Information Technology Reference
In-Depth Information
Fig. 3.
Birth-death Markov process related to the buffer
B
j
P
K−
1
0
α
K−
1
=
ω
K−
1
×
(1
−
)
.
(18)
ω
K
P
j−
0
)
ω
i
×
(1
−
∀j
=2
...K −
2
,α
j
=
)
.
(19)
P
j
+1
N
j
+1
ω
i
+1
×
(1
−
Equations (17) and (18) consider, respectively, the particular case of the first
and the last buffer because the first machine cannot be starved and the last one
cannot be blocked.
Based on the analysis presented in Section 2, the probabilities of empty and
full states of each buffer are calculated using Equations (20) and (21).
⎧
⎨
α
j
1
−α
N
j
+1
j
1
−
if
α
j
=1
P
0
=
(20)
⎩
1
N
j
+1
if
α
j
=1
⎧
⎨
α
N
j
j
×
(1
−α
j
)
1
−α
N
j
+1
j
if
α
j
=1
P
N
i
=
(21)
⎩
1
N
j
+1
if
α
j
=1
The resolution of Equations (17) to (21) allows the determination of the process-
ing rates ratio
α
j
and empty and full states probabilities (
P
0
,P
N
j
) of each buffer.
So, based on this information and the two-machine-one-buffer model presented
above, we can calculate the effective production rate of each machine considering
the influence of the buffers availabilities using Equation (22).
ξ
j
μ
i
+
ξ
j
×
μ
i
×
∀
i
=1
...K, ρ
i
=
ω
i
×
(22)
λ
i
