Information Technology Reference
In-Depth Information
Tabl e 1.
Notation
λ
i
Equivalent failure rate of the machine
M
i
μ
i
Equivalent repair rate of the machine
M
i
ω
i
Processing rate of the machine
M
i
N
j
,j ∈{
1
...K −
1
}
Capacity of the intermediate buffer
B
j
ξ
i
,i∈{
1
...K −
1
}
Availability of the intermediate buffer
B
j
regarding machine
M
i
α
j
,j ∈{
1
...K −
1
}
Processing rates fraction related to the buffer
B
j
ρ
i
,i∈{
1
...K}
Equivalent production rate of the machine
M
i
P
j
,s∈{
0
...N
j
}
Steady probability to have
s
products in the buffer
B
j
ψ
Production line throughput
3 Single-failure Mode Transformation
In this section, we present the transformation proposed by Belmansour and
Nourelfath [2,3]. Each machine
M
i
can be defined as a single failure mode ma-
chine with an equivalent failure rate and an equivalent repair rate.
The equivalent failure rate is defined as the sum of the different rates of the
different failure modes (see Equation (1)).
F
i
∀
i
=1
...K, λ
i
=
λ
im
(1)
m
=1
Based on the assumption that the equivalent machine should have the same
availability, the equivalent repair rate is given by Equation (2).
1
∀
i
=1
...K, μ
i
=
(2)
F
m
=1
λ
i
1
μ
im
λ
im
×
Case of a machine with two failure modes:
If we consider a machine
M
i
with two failure modes represented by
λ
i
1
,λ
i
2
and
μ
i
1
,μ
i
2
, this machine corresponds to a single failure mode machine with
failure
λ
i
and repair rates
μ
i
such as both machines have the same availability
asshowninEquation(3).
1
1+
λ
μ
i
1
1+
λ
i
1
μ
i
1
=
(3)
+
λ
i
2
μ
i
2
BasedonEquation(3)weobtain:
1
μ
i
=
(4)
λ
i
1
1
μ
i
1
λ
i
2
1
μ
i
2
(
λ
i
1
+
λ
i
2
×
)+(
λ
i
1
+
λ
i
2
×
)
