Hospitals are complex systems with essential societal benefits and huge mounting costs. These costs are exacerbated by inefficiencies in hospital processes, which are often manifested by congestion and long delays in patient care. Thus, a queueing-network view of patient flow in hospitals is natural for studying and improving its performance. The goal of our research is to explore patient flow data through the lens of a queueing scientist. The means is exploratory data analysis (EDA) in a large Israeli hospital, which reveals important features that are not readily explainable by existing models. Questions raised by our EDA include: Can a simple (parsimonious) queueing model usefully capture the complex operational reality of the Emergency Department (ED)? What time scales and operational regimes are relevant for modeling patient length of stay in the Internal Wards (IWs)? How do protocols of patient transfer between the ED and the IWs influence patient delay, workload division and fairness? EDA also underscores the importance of an integrative view of hospital units by, for example, relating ED bottlenecks to IW physician protocols. The significance of such questions and our related findings raise the need for novel queueing models and theory, which we present here as research opportunities. Hospital data, and specifically patient flow data at the level of the individual patient, is increasingly collected but is typically confidential and/or proprietary. We have been fortunate to partner with a hospital that allowed us to open up its data for everyone to access. This enables reproducibility of our findings, through a user-friendly platform that is accessible via the Technion SEELab.

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W e analyze a queueing model that we call Erlang-R, where the "R" stands for reentrant customers. Erlang-R accommodates customers who return to service several times during their sojourn within the system, and its modeling power is most pronounced in time-varying environments. Indeed, it was motivated by healthcare systems, in which offered-loads vary over time and patients often go through a repetitive service process. Erlang-R helps answer questions such as how many servers (physicians/nurses) are required to achieve predetermined service levels. Formally, it is merely a two-station open queueing network, which, in a steady state, evolves like an Erlang-C (M/M/s) model. In time-varying environments, on the other hand, the situation differs: here one must account for the reentrant nature of service to avoid excessive staffing costs or undesirable service levels. We validate Erlang-R against an emergency ward (EW) operating under normal conditions as well as during a mass casualty event (MCE). In both scenarios, we apply time-varying fluid and diffusion approximations: the EW is critically loaded and the MCE is overloaded. In particular, for the EW we propose a time-varying square-root staffing policy, based on the modified offered-load, which is proved to perform well over small-to-large systems.

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In a number of service systems, there can be substantial latitude to vary service rates. However, although speeding up service rate during periods of congestion may address a present congestion issue, it may actually exacerbate the problem by increasing the need for rework. We introduce a state-dependent queuing network where service times and return probabilities depend on the "overloaded" and "underloaded" state of the system. We use a fluid model to examine how different definitions of "overload" affect the long-term behavior of the system and provide insight into the impact of using speedup. We identify scenarios where speedup can be helpful to temporarily alleviate congestion and increase access to service. For such scenarios, we provide approximations for the likelihood of speedup to service. We also identify scenarios where speedup should never be used; moreover, in such a situation, an interesting bi-stability arises, such that the system shifts randomly between two equilibria states. Hence, our analysis sheds light on the potential benefits and pitfalls of using speedup when the subsequent returns may be unavoidable.