This document analyzes wait times at two Starbucks locations to determine if the beverage delivery process is reliable. Wait time data was collected from each store and analyzed to determine if it followed a Weibull, gamma, or normal distribution. The data did not follow a normal distribution but did fit a Weibull or gamma model. Process capability calculations showed the process was not capable of meeting the target wait time less than 5 minutes at the New Brunswick location based on either distribution. The document concludes an analysis of the beverage making process is also needed.

1. Starbucks Wait Time Analysis
Brandon R. Theiss
Mathew Brown

2. 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.

3. 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.

4. 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

5. Representative Stores
• Two of the 6,075 company operated
stores were selected by geographical
convenience
– Marlboro NJ
– New Brunswick NJ

6. 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

7. 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

8. Measurement System

9. 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

10. Marlboro NJ Location

11. Marlboro Wait Time Data

12. Does the Data Follow a Weibull
Distribution?
Hi st ogr am of Ti me
Weibull
25 Shape 2.007
Scale 216106
N 94
20
15
Fr equency
10
5
0
0 100000 200000 300000 400000 500000
Time

13. Does the Data Follow a Gamma
Distribution?
Hi st ogr am of Ti me
Gamma
25 Shape 3.977
Scale 47936
N 94
20
15
Fr equency
10
5
0
0 100000 200000 300000 400000 500000
Time

14. Can the arrivals
of customers
be Modeled as
a Poisson
Process?
Goodness-of-Fit Test for Poisson Distribution
Data column: Marlboro
Poisson mean for Marlboro = 5.22222
Poisson Contribution
Marlboro Observed Probability Expected to Chi-Sq
<=3 7 0.235206 4.23371 1.80748
4 2 0.167197 3.00954 0.33865
5 3 0.174628 3.14330 0.00653
6 1 0.151991 2.73583 1.10135
7 1 0.113390 2.04102 0.53097
>=8 4 0.157589 2.83660 0.47716
N N* DF Chi-Sq P-Value
18 0 4 4.26215 0.372

15. Formal Test for the Data Being
Normally Distributed
Pr obabi l i t y Pl ot f or Ti me
Normal - 95% CI
99.9
Goodness of Fit Test
99
AD = 2.887
P-Value < 0.005
95
90
80
70
Per cent
60
50
40
30
20
10
5
1
0.1
-200000 -100000 0 100000 200000 300000 400000 500000 600000
Time

16. Formal Test for the Data Being
Gamma Distributed
Pr obabi l i t y Pl ot f or Ti me
Gamma - 95% CI
99.9
Goodness of Fit Test
99
95 AD = 0.699
90 P-Value = 0.075
80
70
60
50
40
Per cent
30
20
10
5
1
0.1
10000 100000 1000000
Time

17. Formal Test for the Data Being
Weibull Distributed
Pr obabi l i t y Pl ot f or Ti me
Weibull - 95% CI
99.9
99 Goodness of Fit Test
90
AD = 1.509
80
70 P-Value < 0.010
60
50
40
30
20
Per cent
10
5
3
2
1
0.1
10000 100000 1000000
Time

18. 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

19. Is the Process Capable Based
Upon a Gamma Model?
Pr ocess Capabi l i t y of Ti me
Calculations Based on Gamma Distribution Model
LB USL
Process Data O v erall Capability
LB 0 Pp *
Target * PPL *
USL 300000 PPU 0.29
Sample Mean 190653 Ppk 0.29
Sample N 94
Exp. O v erall Performance
Shape 3.97724
PPM < LB *
Scale 47936
PPM > USL 127306.05
O bserv ed Performance PPM Total 127306.05
PPM < LB 0.00
PPM > USL 95744.68
PPM Total 95744.68
0 100000 200000 300000 400000 500000

