Summary of Results for Other Queueing Models

The reader should remember that as well as the results displayed in this table for each queueing model, the general results for all queueing models also apply. The results for the model M / M / 1, developed in more detail previously, are also summarised. The reader may find it useful to refer to the section on notation when reading this table.

In the general distribution models below, 1 over Mu denotes the mean of the distribution and Variance the variance.
 

Kendall Notation for Queue Server Utilisation Average Customer Wait Time (Wq) Average Length of Queue (Lq)
       
M / M / 1 Rho equals lambda over mu M-M-1 Average wait time M-M-1 Lq
M / M / n Rho over N Not available as a general formula. M-M-n Lq
M / D / 1 Not available as a general formula. M-G-1 Wq M-D-1 Lq
M / D / n
Methods too complex for a simple formula. Tabulated results easily available.
M / Ek / 1   M-Ek-1 Wq M-Ek-1 Lq
M / Ek / n
Methods too complex for a simple formula. Tabulated results easily available.
M / G / 1 Not available as a general formula. M-G-1 Wq M-G-1 Lq

Historical Note: The formula for Lq (or Wq) for the M / G / 1 model is called the Pollaczek-Khintchine formula, named after it's discovers. The formula is very useful - it is remarkably simple for such a generalised model. The formula for the M / D / 1 is also derived directly from this formula - it is a special case where the variance is equal to nil.

Other models also exist, but in most other cases apart from those tabulated above, formulae are too complex to consider here. However, methods of obtaining numerical results for most queueing problems exist, even if general formulae do not. [KLEI75] covers many diverse models.
 

Last Updated: 14th June 1999
Written by: Andrew Ferrier