Queue Time vs Utilization Calculator Using the Kingman Formula
Queue time at a loaded machine does not grow in a straight line. It explodes as utilization approaches 100 percent. The Kingman formula, often written as variability times utilization times time, is the classic way to estimate this and to show why a fully loaded schedule is a slow one.
Your numbers
Coefficient of variation of time between arrivals. 1.0 is random arrivals, below 1 is regular.
Coefficient of variation of process time including setups and stoppages. Raw machine time is often 0.25 to 0.5, with failures and setups it is often above 1.
Your results
Kingman's formula is an approximation for a single machine with one job type. Real shops with routing variety, batching or priority rules will differ.
Queue time (hours) vs utilization
Sweeps Work center utilization from 50 to 98, holding your other inputs fixed.
Measure utilization and variability from ERP history
Optional: we can review whether your ERP and shop floor data can produce utilization and variability by work center.
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How to use it
Enter the utilization you expect from the finite or infinite load, the average process time per job and two variability measures. A coefficient of variation is standard deviation divided by mean, for arrival gaps and for process time. If you do not have data, 1.0 for both is a reasonable starting point for a job shop, and lower values suit paced lines. Then change the utilization slider and watch the queue time respond.
- Measure utilization against the capacity you really have, after downtime
- Include setups and stops in the process time variability
- Check the effect of lowering variability, not only load
- Use the same time unit for process time throughout
The formula
Queue time Wq = ((Ca squared + Cs squared) / 2) times (u / (1 - u)) times te. The first term is variability, the second is utilization, the third is the average process time. At 85 percent with both variabilities at 1 and a 2 hour process time, Wq = 1 times 5.67 times 2 = 11.3 hours. At 95 percent it becomes 38 hours. Cycle time at the center is queue plus process time.
What it tells a scheduler
Pushing a bottleneck to 98 percent on paper creates very long and unpredictable queues in practice. Cutting variability is as powerful as cutting load: halving the CVs cuts queue time to a quarter. This is why scheduled maintenance, shorter setups and stable arrivals matter, and why a modest capacity buffer at the constraint often shortens lead time.
Common mistakes
Entering utilization as the planned load and ignoring unplanned downtime understates real utilization. Using the CV of the raw cycle time while leaving out setups and failures understates process variability. And applying the result to a work center that processes many different part families without adjusting for the mix gives a number that is a guide, not a forecast.
Frequently Asked Questions
Why does queue time explode near 100 percent utilization?
Because there is no spare capacity to absorb random bursts of work, so any disruption creates a backlog that takes a long time to clear. The u / (1 - u) term grows without limit as utilization approaches 1.
How do I find the coefficient of variation?
Collect a sample of process times, or time between arrivals, compute the standard deviation and divide it by the average. Anything near 1 behaves like random events, below 0.5 is fairly regular, and above 1.5 is highly irregular.
Is Kingman's formula exact?
No, it is an approximation that works best at high utilization for a single server. It is good enough to compare scenarios and show the direction and rough size of the effect, but not to promise a specific queue time.
Does it apply to multi machine work centers?
Approximately. For several identical machines the queue is shorter than for one machine at the same utilization. Use the single machine figure as a conservative upper bound or move to a multi server queueing model.
If your ERP or MES records actual queue and run times per operation, we can calculate real utilization and variability per work center from that history.
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