Chapter 8
Performance analysis and design
of Bernoulli lines
Learning objectives :
Understanding the mathematical models of production lines
Understanding the impact of machine failures
Understanding the role of buffers
Able to correctly dimension buffer capacities
Textbook :
J. Li and S.M. Meerkov, Production Systems Engineering
1
Plan
• Definitions and justifications
• Two-machine Bernoulli lines
• Long Bernoulli Lines
• Continuous Improvement of Bernoulli Lines
• Constrained Improvability
• Unconstrained Improvability
2
Definitions and justifications
3
Bernoulli lines
Definition
M1
B1
M2
B2
M3
B3
M4
• A Bernoulli line is a synchronuous line with all
machines having identical cycle time.
• It is a slotted time model with time indexed t = 0, 1, 2,
...
Justification: appropriate for high volume assembly
lines.
4
Bernoulli lines
Definition
M1
B1
M2
B2
M3
B3
M4
• Machines are subject to Time Dependent Failures (TDF).
Justifications:
• For most practical cases, the difference of performance
measures with TDF and ODF models is within 1% - 3%
(especially when buffers are not too small).
• The error resulting from the selection of failure model is
small with respect to usual errors in identification of
reliability parameters (rarelly known with accuracy better
than 5% - 10%.
• The TDF model is simpler for analysis
5
Bernoulli lines
Definition
M1
B1
M2
B2
M3
B3
M4
• Each machine is characterized by a Bernoulli reliability model.
• At the beginning of each time slot,
─ the status of each machine Mi - UP or DOWN - is determined by a
chance experiment.
─ It is UP with proba pi and DOWN with proba 1-pi, independent of
its status in all previous time slots and independent of the status of
remaining system.
Justification:
• It is practical for describing assembly operations where the downtime
is typically very short and comparable with the cycle time of the
machine.
6
Bernoulli lines
Operating rules
M1
B1
M2
B2
M3
B3
M4
• A Bernoulli line can be represented by a vector
(p1, ..., pM, N1, ..., NM-1)
of machine reliability parameters and buffer capacities.
• The time is slotted with the cycle time t of the machines.
• The status of each machine is determined at the beginning and the
state of the buffers at the end of each time slot.
• The status of a machine is UP with proba pi and DOWN with
proba (1-pi) and it is independent of past history and the status of
the remaining system
7
Bernoulli lines
Operating rules
M1
B1
M2
B2
M3
B3
M4
• Blocking Before Service:
─ an UP machine is blocked if its downstream buffer is full at
the end of previous time slot and the downstream machine
cannot produce.
─ It is starved if its upstream buffer is empty at the end of the
previous time slot.
• At the end of a time slot, an UP machine that is neither blocked
nor starved removes one part from its upstream buffer and adds
one part in its downstream buffer.
• The first machine is never starved; the last machine is never
blocked.
8
Transformation of
a failure-prone line into a Bernoulli line
M1
B1
M2
B2
M3
B3
M4
• A failure-prone line with parameters :
ti = 1/Ui, li, mi, hi
• Bernoulli Line transformation
t = min{ti, "i}
pi = tei/ti, with ei = 1/(1+ li/mi)
