Data-Driven Bed Capacity Planning Using $M_t/G_t/\infty$ Queueing Models with an Application to Neonatal Intensive Care Units

This paper proposes a data-driven framework using time-varying $M_t/G_t/\infty$ queueing models to improve long-term ICU capacity planning by capturing fluctuating admission rates and heterogeneous length-of-stay distributions, demonstrating that static heuristics like the 85% occupancy rule are inadequate for managing real-world demand variability in neonatal intensive care units.

The Gist

Imagine a hospital's Neonatal Intensive Care Unit (NICU) as a busy, high-stakes hotel for premature babies. The "guests" are infants who need special care, and the "rooms" are the hospital beds.

The big problem hospitals face is: How many rooms do we need to build to make sure no baby is turned away, without building so many that the hotel sits empty and wastes money?

For a long time, hospital planners used a simple, old rule of thumb: "Keep the hotel 85% full." They thought, "If we keep the average occupancy at 85%, we'll have enough empty rooms for emergencies."

But this paper argues that this old rule is broken, like trying to predict the weather by only looking at the average temperature of the last 10 years. It ignores the fact that demand is wild and unpredictable. Sometimes, a "storm" of sick babies arrives all at once, and the 85% rule leaves the hospital overflowing.

Here is how the authors fixed the problem, explained in simple terms:

1. The "Weather Forecast" vs. The "Almanac"

The old way of planning was like using an Almanac: "On average, 10 babies arrive a day, and they stay for 10 days. So, we need 100 beds."

The new method in this paper is like a dynamic Weather Forecast. The authors realized that:

• Arrivals aren't steady: More babies might be born in winter due to seasonal illnesses, or fewer in summer.
• Stays aren't steady: Some babies get better quickly; others stay for months.
• The "Shape" of the stay matters: It's not just how long they stay, but how much that time varies. If everyone leaves on the exact same day, you need fewer beds. If everyone leaves on random days, you need a lot more beds to handle the chaos.

2. The "Infinite Server" Hotel

The authors used a special math model called Mt/Gt/∞. Let's break that scary name down:

• Mt (Time-Varying Arrivals): The number of babies checking in changes every day, like traffic on a highway.
• Gt (Time-Varying Stays): The length of stay changes based on the day they arrived and the baby's condition.
• ∞ (Infinite Servers): This is the most important part. In a normal hotel, if all rooms are full, you have a line of people waiting. But in a NICU, you cannot make a sick baby wait. They must be admitted immediately.

So, the model assumes the hospital has "infinite" rooms for the sake of calculation. It doesn't ask, "How many babies are waiting?" It asks, "How many babies are inside the hotel at the exact same time?" If the answer is higher than the actual number of beds, the hospital is in trouble (overloaded).

3. The "Safety Buffer" Analogy

The authors tested two ways to decide how many beds to build:

• The "Average" Approach (The Old Way): "We need enough beds for the average day."
  • Result: On a normal day, the hotel is 85% full. But on a "stormy" day, the hotel is 120% full. Babies are stuck in hallways. This is dangerous.
• The "Safety Buffer" Approach (The New Way): "We need enough beds so that even on a bad day, we only exceed capacity 1% or 5% of the time."
  • Result: To achieve this safety, the hospital needs more beds than the average suggests. This means the hotel might sit at 60% full on a normal Tuesday.
  • The Trade-off: You have "wasted" empty beds on quiet days, but you have saved lives on the chaotic days.

4. The "Surprise" Finding: Variability is a Double-Edged Sword

One of the coolest discoveries in the paper is about variability.

• Old Logic: "If the length of stay is unpredictable (high variance), we need more beds because it's chaotic."
• New Logic: "Actually, if everyone stays for a random amount of time, they leave at random times, which spreads out the crowd. But if everyone stays for the exact same amount of time, they all leave at the same time, creating a massive gap, and then a massive new crowd arrives all at once."

It turns out, predictable chaos is sometimes worse than unpredictable chaos in this specific math model. If you can't predict exactly when a baby will leave, the "crowd" smooths itself out. If you force everyone to leave at the same time, you need more beds to handle the sudden rush.

