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Transcript of 1 Chapter 8 Performance analysis and design of Bernoulli lines Learning objectives : Understanding...
1
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
2
PlanPlan
• Definitions and justifications
• Two-machine Bernoulli lines
• Long Bernoulli Lines
• Continuous Improvement of Bernoulli Lines
• Constrained Improvability
• Unconstrained Improvability
3
Definitions and justifications
4
Bernoulli linesBernoulli linesDefinitionDefinition
• 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, ...
M1 B1 M2 B2 M3 B3 M4
Justification: appropriate for high volume assembly lines.
5
Bernoulli linesBernoulli linesDefinitionDefinition
• Machines are subject to Time Dependent Failures (TDF).
M1 B1 M2 B2 M3 B3 M4
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
6
Bernoulli linesBernoulli linesDefinitionDefinition
• 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.
M1 B1 M2 B2 M3 B3 M4
Justification:
•It is practical for describing assembly operations where the downtime is typically very short and comparable with the cycle time of the machine.
7
Bernoulli linesBernoulli linesOperating rulesOperating 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 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
8
Bernoulli linesBernoulli linesOperating rulesOperating 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.
9
Transformation of Transformation of a failure-prone line into a Bernoulli linea failure-prone line into a Bernoulli line
M1 B1 M2 B2 M3 B3 M4
• A failure-prone line with parameters :
i = 1/Ui, i, i, hi
• Bernoulli Line transformation
= min{ii}
pi = ei/i, with ei = 1/(1+i/i)
Ni = min{hiii+1, hii+1i} + 1Justifications:•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.
10
Transformation of Transformation of a failure-prone line into a Bernoulli linea failure-prone line into a Bernoulli line
M1 B1 M2 B2 M3 B3 M4
Why Ni = Ni = min{hiii+1, hii+1i} +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
•hii+1 = largest time during which the downstream machine is protected from failure of upstream machine
•hiii+1= fraction of average downtime of the upstream machine that can be accommodated by the buffer.
•hii+1i= fraction of average downtime of the downstream machine that can be accommodated by the buffer.
•Fractional buffer sizes are allowed in this chapter
11
Transformation of Transformation of a failure-prone line into a Bernoulli linea failure-prone line into a Bernoulli line
Examples to work out:Line 1e = {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 2e = {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 3e = {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}.
12
Two-machine Bernoulli linesTwo-machine Bernoulli lines
13
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
M1 B M2
p1 p2N > 0
0 1 N-1 N
p01
…p12
pN-2,N-1 pN-1,N
p10 p21pN-1,N-2
pN,N-1
p00p11 pNN
pN-1,N-1
DTMC modelDTMC model
14
Blockage of M1 in period n+1•xn = N•M1 is UP•M2 is DOWN
0 1 N-1 N
p01
…p12
pN-2,N-1 pN-1,N
p10 p21pN-1,N-2
pN,N-1
p00p11 pNN
pN-1,N-1
DTMC modelDTMC model
Starvation of M2 in period n+1•xn = 0•M2 is UP
M1 B M2
p1 p2N
15
DTMC modelDTMC modelTransition probabilitiesTransition probabilities
p00 = 1 - p1
p01 = p1
p10 = (1 - p1)p2
M1 B M2
p1 p2N
0 1 N-1 N
p01
…p12
pN-2,N-1 pN-1,N
p10 p21pN-1,N-2
pN,N-1
p00p11 pNN
pN-1,N-1
pii = p1p2 + (1 - p1) (1 - p2)
pi,i+1 = p1(1 - p2), i = 1, ..., N-1
pi+1,i = (1 - p1)p2
pNN = p1p2 + (1 - p1) (1 - p2) + p1(1 - p2) = p1p2 + 1 - p2
16
DTMC modelDTMC modelSteady state distributionSteady state distribution
0 1 N-1 N
p01
…p12
pN-2,N-1 pN-1,N
p10 p21pN-1,N-2
pN,N-1
p00p11 pNN
pN-1,N-1
Equilibrium equation
•states {0,1, ..., i} : i+1pi+1,i = ipi,i+1, i < N
Normalization equation
•0+ 1 + ... + N = 1
M1 B M2
p1 p2N
17
DTMC modelDTMC modelSteady state distributionSteady state distribution
0 1 N-1 N
p01
…p12
pN-2,N-1 pN-1,N
p10 p21pN-1,N-2
pN,N-1
p00p11 pNN
pN-1,N-1
To be shown :
M1 B M2
p1 p2N
20 2
2
02
1 21 2
2 1
1
1 ...