Ni = min{himiti+1, himi+1ti} + 1
Justifications:
• From numerical results with real data, the error between the two
models is quite small (less than 4%) for the case Ni ≥ 2 and is up to
7% - 8% for the case Ni < 2.
• The theory and results work for fractional buffer sizes as well.
9
Transformation of
a failure-prone line into a Bernoulli line
M1
B1
M2
B2
M3
B3
M4
Why Ni = Ni = min{himiti+1, himi+1ti} +1:
• A Bernoulli buffer can prevent starvation of the downstream
machine and the blockage of upstream machine for a
number of time slots at most equal to Ni
• hiti+1 = largest time during which the downstream machine
is protected from failure of upstream machine
• himiti+1= fraction of average downtime of the upstream
machine that can be accommodated by the buffer.
• himi+1ti= fraction of average downtime of the downstream
machine that can be accommodated by the buffer.
• Fractional buffer sizes are allowed in this chapter
10
Transformation of
a failure-prone line into a Bernoulli line
Examples to work out:
Line 1
e = {0.867; 0.852; 0.925; 0.895; 0.943; 0.897; 0.892; 0.935; 0.903; 0.870};
Tdown = {14.23; 16.89; 18.83; 16.08; 7.65; 11.09; 19.05; 18.76; 11.15; 18.42};
N = {7.026; 17.350; 33.461; 5.345; 9.861; 12.097; 11.955; 26.133; 14.527};
U = {1.950; 1.231; 1.607; 1.486; 1.891; 1.762; 1.457; 1.019; 1.822; 1.445}.
Line 2
e = {0.945; 0.873; 0.911; 0.899; 0.939; 0.926; 0.896; 0.852; 0.932; 0.895};
Tdown = {14.22; 16.89; 18.83; 16.08; 7.65; 11.09; 19.05; 18.76; 11.15; 18.42};
N = {5.535; 31.138; 20.578; 37.614; 21.310; 19.653; 34.618; 23.380; 12.093};
U = {1.672; 1.838; 1.020; 1.681; 1.380; 1.832; 1.503; 1.709; 1.429; 1.305}.
Line 3
e = {0.869; 0.869; 0.918; 0.880; 0.904; 0.865; 0.920; 0.888; 0.936; 0.935};
Tdown = {13.91; 12.45; 18.48; 17.33; 14.68; 17.27; 14.90; 10.13; 9.35; 10.12};
N = {26.746; 32.819; 38.490; 23.291; 35.805; 11.054; 39.291; 14.501; 13.832};
U = {1.534; 1.727; 1.309; 1.839; 1.568; 1.370; 1.703; 1.547; 1.445; 1.695}.
11
Two-machine Bernoulli lines
12
DTMC model
p1
N>0
p2
M1
B
M2
States of the system:
• Bernoulli machines are memoryless
• System state = Buffer state xn at the end of time slot n
• xn is a discrete time Markov chain
State transition diagram
p01 p11
p00
0
pN-2,N-1
p12
…
1
p10
pN-1,N-1
p21
N-1
pN-1,N-2
pNN
pN-1,N
N
pN,N-1
13
DTMC model
p1
N
p2
M1
B
M2
Blockage of M1 in period n+1
• xn = N
• M1 is UP
• M2 is DOWN
Starvation of M2 in period n+1
• xn = 0
• M2 is UP
pN-1,N-1
p01 p11
p00
0
…
1
p10
pN-2,N-1
p12
p21
N-1
pN-1,N-2
pNN
pN-1,N
N
pN,N-1
14
DTMC model
Transition probabilities
p1
N
p2
M1
B
M2
p00 = 1 - p1
pii = p1p2 + (1 - p1) (1 - p2)
p01 = p1
pi,i+1 = p1(1 - p2),
p10 = (1 - p1)p2
pi+1,i = (1 - p1)p2
i = 1, ..., N-1
pNN = p1p2 + (1 - p1) (1 - p2) + p1(1 - p2) = p1p2 + 1 - p2
pN-1,N-1
p01 p11
p00
0
…
1
p10
pN-2,N-1
p12
p21
N-1
pN-1,N-2
pNN
pN-1,N
N
pN,N-1
15
DTMC model
Steady state distribution
p1
N
p2
M1
B
M2
Equilibrium equation
• states {0,1, ..., i} : pi+1pi+1,i = pipi,i+1, " i < N
Normalization equation
• p0+ p1 + ... + pN = 1
pN-1,N-1
p01 p11
p00
0
…
1
p10
pN-2,N-1
p12
p21
N-1
pN-1,N-2
pNN
pN-1,N
N
pN,N-1
16
DTMC model
Steady state distribution
p1
N
p2
M1
B
M2
To be shown :
p0 
pi 
1  p2
1  p 2      ...  
2

