1

Introduction

Hospitals worldwide are tasked with delivering high-quality medical services to an increasing patient population, all while navigating the constraints of limited resources and time. Central to hospital operations are Operating Rooms (ORs), which significantly influence patient throughput, surgical outcomes, and overall institutional efficiency. Effective OR management is essential but complicated by the unpredictable nature of emergency surgeries. These unforeseen demands often disrupt scheduled procedures, leading to cancellations, resource misallocation, and heightened operational costs.

Surgical cancellations present a substantial challenge in healthcare systems globally. Studies indicate that cancellation rates can vary widely, ranging from 5 % to 40 % depending on the institution and setting. Such cancellations not only escalate healthcare costs—estimated to reach millions of dollars annually for large hospitals —but also adversely affect patient outcomes, staff morale, and hospital reputation. Emergency surgeries exacerbate these issues by requiring immediate attention, thereby disrupting elective surgery schedules and intensifying resource strain.

At our major regional hospital in Australia, which operates 12 ORs across five main specialties—General Surgery, Gynaecology, Orthopaedics, Urology, and Vascular Surgery—the challenges reflect broader systemic issues. Between September 2017 and December 2022, the hospital performed over 40,000 surgeries, with emergency cases constituting approximately 36.4 % of the total. Despite the critical nature of these cases, data analysis revealed that about 9.5 % of emergency surgeries were cancelled due to factors such as patient-related issues, operational constraints, resource shortages, and administrative errors.

Importantly, 14.9 % of these cancellations were directly attributed to the unavailability of OR time, underscoring the difficulty in balancing emergency and elective surgeries during periods of unpredictable demand. Fig. 1 illustrates the substantial number of emergency surgery cancellations due to the lack of OR time across various specialties from May to October 2022, with a notable increase during weekdays when demand unpredictably peaks.

Total Number of Emergency cancellations due to the lack of OR time (May to October 2022).

However, physical OR space is just one facet of the scheduling challenge. Staffing constraints—including the availability of anaesthetists, surgeons, and nursing staff—play a crucial role in surgical capacity. Accurate forecasting of emergency surgery demand is therefore critical. It can inform better resource allocation across various dimensions, including staffing and OR availability, to reduce cancellations and enhance patient care outcomes.

In addition to these, waiting times for emergency surgeries significantly impact patient morbidity and mortality, as delays can increase the risk of severe complications or death. Hospital data indicate that emergency surgeries are categorized by urgency levels, each with a maximum allowable waiting time—from requiring surgery within 1 hour to within 72 h of arrival. Approximately 15.2 % of emergency cases require surgery within 1 hour, highlighting the necessity for immediate resource availability.

Forecasting emergency surgery demand is inherently challenging due to its random nature. While predicting the exact number of daily emergency surgeries is unfeasible, developing models that provide reasonable approximations based on historical trends can offer valuable insights for planning and resource allocation. However, forecasting at the daily level may not suffice. Given the time-sensitive nature of emergency surgeries, accurate hourly forecasting becomes essential to ensure that resources—including ORs and staffing—are available when needed, thereby minimizing waiting times and reducing the risk of adverse patient outcomes.

Previous research on demand forecasting has primarily focused on Emergency Medical Services (EMS) and emergency department arrivals, utilizing methodologies like time series analysis, artificial intelligence techniques, and simulation models. For instance, Sun (2009) demonstrated that time series analysis is effective for predicting emergency department workload. Duarte et al. (2021) found that Prophet, a decomposable forecasting model, outperformed traditional ARIMA models in predicting emergency department demand. Artificial intelligence methods, such as Long Short-Term Memory (LSTM) networks, have also shown promise in capturing temporal dependencies for short-term forecasts.

However, these studies often focus on patient arrivals to emergency departments rather than emergency surgeries in ORs. The perioperative setting differs significantly, requiring coordination of specialized surgical teams, OR availability, and postoperative care resources. While agent-based simulations have been used to model emergency department dynamics, they do not directly address the complexities of OR management. Moreover, modelling EMS demand as a non-homogeneous Poisson process has been effective in capturing time-varying arrival rates, but applications in the perioperative context remain limited.

To bridge this gap, our study aims to develop and evaluate advanced forecasting methods tailored specifically to the dynamic nature of emergency surgery demand in the perioperative setting. We employ a two-step approach: first, we assess several daily forecasting methods—including Prophet, ARIMA, SARIMAX, LSTM, and agent-based simulation—using accuracy metrics like Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE). The most effective method is then applied to predict daily emergency surgery numbers. Recognizing the critical need for timely interventions, we supplement these forecasts with a non-homogeneous Poisson process for precise hourly predictions.

By accurately forecasting emergency surgery demand at both daily and hourly levels, hospitals can better anticipate periods of high and low demand. This enables proactive management of staffing, scheduling, and resource allocation—such as reserving ORs for emergencies during predicted peak times and scheduling elective surgeries during quieter periods. While forecasting does not guarantee that all cases will be completed without delays or cancellations, it serves as a strategic tool to enhance overall surgical department efficiency and patient outcomes.

