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Variability and Queuing Models in Manufacturing
2
Read Tour A in Schmenner
• Can you draw the process flow graph?• What happens “upstream” and
“downstream” from the paper making machines?
3
Review
• Network of resources• Products with processing requirements
– process flow graphs– process flow charts
• Little’s Law: I = R x T• Capacity requirements• Strategies for increasing utilization• Strategies for decreasing flow time
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Overview of Next Segment
• Understanding the impact of variability on flow time & utilization
• Predictive models– for design– for process management
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Suppose we observe a grinder
What do we actually observe?
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We can observe:System
Events:job arrives at IBjob starts operation, leaves IBjob finishes operation, enters OBjob leaves OB
Arrival Process
Service Process
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We can observe:
• Time between arrivals (inter-arrival time)
• Length of operations (service time)
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Intro to Queuing Models
• Read the material in the textbookPages 155 – 181 of Modeling in IE
• Be responsible for understanding the concepts and models presented there
• Notation is different from that in the first section of the text.
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Notation
[λ]R : the rate at which jobs arrivea[1/λ]T : the average inter-arrival timea
R = aTa
R : the service rate s
1
[µ][1/µ]T : the average service times
1R = sTs
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More Notation[Wq]T : the average time a job spends in the queueq
T : the average time a job spends in services
T = T +q Ts
[Lq]I : the average number of jobs in the queueq
I : the average number of jobs in services
I = I +q Is
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Still More Notation[L = λ W]I = R x Ta
= R xa (T +q T )s
= I +q Is
Ra Tsu: utilization, = = < 1.0 Rs Ta
[u = λ/µ]
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Consider a (hypothetical) sample of 50 jobsstarting with no jobs in the system
Job ArrTime StTime FinTime WaitTime
Inter-Arrival Time
Service Time
1 8 8 34 0 8 272 17 34 64 18 9 293 59 64 84 5 43 214 137 137 234 0 78 975 143 234 464 91 6 2306 211 464 554 252 69 907 284 554 562 270 73 88 301 562 629 261 16 679 458 629 681 171 157 5210 499 681 765 182 41 8411 629 765 810 136 130 4612 654 810 820 156 25 913 715 820 872 105 61 5214 734 872 904 137 20 33
In the buffer
50 2821 2920 2963 99 52 43average 68.5 56.4 48.6max 269.8 172.6 229.8st.dev. 76.3 47.2 43.3
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Inter-arrival time histogram
024681012141618
3.710
4625
8127
.8373
6442
51.96
4266
2776
.0911
6811
100.2
1807
124.3
4497
1814
8.471
8736
More
Series1
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Service time histogram
0
5
10
15
20
25
1.579
8466
534
.1832
3505
66.78
6623
4499
.3900
1184
131.9
9340
0216
4.596
7886
197.2
0017
7
More
Series1
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Waiting time histogram
0246810121416
038
.5370
4329
77.07
4086
5811
5.611
1299
154.1
4817
3219
2.685
2164
231.2
2225
97
More
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Conclusions
• Average wait is over an hour• 20% of jobs wait more than 2 hours• Do we have enough staging space?• How many “active” jobs do we have,
on average?• What is the average manufacturing
cycle time?• What can we do to improve?
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Queuing Models
• This is mathematics, not reality!• Assumptions
– arrival process, service process– queue size and discipline– time horizon– calling population
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For the M/M/1 Queue
TsT (M/M/1) =qu
1-u
“theory”Rs
Ra
RaRs -=
In other words, the average time in queue can be estimated from just the average inter-arrival time and average
service time!
We can use the theoretical model to make estimates about the behavior of a real system
But we must keep in mind that these are always approximations
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Reality Model
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Actual vs theoretical
024681012141618
3.710
4625
8127
.8373
6442
51.96
4266
2776
.0911
6811
100.2
1807
124.3
4497
1814
8.471
8736
More
Exponential inter-arrival times
Assume an exponential distribution with Ra equal to (1/56.4) [if you want to see how I did this, look at MM1Sample.xls]
Sample data
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Can we make the assumption?
• The distributions “look” very similar.• There could well be some error
introduced by making the assumption• We’ll need some way to “test” any
conclusions we draw from the theoretical models.
You learn how to do statistical test in 2028
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Applying to our exampleu = 48.6/56.4 = 0.86
Tsu
1-u= (0.86/0.14) 48.6= 298.5
T (M/M/1) =q
This a LOT larger than the sample average!What gives?
Examples from on-line “word”
Be sure you can associate the correct parameter with the
given or observed data
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Fundamental Phenomenon
Ave
rage
MC
T
utilization 100%
See MM1Sample.xls
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Exponential DistributionExponential Distribution
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10 12
Time
Den
sity
/Cum
ulat
ive
Exponential distribution has a coefficient of variation (CV) of 1--the standard deviation is equal to the mean. Variability plays a major role in queuing phenomena.
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Queues are a model archetype
Arrivalprocess
System boundary
Flow unit
queue server
Arrival rate
Throughput,output rate
Inventory, WIP
The entities in a queuing system are the flow units, the queue, and the server. Each entity has its own state
space, and corresponding transitions.
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Notes
• These are basic queuing theory results
• Always be sure the units are correct– R has dimension [units/time]– T has dimension [time]– I has dimension [units]
• I=RT <-> [units] = [units/time][time]
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Summary
• M/M/1 model for predicting inventory and waiting time
• Impact of utilization on waiting time
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For Next Time
• READ THE BOOK!• Deal with the differences in notation
between section 1 and section 2--I will stick with the notation in section 1 as much as possible, for consistency
• Look at the spreadsheets