i
1  p2
p 0 ,"i  0
  p1 , p 2  
p01 p11
p00
0
p1  1  p 2 
p 2  1  p1 
pN-1,N-1
pN-2,N-1
p12
…
1
p10
N
p21
N-1
pN-1,N-2
pNN
pN-1,N
N
pN,N-1
17
DTMC model
Steady state distribution
p1
N
p2
M1
B
M2
Case of identical machines, p1 = p2 = p
p0 
pi 
1 p
For practical case with p  1,
p0  0
pi  1/N, "i > 0
N 1 p
1
N 1 p
,"i  0
  p1 , p 2   1
pN-1,N-1
p01 p11
p00
0
…
1
p10
pN-2,N-1
p12
p21
N-1
pN-1,N-2
pNN
pN-1,N
N
pN,N-1
18
DTMC model
Steady state distribution
p1
N
p2
M1
B
M2
Case of nonidentical machines, i.e. p1 ≠p2
p0 
1 
p1   1  
1
p1


N
p2
pN-1,N-1
p01 p11
p00
0
…
1
p10
pN-2,N-1
p12
p21
N-1
pN-1,N-2
pNN
pN-1,N
N
pN,N-1
19
DTMC model
Steady state distribution
p1 = 0.8, p2 = 0.82, N = 5
p1 = 0.6, p2 = 0.9, N = 5
p1 = 0.82, p2 = 0.8, N = 5
p1 = 0.9, p2 = 0.6, N = 5
20
DTMC model
Steady state distribution
Theorem: Function Q(x, y, N) defined below, with 0<x<1,
0<y<1, and N ≥ 1, takes values on (0,1) and is
• strictly decreasing in x,
• strictly increasing in y
• strictly decreasing in N
where
 1  x  1    x , y  
,

x
N

 1    x, y 
Q  x, y, N   
y

 1 x ,

 N 1 x
if x  y
if x  y
21
DTMC model
Steady state distribution
Theorem:
• p0 = Q(p1, p2, N)
• pN = Q(p2, p1, N)/(1-p2)
• (y, x) = 1/(x, y)
Meaning of Q(p1, p2, N) :
The intermediate buffer is empty
Implication : M2 is starved if it is UP
Meaning of Q(p2, p1, N) :
The intermediate buffer is full & its downstream machine
does not produce
Implication : M1 is blocked if it is UP
22
DTMC model
Performance measures
Production rate (PR)
• PR = p2(1 - p0)
• PR = p1(1 - pN(1-p2))
• PR = p2(1 - Q(p1, p2, N))
• PR = p1(1 - Q(p2, p1, N))
23
DTMC model
Performance measures
Work In Process (WIP)
N
W IP 

i 1
 N  N  1
,
if p1  p 2  p

2N 1 p
ip i  
N

1


 p1 , p 2 
p

1
 N
 p  p  N  p , p   1  p , p 

1
2
1
1
2
 2
N


p1 , p 2   , otherw ise

24
DTMC model
Blockage and Starvation
Blocking probability of M1 (BL1)
BL1 = p1pN(1-p2) = p1 Q(p2, p1, N)
Starvation probability of M2 (ST2)
ST2 = p2p0 = p2 Q(p1, p2, N)
Relation with PR
PR = p1 - BL1
PR = p2 - ST2
25
DTMC model
L1: p1 = p2 = 0.9
L2: p1 = 0.9, p2 = 0.7
L3: p1 = 0.7, p2 = 0.9
26
DTMC model
Theorem:
lim P R  m in ( p1 , p 2 )
N

 ,
p1  p 2

 p1  1  p1 
lim W IP  
,
N
 p 2  p1

N
, p1  p 2
 lim
N



2
p2 

 p1 

lim B L1 
 p1 
lim S T 2 
 p2
N
N
p1  p 2
27
Long Bernoulli Lines
28
DTMC model
p1
N1
p2
N2
p3
N3
p4
M1
B1
M2
B2
M3
B3
M4
• The vector of buffer states
(x1(n), x2(n), ..., xM-1(n))
is a discrete time Markov chain.
• Unfortunately, the state space is large with (N1+1) (N2+1)...
(NM-1+1) states.
• Analytical formula are not available for performance
measures of long Bernoulli lines.
• Focus on an aggregation approach.
29
Idea of the aggregation
Backward aggregration
M1
B1
M2
B2
M3
M1
B1
M2
B2
Mb 3
M1
B1
Mb 2
B3
M4
• pb3 = production rate of the 2-machine
line (M3, B3, M4)
• Repeating the aggregation process
Mb
1
• pbi = production rate of the 2-machine
line (Mi, Bi, Mbi+1)
• Drawback : is quite different from
the production rate of the M-machine
line
30
Idea of the aggregation
Forward aggreation
M1
B1
Mb 2
Mf 2
• Forward aggreation is introduced to
improve the aggregration.
• pfi is determine to take into account
the starvation of Bi-1 in the 2machine line (Mfi-1, Bi-1, Mbi)
B2
Mb 3
Mf 3
B3
Mb 4
Mf 4
• The whole process repeats to futher
improved the aggregation
31
Aggregation procedure
Formal definition
The recursive aggregation procedure is as follow (Why?)
pi