In conclusion, this study contributes to the existing body of knowledge by focusing on the underrepresented area of emergency surgery demand forecasting in the perioperative setting. By integrating advanced forecasting models and emphasizing the importance of hourly predictions, we aim to provide hospitals with actionable insights to reduce emergency surgery cancellations, optimize resource utilization, and improve patient care.

2

Materials and methods

This study adheres to the Standards for Quality Improvement Reporting Excellence (SQUIRE) guidelines to ensure clear and transparent reporting of methods, findings, and implications.

It employs a two-step approach to forecast emergency surgical cases in the operating room (OR) at a major regional hospital in Australia. The first step involves evaluating several forecasting methods to predict daily emergency case arrivals. These methods include Prophet, ARIMA, SARIMAX, Long Short-Term Memory (LSTM) networks, and Agent-Based Simulation (ABS). Fig. 2 categorizes the selected daily forecasting methodologies based on their effectiveness in predicting emergency cases according to the literature.

Selecting forecasting methodologies for emergency case arrivals.

The second step refines these daily forecasts using a non-homogeneous Poisson process to provide more granular, hourly predictions.

The input data for the forecasting models consisted of historical ED arrival records, including arrival dates and times, as well as external factors such as weekends, holidays, and public events that could influence patient influx. Each model was trained on 80 % of data from October 2020 to April 2022 and tested on 20 % of data from May to October 2022 to assess their performance in real-world scenarios. This approach allows for the evaluation of the models’ predictive performance under realistic conditions, which is particularly helpful for readers less familiar with forecasting models.

Traditional statistical methods like ARIMA and SARIMAX analyze time series data by capturing underlying patterns such as trends and seasonality. ARIMA models combine autoregressive terms, differencing to achieve stationarity, and moving average components. SARIMAX extends ARIMA by incorporating seasonal effects and external variables, making it suitable for data exhibiting regular patterns influenced by external factors.

Prophet, developed by Facebook, is designed to handle time series data with strong seasonal effects and irregular events. It automatically detects and models trends and seasonality and allows the inclusion of holidays or special events specified by the user. This flexibility makes it particularly useful in healthcare settings where patient arrivals vary by time of day, day of the week, or season.

LSTM networks, a type of recurrent neural network (RNN), are effective at capturing long-term dependencies in sequential data. They can model complex patterns, including non-linear relationships, making them suitable for time series forecasting in scenarios where patterns may not be immediately apparent.

Agent-based simulation models the actions and interactions of autonomous agents, such as patients and healthcare providers, to assess their effects on the system as a whole. ABS is valuable for simulating complex systems like healthcare, where interactions can lead to emergent behaviours difficult to predict with traditional models.

The forecasting methods were evaluated using error metrics to determine the most suitable approach for predicting daily emergency case arrivals. The metrics used were Mean Absolute Error (MAE), Mean Squared Error (MSE), and Root Mean Squared Error (RMSE), calculated as follows:

• •
  

  Mean Absolute Error (MAE):

  
  
  <SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='MAE=1n∑i=1n|yi−y^i|’>𝑀𝐴𝐸=1𝑛∑𝑛𝑖=1∣∣𝑦𝑖−𝑦ˆ𝑖∣∣MAE=1n∑i=1n|yi−y^i|
  M A E = 1 n ∑ i = 1 n | y i − y ^ i |
  
  
  

• •
  

  Mean Squared Error (MSE):

  
  
  <SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='MAE=1n∑i=1n(yi−y^i)2′>𝑀𝐴𝐸=1𝑛∑𝑛𝑖=1(𝑦𝑖−𝑦ˆ𝑖)2MAE=1n∑i=1n(yi−y^i)2
  M A E = 1 n ∑ i = 1 n ( y i − y ^ i ) 2
  
  
  
  
  
• •
  

  Root Mean Squared Error (RMSE):

  
  
  <SPAN role=presentation tabIndex=0 id=MathJax-Element-5-Frame class=MathJax style="POSITION: relative" data-mathml='RMSE=1n∑i=1n(yi−y^i)2′>𝑅𝑀𝑆𝐸=1𝑛∑𝑛𝑖=1(𝑦𝑖−𝑦ˆ𝑖)2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√RMSE=1n∑i=1n(yi−y^i)2
  R M S E = 1 n ∑ i = 1 n ( y i − y ^ i ) 2
  
  
  

Models with lower values of these metrics were considered to have better predictive performance.

To validate the similarity between the forecasted and actual arrival time distributions, we employed the Kolmogorov-Smirnov (KS) test. This non-parametric test compares the cumulative distributions of two datasets to determine whether they differ significantly. A KS statistic close to zero with a high p-value indicates no significant difference between the forecasted and actual distributions, validating the accuracy of the model in replicating the real-world arrival patterns.

To provide time-specific (hourly) forecasts, we employed a non-homogeneous Poisson process, which allows the event rate λ(t) to vary over time, reflecting dynamic arrival rates throughout the day. The number of arrivals in the interval [t,t + s] follows a Poisson distribution with mean equal to the integral of the rate function over the interval:

The probability that the time until the first event X ₁ ​ exceeds t is given by:

To simulate inter-arrival times:

• 1.
  

