Model waiting time, throughput, and utilization in minutes. Choose model and set service levels quickly. Get clear results, export data, and optimize resources today.
Use consistent units for λ and μ (per minute, per hour, or per second). Results are shown in minutes for easier planning.
This calculator supports M/M/1 and M/M/c queues. Let λ be the arrival rate and μ the service rate per server.
Sample scenarios (rates per minute). These rows are illustrative.
| Scenario | Model | λ (arrivals/min) | μ (services/min/server) | c | ρ | Wq (min) | W (min) |
|---|---|---|---|---|---|---|---|
| A | M/M/1 | 8 | 10 | 1 | 0.8000 | 0.3200 | 0.4200 |
| B | M/M/1 | 12 | 15 | 1 | 0.8000 | 0.2133 | 0.2800 |
| C | M/M/3 | 30 | 12 | 3 | 0.8333 | 0.1130 | 0.1963 |
| D | M/M/4 | 40 | 12 | 4 | 0.8333 | 0.0740 | 0.1573 |
In M/M/1 and M/M/c systems, utilization (ρ) is the primary congestion indicator. When ρ rises from 0.70 to 0.85, the queue grows faster than linearly because the safety margin between capacity and demand shrinks. Even small demand spikes can push the effective load toward instability, increasing Wq sharply. As ρ approaches 0.95, delays escalate rapidly in these models.
Wq is the expected pre-service delay, while W includes service time (1/μ). For field dispatch or call handling, W maps to end-to-end response time and Wq maps to “time to first touch.” Little’s Law links time and inventory: Lq = λ·Wq and L = λ·W, enabling staffing decisions from a target backlog. A simple check is Wq ≈ Lq/λ for your unit.
A single constrained workstation behaves like M/M/1 when jobs arrive randomly and processing variability is high. With λ = 12/min and μ = 15/min, ρ = 0.80 and the expected waiting time is about 0.213 minutes. If μ drops to 14/min at the same λ, utilization rises and the predicted queue delay increases noticeably, signaling sensitivity to small cycle-time losses. This helps quantify the value of setup reduction and maintenance.
M/M/c uses Erlang C to estimate the probability that an arrival must wait (Pw). For shared resources like help desks, adding a server increases total capacity (c·μ) and reduces Pw, which reduces Lq and Wq. Compare scenarios C and D in the example table: keeping ρ similar while raising c typically lowers waiting because work is distributed across more servers. Pw also approximates the share of arrivals that see any delay.
Start by measuring arrivals per unit time and average completions per server. Choose a model, confirm λ < c·μ, then iterate c or μ until Wq meets your service-level objective. Export results to document assumptions, share with stakeholders, and revisit inputs after process changes, seasonality shifts, or automation improvements. Test both typical and peak λ values. Record the chosen unit, data window, and any exclusions for auditability.
It models M/M/1 and M/M/c queues with Poisson arrivals and exponential service times, producing utilization, waiting probability, and expected Wq and W values.
Unstable means arrival rate meets or exceeds total capacity, λ ≥ c·μ. In that condition, the expected queue length and waiting time grow without bound in these models.
Many staffing and operations decisions use minute-level thresholds. You can enter rates per hour or second, and the calculator converts Wq and W into minutes for readability.
Use arrivals counted over a stable window for λ. For μ, use average completions per server over the same unit. Remove outages and clearly separate active service time from idle time.
Use M/M/1 for a single bottleneck resource. Use M/M/c when multiple identical servers share the same queue, such as agents, technicians, or parallel machines.
Check assumptions: arrival bursts, priority rules, batch service, and non-exponential times can shift outcomes. Re-estimate λ and μ, validate data windows, and consider more detailed simulation if variability is high.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.