b

f
b
 s  1 
p i 1  Q  pi 1  s  1 , p i
f
 s  1 
p i 1  Q  p i 1  s  1  , p i
pi
f
b
 s  , Ni  , i  M
 1, ...,1
 s  1  , N i 1   , i  2, ..., M
with initial condition
pi
f
0 
p i , i  1, ..., M
and boundary conditions
f
p1
p
b
M
s 
s 
p1
pM
 1  x  1    x , y  
,

x
N

 1    x, y 
Q  x, y, N   
y

 1 x ,

 N 1 x
if x  y
if x  y
32
Aggregation procedure
Example to workout with Excel
A 3-machine line L = (0.9, 0.9, 0.9, 2, 2)

1   p 1  Q  p 1  , p  0  , N
1   p 1  Q  p 1  , p 1  , N
1   p 1  Q  p 1  , p 1  , N

   0.9 1  Q  0.8571, 0.9, 2    0.8257
   0.9 1  Q  0.9, 0.8571, 2    0.8670
   0.9 1  Q  0.8670, 0.9, 2    0.8333
p 2  1   p 2 1  Q  p 3  1  , p 2  0  , N 2   0.9 1  Q  0.9, 0.9, 2    0.8571
b
b
p1
f
p2
f
p3
1
b
f
b
2
f
1
1
2
f
1
b
2
2
3
f
2
b
3
2


 2   p 1  Q  p  2  , p 1  , N    0.9 1  Q  0.8650, 0.9, 2    0.8318
 2   p 1  Q  p  2  , p  2  , N    0.9 1  Q  0.9, 0.8650, 2    0.8654
 2   p 1  Q  p  2  , p  2  , N    0.9 1  Q  0.8654, 0.9, 2    0.8321
p 2  2   p 2 1  Q  p 3  2  , p 2  1  , N 2   0.9 1  Q  0.9, 0.8670, 2    0.8650
b
b
p1
f
p2
f
p3
...
1
b
f
b
2
f
1
1
2
f
1
b
2
2
3
f
2
b
3
2
33
Aggregation procedure
Convergence
Theorem.
Both sequence pfi(s) and pbi(s) are converging, i.e. the
following limits exist :
p i : lim p i
b
b
s 
s,
p i : lim p i
f
s 
f
s
For each i, the sequence pfi(s) is monotonically decreasing
and the sequence pfi(s) is monotonically increasing.
Moreover,
p1  p M
b
f
Interpretation
b
pi
pi
f
the downstream subline of buffer Bi-1
the upstream subline of buffer Bi
34
Aggregation procedure
Exercice
L1 : (0.9, 0.9, 0.9, 0.9, 0.9; 3, 3, 3, 3)
L2 : (0.7; 0.75; 0.8; 0.85; 0.9; 3, 3, 3, 3)
L3: (0.7; 0.85; 0.9; 0.85; 0.7; 3, 3, 3, 3)
L4: (0.9; 0.85; 0.7; 0.85; 0.9; 3, 3, 3, 3)
How the production capacity is distributed in above lines?
35
Aggregation procedure
Performance measures
Production rate estimation:
b
f
p1 or p M
WIP estimation
estimated directly for the corresponding 2-machine line
 M i ,B i , M i 1 
f
b
Blockage estimation
B Li  p i Q  p i  1 , p i , N i  , p i  p i  B Li
b
f
b
Starvation estimation
STi  p i Q  p i  1 , p i , N i  1  , p i  p i  STi
f
b
f
36
Aggregation procedure
Numerical evidence on the accuracy of the estimates
• In general, the PR estimate is relatively accurate with the
error within 1% for most cases and 3% for the largest
error
• The accuracy of WIP, ST and BL estimates is typically
lower
• The highest accuracy of all estimates is for the uniform
machine efficiency pattern
• The lowest accuracy is for the inverted bowl and
"oscillating" pattern
37
Aggregation procedure
Home work examples
Eight 5-machines with with identical buffer capacity Ni = N
varying from 1 to 20
L1 : p = [0.9; 0.9; 0.9; 0.9; 0.9] :uniform pattern
L2 : p = [0.9; 0.85; 0.8; 0.75; 0.7] : decreasing efficiency
L3 : p = [0.7; 0.75; 0.8; 0.85; 0.9] : increasing efficiency
L4 : p = [0.9; 0.85; 0.7; 0.85; 0.9] : bowl pattern
L5 : p = [0.7; 0.85; 0.9; 0.85; 0.7] : inverted bowl pattern
L6 : p = [0.7; 0.9; 0.7; 0.9; 0.7] : oscillating
L7 : p = [0.9; 0.7; 0.9; 0.7; 0.9] : oscillating
L8 : p = [0.75; 0.75; 0.95; 0.75; 0.75] : single bottleneck
38
Aggregation procedure
Properties
Static law of production systems