, 01
1,
1
N
i
i
p
p
ip
p pp p
p p
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DTMC modelDTMC modelSteady state distributionSteady state distribution
0 1 N-1 N
p01
…p12
pN-2,N-1 pN-1,N
p10 p21pN-1,N-2
pN,N-1
p00p11 pNN
pN-1,N-1
Case of identical machines, p1 = p2 = p
M1 B M2
p1 p2N
0
1 2
1
1
1, 0
1
, 1
i
p
N p
iN p
p p
For practical case with p 1,0 0i 1/N, i > 0
19
DTMC modelDTMC modelSteady state distributionSteady state distribution
0 1 N-1 N
p01
…p12
pN-2,N-1 pN-1,N
p10 p21pN-1,N-2
pN,N-1
p00p11 pNN
pN-1,N-1
Case of nonidentical machines, i.e. p1 ≠p2
M1 B M2
p1 p2N
10
1
2
1 1
1 N
p
p
p
20
DTMC modelDTMC modelSteady state distributionSteady state distribution
p1 = 0.8, p2 = 0.82, N = 5 p1 = 0.82, p2 = 0.8, N = 5
p1 = 0.6, p2 = 0.9, N = 5 p1 = 0.9, p2 = 0.6, N = 5
21
DTMC modelDTMC modelSteady state distributionSteady 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 1 ,, if
1 ,, ,
1, if
1
N
x x yx y
xx y
Q x y N y
xx y
N x
22
DTMC modelDTMC modelSteady state distributionSteady state distribution
Theorem:
•0 = Q(p1, p2, N)
•N = 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
23
DTMC modelDTMC modelPerformance measuresPerformance measures
Production rate (PR)
•PR = p2(1 - 0)
•PR = p1(1 - N(1-p2))
•PR = p2(1 - Q(p1, p2, N))
•PR = p1(1 - Q(p2, p1, N))
24
DTMC modelDTMC modelPerformance measuresPerformance measures
Work In Process (WIP)
1 2
1 1 211 2
1 22 1 1 2
1, if
2 1
1 ,, , otherwise
1 ,,
N
i Ni N
N
N Np p p
N pWIP i
p ppN p p
p pp p p p
25
DTMC modelDTMC modelBlockage and StarvationBlockage and Starvation
Blocking probability of M1 (BL1)
BL1 = p1N(1-p2) = p1 Q(p2, p1, N)
Starvation probability of M2 (ST2)
ST2 = p20 = p2 Q(p1, p2, N)
Relation with PR
PR = p1 - BL1
PR = p2 - ST2
26
DTMC modelDTMC model
L1: p1 = p2 = 0.9 L2: p1 = 0.9, p2 = 0.7
L3: p1 = 0.7, p2 = 0.9
27
DTMC modelDTMC model
Theorem:
1 2
1 2
1 11 2
2 1
1 2
1 1 2
2 2 1
lim min( , )
,
1lim ,
lim ,2
lim
lim
N
N
N
N
N
PR p p
p p
p pWIP p p
p p
Np p
BL p p
ST p p
28
Long Bernoulli LinesLong Bernoulli Lines
29
DTMC model
• 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.
M1 B1 M2 B2 M3 B3 M4
p2 p3 p4N2 N3p1 N1
30
Idea of the aggregationBackward aggregration
• pb3 = production rate of the 2-machine
line (M3, B3, M4)
• Repeating the aggregation process
• 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
M1 B1 M2 B2 M3 B3 M4
M1 B1 M2 B2 Mb3
M1 B1 Mb2
Mb1
31
Idea of the aggregationForward aggreation
• Forward aggreation is introduced to improve the aggregration.
• pfi is determine to take into account
the starvation of Bi-1 in the 2-machine line (Mf
i-1, Bi-1, Mbi)
• The whole process repeats to futher improved the aggregation
Mf2 B2 Mb
3
M1 B1 Mb2
Mf3
B3 Mb4
Mf4
32
Aggregation procedureFormal definition
The recursive aggregation procedure is as follow (Why?)
