KEVOS® MAKEOS · Operations
Queue Wait Check (Utilisation, Chance of Waiting, Average Wait and Queue, Servers Needed)
Jobs arrive at a shared resource such as an inspection bench, a crane or a CNC programmer: how busy is it, how long do jobs wait on average, and how many servers are needed to keep the wait under my target?
Multi-server queue with random arrivals and service times (M/M/c): utilisation, the probability that an arriving job must wait (Erlang C), the average wait in the queue, the average time in the system and the average queue length (by Little’s law), and the fewest servers that keep the average wait within your target, against that target.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Arrival rate | jobs/h | Average number of jobs arriving per hour (count over a representative period). |
| Average service time per job | min | Average time one server spends on one job. |
| Number of servers | servers | Number of people or machines serving the queue in parallel. |
| Average wait accepted | min | Longest average wait in the queue you accept. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Arrival rate
- What it is
- The average number of jobs arriving at the shared resource per hour, such as parts for inspection or crane lifts requested.
- Units and format
- Enter a number in jobs/h: jobs/h. The form accepts 0.000001 jobs/h to 1000000 jobs/h. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 jobs/h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More arrivals raise utilisation, and the wait rises steeply as it nears 100 %.
- Get it right
- Count arrivals over a representative period, including busy and quiet hours.
What this means: Average service time per job
- What it is
- The average time one server spends serving one job, from the start of service to its end.
- Units and format
- Enter a number in min: minutes. The form accepts 0.000001 min to 1000000 min. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 min as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Longer service raises utilisation and the wait.
- Get it right
- Measure actual service times, not the planned or standard time.
What this means: Number of servers
- What it is
- The number of people, machines or stations serving the queue in parallel, each able to take any job.
- Units and format
- Enter a number in servers: servers. The form accepts 1 servers to 500 servers. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3 servers as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adding a server cuts the wait sharply when utilisation is high.
- Get it right
- Count only servers available at the same time on the shift being studied.
What this means: Average wait accepted
- What it is
- The longest average wait in the queue you accept before service starts, compared with the calculated wait.
- Units and format
- Enter a number in min: minutes. The form accepts 0 min to 1000000 min. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 15 min as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Only moves the pass or fail line and the servers needed.
- Get it right
- Set it from what the wait costs: idle operators, held-up machines or late orders.
Method and formulas
- Service rate μ = 60 ÷ service time per hour; offered load a = λ ÷ μ; utilisation ρ = a ÷ c (must be below 1).
- Erlang C probability of waiting P_w = [aᶜ ÷ c! × 1 ÷ (1 − ρ)] ÷ [Σ_{k=0}^{c−1} aᵏ ÷ k! + aᶜ ÷ c! × 1 ÷ (1 − ρ)] (M/M/c queue, Kleinrock).
- Average wait W_q = P_w ÷ (c μ − λ); time in system W = W_q + 1 ÷ μ; queue length L_q = λ W_q (Little’s law).
- Example: 55.556 % busy; jobs wait 4.4964 min on average; 3 server(s) meet the target.
Limitations
- Random (Poisson) arrivals and exponential service times; very regular arrivals or service give shorter waits, and bunched arrivals longer ones.
- Steady state only: a queue that builds up during a shift peak behaves worse than the daily average suggests.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Decide whether a second inspector or test bench is needed
- Show why a resource at 90 % utilisation causes long waits
- Size a tool crib, crane or maintenance crew
Who it is for
- Operations and production managers
- Industrial engineers
- Maintenance planners
What you get
- Utilisation
- Chance an arriving job waits
- Average wait, time in system and queue length
- Fewest servers for the target wait
Interpreting the result
- Waits grow sharply as utilisation approaches 100 %: going from 80 % to 95 % busy can multiply the wait several times.
References
- Kleinrock L., Queueing Systems, Volume 1: Theory (Wiley, 1975): the M/M/c queue and the Erlang C formula.
- Little J. D. C., A Proof for the Queuing Formula: L = λW, Operations Research 9 (3), 383–387 (1961).
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Explains the concept · KEVOS® original book
Queues in the test lab: utilisation, waiting and Little's law
In Prototypes and Simulation by the Numbers
What it covers: One climatic chamber at 79% utilisation: M/M/1 and Kingman waits, the utilisation curve, a peak checked by simulation, Little's law on the lab log, and a week lost to a queue.
Method explained here · KEVOS® original book
Sizing the maintenance crew
In Breakdowns and Buffers by the Numbers
What it covers: Breakdown calls from many machines as a queue: the M/M/c model and Erlang C, utilisation and waiting, Little's law, when an infinite-source queue overstates waits, and crew size.
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