p i  p i 1  Q  p i 1 , p i , N i 
b
b

f

p i  p i 1  Q  p i 1 , p i , N i 1 
f
f
b

Monotonicity :
The production rate PR(p1, ..., pM, N1, ..., NM-1) is
• strictly increasing in Ni
• strictly increasing in pi
39
Aggregation procedure
Properties
Reversibility : Consider a line L and its reverse Lr with
opposite flow direction. Then,
PR
L
 PR
Lr
, B L i  ST M  i  1
L
Lr
Implications:
1.
More capacity at the end of line is not appropriate for buffer capacity
assignment
2.
If only one buffer is possible and all machines are identical, then it
should be in the middle of the line
3.
If all machines are identical and a total buffering capacity N* must be
allocated, reversibility implies "symmetric assignment".
4.
For 3/, the optimal buffer assignment is of the "inverted bowl" pattern.
However, the difference with respect to "equal capacity" assignment is
not significant.
40
Continuous Improvement
of Bernoulli Lines
41
Two improvability concepts
Constrained improvability :
Can a production system be improved by redistributing its
limited buffer capacity and workforce resources?
Unconstrained Improbability :
Identify the bottleneck resource (buffer capacity or machine
capability) such that its improvement best improves the system?
42
Constrained
Improvability
43
Resource constraints
Buffer capacity constraint (BC):
M 1

Ni  N *
i 1
Workforce constraints (WF):
M

pi  p *
i 1
Production rates of the machines depend on workforce
assignment
44
Definitions
Definition: A Bernoulli line is
• improvable wrt BC if there exists a buffer assignment N'i
such that SiN'i = N* and
PR(p1, ..., pM, N'1, ..., N'M-1) > PR(p1, ..., pM, N1, ..., NM-1)
• improvable wrt WF if there exists a workforce assignment p'i
such that Pi p'i = p* and
PR(p'1, ..., p'M, N1, ..., NM-1) > PR(p1, ..., pM, N1, ..., NM-1)
• improvable wrt BC and WF simultaneously if there exist
sequences N'i and p'i such that Si N'i = N*, Pi p'i = p* and
PR(p'1, ..., p'M, N'1, ..., N'M-1) > PR(p1, ..., pM, N1, ..., NM-1)
45
Improvability with respec to WF
Theorem: A Bernoulli line is unimprovable wrt WF iff
p i  p i  1 , i  1, ..., M  1
f
b
(W F 1)
where p f , p b are the steady states of the recursive aggregation
i
i 1
procedure.
Corollary. Under condition (WF1),
W IPi 
N i  N i  1
2  N i  1  pi
f

which implies
Ni
2
 W IPi 
Ni 1
2
, "i
Half buffer capacity usage
46
Improvability with respec to WF
WF-improvability indicator:
A Bernoulli line is practically unimprovable wrt workforce if
each buffer is, on the average, close to half full.
47
WF unimprovable allocation
Unimprovable allocation
PR * 
m ax PR  p1 , ... p M , N 1 , ..., N M 1 
p i : p i  p *
i
Theorem. If Si Ni-1 ≤ M/2, then the series x(n) defined below
converges to PR* where
2
1
x  n  1   p *  M
M 1

i 1
 Ni  x n M

 , x  0   (0,1)
 Ni 1 
48
WF unimprovable allocation
Theorem. The sequence p*i such that Pip*i = p*, which renders
the line unimprovable wrt WF, is given by
 N1  1 
p 
 PR *
 N1  PR * 
*
1
 N i 1  1   N i  1 
p 