20. Is the Process Capable Based
Upon a Weibull Model?
Pr ocess Capabi l i t y of Ti me
Calculations Based on Weibull Distribution Model
LB USL
Process Data O v erall Capability
LB 0 Pp *
Target * PPL *
USL 300000 PPU 0.32
Sample Mean 190653 Ppk 0.32
Sample N 94
Exp. O v erall Performance
Shape 2.00713
PPM < LB *
Scale 216106
PPM > USL 144910.81
O bserv ed Performance PPM Total 144910.81
PPM < LB 0.00
PPM > USL 95744.68
PPM Total 95744.68
0 100000 200000 300000 400000 500000

21. Is the Beverage Delivery
Process in Control?
I -MR Char t of Mar l bor o I -MR Char t of Mar l bor o
Using Box-Cox Transformation With Lambda = 0.50
600000
1
1 1 1 800
1 1 1
450000 1 1 1
I n d i v i d u a l V a lu e
UCL= 407256 UCL= 679.6
I ndiv idual Value
600
300000
_
_
X= 190653 X= 422.7
150000 400
0
LCL= -25950 200
LCL= 165.8
1 10 19 28 37 46 55 64 73 82 91
O b se r v a t io n 1 10 19 28 37 46 55 64 73 82 91
Observ at ion
1
11 11 1
400000 450
M o v in g Ra n g e
300000
UCL= 315.6
Mov ing Range
UCL= 266097 300
200000
__ 150 __
100000
MR= 81443 MR= 96.6
0 LCL= 0 0 LCL= 0
1 10 19 28 37 46 55 64 73 82 91 1 10 19 28 37 46 55 64 73 82 91
O b se r v a t io n Observ at ion

22. New Brunswick NJ Location

23. New Brunswick Wait Time Data

24. Does the Data Follow a Weibull
Distribution?
Hi st ogr am of Ti me
Weibull
40 Shape 1.994
Scale 273830
N 198
30
Fr equency
20
10
0
0 100000 200000 300000 400000 500000 600000
Time

25. Does the Data Follow a Gamma
Distribution?
Hi st ogr am of Ti me
Gamma
40 Shape 3.080
Scale 78771
N 198
30
Fr equency
20
10
0
0 100000 200000 300000 400000 500000 600000
Time

26. Can the arrivals
of customers
be Modeled as
a Poisson
Process?
Goodness-of-Fit Test for Poisson Distribution
Data column: New Brunswick
Poisson mean for New Brunswick = 9.9
New Poisson Contribution
Brunswick Observed Probability Expected to Chi-Sq
<=6 4 0.136574 2.73148 0.589107
7 - 8 3 0.207617 4.15235 0.319795
9 - 10 5 0.251357 5.02715 0.000147
11 - 12 4 0.205390 4.10780 0.002829
>=13 4 0.199062 3.98123 0.000088
N N* DF Chi-Sq P-Value
20 0 3 0.911967 0.823

27. Formal Test for the Data Being
Normally Distributed
Pr obabi l i t y Pl ot f or Ti me
Normal - 95% CI
99.9
Goodness of Fit Test
99
AD = 1.680
95 P-Value < 0.005
90
80
70
Per cent
60
50
40
30
20
10
5
1
0.1
00 00 0 00 00 00 00 00 00 00
000 000 00 00 00 00 00 00 00
-2 -1 10 20 30 40 50 60 70
Time

28. Formal Test for the Data Being
Gamma Distributed
Pr obabi l i t y Pl ot f or Ti me
Gamma - 95% CI
99.9
Goodness of Fit Test
99
95 AD = 0.911
90 P-Value = 0.023
80
70
60
50
40
30
Per cent
20
10
5
1
0.1
10000 100000 1000000
Time

29. Formal Test for the Data Being
Weibull Distributed
Pr obabi l i t y Pl ot f or Ti me
Weibull - 95% CI
99.9
99 Goodness of Fit Test
90
AD = 0.441
80
70 P-Value > 0.250
60
50
40
30
20
Per cent
10
5
3
2
1
0.1
10000 100000 1000000
Time

