15th QMOD conference on Quality and Service Sciences 9/07/2012
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Transcript of 15th QMOD conference on Quality and Service Sciences 9/07/2012
Starbucks Wait Time Analysis
Brandon R. Theiss
Mathew Brown
Motivation
• Reliability is defined as:– the probability of a product performing its intended
function under stated conditions for a defined period of time.
• This definition unfortunately too narrowly defines the term in the context of a tangible product.
• Services represent 76.8% of the overall Gross Domestic Product of the United States or 11.9 Trillion dollars.
• A more applicable definition is therefore– The ability of process to perform its intended function
under customer specified conditions for a customer defined period of time.
Objective
• To study the reliability of the Starbucks beverage delivery system to provide a beverage to a customer prior to reaching their critical wait time.
About Starbucks• Founded 1971, in Seattle’s Pike Place Market. Original
name of company was Starbucks Coffee, Tea and Spices, later changed to Starbucks Coffee Company.
• In United States:– 50 states, plus the District of Columbia– 6,075 Company-operated stores– 4,082 Licensed stores
• Outside US– 2,326 Company Stores– 3,890 Licensed stores
Representative Stores• Two of the 6,075 company operated
stores were selected by geographical convenience– Marlboro NJ– New Brunswick NJ
About Marlboro NJ
Marlboro is a Township in Monmouth County, New Jersey. It has a population of 40,191 with a median household income of $101,322
About New Brunswick
New Brunswick is a city in Middlesex County, New Jersey. It has a population of 55,181 with a median household income of $36,080
Measurement System
Measurement Procedure
1. Click Start on 1 of 10 timers in the Custom Application
2. Enter Identifying characteristic in textbox
3. Click Stop when the customer receives their beverage or leaves the store. Data is automatically recorded with times measured in milliseconds
4. Click Reset for the next customer
Marlboro NJ Location
Marlboro Wait Time Data
Does the Data Follow a Weibull Distribution?
5000004000003000002000001000000
25
20
15
10
5
0
Time
Frequency
Shape 2.007Scale 216106N 94
Histogram of TimeWeibull
Does the Data Follow a Gamma Distribution?
5000004000003000002000001000000
25
20
15
10
5
0
Time
Frequency
Shape 3.977Scale 47936N 94
Histogram of TimeGamma
Can the arrivals of customers
be Modeled as a Poisson Process?
Goodness-of-Fit Test for Poisson Distribution Data column: MarlboroPoisson mean for Marlboro = 5.22222 Poisson ContributionMarlboro Observed Probability Expected to Chi-Sq<=3 7 0.235206 4.23371 1.807484 2 0.167197 3.00954 0.338655 3 0.174628 3.14330 0.006536 1 0.151991 2.73583 1.101357 1 0.113390 2.04102 0.53097>=8 4 0.157589 2.83660 0.47716 N N* DF Chi-Sq P-Value18 0 4 4.26215 0.372
Formal Test for the Data Being Normally Distributed
6000005000004000003000002000001000000-100000-200000
99.9
99
95
90
80706050403020
10
5
1
0.1
Time
Perc
ent
Goodness of Fit Test
AD = 2.887 P-Value < 0.005
Probability Plot for TimeNormal - 95% CI
Formal Test for the Data Being Gamma Distributed
100000010000010000
99.9
99
9590
80706050403020
10
5
1
0.1
Time
Perc
ent
Goodness of Fit Test
AD = 0.699 P-Value = 0.075
Probability Plot for TimeGamma - 95% CI
Formal Test for the Data Being Weibull Distributed
100000010000010000
99.999
9080706050403020
10
5
32
1
0.1
Time
Perc
ent
Goodness of Fit Test
AD = 1.509 P-Value < 0.010
Probability Plot for TimeWeibull - 95% CI
Mean Time To Beverage and “Reliability” at Marlboro
Biased Unbiased
190652.872424565 ms 190652.916039948 ms
3.17754787374275 min 3.1775486006658 min
Biased Unbiased
0.8727 0.8754
Is the Process Capable Based Upon a Gamma Model?