5. Looking into the Crystal Ball

Finally, the authors built a tool to look into the future. They combined:

• Demographics: How many babies will be born in Calgary in 2030?
• History: How did the hospital behave in the past?

They created a "crystal ball" that says: "If the birth rate goes up by 5%, and we want to stay safe (only 1% chance of overflow), here is exactly how many extra beds you need to build."

The Big Takeaway

The paper tells us that you cannot plan for the future using only the past average.

If you run a hospital (or a bus system, or a call center) based on "average" numbers, you will fail when the unexpected happens. You need to plan for the worst-case scenarios and accept that you will have some empty resources on quiet days. It's the price of safety.

In short: Don't build a hotel for the average guest. Build it for the stormy night, even if it means having empty rooms on sunny days. That's how you keep the babies safe.

Technical Summary

Here is a detailed technical summary of the paper "Data-Driven Bed Capacity Planning Using Mt/Gt/∞Queueing Models with an Application to Neonatal Intensive Care Units."

1. Problem Statement

Hospitals face significant challenges in long-term Intensive Care Unit (ICU) capacity planning due to non-stationary demand. Traditional planning relies on steady-state queueing models (assuming constant arrival rates and service times) or heuristic rules (e.g., the "85% occupancy rule"). These approaches fail to capture:

• Temporal Variability: Admissions fluctuate due to seasonality, population dynamics, and clinical shifts.
• Length of Stay (LOS) Heterogeneity: LOS distributions vary by patient severity, case mix, and clinical protocols, and are not merely defined by their mean.
• The "85% Rule" Limitation: Relying on long-run averages masks daily congestion. The authors demonstrate that even when long-run utilization targets are met, daily occupancy frequently exceeds 100% capacity due to transient surges.

The specific context is Neonatal Intensive Care Units (NICUs) in Calgary, Canada, where patient transfers between sites are often capacity-driven, obscuring the true demand at individual facilities. The goal is to estimate bed requirements that ensure resilience against overflow without relying on static averages.

2. Methodology

The authors propose a data-driven framework based on the $M_t/G_t/\infty$ queueing model (Non-homogeneous Poisson arrivals, General time-varying service times, Infinite servers). The infinite-server assumption reflects the clinical reality that critically ill patients are admitted immediately; strain manifests as overload rather than waiting lines.

The framework consists of four main components:

A. Arrival Rate Estimation ($\lambda_t$)

• Technique: Seasonal-Trend decomposition using Loess (STL) is applied to historical daily admission data.
• Process: A grid search optimizes seasonal (7, 15, 31 days) and trend (15, 31, 61 days) windows to extract a smooth, time-varying arrival intensity $\lambda_t$.
• Justification: STL is preferred over black-box ML or SARIMA because it provides an interpretable separation of trend and seasonality, essential for scenario planning.

B. LOS Distribution Modeling

• Moments Estimation: STL is also applied to LOS data to estimate time-varying mean ($\mu_t$) and variance ($\sigma^2_t$).
• Parametric Fitting: Five candidate distributions (Weibull, Lognormal, Gamma, Fisk/Burr, Exponential) are fitted to empirical data using Maximum Likelihood Estimation (MLE).
• Stability Assumption: The shape parameter ($\kappa$) of the distributions is found to be stable over time. Therefore, $\kappa$ is fixed per site, while scale parameters are dynamically re-parameterized to match the time-varying $\mu_t$ and $\sigma^2_t$ derived from STL.
• Selection: The best-fitting distribution is selected based on the Root Mean Squared Error (RMSE) against the empirical Kaplan-Meier survival curve.

C. Occupancy Estimation

• Convolution Formula: The expected occupancy $\rho_t$ on day $t$ is calculated by convolving past arrivals with the conditional survival probability of LOS:
  $$\rho_t = \sum_{u=0}^{S_{max}} \lambda_{t-u} \cdot P(S > u \mid \tau = t-u)$$
  Where $P(S > u \mid \tau)$ is the probability a patient admitted $u$ days ago is still present, derived from the time-varying parametric distributions.
• Assumptions: Arrivals follow a non-homogeneous Poisson process; LOS is independent of other patients and the arrival process.

