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

4

Overview of Next Segment

• Understanding the impact of variability on flow time & utilization

• Predictive models– for design– for process management

5

Suppose we observe a grinder

What do we actually observe?

6

We can observe:System

Events:job arrives at IBjob starts operation, leaves IBjob finishes operation, enters OBjob leaves OB

Arrival Process

Service Process

7

We can observe:

• Time between arrivals (inter-arrival time)

• Length of operations (service time)

8

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.

9

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

10

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

11

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]

12

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

13

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

14

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

15

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

16

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?

17

Queuing Models

• This is mathematics, not reality!• Assumptions

– arrival process, service process– queue size and discipline– time horizon– calling population

18

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

20

Reality Model

21

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

22

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

23

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

25

Fundamental Phenomenon

Ave

rage

MC

T

utilization 100%

See MM1Sample.xls

26

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.

27

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.

28

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]

29

Summary

• M/M/1 model for predicting inventory and waiting time

• Impact of utilization on waiting time

30

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