5000004000003000002000001000000
LB USL
LB 0
Target *USL 300000Sample Mean 190653
Sample N 94Shape 3.97724Scale 47936
Process DataPp *
PPL *PPU 0.29Ppk 0.29
Overall Capability
PPM < LB 0.00
PPM > USL 95744.68PPM Total 95744.68
Observed Performance
PPM < LB *
PPM > USL 127306.05PPM Total 127306.05
Exp. Overall Performance
Process Capability of TimeCalculations Based on Gamma Distribution Model
Is the Process Capable Based Upon a Weibull Model?
5000004000003000002000001000000
LB USL
LB 0
Target *USL 300000Sample Mean 190653
Sample N 94Shape 2.00713Scale 216106
Process DataPp *
PPL *PPU 0.32Ppk 0.32
Overall Capability
PPM < LB 0.00
PPM > USL 95744.68PPM Total 95744.68
Observed Performance
PPM < LB *
PPM > USL 144910.81PPM Total 144910.81
Exp. Overall Performance
Process Capability of TimeCalculations Based on Weibull Distribution Model
Is the Beverage Delivery Process in Control?
918273645546372819101
800
600
400
200
Observation
Indiv
idual V
alu
e
_X=422.7
UCL=679.6
LCL=165.8
918273645546372819101
450
300
150
0
Observation
Movin
g R
ange
__MR=96.6
UCL=315.6
LCL=0
1111
111
I-MR Chart of MarlboroUsing Box-Cox Transformation With Lambda = 0.50
918273645546372819101
600000
450000
300000
150000
0
Observation
Indiv
idual V
alu
e
_X=190653
UCL=407256
LCL=-25950
918273645546372819101
400000
300000
200000
100000
0
Observation
Movin
g R
ange
__MR=81443
UCL=266097
LCL=0
111
111
111
I-MR Chart of Marlboro
New Brunswick NJ Location
New Brunswick Wait Time Data
Does the Data Follow a Weibull Distribution?
6000005000004000003000002000001000000
40
30
20
10
0
Time
Frequency
Shape 1.994Scale 273830N 198
Histogram of TimeWeibull
Does the Data Follow a Gamma Distribution?
6000005000004000003000002000001000000
40
30
20
10
0
Time
Frequency
Shape 3.080Scale 78771N 198
Histogram of TimeGamma
Goodness-of-Fit Test for Poisson Distribution Data column: New BrunswickPoisson mean for New Brunswick = 9.9New Poisson ContributionBrunswick Observed Probability Expected to Chi-Sq<=6 4 0.136574 2.73148 0.5891077 - 8 3 0.207617 4.15235 0.3197959 - 10 5 0.251357 5.02715 0.00014711 - 12 4 0.205390 4.10780 0.002829>=13 4 0.199062 3.98123 0.000088 N N* DF Chi-Sq P-Value20 0 3 0.911967 0.823
Can the arrivals of customers
be Modeled as a Poisson Process?
Formal Test for the Data Being Normally Distributed
7000
00
6000
00
5000
00
4000
00
3000
00
2000
00
1000
000
-100
000
-200
000
99.9
99
9590
80706050403020
105
1
0.1
Time
Perc
ent
Goodness of Fit Test
AD = 1.680 P-Value < 0.005
Probability Plot for TimeNormal - 95% CI
Formal Test for the Data Being Gamma Distributed
100000010000010000
99.9
99
959080706050403020
10
5
1
0.1
Time
Perc
ent
Goodness of Fit Test
AD = 0.911 P-Value = 0.023
Probability Plot for TimeGamma - 95% CI
Formal Test for the Data Being Weibull Distributed
100000010000010000
99.999
9080706050403020
10
5
32
1
0.1
Time
Perc
ent
Goodness of Fit Test
AD = 0.441 P-Value > 0.250
Probability Plot for TimeWeibull - 95% CI
Why Might the Data Not Follow a Gamma?
Wait in LineMake Drink
Process
Poisson
Arrival To Store
Gamma ?
Deliver Drink
Gamma * ? =?
