Page 81 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

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JWBK119-05
66 August 31, 2006 2:55 Fortifying Six Sigma with OR/MS Tools
¯
waiting times of an M/M/s queuing system (W M/M/s ):
i
L ¯ M/M/s
M/M/s
¯
i
W =
i
λ i
where the mean queue length of the M/M/s system, L ¯ M/M/s , is given by:
i
s i
(s i ρ i ) ρ i P i0
L ¯ M/M/s = 2 .
i
s i !(1 − ρ i )
In this expression P i0 is the probability of station i begin empty,

−1

s i −1 n
(s i ρ i ) (s i ρ i ) s i
P i0 = + ,
n=0 n! s i !(1 − ρ i )
and ρ i is the utilization of service station i, this time multiple servers,

λ i
ρ i = .
s i μ i
Next, we compute the mean waiting times of an M/D/s queuing system, that is,
¯
for s servers with constant (i.e deterministic) service times (W i M/D/s ), as follows:

1 1 M/M/s
M/D/s
¯
¯
W = W ,
i i
2 K i
where
√ −1
4 + 5s i − 2

K i = 1 + (1 − ρ i )(s i − 1) .
16ρ i s i
The approximate mean waiting times for an M/G/s system can finally be computed
from:

¯
¯
¯
2

W M/G/s ≈ CV W M/M/s + 1 − CV 2 W M/D/s .
i i i i i
Figure 5.6 shows the difference in mean waiting times computed with and without
exponential service times assumptions. It was observed that the mean total waiting
time was higher when service times were assumed to be exponentially distributed
than to when no such assumption was made. In many service processes, the exponen-
tial assumption of the distribution of service times usually results in more conservative
queuing system designs. This is because, given the same system configurations, the