Geoscience Reference
In-Depth Information
where W
q
is the expected time in queue,
D
= is the utilization ratio and
2
is
the squared coefficient of variation for service times, given by
2
D
2
2
,where
2
is the variance of service times. Each term in the expression for W
q
has an
intuitive interpretation. Recall that we are assuming Poisson arrivals, which have
coefficient of variation equal to 1, and thus the term
1
C
2
2
represents the average
squared coefficient of variation for arrival and service processes, often called the
“variability factor” (for exponential service this term equals to 1). The second term,
=.1
/ can be interpreted by recalling that is the probability that the server is
busy and thus .1
/ is the probability that an arriving demand goes straight into
service. The ratio can thus be interpreted as the length of the busy period measured
in units of the length of the free period. The last term is simply the average service
time per customer, sometimes known as the “scale effect” to recognize that as more
capacity is assigned to the system, the average service time per customer declines.
Thus
W
q
D
ŒVariability Factor
Prob system busy
Prob system free
ŒScale Effect:
(17.13)
The expression for W simply adds the expected service time to the above.
Remark
As noted earlier, two popular ways to represent the queueing system at a
given facility are as either single-server M=G=1 queue with capacity ,where is
a decision variable, or as a multi-server M=G= system where each of the servers
has capacity
0
and is the decision variable. If we set
0
D
, i.e., require both
systems to have the same processing capacity, we can ask to what extent are these
systems “equivalent”? Can the simpler M=G=1 system be used as an approximation
of harder-to-analyze M=G= one?
Equations (
17.12
)and(
17.13
) can be used to analyze the relationship between
these two systems. First note that the coefficient
of
utilization is the same when
D
0
. While no closed-form expression for W is known for the multi-server
M=G= case, a popular approximation (see e.g., Hopp and Spearman
2000
, p. 273)
is:
p
2.C1/1
1
0
1
C
2
W
D
W
q
C
1
0
C
1
1
0
;
(17.14)
2
which is very similar to (
17.12
): focusing on the expression for W
q
, we
see th
at
the only difference is that in the numerator of (
17.12
) is replaced with
p
2.C1/1
in (
17.14
). In fact, the latter approximates the probability that all servers are
bu
sy in
the M=G= system. Thus, each term in the intuitive interpretation (
17.13
)ofW
q
has
the same interpretation for both systems. The only difference in the expecte
d wai
ting
times is that M=G=1 system
is
busy more frequently (since 1>>
p
2.C1/1
),
thus yielding larger values of W
q
. On one hand, the relative difference in W
q
can be
quite large (it approaches100 %as
!
0). On the other hand, this difference should
be small when is close to 1 and waiting times in both systems are significant,