Order Drink
What We Measured
Is the Process Capable Based Upon a Weibull Model?
6000005000004000003000002000001000000
LB USL
LB 0Target *USL 300000Sample Mean 242647Sample N 198Shape 1.99408Scale 273830
Process DataPp *PPL *PPU 0.15Ppk 0.15
Overall Capability
PPM < LB 0.00PPM > USL 303030.30PPM Total 303030.30
Observed Performance
PPM < LB *PPM > USL 301307.05PPM Total 301307.05
Exp. Overall Performance
Process Capability of TimeCalculations Based on Weibull Distribution Model
Is the Process Capable Based Upon a Gamma Model?
6000005000004000003000002000001000000
LB USL
LB 0Target *USL 300000Sample Mean 242647Sample N 198Shape 3.0804Scale 78771.2
Process DataPp *PPL *PPU 0.13Ppk 0.13
Overall Capability
PPM < LB 0.00PPM > USL 303030.30PPM Total 303030.30
Observed Performance
PPM < LB *PPM > USL 283036.30PPM Total 283036.30
Exp. Overall Performance
Process Capability of TimeCalculations Based on Gamma Distribution Model
Mean Time To Beverage and “Reliability” at New Brunswick
Biased Unbiased
242688.9419 ms 242371.0724 ms
4.0448 mins 4.0395 mins
Biased Unbiased
0.6987 0.6993
Is the Beverage Delivery Process in Control?
181161141121101816141211
800
600
400
200
Observation
Indiv
idual V
alu
e
_X=473.9
UCL=733.1
LCL=214.7
181161141121101816141211
600
400
200
0
Observation
Movin
g R
ange
__MR=97.4
UCL=318.4
LCL=0
1
1
11
1
1111
11111
11
1111
111
1
I-MR Chart of New BrunswickUsing Box-Cox Transformation With Lambda = 0.50
181161141121101816141211
600000
450000
300000
150000
0
Observation
Indiv
idual V
alu
e
_X=242647
UCL=485623
LCL=-330
181161141121101816141211
480000
360000
240000
120000
0
Observation
Movin
g R
ange
__MR=91359
UCL=298497
LCL=0
111
111
1
1
1
11
1
1111
1
I-MR Chart of New Brunswick
COMBINEDStarbucks Wait Time Analysis
Marlboro New Brunswick
Combined Wait Time Data
Is there a difference between Marlboro and New Brunswick?
6000005000004000003000002000001000000
40
30
20
10
0
Data
Frequency
3.977 47936 943.080 78771 198
Shape Scale N
MarlboroNew Brunswick
Variable
Histogram of Marlboro, New BrunswickGamma
Is there a difference between Marlboro and New Brunswick?
Kruskal-Wallis Test: Wait Times versus Location
Kruskal-Wallis Test on C2
Subscripts N Median Ave Rank Z
Marlboro 94 173350 121.6 -3.47
New Brunswick 198 216245 158.3 3.47
Overall 292 146.5
H = 12.04 DF = 1 P = 0.001
H = 12.04 DF = 1 P = 0.001 (adjusted for ties)
Does the Data Follow a Weibull Distribution?
6000005000004000003000002000001000000
35
30
25
20
15
10
5
0
Combined
Frequency
Shape 1.954Scale 255391N 292
Histogram of CombinedWeibull
Does the Data Follow a Gamma Distribution?
6000005000004000003000002000001000000
35
30
25
20
15
10
5
0
Combined
Frequency
Shape 3.201Scale 70580N 292
Histogram of CombinedGamma
Are the Arrival Rates the Same?
161412108642
9
8
7
6
5
4
3
2
1
0
161412108642
Marlboro
Frequency
New Brunswick
Histogram of Marlboro, New Brunswick
Are the Arrival Rates the Same?
