🏥 Emergency Department Patient Flow — Monte Carlo Report

Generated: 15-Jun-2026 13:56 | Python engine

9000

Total Simulations

250

Replications / Combo

48 h

Sim Period (24 h warm-up)

Factor Combinations

53 s

Total Compute Time

Executive Summary

✅ Best staffing plan: Surge yields the lowest average wait time (46.6 min) with 58.1% ESI target compliance.
❌ Mass-casualty impact: Normal → MCI increases wait from 45.0 to 83.0 min (Δ=38.1 min) and LWBS from 8.3% to 49.4%.
📈 Wait–Satisfaction link: r = -0.787. Longer waits strongly erode patient satisfaction.
🔗 Wait–LWBS correlation: r = 0.773. Reducing wait time directly reduces LWBS rates.

Jump to full analysis ↓

▶ Watch the animated simulation

A frame-by-frame canvas animation of this scenario, side-by-side under contrasting operating policies.

1. Design of Experiments

A full-factorial design explores how staffing levels, room configurations, and arrival patterns interact to shape Emergency Department performance.

Factors

Factor	Levels	Description
Staffing Plan	4	Minimal (3D/6N/1T), Standard (5D/10N/2T), Enhanced (7D/14N/3T), Surge (9D/18N/4T)
Room Config	3	Conservative (15S/5F/1T), Balanced (12S/8F/2T), FT-Heavy (10S/12F/2T)
Arrival Pattern	3	Normal (1×), Busy (1.5×), Mass-Casualty (2× + surge)

ESI Triage Levels

ESI	Category	Probability	Wait Target	Treatment
1	🚨 Resuscitation	2%	Immediate	60–120 min
2	⚠️ Emergency	10%	<10 min	45–90 min
3	⏱ Urgent	35%	<30 min	30–60 min
4	🔼 Less Urgent	35%	<60 min	15–30 min
5	🟢 Non-Urgent	18%	<120 min	10–20 min

2. Key Performance Indicators

61.3min

Avg Wait Time (±29.4)

52.0min

Median Wait Time (±20.8)

67.4min

Avg Door-to-Doctor (±30.1)

220.7min

Avg Length of Stay (±89.1)

26.1%

LWBS Rate (±27.1)

76/24h

Patient Throughput (±29)

66.4%

Doctor Utilisation (±20.8)

38.4%

Nurse Utilisation (±12.9)

16.4%

Room Utilisation (±5.4)

46.6%

Target Compliance (±18.2)

35.7

Patient Satisfaction (±21.6)

1328€

Cost / Patient (±213)

27.2min

Avg Idle Wait (±29.1)

3. Monte Carlo Summary Dashboard

20-panel overview: distributions, heatmaps, main effects, Pareto front, time series, and queue dynamics.

3.1 Wait Time & Length of Stay

Patient wait time distributions by staffing plan and room configuration, length-of-stay profile, and left-without-being-seen rates.

3.2 Trade-offs & Patient Experience

How wait times relate to throughput and cost, door-to-doctor times by arrival pattern, and overall patient satisfaction.

3.3 Utilisation & Queue Dynamics

Resource utilisation heatmaps, wait-time interaction surfaces, ESI-level analysis, and 24-hour queue profiles.

3.4 Factor Effects & Pareto

Main-effect decomposition for wait time and LWBS, cost-vs-wait Pareto frontier, and target compliance metrics.

3.5 Temporal Patterns & Queue Peaks

Arrival vs discharge dynamics, doctor utilisation over 24 hours, and peak triage/treatment queue depths.

4. Key Insights

✅ Best staffing plan: Surge yields the lowest average wait time (46.6 min) with 58.1% ESI target compliance.

❌ Mass-casualty impact: Normal → MCI increases wait from 45.0 to 83.0 min (Δ=38.1 min) and LWBS from 8.3% to 49.4%.

📈 Wait–Satisfaction link: r = -0.787. Longer waits strongly erode patient satisfaction.

🔗 Wait–LWBS correlation: r = 0.773. Reducing wait time directly reduces LWBS rates.

💰 Cost trade-off: Cheapest staffing (Minimal) costs 1430€/pt but waits 85 min; best (Surge) costs 1336€/pt with 47 min wait.

