M/M/s Queue

The M/M/s queue has exponentially distributed interarrival and process times, and s servers. In a wafer fab, this might correspond to a workstation with highly variable arrivals and highly variable process times, with multiple identical tools in the workstation. We should note that processing times in a wafer fab are not usually variable enough to follow an exponential distribution. However, the M/M/s formulas are a good place to start, and can apply in some cases.

Let λ be the arrival rate, μ the service rate, and ρ = λ/(s μ) the traffic intensity (server loading). For either first-in-first-out (FIFO), or last-in-first-out (LIFO) dispatch rules, the mean of the long-term workstation cycle time distribution is given by

  • υs (s μ (1 - ρ)2)-1 + μ-1,
  • where
  • us = (λ/μ)s (s!)-1 [ ( Σ (λ/μ)i (i!)-1) + (λ/μ)s (s!(1 - ρ))-1]-1,
  • where the summation runs from 0 to (s -1). When the dispatch rule is FIFO, the variance of the limiting cycle time distribution is given by
  • υs((s μ)2 (1 - ρ)4)-1 (2 - 2ρ - us) + μ-2.
  • When the rule is LIFO, the variance is given by
  • υs((s μ)2 (1 - ρ)4)-1 (2 - υs) + μ-2.
  • In each of these variance calculations, us is as given above.