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 l
be the arrival rate, m
the service rate, and r = l/(s
m)
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
us (s m
(1 - r)2)-1 + m-1,
where
us
= (l/m)s
(s!)-1
[ ( S
(l/m)i
(i!)-1)
+ (l/m)s
(s!(1 - r))-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 us((s
m)2 (1 - r)4)-1
(2 - 2r - us)
+ m-2.
When the rule is LIFO, the variance
is given by
us((s m)2
(1 - r)4)-1 (2
- us) + m-2.
In each of these variance calculations, us
is as given above. |
