Data-Driven Bed Occupancy Planning in Intensive Care Units Using $M_t/G_t/\infty$ Queueing Models

TL;DR

This paper tackles long-term ICU bed capacity planning under time-varying demand and LOS by adopting a data-driven $M_t/G_t/\infty$ queueing framework. It integrates STL-based estimation of time-varying arrivals and LOS distributions with a non-stationary convolution to compute daily occupancy $\rho_t=\sum_{u=0}^{S_{\max}}\lambda_{t-u}\,\mathbb{P}(S>u\mid A=t-u)$, enabling scenario-based planning. Key contributions include site-specific LOS modeling with dynamic mean and variance, parametric distribution fitting, and multiple planning thresholds (e.g., $B_{average}$, $B_{0.05}$, $B_{0.01}$, and $B_{max}$) that quantify overflow risk and resilience. The findings show static heuristics like the 85% rule understate overflow risk in non-stationary settings and demonstrate how LOS variance materially influences required capacity. The framework, validated on Calgary NICUs, offers interpretable, forward-looking capacity planning applicable to other ICU settings and informs resilience-focused resource management amid rising demand.

Abstract

Hospitals struggle to make effective long-term capacity planning decisions for intensive care units (ICUs) under uncertainty in future demand. Admission rates fluctuate over time due to temporal factors, and length of stay (LOS) distributions vary with patient heterogeneity, hospital location, case mix, and clinical practices. Common planning approaches rely on steady-state queueing models or heuristic rules that assume fixed parameters, but these methods often fall short in capturing real-world occupancy dynamics. One widely used example is the 85\% occupancy rule, which recommends maintaining average utilization below this level to ensure responsiveness; however, this rule is based on stationary assumptions and may be unreliable when applied to time-varying systems. Our analysis shows that even when long-run utilization targets are met, day-to-day occupancy frequently exceeds 100\% capacity. We propose a data-driven framework for estimating ICU bed occupancy using an $M_t/G_t/\infty$ queueing model, which incorporates time-varying arrival rates and empirically estimated LOS distributions. The framework combines statistical decomposition and parametric distribution fitting to capture temporal patterns in ICU admissions and LOS. We apply it to multi-year data from neonatal ICUs (NICUs) in Calgary as a case study. Several capacity planning scenarios are evaluated, including average-based thresholds and surge estimates from Poisson overflow approximations. Results demonstrate the inadequacy of static heuristics in environments with fluctuating demand and highlight the importance of modeling LOS variability when estimating bed needs. Although the case study focuses on NICUs, the methodology generalizes to other ICU settings and provides interpretable, data-informed support for healthcare systems facing rising demand and limited capacity.

Figures (7)

• Figure 1: Comparison of LOS distribution and admission counts by institution
• Figure 2: Time-varying admission rate and mean LOS at an example NICU site
• Figure 3: Smoothed monthly trends for admission rate $\lambda_t$ (top) and mean LOS $\mu_t$ (bottom) for each NICU site. Trends reflect seasonal and long-term changes in admission volume and LOS duration.
• Figure 4: Comparison of empirical LOS tail probabilities $\mathbb{P}(S > u)$ with candidate parametric distributions across sites.
• Figure 5: Daily bed utilization (%) under overflow-constrained threshold $B_{0.05}$. Dashed lines show 85% and 100% thresholds.