5. Patient Flow Analysis

Average Hourly Flow by Arrival Pattern

Pattern	Avg Arrivals/24 h	Avg Throughput	Avg Wait [min]	LWBS [%]	Avg LOS [min]
Normal	90.4	66.6	45.0	8.3	168.1
Busy	135.7	85.5	55.7	20.5	196.3
Mass-Casualty	201.2	76.5	83.0	49.4	297.7

6. Staffing Impact Analysis

Performance by Staffing Plan

Staffing	Doctors	Nurses	Triage St.	Avg Wait [min]	D2D [min]	LWBS [%]	Throughput	Cost/Pt [€]	Satisfaction
Minimal	3	6	1	85.3	92.9	55.5	41	1430	13.5
Standard	5	10	2	62.0	67.8	27.4	72	1276	33.4
Enhanced	7	14	3	51.2	56.8	14.0	91	1270	45.0
Surge	9	18	4	46.6	52.1	7.4	101	1336	50.8

7. KPI Correlation Analysis

💡 Correlation Insights

What these emergency-medicine relationships reveal — with clinical operations research context.

KPI Pair	r	Strength	Actionable Insight
Wait ↔ D2D	1.000	Strong	Wait and D2D show a positive relationship (r=1.00). Changes in one tends to reinforce the other.
Doc% ↔ Nurse%	0.996	Strong	Doc% and Nurse% show a positive relationship (r=1.00). Changes in one tends to reinforce the other.
Compliance ↔ Satisfaction	0.954	Strong	Compliance and Satisfaction show a positive relationship (r=0.95). Changes in one tends to reinforce the other.
LWBS% ↔ Satisfaction	-0.935	Strong	LWBS% and Satisfaction show a negative relationship (r=-0.94). Improving one may come at the expense of the other.
Nurse% ↔ Satisfaction	-0.932	Strong	Nurse% and Satisfaction show a negative relationship (r=-0.93). Improving one may come at the expense of the other.
Med.Wait ↔ D2D	0.932	Strong	Med.Wait and D2D show a positive relationship (r=0.93). Changes in one tends to reinforce the other.
Wait ↔ Med.Wait	0.931	Strong	Wait and Med.Wait show a positive relationship (r=0.93). Changes in one tends to reinforce the other.
Doc% ↔ Satisfaction	-0.925	Strong	Doc% and Satisfaction show a negative relationship (r=-0.92). Improving one may come at the expense of the other.

8. Full Statistical Summary

KPI	Mean	Std	Min	Median	Max	95% CI
Avg Wait [min]	61.25	29.41	23.93	54.19	342.83	[57.61, 64.90]
Median Wait [min]	51.97	20.79	19.00	47.00	334.00	[49.40, 54.55]
Door-to-Doctor [min]	67.40	30.15	29.74	60.24	360.13	[63.66, 71.14]
LOS [min]	220.70	89.08	90.27	193.53	794.85	[209.66, 231.74]
LWBS Rate [%]	26.07	27.07	0.00	17.01	89.22	[22.72, 29.43]
Throughput [pts/24h]	76.21	28.76	17.00	74.00	165.00	[72.64, 79.77]
Doctor Util [%]	66.36	20.80	18.59	70.21	100.00	[63.78, 68.93]
Nurse Util [%]	38.42	12.92	10.86	39.85	64.21	[36.82, 40.02]
Room Util [%]	16.38	5.44	6.57	14.84	36.53	[15.71, 17.06]
Triage Q Peak	3.52	4.88	0.00	2.00	37.00	[2.91, 4.12]
Treatment Q Peak	14.93	13.17	0.00	12.00	72.00	[13.30, 16.57]
Target Compliance [%]	46.59	18.18	0.00	48.08	84.85	[44.33, 48.84]
Satisfaction	35.69	21.57	0.00	35.45	77.17	[33.01, 38.36]
Cost/Patient [€]	1327.77	212.62	899.40	1277.28	2627.22	[1301.42, 1354.13]
Idle Wait [min]	27.17	29.13	0.00	20.83	307.74	[23.56, 30.78]

9. Methodology & Model Description

9.1 Architecture

Discrete-event microsimulation with minute-by-minute time steps over a 48-hour period (first 24 h warm-up, second 24 h KPI collection). Self-contained simulation engine running inside ProcessPoolExecutor for embarrassingly parallel Monte Carlo execution.

9.2 Patient Model

Arrivals follow a time-varying Poisson process with hour-dependent rates (bimodal peaks at 10–14 h and 18–22 h). Each patient receives an ESI triage level (1–5) with defined probabilities, treatment durations, wait-time targets, and patience thresholds for LWBS behaviour.

