Urban Operations Research
Supplementary Notes
Pinker, 1994
Updated by Kang, 2001
The M/G/1 Queueing System
For
the
M/G/1
queueing
system
being
operated
under
the
FIFO
service
rule,
we
derive
the
expres
sions
of the
following
quantities
in
terms
of the
arrival
rate
λ
,
the
mean
service
time
E
[
S
],
and
the
variance
of service
time
σ
2
S
.
•
W
:
the
average
time
a
randomly
arriving
customer
will
spend
in
the
system,
which
is
com
posed
of the
waiting
time
in
the
queue
and
the
service
time.
•
L
:
the
average
number
of people
in
the
system
that
a
randomly
arriving
customer
±nds,
which
is
composed
of the
number
of people
in
the
queue
and
the
person
in
service.
•
ρ
:
the
long
run
fraction
of
time
the
server
is
busy,
which
is
equivalently
the
probability
that
the
server
is
busy
at
a
random
point
in
time.
•
B
:
the
long
run
average
duration
of a
server
busy
period.
Recall
that
the
M/G/1
queueing
system
is
a
single
server
queueing
system
in
which
the
customer
arrival
process
is
Poisson
with
rate
λ
and
the
service
time,
S
,
for
each
customer
follows
a
general
distribution
with
PDF
f
S
(
s
),
mean
E
[
S
],
and
variance
σ
2
S
.
Let
us
±rst
compute
ρ
and
B
.
∗
Suppose
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 Fall '06
 hansman
 Probability theory, Randomness, randomly arriving customer

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