Information Technology Reference
In-Depth Information
More implicitly:
⎧
⎨
1
1
ω
1
×
min
{
;
}
α
N
+1
1
−α
N
+
λ
1
α
N
+1
1
−α
N
+
α×
1
−
1
−
λ
μ
μ
1
2
if
α
=1
ψ
=
(16)
⎩
1
1
ω
×
min
{
;
2
}
+
λ
μ
1
+
λ
2
μ
N
N
+1
N
N
+1
if
α
=1
4.2 General Model with
1) Buffers
The two-machine-one-buffer presented above is used as a building block to anal-
yse larger production lines. Therefore, the states of each intermediate buffer
B
j
are analyzed using a dedicated birth-death Markov process. These Markov pro-
cesses differ in terms of number of states because the buffers are not identical.
They also differ in terms of birth and death transition rates because each buffer
B
j
is differently influenced by the machines and the other buffers (see Figure 2).
K
Machines and (
K −
Fig. 2.
Sub-system of the original production line
In the simple case of two machines and one buffer, each available machine
processes
ω
i
products per time unit. For this reason,
ω
1
and
ω
2
are, respectively,
the birth and death transition rates. But in the general case, the machines
M
i
and
M
i
+1
related to the buffer
B
j
are subject to starvation and blockage. So
their effective processing rates are affected by the availabilities of the buffers
B
j−
1
and
B
j
+1
. The upstream machine
M
i
can process products if the buffer
B
i−
1
is not empty and the downstream machine
M
i
+1
can process products
when the buffer
B
j
+1
is not full.
The first buffer and the last one should be considered as particular cases
because the first machine cannot be starved and the last machine cannot be
blocked. The birth-death Markov process, related to the buffer
B
j
, is represented
in Figure 3.
The different states of the (
K
1) related but
different birth-death Markov processes. Each stochastic process is defined by its
processing rates ratio
α
j
. The different ratios
α
j
are defined by the following
equations:
−
1) buffers are modeled by (
K
−
P
N
2
)
α
1
=
ω
1
×
(1
−
.
(17)
ω
2
