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Queueing Delay (M/M/1) Calculator

Work out queueing delay (m/m/1) instantly with clear inputs, formula shown and shareable results.

About Queueing Delay (M/M/1) Calculator

For an M/M/1 queue with arrival rate lambda and service rate mu, total time in system is 1/(mu - lambda) and queueing delay is rho/(mu - lambda). The key behaviour is the denominator: as lambda approaches mu the delay diverges, so a system at 80 percent utilisation waits four times its service time and at 95 percent nineteen times.

Formula

M/M/1 results

rho = lambda/mu; W = 1/(mu - lambda); Wq = rho/(mu - lambda); L = rho/(1 - rho); p-th percentile of W = ln(1/(1-p))/(mu - lambda)

Worked example

With the default values below, Queueing Delay (M/M/1) Calculator returns Average queueing delay of 40.00. Change any input above and every figure here is recalculated in your browser.

Inputs used

Arrival rateminimum 0.001 req/s
80 req/s
Service rateminimum 0.001 req/s
100 req/s
Percentile to reportaccepts 50–99.99 %
95 %

Results

Average queueing delay
40.00 ms
Average total time in system
50.00 ms
Average items in system
4.00
Response time at the chosen percentile
149.79 ms

Step by step

  1. 1Values used
  2. 2M/M/1 results
  3. 3Average queueing delay= 40.00 ms
  4. 4Average total time in system= 50.00 ms
  5. 5Average items in system= 4.00
  6. 6Response time at the chosen percentile= 149.79 ms

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 2 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Is M/M/1 realistic for a web service?

It is a simplification — real service times are not exponential and there is usually more than one server — but it captures the crucial non-linear relationship between utilisation and delay.

Why does the percentile grow with the same denominator?

Because response time in M/M/1 is exponentially distributed with rate mu - lambda, so every percentile scales with 1/(mu - lambda).

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