9.3 Resource Model

Three resource types: Doctors (1 per treatment), Nurses (1 per triage + 1 per treatment), and Treatment Rooms (Standard for ESI 1–3, Fast-track for ESI 4–5, Trauma bays preferentially for ESI-1). Rooms undergo 15–25 min cleaning between patients.

9.4 Monte Carlo Design

4 staffing plans × 3 room configs × 3 arrival patterns = 36 combinations × 250 reps = 9000 runs. 48-hour (2 880-minute) simulation with 24 h warm-up each. Executed with 12 CPU cores in 53.2 seconds.

10. Interactive Analysis

11. Strategic Insights by Persona

Data-driven recommendations derived from simulation results, tailored for four key stakeholder groups.

💰

Cost Efficiency

Cost per patient ranges from 1270€ (Enhanced) to 1430€ (Minimal). Enhanced staffing offers the best cost-to-quality ratio.

📈

Throughput ROI

Surge processes 101 patients/24 h vs Minimal at 41. Each additional doctor adds ~10 patients/day.

🏆

Resource Utilisation

Doctor utilisation: 88% (Minimal) to 48% (Surge). Over-staffing reduces utilisation but improves patient outcomes.

🚨

Mass-Casualty Readiness

MCI scenarios generate 77 patients and push LWBS to 49.4%. Only Surge staffing maintains acceptable performance under MCI conditions.

⚡

Understaffing Risk

Minimal staffing shows wait times of 85 min and LWBS of 55.5%. Risk of patient harm and regulatory penalties is high.

⏱

Door-to-Doctor Performance

Average D2D time: 52.1 min (Surge) vs 92.9 min (Minimal). CMS benchmark is <30 min for ESI-2.

🎯

ESI Target Compliance

Overall compliance: 46.6%. ESI-1: 22.8%, ESI-2: 13.0%. Critical-acuity compliance must exceed 90%.

🚨

MCI Preparedness

Mass-casualty events increase wait by 38 min. The surge protocol with 9 doctors maintains D2D under 52 min even during MCI.

📊

Room Utilisation Gap

Room config matters: Conservative configuration minimises wait. Fast-track rooms handle 53% of patients (ESI 4–5) more efficiently.

🔬

Quality Metrics

Wait–LWBS correlation: r = 0.773. Every 10 min reduction in wait decreases LWBS by ~7.7%. Patient satisfaction peaks at 50.8 with Surge staffing.

📉

Nurse Utilisation Balance

Nurse util ranges from 27% to 53%. Optimal zone is 70–85%. Minimal exceeds safe thresholds.

⚖️

Triage Efficiency

Peak triage queue: 9 (Minimal) vs 1 (Surge). More triage stations dramatically reduce initial bottleneck.

🕐

Peak Demand Periods

Arrivals peak at hours 10–14 and 18–22 (up to 6 patients/h). Nurse staffing should be front-loaded to morning and evening shifts.

💪

Workload by Config

Nurses assist treatment at 1:2.5 ratio. With Surge plan (18 nurses), workload per nurse drops 2.0× compared to Minimal.

🔥

Burnout Risk

Nurse utilisation above 85% correlates with burnout. Minimal staffing puts nurses at 53% utilisation — unsustainable for sustained operations.

⏱

Wait Time Equity

ESI-1 waits: 44.4 min, ESI-5: 48.4 min. While triage prioritisation is appropriate, low-acuity patients face disproportionately long waits.

🚶

LWBS Impact

Overall LWBS rate: 26.1%. These patients may delay care, leading to worse outcomes. Surge staffing reduces LWBS to 7.4%.

💖

Satisfaction Drivers

Patient satisfaction: 35.7/100. Wait–satisfaction correlation: r = -0.787. Wait time is the single largest driver of patient experience.

🕔

Wait by Time of Day

Waits are longest during peak arrival hours (10–14 h, 18–22 h). Patients arriving at night (02–06 h) experience minimal delays.

🌟

Recommendations

Invest in Enhanced or Surge staffing, expand fast-track for low-acuity patients, and implement real-time wait-time displays to manage expectations and reduce LWBS.

12. Boarding Analysis

After treatment, admitted patients board in the ED while waiting for an inpatient bed. Boarding time depends on time of day (inpatient bed availability factors: overnight 0.3, morning 0.8, afternoon 0.6, evening 0.4). Higher factor = faster placement.

369 min

Avg Boarding Time

Avg Boarders / 24 h

69.7%

Boarding > 4 hrs

14:00

Peak Boarding Hour

25.3%

Overnight Boarding %

🚨 Boarding impact: ESI-1 patients board an average of 289 min, ESI-2: 362 min. 69.7% of boarders exceed the 4-hour threshold, indicating significant inpatient bed constraints overnight (factor 0.3).

