Variability & Queuing IV

Review• Little’s Law• Basic Queuing Models/Approximations

– M/M/1, M/M/m, G/G/1, G/G/m• Variability

– Quantifying (CV & SCV)– Sources:

• natural• incidental

– setups– breakdowns

• Applications– Performance modeling, “cost” analysis, decision making

Qcalc.xls

• Spreadsheet that contains all the models we’ve discussed

• To use it, you will need to be able to convert parameters given in problem to parameters used in the spreadsheet

• You will have to make sure your units are consistent

“Economic” Analysis of Queues

• Cost of service– fixed cost (equipment, space, etc)– variable cost (materials, wages, etc)

• “Cost” of waiting– opportunity cost of variable “investment”– opportunity cost of delay

• obsolescence, price erosion, lost sales, engineering changes

Textbook Approach

• Hypothesize a “cost of waiting”• Formulate a “total cost” model• Vary waiting cost parameter to see

how it affects total cost

• I’m not crazy about this approach...

You can never know the value of the waiting cost parameter

• But you can observe the impact of waiting time

• My suggestion: vary what you can control, observe the resulting performance, and identify an appropriate “performance target.”

Example, pp. 166-170

• Ra=5, Rs=8• How many servers?

• Number of servers is what you can control; average waiting time is the performance variable of interest

Use Qcalc.xlsWaiting Time vs. # Servers

0

5

10

15

20

0 1 2 3 4 5 6

Number of Servers

Expe

cted

Wai

ting

Tim

e

Target service level

General Strategy

• Minimize cost of service• Subject to service level constraint(s)

• Why do I prefer this to a ‘waiting cost’ model?

Big PictureDescriptive Models•Little’s law•Process flow graph•Process flow chart•From/to matrix

Analytic Models•Queuing•Simulation

Decision Support

Monte Carlo Sampling

And Simulation

Sample data generator

• Inter-arrival times and service times are drawn from exponential distributions.

• Event times are computed from “observations”

• Get a new sample every time the spreadsheet is recalculated

Sampling ProcedureExponential Distribution

0

0.2

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0.6

0.8

1

1.2

0 2 4 6 8 10 12

Time

Dens

ity/C

umul

ativ

e

Sampling Procedure

• Requires “random” numbers in (0, 1)• Requires the inverse of F(x)• F(x) = 1-exp(-λx) = r [random number]• exp(-λx) = 1-r• ln(exp(-λx) ) = -λx = ln(1-r)• x = -(1/λ)ln(1-r) = -(1/λ)ln(r)

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12

0.6873

.98 2.35

1. Generate the “random” number2. Find the inter-arrival time that

corresponds to that probability in the cumulative density function

0.9250

Random Numbers

• There are mathematical methods that generate sequences of “pseudo-random” numbers.

• Excel has a function, rand()

MM1Sim.xls

How I generated samples of 1000 jobs for the Norcen

grinder!

How “random” are Excel’s random numbers?

Sample of 1000 Rand() ValuesHistogram of 1000 random numbers

05

101520253035404550

0.000

2532

260.0

6470

4553

0.129

1558

8

0.193

6072

080.2

5805

8535

0.322

5098

630.3

8696

119

0.451

4125

170.5

1586

3845

0.580

3151

720.6

4476

6499

0.709

2178

270.7

7366

9154

0.838

1204

820.9

0257

1809

0.967

0231

36

Bin Ranges

Cou

nt

Chart of 1000 random points in the unit square

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0 0.2 0.4 0.6 0.8 1 1.2

Chart of 10000 random points in the unit square

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1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Monte Carlo Sampling

• Traditional name for this method• Very flexible• I can generate as many samples as I

want• I can generate larger or smaller

samples

Monte Carlo sampling is a tool that allows us to create

“synthetic” data. We assumesome distribution for the

random process of interest, and use Monte Carlo sampling to create “observations” of this

random variable. All we need is a distribution function.

Empirical Distributions

• Monte Carlo sampling also works with empirical distributions

• Can use the discrete distribution (the histogram), or

• Can interpolate to get values in between the bin boundaries

• Need the vlookup() function in Excel

Empirical HistogramHistogram of 1000 Inter-Arrival Times

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200

0.00

28.0

6

56.1

1

84.1

7

112.

23

140.

28

168.

34

196.

39

224.

45

252.

51

280.

56

308.

62

336.

67

364.

73

392.

79

420.

84

Bin Range

Cou

nt

CDF for Sample

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0.40.6

0.8

1

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0.00

28.0

6

56.1

1

84.1

7

112.

23

140.

28

168.

34

196.

39

224.

45

252.

51

280.

56

308.

62

336.

67

364.

73

392.

79

420.

84

Bin Range

Prob

abili

ty

NB: I let Excel pick the bin ranges

Sampling from discrete CDFCDF for Sample

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0.2

0.40.6

0.8

1

1.20.

00

28.0

6

56.1

1

84.1

7

112.

