Enrichment · STEM Explorations · Probability

One more register,
and the lunch line
almost disappears.

Type how many people arrive an hour, how long each takes and how many servers are open. The calculator gives the average wait, the line length and the fewest servers that meet your target.

Model a lineWhat you get

The queueing calculator

The worked example is three cafeteria registers serving 180 students an hour at 45 seconds each. Nothing you type leaves this browser.

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What you get

Enrichment · STEM ExplorationsFree

What you finish with
Utilisation, the chance of waiting, average wait and line length for the servers you have, the fewest servers that meet the target, and a staffing table. It prints as a summary and downloads as a spreadsheet.
Who it is for
Statistics, math and computer science students learning probability models, and student organisers planning lines at events.
Format
Browser calculator for a phone, tablet or computer, by touch or keyboard. Prints a summary; downloads a spreadsheet (.csv); export and import a file.
Access
Free. No account, no sign-in and nothing to install. Everything you type stays in this browser; nothing is sent to Caveat.
Where it lives
The STEM Explorations space in Caveat Enrichment.
Support
Questions, a correction or a request: team@caveatacademic.com.

How to use it

  1. Type the arrivals per hour and the average service time (time a few at the real counter).
  2. Type the servers open and your target average wait.
  3. Read the wait, the line and the servers needed.
  4. Compare staffing levels in the table.
  5. Print the summary for a project or a planning meeting.

Model a line

How the wait is worked out

The M/M/c queue and Erlang C.

The worked example: three registers at lunchOffered load2.25180 an hour × 0.75 min ÷ 60Erlang C, 3 servers56.8%chance an arrival waitsAverage wait34 s0.568 ÷ (3 × 80 − 180) per hourIn line1.70180 × wait (Little's law)The worked example: three registers at lunchOffered load2.25180 an hour × 0.75 min ÷ 60Erlang C, 3 servers56.8%chance an arrival waitsAverage wait34 s0.568 ÷ (3 × 80 − 180) per hourIn line1.70180 × wait (Little's law)
Waiting is driven by how close the servers are to full use. At 75% busy, three registers keep the line short; at two they would never catch up.

The model

With arrival rate λ, service rate μ per server (60 ÷ service minutes an hour) and c servers, the offered load is a = λ ÷ μ and utilisation is a ÷ c, which must be under 1 or the line grows forever. Erlang C gives the chance an arrival has to wait. The average wait is that chance ÷ (cμ − λ), the chance of waiting longer than t is that chance × e^(−(cμ − λ)t), and Little's law gives the average number waiting as λ × average wait.

When it fits

The model assumes arrivals at random (a Poisson process), service times that vary at random around their average (exponential), one shared line, and nobody leaving it. Lines fed by a bell or a bus arrive in bursts, and very regular service makes waits shorter than this. The results are averages over a steady busy period.

Sources

  1. Hillier, F. S., and Lieberman, G. J. Introduction to Operations Research. McGraw-Hill: chapter on queueing theory, the M/M/s model and Erlang C.
  2. Little, J. D. C. (1961). A proof for the queuing formula L = λW. Operations Research 9(3), 383 to 387.
  3. National Institute of Standards and Technology. NIST/SEMATECH e-Handbook of Statistical Methods: the Poisson and exponential distributions (itl.nist.gov/div898/handbook).
  4. Massachusetts Institute of Technology OpenCourseWare. Probabilistic Systems Analysis: Poisson processes and queues (ocw.mit.edu).

The worked example is made up. Results are model averages for a steady busy period. Your figures stay in this browser. Support: team@caveatacademic.com.