with initial condition
and boundary conditions
11 1 1 , , , 1,...,1b b fi i i i ip s p Q p s p s N i M
1 11 1 1 , 1 , , 2,...,f f bi i i i ip s p Q p s p s N i M
0 , 1,...,fi ip p i M
1 1f
bM M
p s p
p s p
1 1 ,, if
1 ,, ,
1, if
1
N
x x yx y
xx y
Q x y N y
xx y
N x
33
Aggregation procedureExample to workout with Excel
A 3-machine line L = (0.9, 0.9, 0.9, 2, 2)
2 2 3 2 2
1 1 2 1 1
2 2 1 2 2
3 3 2 3 2
2 2
1 1 1 , 0 , 0.9 1 0.9,0.9,2 0.8571
1 1 1 , 0 , 0.9 1 0.8571,0.9,2 0.8257
1 1 1 , 1 , 0.9 1 0.9,0.8571,2 0.8670
1 1 1 , 1 , 0.9 1 0.8670,0.9,2 0.8333
2 1
b b f
b b f
f f b
f f b
b
p p Q p p N Q
p p Q p p N Q
p p Q p p N Q
p p Q p p N Q
p p Q p
3 2 2
1 1 2 1 1
2 2 1 2 2
3 3 2 3 2
2 , 1 , 0.9 1 0.9,0.8670,2 0.8650
2 1 2 , 1 , 0.9 1 0.8650,0.9,2 0.8318
2 1 2 , 2 , 0.9 1 0.9,0.8650,2 0.8654
2 1 2 , 2 , 0.9 1 0.8654,0.9,2 0.8321
...
b f
b b f
f f b
f f b
p N Q
p p Q p p N Q
p p Q p p N Q
p p Q p p N Q
34
Aggregation procedureConvergence
Theorem.
Both sequence pfi(s) and pb
i(s) are converging, i.e. the following limits exist :
For each i, the sequence pfi(s) is monotonically decreasing
and the sequence pfi(s) is monotonically increasing.
Moreover,
: lim , : limb b f fi i i is s
p p s p p s
1b f
Mp pInterpretation
the downstream subline of buffer Bi-1
the upstream subline of buffer Bi
bipf
ip
35
Aggregation procedureExercice
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?
36
Aggregation procedurePerformance measures
Production rate estimation:
WIP estimation
estimated directly for the corresponding 2-machine line
Blockage estimation
Starvation estimation
1 or b fMp p
i 1,B , f bi iM M
1, , ,b f bi i i i i i i iBL p Q p p N p p BL
1 1, , ,f b fi i i i i i i iST p Q p p N p p ST
37
Aggregation procedureNumerical 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
38
Aggregation procedureHome 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 patternL2 : p = [0.9; 0.85; 0.8; 0.75; 0.7] : decreasing efficiencyL3 : p = [0.7; 0.75; 0.8; 0.85; 0.9] : increasing efficiencyL4 : p = [0.9; 0.85; 0.7; 0.85; 0.9] : bowl patternL5 : p = [0.7; 0.85; 0.9; 0.85; 0.7] : inverted bowl patternL6 : p = [0.7; 0.9; 0.7; 0.9; 0.7] : oscillatingL7 : p = [0.9; 0.7; 0.9; 0.7; 0.9] : oscillatingL8 : p = [0.75; 0.75; 0.95; 0.75; 0.75] : single bottleneck
39
Aggregation procedureProperties
Static law of production systems
11 , ,b b fi i i i ip p Q p p N
1 11 , ,f f bi i i i ip p Q p p N
Monotonicity :
The production rate PR(p1, ..., pM, N1, ..., NM-1) is
•strictly increasing in Ni
•strictly increasing in pi
40
Aggregation procedureProperties
Reversibility : Consider a line L and its reverse Lr with opposite flow direction. Then,
1,L Lr L Lri M iPR PR BL ST
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.
41
Continuous Improvement of Bernoulli Lines
42
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?
43
Constrained Improvability
44
Resource constraints
Buffer capacity constraint (BC):
Workforce constraints (WF):
1
1
*M
ii
N N
1
*M
ii
p p
Production rates of the machines depend on workforce assignment
45
Definitions
Definition: A Bernoulli line is
•improvable wrt BC if there exists a buffer assignment N'i such that iN'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 i 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 i N'i = N*, i p'i = p* and
PR(p'1, ..., p'M, N'1, ..., N'M-1) > PR(p1, ..., pM, N1, ..., NM-1)
46
Improvability with respec to WF
Theorem: A Bernoulli line is unimprovable wrt WF iff
where are the steady states of the recursive aggregation procedure.
Corollary. Under condition (WF1),
which implies
1
2 1i i
i fi i
N NWIP
N p
1, 1,..., 1 ( 1)f bi ip p i M WF
1,f bi ip p
1,
2 2i i
i
N NWIP i
Half buffer capacity usage
47
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.
