide evidence that the proposed methodology is better than other commonly used queueing theory estimators such as last-to-enter-service (which is based on snapshot-principle arguments) and queue length, and we replicate previous results showing that the most accurate estimations are obtained when using our model result as a feature in state-of-the-art machine learning estimation methods. Managerial implications: Our results allow management to implement individual, real-time, state-dependent delay announcements in complex FJ networks. We also provide rules of thumb with which one could decide whether to use a model with an explicit FJ structure or to reduce it to a simpler model requiring less computational effort. Funding: This work was supported by the Dutch Research Council (NWO) Gravitation Programme NETWORKS [Grant 024.002.003], the Israel Ministry of Science and Technology [Grant 880011], and the Israel Science Foundation [Grant 1955/15]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.1167 .