Throughput Evaluation of Buffered Unreliable Production Lines with Machines Having Multiple Failure Modes - Computational Logistics - page 234

In the steady state of the system, all the differential terms are equal to zero
(see Equation (9)).
⎧
⎨

0= − ω 1 × P 0 + ω 2 × P 1

0= ω 1 ×

P j− 1 −

( ω 1 + ω 2 )

× P j + ω 2 × P j +1

(9)

⎩

0= ω 1 × P N− 1 + ω 2 × P N

So, by simplifying the system above and considering the normalization
equation: j =0 P j = 1 we obtain the steady probabilities of each buffer state
(10).

⎧
⎨

α j

× (1 −α )

if α =1

1 − α N +1

P j =

(10)

⎩

1
N +1

if α =1

We are especially interested in the starvation and blockage probabilities, re-
spectively, represented by empty and full buffer states given by the following
equations:

⎧
⎨
⎩

1 −α

1 −α N +1 if α =1

P 0 = (11)

1
N +1

if α =1

⎧
⎨
⎩

α N

× (1 −α )
1 −α N +1

if α =1

P N = (12)

1
N +1

if α =1

Based on these two probabilities, the effective production rate of each work
station is defined as function of machine processing rate, machine and the buffer
availabilities (13).

ξ i
μ i + ξ i ×

μ i ×

ρ i = ω i × (13)

λ i

P 0 .
The system throughput ψ is defined as the bottleneck between the two effective
production rates ρ 1 and ρ 2 .
ψ = min ω 1 ×

Such as: ξ 1 =1 − P N and ξ 2 =1 −

μ 1 × − P N ) μ 2 × − P 0 )

(1 (1

P N ) ,ω 2 ×

(14)

μ 1 + λ 1 × (1 − μ 2 + λ 2 × (1 − P 0 )

After some transformations, the system throughput can be re-written as follows:

⎧
⎨

μ 1 × (1 −α N )
μ 1 × (1 −α N +1 )+ λ 1 × (1 −α N ) ;
μ 2 × (1 −α N )

ω 1 × {

min

α N ) } μ 2 × (1 − α N +1 )+ λ 2 × α × (1 −

if α =1

ψ = (15)

⎩

N×μ 1
N×λ 1 +( N +1) ×μ 1

N×μ 2

ω × min { ,

N×λ 2 +( N +1) ×μ 2 }

if α =1