Queue Waiting Time Calculator
Estimate expected waiting times using queueing theory models
Queueing Theory
Operations Research
Capacity Planning
Queue Parameters
How many customers arrive on average per time unit
How many customers one server can handle per time unit
About Queue Waiting Time Calculator
This calculator uses queueing theory to estimate waiting times based on arrival patterns and service capacity. It implements the classic M/M/1 and M/M/c models commonly used in operations research and capacity planning.
Key formulas used:
- Utilization (ρ): λ / (c × μ) — fraction of time servers are busy
- Average Queue Length (Lq): Expected number of customers waiting
- Average Wait Time (Wq): Lq / λ (Little's Law)
- Total System Time (W): Wq + 1/μ — wait time plus service time
For planning events or venues, try our Crowd Density Calculator. For financial planning, check out our Time Card Calculator for tracking work hours.
All calculations happen in your browser — no data is sent to any server. Use this tool for planning staffing levels, estimating customer wait times, or optimizing service operations.
Frequently Asked Questions (FAQ)
What is queueing theory?
Queueing theory is a branch of mathematics that studies waiting lines (queues). It helps predict waiting times, queue lengths, and system efficiency. Key parameters include arrival rate (λ), service rate (μ), and number of servers (c). This calculator uses queueing theory models to estimate how long customers will wait.
What do M/M/1 and M/M/c mean?
M/M/1 and M/M/c are standard queue notation. The first M stands for Markovian (random) arrivals, the second M for Markovian service times, and the number (1 or c) indicates server count. M/M/1 has one server; M/M/c has multiple servers. These models assume arrivals follow a Poisson distribution and service times are exponentially distributed.
What is server utilization?
Server utilization (ρ) is the percentage of time servers are busy. It equals arrival rate divided by total service capacity. For example, 80% utilization means servers are busy 80% of the time. Utilization above 100% means the queue will grow infinitely — more capacity is needed. Aim for 70-85% utilization for efficiency with acceptable wait times.
When is a queue stable vs unstable?
A queue is stable when arrival rate is less than service capacity (λ < cμ). This means servers can handle the workload on average. An unstable queue (λ ≥ cμ) will grow infinitely over time because customers arrive faster than they can be served. If your calculation shows "unstable," you need more servers or faster service.
How can I reduce waiting time?
You can reduce waiting time by: (1) Adding more servers to increase capacity, (2) Reducing service time per customer through training or automation, (3) Smoothing arrival patterns to avoid peak congestion, (4) Implementing appointment systems, or (5) Using priority queuing for different customer types. Even small improvements in service rate can significantly reduce wait times.
What are typical applications of queue calculators?
Queue waiting time calculators are used in: call centers for staffing decisions, banks for teller scheduling, hospitals for emergency room planning, retail stores for checkout optimization, restaurants for table management, airports for security lane staffing, and manufacturing for production line balancing. Try our Meeting Cost Calculator for another operations planning tool.