Question

Delta Sonic is a car wash provider in Western New York. VIP Customers at their Buffalo, NY location sign up for unlimited car washes and a separate line & dedicated car wash services those customers (i.e. a single-server single-queue model). Assume VIP customers arrive every 10 minutes on average and that their inter-arrival time is exponentially distributed. Also, assume that processing (washing) time is the sum of two components:

A constant (i.e. not random) basic washing time that is exactly 4 minutes.

A random extra-service time that is exponentially distributed with mean time of 2 minutes.

In Excel, simulate the arrival times and processing times of VIP customers at this car wash using 2,000 sample customers. Using the results of your simulation, calculate the percentage of VIP customers that were in the process (i.e. waiting+washing) for longer than 12 minutes. Press F9 to rerun your simulation several times and record the results for the percentage of customers who wait longer than 12 minutes. Using the median of these recorded percentages as your estimate of the percentage of customers expected to wait longer than 12 minutes, enter that probability here as a two digit decimal e.g. 0.25, 0.45, 0.99, etc.)

Answer 1

Answer:

First, I have Prepared Simulation data for 2000 Sample with Constant Arrival time, Constant Wash Time and Random Extra Service Time

Random Extra Service Time Calculation = In MS Excel, NORMINV(RAND(),$I$1,$I$2)

$I$1 = Mean value given 2 Minutes,

$I$2= Standard Deviation Value 1.5 Minutes – Considered, for Generating Random Number,

1’st Random number generated in one cell, then dragged to all cell (total 2000 cell)

Let’s do Normality test for the Total Time include Arrival Time Plus Constant Wash Time and Extra Service Time, then, for total time as P (Probability) value is more than 0.05 , therefore Total time data is normally distributed,

Normality Check for Total Time Data in MiniTab 2017 – Stat > basic Statistics >Normality Test > select Variable Total Time , Anderson – darling Normality Test will come below like

Now Calculated Mean Time and Standard deviation for Total Time data where In MiniTab 2017, Path Stat>Basic Statistics>Display Descriptive Statistics

I will get result like

Mean Time for Total Time data = 15.994

Standard deviation for Total Time data = 1.522

Now In Minitab 2017 Go to Graph Tool Bar> Probability Distribution Plot>View Probability>Distribution Normal as Total time is Normally Distributed, Select Mean = 15.994, Standard deviation = 1.522, Keeping X Value Right Tail = 12 will get Result like below

So, The percentage of VIP customers that were in the process (i.e. waiting+washing) for longer than 12 minutes is 0.9957 X 100 = 99.57% and The Probability is .99%

Simulation data of 2000 Sample in MS Excel format here can't be uploaded so, only 500 samples data presented for reference , which used for above calculation