30. Why Might the Data Not Follow
a Gamma?
Poisson Gamma ?
Gamma * ? =?
Make Drink
Wait in Line
Process
Arrival Deliver
To Store Order Drink
Drink
What We Measured

31. Is the Process Capable Based
Upon a Weibull Model?
Pr ocess Capabi l i t y of Ti me
Calculations Based on Weibull Distribution Model
LB USL
Process Data O v erall Capability
LB 0 Pp *
Target * PPL *
USL 300000 PPU 0.15
Sample Mean 242647 Ppk 0.15
Sample N 198
Exp. O v erall Performance
Shape 1.99408
PPM < LB *
Scale 273830
PPM > USL 301307.05
O bserv ed Performance PPM Total 301307.05
PPM < LB 0.00
PPM > USL 303030.30
PPM Total 303030.30
0 100000 200000 300000 400000 500000 600000

32. Is the Process Capable Based
Upon a Gamma Model?
Pr ocess Capabi l i t y of Ti me
Calculations Based on Gamma Distribution Model
LB USL
Process Data O v erall Capability
LB 0 Pp *
Target * PPL *
USL 300000 PPU 0.13
Sample Mean 242647 Ppk 0.13
Sample N 198
Exp. O v erall Performance
Shape 3.0804
PPM < LB *
Scale 78771.2
PPM > USL 283036.30
O bserv ed Performance PPM Total 283036.30
PPM < LB 0.00
PPM > USL 303030.30
PPM Total 303030.30
0 100000 200000 300000 400000 500000 600000

33. 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

34. Is the Beverage Delivery
Process in Control?
I -MR Char t of New Br unsw i ck I -MR Char t of New Br unsw i ck
1 1
Using Box-Cox Transformation With Lambda = 0.50
600000 11
1 1 1
1 1 800 11 1
UCL= 485623 UCL= 733.1
I n d iv i d u a l V a l u e
450000
I ndiv idual Value
600
300000 _ _
X= 242647 X= 473.9
400
150000
0 LCL= -330 200 LCL= 214.7
1 1 1 1 1 1
1 1
1 21 41 61 81 101 121 141 161 181 1
O b se r v a t io n 1 21 41 61 81 101 121 141 161 181
Observ at ion
1
480000 1 11
1 1 600
1 1 1
1
360000 1 1
M o v in g Ra n g e
1 1
11 1 1 1
Mov ing Range
UCL= 298497 400 1
240000 UCL= 318.4
__ 200
120000 __
MR= 91359
MR= 97.4
0 LCL= 0 0 LCL= 0
1 21 41 61 81 101 121 141 161 181 1 21 41 61 81 101 121 141 161 181
O b se r v a t io n Observ at ion

35. Marlboro New Brunswick
Starbucks Wait Time Analysis
COMBINED

36. Combined Wait Time Data

37. Is there a difference between
Marlboro and New Brunswick?
Hi st ogr am of Mar l bor o, New Br unsw i ck
Gamma
40 Variable
Marlboro
New Brunswick
Shape Scale N
30 3.977 47936 94
3.080 78771 198
Fr equency
20
10
0
0 100000 200000 300000 400000 500000 600000
Dat a

38. 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)

39. Does the Data Follow a Weibull
Distribution?
Hi st ogr am of Combi ned
Weibull
35 Shape 1.954
Scale 255391
N 292
30
25
Fr equency
20
15
10
5
0
0 100000 200000 300000 400000 500000 600000
Combined

40. Does the Data Follow a Gamma
Distribution?
Hi st ogr am of Combi ned
Gamma
35 Shape 3.201
Scale 70580
N 292
30
25
Fr equency
20
15
10
5
0
0 100000 200000 300000 400000 500000 600000
Combined