D. Capacity Planning Strategies

The framework translates occupancy estimates into capacity targets ($B$) using two strategies:

1. Average-Based ($B_{average}$): $B = \bar{\rho} + \sqrt{\bar{\rho}}$. A baseline using long-run averages plus a square-root buffer (standard in call center staffing).
2. Overflow-Constrained ($B_\alpha$): Solves for the minimum $B$ such that the probability of exceeding a utilization threshold $\gamma$ (e.g., 100% or 85%) is below a risk tolerance $\alpha$:
   $$P(L_t > \gamma B) \leq \alpha$$
   Since $L_t$ follows a Poisson distribution with mean $\rho_t$, this is solved using the Poisson tail probability.

E. Forward-Looking Projections

A births-driven projection module links demographic forecasts to capacity needs. It scales historical admission patterns by projected birth rates ($\tilde{K}_y$) and resamples historical daily arrival/LOS patterns to generate 300 scenario-based trajectories for future years (2024–2030).

3. Key Contributions

1. Non-Stationary Modeling: Moves beyond steady-state assumptions by integrating time-varying arrival rates and LOS distributions into an $M_t/G_t/\infty$ framework.
2. Data-Driven Decomposition: Uses STL decomposition to separate trend and seasonality for both admissions and LOS, capturing temporal dynamics often ignored in planning.
3. Risk-Explicit Capacity: Shifts planning from utilization heuristics (e.g., "keep at 85%") to probabilistic resilience targets (e.g., "ensure <1% chance of overflow").
4. Insight on Variance: Demonstrates that in infinite-server systems, higher LOS variance can actually reduce peak occupancy (by dispersing discharges), while lower variance concentrates departures, increasing congestion risk—a counter-intuitive finding for capacity planners.
5. Adjustment for Transfers: Develops a method to "adjust" historical data by attributing capacity-driven transfers back to the originating site, revealing true structural demand imbalances across a hospital network.

4. Key Results

• Inadequacy of Static Rules: The study applied to 5 Calgary NICU sites showed that the traditional "85% rule" or average-based estimates significantly underestimated capacity needs. For example, at Site 3, the average-based rule suggested 14 beds, but occupancy exceeded 100% on 23% of days.
• Capacity Requirements:
  • Site 2 (High Demand): Required 67 beds under the average rule, but 90 beds to limit overflow risk to 1% ($B_{0.01}$).
  • Trade-off: Moving from $B_{0.05}$ (5% overflow risk) to $B_{0.01}$ (1% overflow risk) increased capacity requirements by 8–13 beds per site but reduced average utilization from ~65% to ~56%.
• Impact of LOS Variance: Sensitivity analysis showed that reducing LOS variance (making stays more deterministic) increased the required bed count to meet overflow targets. Conversely, higher variance dispersed patient exits, lowering peak congestion.
• Future Projections: Projections to 2030, driven by rising birth rates, indicate a need for 4–6 additional beds (for 5% risk) and 8–13 additional beds (for 1% risk) compared to average-based estimates.
• Network Imbalance: When transfers were accounted for, Site 2 was found to be severely under-capacity relative to its intrinsic demand, while Sites 3–5 appeared over-capacity, highlighting a structural regional imbalance masked by operational transfers.

5. Significance and Implications

• Operational Resilience: The framework provides a quantitative basis for setting bed levels that explicitly manage the risk of overflow, crucial for critical care where delays in admission can be fatal.
• Policy Guidance: It challenges the universal application of the 85% occupancy rule, suggesting that in non-stationary environments, higher capacity buffers are necessary to maintain responsiveness.
• Strategic Planning: The ability to model "what-if" scenarios (e.g., changes in LOS variance due to new clinical protocols or demographic shifts) allows hospital administrators to make informed decisions about infrastructure investment.
• Generalizability: While applied to NICUs, the $M_t/G_t/\infty$ framework is applicable to other ICU settings and hospital units where immediate admission is prioritized and demand is time-varying.

The study concludes that robust capacity planning requires moving from static averages to dynamic, probabilistic models that account for the full distribution of service times and the temporal variability of demand.