CS4 Modelling and Simulation
LN10
10
Solving Queueing Models
10.1
Introduction
In this note we look at the solution of systems of queues, starting with simple isolated
queues.
The benefits of using predefined, easily classified queues will become appar
ent: many performance measures can be calculated directly from the parameters of the
model. Obviously the situation becomes more complicated when queues are connected to
gether. However we see that even in this case deriving performance measures can be very
straightforward as we are able to consider each queue in isolation. Finally we summarise
the assumptions which we need to make in order to obtain these solutions.
10.2
Single Queues
Since we assume that all the queues which we consider have a Markovian arrival process
and a Markovian service process the queue can be modelled as a Markov process. The state
transition diagram for a singleserver queue with infinite capacity is shown in Figure 22.
Note that unlike the Markov processes which we have considered earlier in the course this
process has an infinite state space.
x
0
x
1
x
3
x
2
. . .
µ
µ
µ
µ
λ
λ
λ
λ
Figure 22: The state transition diagram for a simple
M/M/
1 queue
If we write the global balance equations for this system we can soon recognise a regular
pattern emerging.
λ
π
0
=
µ
π
1
(
λ
+
µ
)
π
1
=
λ
π
0
+
µ
π
2
(
λ
+
µ
)
π
2
=
λ
π
1
+
µ
π
3
.
.
.
Using simple algebra we can rewrite these as shown below:
π
1
=
λ
µ
π
0
π
2
=
λ
µ
π
1
=
λ
µ
2
π
0
π
3
=
λ
µ
π
2
=
λ
µ
3
π
0
.
.
.
68
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
CS4 Modelling and Simulation
LN10
Remembering that
traﬃc intensity
,
ρ
is defined to be
λ/µ
(for the single server case,
λ/
(
c
×
µ
) in the general case) we can see that for an arbitrary state
x
i
π
=
ρ
i
π
0
and using the normalisation condition we see that
1 =
∞
i
=0
π
=
π
0
∞
i
=0
ρ
i
=
π
0
1
1
−
ρ
if
ρ <
1
From this we can deduce that
π
0
= 1
−
ρ
and
π
i
= (1
−
ρ
)
ρ
i
for all
i >
0.
Thus we have a symbolic evaluation of the steady state distribution for any
M/M/
1
queue, i.e. the steady state distribution expressed in terms of the parameters of the model,
λ
and
µ
. This means that for any particular model we can find the steady state distribution
simply by evaluating the expressions above with our particular value of
ρ
=
λ/µ
. More
importantly, we can derive symbolic expressions for the important performance measures
we are likely to want to derive from a queue, and then evaluate those measures for a
particular model
without having to carry out any numerical solution of global balance
equations
.
Utilisation,
U
The queue is being utilised whenever it is nonempty; in other words
the utilisation,
U
, is 1
−
π
0
.
Utilisation
U
=
ρ
Mean number of customers in the queue,
N
This is the expectation of the number
of customers in the service facility as a whole, i.e.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 MohammadAbdolahiAzgomiPh.D
 Poisson Distribution, Steady State, Queueing theory, Service Centre

Click to edit the document details