Stochastic analysis of an emergency department with two levels sharing the servers
Abstract
In this paper, a stochastic model of an overcrwoded emergency department (ED) with two levels is analysed. A patient care pathway consists of a first consultation, followed by diagnostic tests and then a discharge consultation. The first level includes the patients before or during the first consultation. The second level applies to patients currently undergoing diagnostic tests, and patients waiting for or attending a second consultation. Each level has limited capacity; thus, when the second level is full, patients must remain at the first level. One of the key features of this model is the fact that first and second consultations are handled by the same pool of emergency physicians, which means that their decision to prioritize level 1 or level 2 affects the behavior of the system. Under a scaling assumption, the evolution of the number of patients in each step of the pathway is investigated. It is shown that, depending on the parameters, there is a phase transition between two different regimes: a regime limited by the number of emergency physicians and a regime limited by the second level capacity. Stochastic calculus gives the convergence of the state Markov process to a deterministic solution of a differential inclusion, which is proved to be unique and is explicitly characterized. The equilibrium points of the solution are determined depending on the service policies. It allows to compute the exit rate of the ED and the average length of stay, and find the policies that optimize them.
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BibTeX
@misc{ballif2026stochastic,
title = {{Stochastic analysis of an emergency department with two levels sharing the servers}},
author = {Guillaume Ballif and Christine Fricker},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35061},
eprint = {2610.35061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35061},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stochastic analysis of an emergency department with two levels sharing the servers
AU - Guillaume Ballif
AU - Christine Fricker
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35061
UR - https://arxiv.org/abs/2610.35061
AB - In this paper, a stochastic model of an overcrwoded emergency department (ED) with two levels is analysed. A patient care pathway consists of a first consultation, followed by diagnostic tests and then a discharge consultation. The first level includes the patients before or during the first consultation. The second level applies to patients currently undergoing diagnostic tests, and patients waiting for or attending a second consultation. Each level has limited capacity; thus, when the second level is full, patients must remain at the first level. One of the key features of this model is the fact that first and second consultations are handled by the same pool of emergency physicians, which means that their decision to prioritize level 1 or level 2 affects the behavior of the system. Under a scaling assumption, the evolution of the number of patients in each step of the pathway is investigated. It is shown that, depending on the parameters, there is a phase transition between two different regimes: a regime limited by the number of emergency physicians and a regime limited by the second level capacity. Stochastic calculus gives the convergence of the state Markov process to a deterministic solution of a differential inclusion, which is proved to be unique and is explicitly characterized. The equilibrium points of the solution are determined depending on the service policies. It allows to compute the exit rate of the ED and the average length of stay, and find the policies that optimize them.
ER -