48
WF unimprovable allocation
Unimprovable allocation
Theorem. If i Ni-1 ≤ M/2, then the series x(n) defined below
converges to PR* where
1 1 1: *
* max ,... , ,...,i i
i
M Mp p p
PR PR p p N N
2
11
1
1 * , 0 (0,1)1
M Mi
M
i i
N x nx n p x
N
49
WF unimprovable allocation
Theorem. The sequence p*i such that ip*i = p*, which renders the line unimprovable wrt WF, is given by
* 11
1
* 1
1
* 1
1
1*
*
1 1*, 2,..., 1
* *
1*
*
i ii
i i
MM
M
Np PR
N PR
N Np PR i M
N PR N PR
Np PR
N PR
Corollary. If all buffers are of equal capacity, i.e. Ni = N, then
which is a "flat" inverted bowl allocation.
Example : M = 5, Ni = 2, p* = 0.95. Compare with equal capacity.
* * * * *1 2 3 1...M Mp p p p p
50
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, pk, from Mk to Mk+1; If WIPk - Nk/2 <0, re-allocate pk+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 = 0.01. Initially, p = (0.9675, 0.9225, 0.8780, 0.8372)
51
Improvability wrt WF and BC simultaneously
Theorem: A Bernoulli line is unimprovable wrt WF and BC simultaneously iff
Corollary. Under condition (WF&BC1),
and, moreover
where N is the capacity of each buffer, i.e. equal capacity buffers.
1, 2,..., 1iWIP WIP i M
1 , 2,..., 1 ( & 1)f bi i Mp p p p i M WF BC
1
1,
2 1i
N NWIP i
N p
52
Unimprovabe allocation wrt WF and BC
Unimprovable allocation
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
1 1 1: *
: *
** max ,... , ,...,i ii
i ii
M MN N N
p p p
PR PR p p N N
* *
** *1 *
2**
*
*
1
1**
**
1**, 2,..., 1
**
i opt
optM
opt
opti
opt
NN N
M
Np p PR
N PR
Np PR i M
N PR
PR** can be determined as PR* with N*i.
flat inverted bowl WF dist.
uniform BC dist.
Improvabiblity wrt BC
Theorem: A Bernoulli line is unimprovable wrt BC iff the quantity
is maximized over all sequences N'i such that iN'i = N*.
Condition of little practical importance.
1,...,min min , ( 1)
f bi i
i b fi Mi i
p pp BC
p p
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
over all sequences N'i such that iN'i = N*.
Implication:
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.
12,..., 1max ( 2)i i i
i MWIP N WIP BC
MiBi-1 Bi
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
,i j
PR PRj i
p p
Problems with this definition:
1/ Gradient information cannot be measured on shopfloor2/ No analytical methods for evaluation of the gradients
Remark: gradient estimation is possible with sample path approaches (to be addressed).
Bottleneck machine
The best machine is the bottleneck
The worst machine is not the bottleneck
• Machine with the smalllest pi is not always the BN-m
• Machine with the largest WIP in front is not always the BN-m
0.9 6 0.7 6 0.8 1 0.7 1 0.75 4 0.6 6 0.7 2 0.85
PR/p 0.05 0.06 0.28 0.38 0.31 0.17 0.06 0.05WIPi 5.59 5.39 0.87 0.69 1.68 1.18 0.66
0.8 2 0.83 2 0.77 3 0.8
PR/p 0.369 0.452 0.443 0.022
Bottleneck buffer
Definition:
•A buffer Bi is the bottleneck buffer (BN-b) of a Bernoulli line if
1 1
1 1
,..., , ,..., 1,...,
,..., , ,..., 1,..., ,
M i M
M j M
PR p p N N N
PR p p N N N j i
Buffer with the smallest Ni is not necessarily the BN-b
0.8 3 0.85 3 0.85 2 0.9
PR(Ni+1) 0.769 0.766 0.763
Bottlenecks in 2-machine lines
Theorem: For a 2-machine Bernoulli line,
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.
1 2 2 1
(respectively, )PR PR PR PR
p p p p
Bottlenecks in 2-machine lines
STi 0 0.0215
BLi 0.1215 0
0.9 2 0.8
• 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.
BN-m
Bottlenecks in long lines
Single Bottleneck
Multiple Bottlenecks
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
BN-bBN-m
0.9 2 0.5 2 0.9 2 0.9 2 0.9 2 0.9 2 0.6 2 0.9
STi 0 0 0.39 0.37 0.33 027 0.11 0.41
BLi 0.41 0.01 0.03 0.05 0.1 0.17 0 0
BN-bPBN-m
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.