2
It holds that
x
0
=
1 when
x
6
=
0.
134
12.3. Congestion: M
/
M
/
1 versus D
/
D
/
1
5.
The expected total sojourn time (queueing time
+
time in service),
W
, and the ex
pected waiting time, W
q
, can be found from application of Little’s formula as
W
=
L
λ
=
λ
/
μ
1

λ
/
μ
λ
=
1
/
μ
1

λ
/
μ
=
1
μ

λ
and
W
q
=
L
q
λ
=
(
λ
/
μ
)
2
1

λ
/
μ
λ
=
λ
/
μ
2
1

λ
/
μ
=
λ
μ
(
μ

λ
)
.
Note that W
=
W
q
+
1
μ
, see the expression (
9.1
).
12.3. Congestion: M/M/1 versus D/D/1
In an M
/
M
/
1 model the expected number of customers in the system equals
L
=
ρ
/
(
1

ρ
)
,
where
ρ
=
λ
/
μ
is the utilization of the server. If the value of
ρ
tends to the value of one
then
L
is getting larger than whatever value considered. For
ρ
≥
1 the expected number of
customers in the system (in the long run) is even infinite. The behavior of
L
as a function
of
ρ
has been sketched in Figure
12.9
. The situation for an M
/
M
/
1 model, or any model
infinity
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L
ρ
=
λ
μ
Figure 12.9.: Congestion in an M
/
M
/
1 model.
with variation in interarrival times and service times, contrasts with a situation in which
interarrival and service times are fixed. A model with fixed lengths of interarrival and
service times, and a single server, is denoted as an D
/
D
/
1 model. If the server has ample
capacity (‘speed’) then there is no waiting line in a D
/
D
/
1 model. For example, under fixed
interarrival intervals of length 15 minutes and a fixed servicetime length of 10 minutes,
each customer has left before the next one arrives. Two thirds of the time the system holds
one customer, and this consequently is the average number of customers in the system.
Two thirds is evidently also the fraction of time that the server is busy. So
L
=
ρ
and this
latter equality does not depend on the values chosen for the interarrival and service times.
The behavior of
L
as a function of
ρ
implied by
L
=
ρ
is totally different from the behavior
135
infinity
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
L
ρ
=
λ
μ
Figure 12.10.: (Absence of) Congestion in a D
/
D
/
1 model.
of
L
as a function of
ρ
in the M
/
M
/
1 model. The function
L
=
ρ
has been sketched in
Figure
12.10
.
For values of
ρ
greater than one the values for
L
are infinite in both the M
/
M
/
1 and the
D
/
D
/
1 model. If the utilization of the server is slightly less than one then the number
of customers in the system is very large in the M
/
M
/
1 model whereas the number of
customers in the system is still very small (less than one) in the D
/
D
/
1 model. In this
respect the predictions of the two models differ greatly. An M
/
M
/
1 predicts congestion
long before the server becomes fully utilized.
12.4. M/M/2
The M
/
M
/
2 models is a system with two identical parallel servers that have exponentially
distributed service times, a single waiting room with infinite capacity (while customers are
not balking), and that accommodates a single queue from which customers are served.
The arrival process is Poisson and the service times are exponentially distributed. A sketch
of the system is in Figure
12.11
. It can serve as a model for a post office with two identical
tellers or a hairdresser’s shop with two identical hairdressers.
arrivals
uncapacitated waiting room
servers
Figure 12.11.: Sketch of a queueing system with two parallel servers.
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