 P R *, i  2, ..., M  1
 N i 1  P R *   N i  P R * 
*
i
p
*
M
 N M 1  1 

 PR *
 N M 1  P R * 
Corollary. If all buffers are of equal capacity, i.e. Ni = N, then
p1  p M  p 2  p 3  ...  p M 1
*
*
*
*
*
which is a "flat" inverted bowl allocation.
Example : M = 5, Ni = 2, p* = 0.95. Compare with equal capacity.
49
WF continuous improvement
WF continuous improvement procedure:
• Determine WIPi, for all i
• Determine the buffer with the largest |WIPi - Ni/2|. Assume
this is buffer k
• If WIPk - Nk/2 > 0, re-allocate a sufficient small amount of
work, epk, from Mk to Mk+1; If WIPk - Nk/2 <0, re-allocate
epk+1 from Mk+1 to Mk.
• Return to step 1)
Example (home work): Continuous improvement of a 4 machine
line with Ni = 5, p* = 0.94 and e = 0.01. Initially, p = (0.9675,
0.9225, 0.8780, 0.8372)
50
Improvability wrt WF and BC simultaneously
Theorem: A Bernoulli line is unimprovable wrt WF and BC
simultaneously iff
p1  p i  p i  p M , i  2, ..., M  1 (W F & B C 1)
f
b
Corollary. Under condition (WF&BC1),
W IPi  W IP1 , i  2, ..., M  1
and, moreover
W IPi 
N  N  1
2  N  1  p1 
, "i
where N is the capacity of each buffer, i.e. equal capacity buffers.
51
Unimprovabe allocation wrt WF and BC
Unimprovable allocation
PR * * 
m ax P R  p1 , ... p M , N 1 , ..., N M  1 
N i : N i  N *
i
p i : p i  p *
i
Theorem. Let N* be a multiple of M-1. Then the series p*i and
N*i, which render the line unimprovable wrt WF and BC, are
given by
N*
N 
 N
uniform BC dist.
M 1
*
i
*
opt
p1  p M
*
*
*

N opt  1 
 *
 PR * *
 N  PR * * 
 opt

2
*

N
1 
opt
*
pi   *
 P R * *, i  2, ..., M  1
 N  PR * * 
 opt

PR** can be determined as PR* with N*i.
flat inverted
bowl WF dist.
52
Improvabiblity wrt BC
Theorem: A Bernoulli line is unimprovable wrt BC iff the
quantity

 p i f p ib  
m in p i  m in  b , f   ( B C 1)


i  1,..., M
p
p
i
i




is maximized over all sequences N'i such that SiN'i = N*.
Condition of little practical importance.
Improvabiblity wrt BC
Numerical Fact.
The production rate ensured by the buffer capacity allocation defined
by (BC1) is almost always the same as the production rate defined by
the allocation that minimizes
m ax
i  2 ,..., M  1
W IPi 1   N i  W IPi 
( BC 2)
over all sequences N'i such that SiN'i = N*.
Implication:
Bi-1
Mi
Bi
A line is practically unimprovable wrt BC if the occupancy of each
buffer Bi-1 is as close to the availability of buffer Bi as possible.
Improvabiblity wrt BC
BC continuous improvement procedure:
• Determine WIPi, for all i
• Determine the buffer with the largest |WIPi - (Ni+1 - WIPi+1)|.
Assume this is buffer k
• If WIPk - (Nk+1 - WIPk+1) > 0, transfer a unit of capacity from
Bk to Bk+1; If WIPk - (Nk+1 - WIPk+1) < 0, re-allocate a unit
from Bk+1 to Bk.
• Return to step 1)
Example (home work): Continuous improvement of a 11 machine
line with pi = 0.8, i = 6, and p6 = 0.6. N* = 24. Determine the
unimprovable buffer allocation (PR = 0.5843).
Unconstrained Improvability
Bottleneck machine
Definition:
• A machine Mi is the bottleneck machine (BN-m) of a
Bernoulli line if
P R
pi

