Inputs
Formula used
This calculator uses classic queueing theory with random arrivals and exponential service times. Results are averages, not guarantees, and work best for steady operations.
- Utilization: ρ = λ / μ
- Queue length: Lq = ρ² / (1−ρ)
- System size: L = ρ / (1−ρ)
- Waiting time: Wq = Lq / λ
- Time in system: W = Wq + 1/μ
- Offered load: a = λ / μ
- Utilization: ρ = λ / (c·μ)
- Probability of waiting: Erlang C with Pw
- Queue length: Lq = (Pw·ρ) / (1−ρ)
- Times: Wq = Lq/λ, W = Wq + 1/μ
Effective rates: λeff = λ·peak, μeff = μ·efficiency.
How to use this calculator
- Pick a time unit and keep all rates consistent.
- Enter arrivals (λ) from logs, counts, or forecasts.
- Enter service rate (μ) per server at normal conditions.
- Use peak and efficiency factors to reflect reality.
- Select M/M/1 for one station, M/M/c for parallel stations.
- Check stability: if ρ ≥ 1, add capacity.
- Export results for reports or daily coordination.
Example data table
| Scenario | Model | λ | μ (per server) | c | Peak | Eff. | Comment |
|---|---|---|---|---|---|---|---|
| Gate security check | M/M/1 | 12 / hour | 18 / hour | 1 | 1.20 | 0.95 | Helps plan staffing at shift start. |
| Material hoist loading | M/M/c | 20 / hour | 14 / hour | 2 | 1.10 | 0.90 | Compare adding a second operator. |
| Concrete truck washout | M/M/c | 9 / hour | 6 / hour | 2 | 1.00 | 1.00 | Checks queue risk during pours. |
Use your own measured rates for the most reliable planning.
Notes for practical planning
- If arrivals come in waves, raise the peak factor to stress test queues.
- If service slows due to safety checks or access constraints, lower the efficiency factor.
- High utilization can be risky even when stable; small disruptions create long queues.
- Use target Lq to size servers for a practical maximum average queue.
Measuring arrival rates on active sites
Use gate logs, turnstile scans, or truck manifests to estimate λ by time block. Separate normal periods from peaks at shift start, pour windows, or delivery convoys. A practical dataset is a 15‑minute tally converted to per‑hour rates, then stress‑tested using a peak factor. Exclude atypical stoppages and document weather or access events consistently.
Estimating service capacity per station
Measure average service completions per hour for one server, μ, using a time study that includes setup, paperwork, and safety checks. If breaks, access constraints, or rework reduce throughput, apply an efficiency factor. Compute μ from median cycle time to reduce outliers. For example: μ=24/h baseline, efficiency 0.90 → μeff=21.6/h.
Interpreting queue length and wait time outputs
Lq is the average number waiting, while Wq is the average waiting time. When utilization ρ approaches 1, both values rise sharply, so small disruptions can create long lines. Use W to estimate total cycle time through the station for schedule coordination. If Wq exceeds allowable idle time for trucks or crews, reschedule deliveries daily.
Sizing parallel resources with M/M/c
For multiple servers, c represents parallel lanes, operators, or inspection bays. Erlang C estimates the probability of waiting and the average queue length. If your target is “no more than two trucks waiting,” set target Lq=2 and increase c until the target is met. Validate c against space, safety spacing, and communication limits.
Example operational scenario for planning
Example inputs (per hour): λ=18, μ=24, c=2, peak=1.20, efficiency=0.90. Effective rates become λeff=21.6 and μeff=21.6, producing ρ≈0.50 and typically short queues. Compare against a peak convoy where λeff rises to 30 to see if an extra lane is justified.
FAQs
1. What does Lq represent on a jobsite?
Lq is the average number of vehicles or tasks waiting, excluding those being served. It helps estimate space needed for stacking lanes, staging areas, or holding zones near the station.
2. What is the difference between L and Lq?
L counts everyone in the system: waiting plus currently being served. Lq counts only the waiting line. Use L for total workload visibility and Lq for congestion and safety spacing.
3. How do I pick the correct time unit?
Choose a unit that matches your data collection. If you count arrivals per hour, keep μ per hour as well. The calculator works as long as λ and μ share the same unit.
4. What is utilization ρ and why does it matter?
ρ is the fraction of capacity being used. As ρ approaches 1, queues and delays rise quickly. Keeping ρ comfortably below 1 improves reliability when weather, access, or rework reduces throughput.
5. What does “unstable” mean in the results?
Unstable means arrivals meet or exceed service capacity, so the average queue does not settle. Reduce arrivals, add servers, or increase μ by streamlining checks, adding tools, or improving layouts.
6. How should I use peak and efficiency factors?
Use peak to model bursts, like convoy deliveries or shift changes. Use efficiency to reflect slowdowns, like safety briefings or restricted access. They adjust λ and μ to realistic effective rates.
7. Can this model handle appointments or scheduled trucks?
It assumes random arrivals and service times, so it is best for mixed, unscheduled flows. For appointment systems, use it as a conservative stress test, then validate with field observations and dispatch rules.