  Segment the Time Horizon: The day is divided into small intervals where λ(t) is approximately constant.

  
  
• 2.
  

  Simulate Event Counts: The number of events in each interval is simulated using the Poisson distribution, with the mean determined by λ(t) multiplied by the interval length.

  
  
• 3.
  

  Determine Event Timing: The exact timing of events within each interval is assumed to be uniformly distributed.

This method allows for accurate modelling of arrival times, reflecting the variability in arrival rates throughout the day, and enables hospitals to plan resources on an hourly basis.

By combining daily forecasts from the best-performing models with the time-specific forecasts from the non-homogeneous Poisson process, we aim to provide accurate and granular predictions of ED arrivals to optimize resource allocation and operational planning.

The SQUIRE guidelines, which have been applied in this study, provide a framework for reporting quality improvement studies in healthcare, emphasizing the systematic and rigorous approach used in this research.

3

Numerical results

The purpose of this section is to assess the performance of various forecasting models in predicting emergency case arrivals and their impact on operating room efficiency. The models were evaluated using multiple metrics, such as prediction errors, scalability, and computational efficiency, with additional validation through statistical tests.

3.1

Daily forecasting

3.1.1

Time series plot

Fig. 3 presents the comparison between the actual emergency case data and the predictions from different forecasting models. The plots offer a clear visual representation of how well each model captures the temporal patterns of emergency case arrivals. Values for Parameters of different models can be found in Table A.1 in Appendix 1.

Forecasting of emergency case arrivals using different methods.

Based on Fig. 3 , SARIMAX aligns closely with the actual data, demonstrating strong performance in capturing both seasonal trends and sudden fluctuations in case volume. The Agent-Based Simulation provides reliable predictions with narrow confidence intervals, though it struggles with certain outlier cases. Prophet captures overall trends but underperforms in detecting short-term variations. Similarly, ARIMA models general trends adequately but struggles to reflect sudden spikes in demand. LSTM performs poorly during periods of sudden demand changes, highlighting its limitations in generalizing short-term variability.

To demonstrate the models’ short-term performance, Table 1 presents the predicted number of emergency cases for the first week of May 2022, alongside the actual observed cases. This comparison highlights the immediate predictive capabilities of each model following the training period.

Table 1

Predictions of emergency cases for the first week of May 2022 using different forecasting methods.

Date	Weekday	Actual cases	Predicted cases
			SARIMAX	Agent Based	Prophet	ARIMA	LSTM
01/05/2022	Sunday	5	4	5	4	3	3
02/05/2022	Monday	6	5	4	3	3	2
03/05/2022	Tuesday	7	7	6	5	4	3
04/05/2022	Wednesday	9	8	7	7	4	3
05/05/2022	Thursday	8	7	7	6	3	2
06/05/2022	Friday	6	5	4	4	3	2
07/05/2022	Saturday	5	5	4	3	2	2

 

3.1.2

Seasonality and day-of-week analysis

Figs. 4 and 5 present the models’ ability to capture recurring seasonal patterns, such as day-of-week and monthly trends.

Day-of-Week Analysis: Actual vs Predicted by Different Models.

Monthly Seasonality Analysis: Actual vs Predicted by Different Models.

Fig. 4 (Day-of-Week Analysis) displays the average number of cases for each day of the week, illustrating how well the models align with weekly patterns. Fig. 5 (Monthly Seasonality Analysis) aggregates case numbers by month, revealing broader seasonal trends while reducing the noise from daily fluctuations.

The SARIMAX model consistently outperforms others in recognizing both weekly and monthly patterns, followed closely by Agent-Based Simulation. Other models struggle with short-term variations and seasonal peaks.

3.1.3

Comparison of accuracy metrics

Table 2 summarizes the accuracy metrics for each model, including MAE, MSE, and RMSE, and evaluates their ability to capture seasonality, general trends, and weekly patterns.

Table 2

Accuracy comparison of various methods for emergency case prediction.

Method	MAE	MSE	RMSE	Monthly Seasonality Capture	General Trend Capture	Week-of-Day Capture
SARIMAX Time Series	1.01	2.21	1.48	Yes	Yes	Yes
Agent-based Simulation	1.07	2.37	1.54	Yes	Partial	Yes
Prophet Time Series	1.79	5.36	2.31	Partial	Partial	No
ARIMA Time Series	1.80	5.50	2.34	Partial	No	No
LSTM	1.84	8.51	2.91	No	No	No

 

3.1.4

Additional performance metrics

Table 3 evaluates the models based on practical considerations, such as computation time, scalability, interpretability, and robustness.

Table 3

Comparative analysis of forecasting models based on additional performance metrics.

Method	Computation Time (s)	Scalability	Interpretability	Complexity	Robustness	Ease of Implementation
SARIMAX Time Series	5.2	High	Moderate	Moderate	High	Moderate
Agent-Based Simulation	10.4	Moderate	Low	High	High	Low
Prophet Time Series	2.1	High	High	Low	Moderate	High
ARIMA Time Series	4.8	Moderate	High	Low	Moderate	High
LSTM	25.7	Low	Low	High	Low	Low

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