41. Are the Arrival Rates the Same?
Hi st ogr am of Mar l bor o, New Br unsw i ck
2 4 6 8 10 12 14 16
Marlboro New Brunswick
9
8
7
6
Fr equency
5
4
3
2
1
0
2 4 6 8 10 12 14 16

42. 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)

43. Can the arrivals
of customers
be Modeled as
a Poisson
Process?
Goodness-of-Fit Test for Poisson Distribution
Data column: Combined
Poisson mean for Combined = 7.68421
Poisson Contribution
Combined Observed Probability Expected to Chi-Sq
<=4 10 0.119196 4.52945 6.60719
5 3 0.102708 3.90291 0.20888
6 4 0.131538 4.99846 0.19945
7 2 0.144396 5.48703 2.21602
8 4 0.138696 5.27044 0.30624
9 3 0.118419 4.49991 0.49995
10 3 0.090995 3.45782 0.06062
11 1 0.063566 2.41551 0.82950
>=12 8 0.090486 3.43846 6.05144
N N* DF Chi-Sq P-Value
38 0 7 16.9793 0.018

44. 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

45. Formal Test for the Data Being
Normally Distributed
Pr obabi l i t y Pl ot f or Combi ned
Normal - 95% CI
99.9
Goodness of Fit Test
99
AD = 4.293
95 P-Value < 0.005
90
80
70
Per cent
60
50
40
30
20
10
5
1
0.1
00 00 0 00 00 00 00 00 00 00
000 000 00 00 00 00 00 00 00
-2 -1 10 20 30 40 50 60 70
Combined

46. Formal Test for the Data Being
Gamma Distributed
Pr obabi l i t y Pl ot f or Combi ned
Gamma - 95% CI
99.9
Goodness of Fit Test
99
95 AD = 0.594
90 P-Value = 0.141
80
70
60
50
40
30
Per cent
20
10
5
1
0.1
10000 100000 1000000
Combined

47. Formal Test for the Data Being
Weibull Distributed
Pr obabi l i t y Pl ot f or Combi ned
Weibull - 95% CI
99.9
99 Goodness of Fit Test
90
AD = 0.959
80
70 P-Value = 0.016
60
50
40
30
20
Per cent
10
5
3
2
1
0.1
10000 100000 1000000
Combined

48. 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

49. Is the Process Capable Based
Upon a Gamma Model?
Pr ocess Capabi l i t y of Combi ned
Calculations Based on Gamma Distribution Model
LB USL
Process Data O v erall Capability
LB 0 Pp *
Target * PPL *
USL 300000 PPU 0.16
Sample Mean 225909 Ppk 0.16
Sample N 292
Exp. O v erall Performance
Shape 3.20075
PPM < LB *
Scale 70580
PPM > USL 237100.41
O bserv ed Performance PPM Total 237100.41
PPM < LB 0.00
PPM > USL 236301.37
PPM Total 236301.37
0 100000 200000 300000 400000 500000 600000

50. Is the Process Capable Based
Upon a Weibull Model?
Pr ocess Capabi l i t y of Combi ned
Calculations Based on Weibull Distribution Model
LB USL
Process Data O v erall Capability
LB 0 Pp *
Target * PPL *
USL 300000 PPU 0.19
Sample Mean 225909 Ppk 0.19
Sample N 292
Exp. O v erall Performance
Shape 1.95393
PPM < LB *
Scale 255391
PPM > USL 254194.23
O bserv ed Performance PPM Total 254194.23
PPM < LB 0.00
PPM > USL 236301.37
PPM Total 236301.37
0 100000 200000 300000 400000 500000 600000

51. Is the Process Capable Based
Upon a Weibull Model?
The corresponds to a Sigma level of 4. The Goal is 6!

52. Is the Process Capable Based
Upon a Gamma Model?
The corresponds to a Sigma level of 2. The Goal is 6!

53. 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

54. ?
Brandon R. Theiss
btheiss@rutgers.edu