Waiting Time Model

Healthcare Systems Engineering

Understanding Queuing Theory & Bottlenecks

Whether it’s patients in an Emergency Department, cars at a toll booth, wait for billing in super market or data packets hitting a server—queues form when demand outpaces service capacity. Operations research uses mathematical models to predict wait times and eliminate bottlenecks before systems collapse.

M/M/1 Model (Single-Server)

A single line feeding into one service point (e.g., a single receptionist or a solo doctor clinic). Highly sensitive to sudden demand spikes.

M/M/c Model (Multi-Server)

A single organized line feeding into multiple parallel servers (e.g., airport security or triage beds or pharamcy with multiple counters). Drastically reduces wait times by absorbing uneven loads.

1

Select Your System Architecture

Which layout best matches your operational workflow?

2

Enter Operational Rates

Adjust variables below to see real-time queue performance.

arrivals/hr
Average patients/customers arriving per hour.
minutes
Average time to serve 1 person (μ = 60 / time).
3

System Performance Output

Calculated metrics based on steady-state probability formulas.

Server Utilization (ρ) 0% Optimal
Avg. Queue Wait (Wq) 0 min Time spent standing in line
Total System Time (W) 0 min Wait time + Service duration
Avg. Queue Length (Lq) 0 People waiting in line
Prob. of Waiting (Pw) 0% Chance arrival encounters line
4

Dynamic Operational Charts

Visualizing non-linear queue explosions and service splits.

Customer Journey Timeline Breakdown

Queue Wait (Delay) 0 min (0%)
Service (Value-Add) 0 min (0%)

Utilization vs. Wait Time Operating Curve

5

Target Wait Time Probability (SLA Analysis)

Evaluate service level compliance and queue delay risks for target wait thresholds.

minutes
Specify maximum acceptable queue waiting time before service.
Wait ≤ 15 mins (Meets Target) 0% Patients/Customers served within target limit
Wait > 15 mins (Exceeds Limit) 0% Patients/Customers experiencing prolonged delay
6

Mathematical Model & Formula Reference

Technical specification of the active queuing model and steady-state formulas used.

Active Analytical Model: M/M/1 (Single-Server Exponential Queue)
Service Rate (μ) μ = 60 / Avg_Service_Time_Mins
Server Utilization (ρ) ρ = λ / μ
Prob. of Waiting (Pw) Pw = ρ
Avg. Queue Length (Lq) Lq = ρ² / (1 - ρ)
Avg. Queue Wait Time (Wq) Wq = Lq / λ
Total System Time (W) W = Wq + (1 / μ)
Target Pure Queue Wait SLA Probability P(Wq > t) P(Wq > t) = Pw × e^[ -μ × (1 - ρ) × t ]