P R
p j
, "j  i
Problems with this definition:
1/ Gradient information cannot be measured on shopfloor
2/ No analytical methods for evaluation of the gradients
Remark: gradient estimation is possible with sample path
approaches (to be addressed).
Bottleneck machine
• Machine with the smalllest pi is not always the BN-m
• Machine with the largest WIP in front is not always the BN-m
2
0.8
0.83
DPR/Dp 0.369
2
0.77
0.452
3
0.443
0.8
0.022
The best machine is the bottleneck
0.9
6
0.7
6
0.8
1
0.7
1
0.75
4
0.6
6
0.7
2
0.85
DPR/Dp 0.05
0.06
0.28
0.38
0.31
0.17
0.06
0.05
WIPi
5.59
5.39
0.87
0.69
1.68
1.18
0.66
The worst machine is not the bottleneck
Bottleneck buffer
Definition:
• A buffer Bi is the bottleneck buffer (BN-b) of a Bernoulli line if
PR  p1 , ..., p M , N 1 , ..., N i  1, ..., N M

 PR  p1 , ..., p M , N 1 , ..., N j  1, ..., N M  , " j  i
0.8
PR(Ni+1)
3
0.769
0.85
3
0.766
0.85
2
0.9
0.763
Buffer with the smallest Ni is not necessarily the BN-b
Bottlenecks in 2-machine lines
Theorem: For a 2-machine Bernoulli line,
 PR
 p1

 PR
p2
(respectively,
 PR
p2

 PR
 p1
)
if and only if
BL1 < ST2 (respectively, BL1 > ST2).
Remarks :
• The theorem reformulates partial derivatives in terms of
"measurable" and "calculable" probabilities.
• It offers the possibility to identify BN-m without knowing the
parameters of the system.
• It offers a simple graphic way of representing the BN-m.
Bottlenecks in 2-machine lines
0.9
2
0.8
STi
0
0.0215
BLi
0.1215
0
• Arrow in the direction of the
inequality of the two probability
• Arrow pointing to the BN-m
Bottlenecks in long lines
Arrow Assignment Rule:
If BLi > STi+1, assign the arrow pointing from Mi to Mi+1.
If BLi < STi+1, assign the arrow pointing from Mi+1 to Mi.
Bottlenecks in long lines
Bottleneck indicator:
• If there is a single machine with no outgoing arrows, it is the BN-m
• If there are multiple machines with no outgoing arrows, the one with
the largest severity is the Primary BN-m (PBN-m), where the severity
of each BN-m is defined by
Si = |STi+1 - BLi| + |BLi-1-STi|, i = 2, ..., M-1
S1 = |ST2 - BL1|
SM = |BLM-1-STM|
• The BN-b is the buffer immediately upstream the BN-m (or PBN-m)
if it is more often starved than blocked, or immediately downstream
the BN-m ( or PBN-m) if it is more often blocked than starved.
Remark : It was shown numerically that Bottleneck Indicator correctly
identifies the BN in most cases.
Bottlenecks in long lines
BN-m BN-b
0.9
6
0.7
6
0.8
1
0.7
1
0.75
4
0.6
6
0.7
2
0.85
STi
0
0
0
0.09
0.23
0.1
0.2
0.36
BLi
0.4
0.2
0.3
0.14
0.03
0
0.01
0
Single Bottleneck
PBN-m BN-b
0.9
2
0.5
2
BN-m
0.9
2
0.9
2
0.9
2
0.9
2
0.6
STi
0
0
0.39
0.37
0.33
027
0.11
BLi
0.41
0.01
0.03
0.05
0.1
0.17
0
Multiple Bottlenecks
2
0.9
0.41
0
Potency of buffering
Motivation:
• When the worst machine is not the BN of the system, the buffer
capacity is often incorrectly set.
• Need to assess the buffering quality
Definition : The buffering of a production system is
• weakly potent if the BN-m is the worst machine; otherwise it is
not potent
• potent if it is weakly potent and its production rate is
sufficiently close to the BN-m efficiency (i.e. within 5%)
• strongly potent if it is potent and the system has the smallest
possible total buffer capacity.