Kruskal-Wallis Test: Arrivals versus Location
Kruskal-Wallis Test on Arrivals
Location N Median Ave Rank Z
Marlboro 18 4.500 12.4 -3.76
New Brunswick 20 10.000 25.9 3.76
Overall 38 19.5
H = 14.11 DF = 1 P = 0.000
H = 14.26 DF = 1 P = 0.000 (adjusted for ties)
Goodness-of-Fit Test for Poisson Distribution
Data column: Combined
Poisson mean for Combined = 7.68421 Poisson ContributionCombined Observed Probability Expected to Chi-Sq<=4 10 0.119196 4.52945 6.607195 3 0.102708 3.90291 0.208886 4 0.131538 4.99846 0.199457 2 0.144396 5.48703 2.216028 4 0.138696 5.27044 0.306249 3 0.118419 4.49991 0.4999510 3 0.090995 3.45782 0.0606211 1 0.063566 2.41551 0.82950>=12 8 0.090486 3.43846 6.05144 N N* DF Chi-Sq P-Value38 0 7 16.9793 0.018
Can the arrivals of customers
be Modeled as a Poisson Process?
Why Might the data set of Combined Arrivals Not Represent a Poisson
Process?
• Not a large enough data set of stores
• Not constant arrival rate– Different demand for Beverages at different
stores at different times
• Other factors are influencing the independence of events– Traffic lights
Formal Test for the Data Being Normally Distributed
7000
00
6000
00
5000
00
4000
00
3000
00
2000
00
1000
000
-100
000
-200
000
99.9
99
9590
80706050403020
105
1
0.1
Combined
Perc
ent
Goodness of Fit Test
AD = 4.293 P-Value < 0.005
Probability Plot for CombinedNormal - 95% CI
Formal Test for the Data Being Gamma Distributed
100000010000010000
99.9
99
9590
80706050403020
10
5
1
0.1
Combined
Perc
ent
Goodness of Fit Test
AD = 0.594 P-Value = 0.141
Probability Plot for CombinedGamma - 95% CI
Formal Test for the Data Being Weibull Distributed
100000010000010000
99.999
9080706050403020
10
5
32
1
0.1
Combined
Perc
ent
Goodness of Fit Test
AD = 0.959 P-Value = 0.016
Probability Plot for CombinedWeibull - 95% CI
Mean Time To Beverage and “Reliability”
Biased Unbiased
225908.8493 ms 226153.1587 ms
3.7651 mins 3.7692 mins
Biased Unbiased
0.7629 0.7617
Is the Process Capable Based Upon a Gamma Model?
6000005000004000003000002000001000000
LB USL
LB 0
Target *USL 300000Sample Mean 225909
Sample N 292Shape 3.20075Scale 70580
Process DataPp *
PPL *PPU 0.16Ppk 0.16
Overall Capability
PPM < LB 0.00
PPM > USL 236301.37PPM Total 236301.37
Observed Performance
PPM < LB *
PPM > USL 237100.41PPM Total 237100.41
Exp. Overall Performance
Process Capability of CombinedCalculations Based on Gamma Distribution Model
Is the Process Capable Based Upon a Weibull Model?
6000005000004000003000002000001000000
LB USL
LB 0
Target *USL 300000Sample Mean 225909
Sample N 292Shape 1.95393Scale 255391
Process DataPp *
PPL *PPU 0.19Ppk 0.19
Overall Capability
PPM < LB 0.00
PPM > USL 236301.37PPM Total 236301.37
Observed Performance
PPM < LB *
PPM > USL 254194.23PPM Total 254194.23
Exp. Overall Performance
Process Capability of CombinedCalculations Based on Weibull Distribution Model
Is the Process Capable Based Upon a Weibull Model?
The corresponds to a Sigma level of 4. The Goal is 6!
Is the Process Capable Based Upon a Gamma Model?
The corresponds to a Sigma level of 2. The Goal is 6!
Conclusions
• The amount of time a customer waits at a Starbucks is dependent on which location they visit.
• Regardless of location, Starbucks is incapable of reliably delivering a beverage in less than 5 minutes
• There is evidence to suggest that the arrivals follow a Poisson distribution which is supported by the literature
• There is evidence to suggest that the wait times follow a gamma distribution which the literature would suggest
?Brandon R. Theiss