13. Patient Pathways by Complaint

Five complaint types with unique ESI distributions, treatment time modifiers, imaging/lab needs, and admission probabilities.

Complaint	Frequency	Avg Wait [min]	Avg LOS [min]	Admission Rate [%]
Chest Pain	41	67.4	295.2	43.6
Trauma	74	66.5	277.7	41.5
Abdominal	54	62.8	182.2	19.0
Respiratory	41	58.2	222.9	29.7
Other	68	48.7	127.1	10.8

14. Shift Handover Impact

At shift changes (07:00, 15:00, 23:00) a 15-minute handover period reduces treatment assignment throughput to 50%. This simulates the real-world impact of clinical handoffs, safety briefings, and staff transitions.

Avg Delayed Assignments / 24 h

3×15 min

Handover Windows

50%

Throughput During Handover

🕑 Handover insight: On average 2 treatment assignments are delayed per 24 h due to 3 shift handovers. Staggered handovers or dedicated handover teams could reduce this bottleneck by 30–50%, improving flow during peak transition hours.

15. Imaging & Lab Resource Utilisation

Discrete resource pools: CT scanners (2), X-ray machines (3), Lab stations (4). Patients queue for resources after triage before entering treatment.

25.9%

Imaging Utilisation

52.9%

Lab Utilisation

6.4 min

Avg Imaging Wait

🔬 Resource insight: Imaging utilisation at 25.9% and lab at 52.9%. Patients wait an average of 6.4 min for imaging. Consider adding a CT scanner if utilisation exceeds 80%.

🏥 KPI Glossary & Definitions

Detailed explanation of every metric used in this analysis, with industry benchmarks and authoritative sources to help all stakeholders interpret results.

KPI	Definition	Why It Matters	Industry Benchmark
Avg Wait [min]	Mean time from arrival to first treatment	Primary patient experience metric. Drives LWBS and satisfaction.	CMS benchmark: ≤28 min median
Door-to-Doctor [min]	Time from arrival to first physician contact	CMS quality metric (ED-1b). Strong predictor of outcomes for ESI 1–2.	CMS: median ≤28 min
LOS [min]	Total time from arrival to discharge/admit	Includes treatment + boarding. Boarding inflates LOS without improving care.	Median: 120–180 min; >240 = concern
LWBS Rate [%]	Patients leaving without being seen	Safety risk: LWBS patients may have serious undiagnosed conditions.	Target: <2% (ACEP)
Throughput	Patients treated per 24-hour period	Departmental capacity metric. Limited by beds, staff, and boarding.	Varies by ED size; compare vs. arrivals
Doctor Utilization [%]	Fraction of time physicians are treating	Below 60%: potential overstaffing. Above 90%: queueing delays.	70–85% optimal (Green, 2006)
Nurse Utilization [%]	Fraction of time nurses are in patient care	Nurses handle triage, vitals, treatment assist, and discharge tasks.	65–80% recommended
Room Utilization [%]	Fraction of time treatment rooms occupied	Physical capacity constraint. Above 85%: hallway boarding starts.	Bernstein et al., 2009
Target Compliance [%]	Fraction meeting ESI-based time targets	Regulatory metric: ESI 1 ≤10 min, ESI 2 ≤15 min, etc.	Target: ≥90% for ESI 1–2
Satisfaction	Composite patient satisfaction score (0–100)	Linked to HCAHPS scores and hospital reimbursement.	Top quartile: >80
Cost/Patient [€]	Total operating cost per patient encounter	Staff + facility + supplies. Key for ED financial sustainability.	National avg: $400–600 per visit

📚 Sources & References

Industry standards, academic research, and data sources backing the model assumptions and benchmarks cited in this report.

#	Source	Used For
1	ACEP — American College of Emergency Physicians	LWBS rate benchmarks, boarding guidelines, triage standards
2	CMS — Emergency Department Quality Reporting	Door-to-Doctor (ED-1b), median wait time, and LOS quality measures
3	Green, L. (2006). Operations Research	Queueing models for ED staffing and capacity planning
4	Welch, S. (2010). Annals of Emergency Medicine	Wait time perception and patient satisfaction relationship
5	Bernstein et al. (2009). Ann. Emergency Medicine	ED crowding metrics and clinical impact of hallway boarding
6	Handel et al. (2011). Academic Emergency Medicine	Simulation-based ED staffing optimization and DES validation