23

140.

28

168.

34

196.

39

224.

45

252.

51

280.

56

308.

62

336.

67

364.

73

392.

79

420.

84

Bin Range

Prob

abili

ty

70.14

.664

NB: Can use the Excel function vlookup() to do this, but there’s a trick

Using vlookupSearches for a value in the leftmost column of a table, and thenreturns a value in the same row from a column you specify in thetable.

VLOOKUP(lookup_value, table_array, col_index_num, range_lookup)

Lookup_value is the value to be found in the first column of the array. Lookup_value can be a value, a reference, or a text string.

Table_array is the table of information in which data is looked up. Use a reference to a range or a range name, such as Databaseor List.

Col_index_num is the column number in table_array from which the matching value must be returned. A col_index_num of 1 returns the value in the first column in table_array; a col_index_num of 2 returns the value in the second column in table_array, and so on. If col_index_num is less than 1, VLOOKUP returns the #VALUE! error value; if col_index_num is greater than the number of columns in table_array, VLOOKUP returns the #REF! error value.

Range_lookup is a logical value that specifies whether you want VLOOKUP to find an exact match or an approximate match. If TRUE or omitted, an approximate match is returned. In other words, if an exact match is not found, the next largest value that is less than lookup_value is returned. If FALSE, VLOOKUP will find an exact match. If one is not found, the error value #N/A is returned. If range_lookup is TRUE, the values in the first column of table_array must be placed in ascending order: ..., -2, -1, 0, 1, 2, ..., A-Z, FALSE, TRUE; otherwise VLOOKUP may not give the correct value

vlookup examples

K L

1 0 14.030062 170 28.058063 342 42.086064 482 56.114075 606 70.142076 697 84.170077 761 98.198078 819 112.2261

VLOOKUP(344, $K$1:$L$31,1,1)=342VLOOKUP(507, $K$1:$L$31,1,1)=482VLOOKUP(600, $K$1:$L$31,1,1)=482

VLOOKUP(750, $K$1:$L$31,2,1)= ?VLOOKUP(800, $K$1:$L$31,2,1)= ?

See the “histograms” tab in MM1Sim.xls, where vlookup is used, both with and without

interpolation

The Trick in Using VLOOKUP

• You have to shift the bin range up one cell, relative to the cum prob, in order to generate the desired target

• See MM1Sim.xls for examples

This works, but ...

• yields only a few unique values• on the other hand, it’s “easy”• we can do a little more work and get

continuous values by interpolating

Interpolating

70.14

56.11

697

606

x

664

X=56.11+ (70.14-56.11)*(664-606)/(697-606) = 65.05

Inter-departure times

• Describe the way jobs are presented to the material handling system, i.e., the “arrival process” for the MHS

• The from/to chart tells the distribution of destinations but not the distribution of inter-arrival times to the MHS

MM1Sim.xlsInter-Departure Times

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0.03

0847

694

26.0

7061

787

52.1

1038

804

78.1

5015

822

104.

1899

284

130.

2296

986

156.

2694

687

182.

3092

389

208.

3490

091

234.

3887

793

260.

4285

494

286.

4683

196

312.

5080

898

338.

5478

6

364.

5876

301

390.

6274

003

Time

Freq

uenc

y

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1

1.2

Cum

ulat

ive

Perc

ent

Now Suppose

• There are two kinds of jobs, appearing at random, with equal probability

• One has a service time that is U(12.5, 37.5)

• One has a service time that is U(62.5, 87.5)

• Departures.xls

Inter-Departure Times

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12.5

0292

165

31.6

8583

38

50.8

6874

596

70.0

5165

811

89.2

3457

027

108.

4174

824

127.

6003

946

146.

7833

067

165.

9662

189

185.

1491

3120

4.33

2043

2

223.

5149

554

242.

6978

675

261.

8807

797

281.

0636

918

300.

2466

04

Time

Freq

uecy

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Cum

ulat

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Perc

ent

Also ...

• What if we don’t have “unlimited” staging capacity for waiting jobs?

• What if we need more reliable estimates of “departure time”?

• What are the relevant cost factors?• What if …?

We may need analysis tools

With a little more “resolution” than the basic queueing models

provide...

Assignment

• Play with the spreadsheet to get a feel for variability and the impact of increasing utilization

• Modify the spreadsheet to generate a larger sample, and compare the distributions of sample averages for different sample sizes.

Final note on MC sampling

It can be useful for lots of other applications as well. For example, the spreadsheet illustrates the use of MC sampling to estimate the area of a circle. You could use MC sampling to estimate the area of other shapes, even non-geometrical shapes. The key is to define a function that evaluates to “true” or “false” when you substitute in the sampled values.

Be Sure You:

• Read the material on Monte Carlo sampling and simulation in the text

• Look at the spreadsheets--play around with them to make sure you understand how they work.

• Look at the Excel function for generating observations from other distributions, in tools -> data_analysis ->random_number_generation