Sl No	Arrival Time(Minutes)	Constant Wash time	Extra Service Time	Total Time
1	10	4	3.56	17.56
2	10	4	-1.05	12.95
3	10	4	3.76	17.76
4	10	4	1.12	15.12
5	10	4	1.94	15.94
6	10	4	2.39	16.39
7	10	4	1.20	15.20
8	10	4	3.46	17.46
9	10	4	3.69	17.69
10	10	4	-0.64	13.36
11	10	4	2.33	16.33
12	10	4	1.86	15.86
13	10	4	2.08	16.08
14	10	4	1.03	15.03
15	10	4	0.09	14.09
16	10	4	1.90	15.90
17	10	4	1.78	15.78
18	10	4	2.13	16.13
19	10	4	2.57	16.57
20	10	4	3.58	17.58
21	10	4	3.25	17.25
22	10	4	3.23	17.23
23	10	4	1.28	15.28
24	10	4	-2.28	11.72
25	10	4	3.17	17.17
26	10	4	0.64	14.64
27	10	4	3.57	17.57
28	10	4	3.29	17.29
29	10	4	-1.90	12.10
30	10	4	0.83	14.83
31	10	4	1.19	15.19
32	10	4	2.09	16.09
33	10	4	2.06	16.06
34	10	4	2.14	16.14
35	10	4	5.47	19.47
36	10	4	0.12	14.12
37	10	4	0.86	14.86
38	10	4	-1.52	12.48
39	10	4	-0.23	13.77
40	10	4	3.10	17.10
41	10	4	-1.19	12.81
42	10	4	1.39	15.39
43	10	4	1.90	15.90
44	10	4	2.73	16.73
45	10	4	2.65	16.65
46	10	4	0.83	14.83
47	10	4	0.40	14.40
48	10	4	0.85	14.85
49	10	4	1.72	15.72
50	10	4	2.14	16.14
51	10	4	1.80	15.80
52	10	4	1.59	15.59
53	10	4	1.67	15.67
54	10	4	2.97	16.97
55	10	4	2.80	16.80
56	10	4	1.93	15.93
57	10	4	3.62	17.62
58	10	4	1.15	15.15
59	10	4	0.00	14.00
60	10	4	0.43	14.43
61	10	4	1.63	15.63
62	10	4	4.59	18.59
63	10	4	1.49	15.49
64	10	4	2.02	16.02
65	10	4	2.66	16.66
66	10	4	1.22	15.22
67	10	4	2.97	16.97
68	10	4	-0.88	13.12
69	10	4	1.73	15.73
70	10	4	1.48	15.48
71	10	4	2.58	16.58
72	10	4	0.76	14.76
73	10	4	1.23	15.23
74	10	4	4.47	18.47
75	10	4	3.48	17.48
76	10	4	2.95	16.95
77	10	4	0.77	14.77
78	10	4	1.14	15.14
79	10	4	1.86	15.86
80	10	4	3.59	17.59
81	10	4	1.15	15.15
82	10	4	1.14	15.14
83	10	4	0.85	14.85
84	10	4	0.83	14.83
85	10	4	2.06	16.06
86	10	4	4.55	18.55
87	10	4	3.46	17.46
88	10	4	3.12	17.12
89	10	4	3.49	17.49
90	10	4	0.50	14.50
91	10	4	0.99	14.99
92	10	4	2.06	16.06
93	10	4	2.33	16.33
94	10	4	4.98	18.98
95	10	4	2.73	16.73
96	10	4	2.17	16.17
97	10	4	2.73	16.73
98	10	4	1.95	15.95
99	10	4	0.24	14.24
100	10	4	0.65	14.65
101	10	4	1.31	15.31
102	10	4	2.14	16.14
103	10	4	1.13	15.13
104	10	4	4.27	18.27
105	10	4	-0.17	13.83
106	10	4	-1.20	12.80
107	10	4	0.89	14.89
108	10	4	-0.79	13.21
109	10	4	3.18	17.18
110	10	4	2.56	16.56
111	10	4	4.17	18.17
112	10	4	2.90	16.90
113	10	4	-0.27	13.73
114	10	4	1.50	15.50
115	10	4	4.19	18.19
116	10	4	3.30	17.30
117	10	4	4.70	18.70
118	10	4	-0.47	13.53
119	10	4	2.15	16.15
120	10	4	1.92	15.92
121	10	4	-0.03	13.97
122	10	4	1.57	15.57
123	10	4	-0.13	13.87
124	10	4	2.75	16.75
125	10	4	0.42	14.42
126	10	4	3.58	17.58
127	10	4	0.92	14.92
128	10	4	2.27	16.27
129	10	4	1.22	15.22
130	10	4	2.55	16.55
131	10	4	0.56	14.56
132	10	4	0.54	14.54
133	10	4	3.82	17.82
134	10	4	2.18	16.18
135	10	4	2.18	16.18
136	10	4	0.37	14.37
137	10	4	1.26	15.26
138	10	4	2.18	16.18
139	10	4	3.44	17.44
140	10	4	1.16	15.16
141	10	4	2.72	16.72
142	10	4	3.78	17.78
143	10	4	0.86	14.86
144	10	4	3.31	17.31
145	10	4	1.10	15.10
146	10	4	0.66	14.66
147	10	4	0.63	14.63
148	10	4	0.04	14.04
149	10	4	3.18	17.18
150	10	4	1.96	15.96
151	10	4	4.90	18.90
152	10	4	1.04	15.04
153	10	4	0.20	14.20
154	10	4	2.45	16.45
155	10	4	2.58	16.58
156	10	4	1.28	15.28
157	10	4	4.40	18.40
158	10	4	0.82	14.82
159	10	4	-0.81	13.19
160	10	4	3.75	17.75
161	10	4	2.41	16.41
162	10	4	0.35	14.35
163	10	4	4.45	18.45
164	10	4	3.48	17.48
165	10	4	6.96	20.96
166	10	4	2.79	16.79
167	10	4	1.08	15.08
168	10	4	4.02	18.02
169	10	4	0.94	14.94
170	10	4	1.70	15.70
171	10	4	4.37	18.37
172	10	4	-0.03	13.97
173	10	4	1.93	15.93
174	10	4	2.27	16.27
175	10	4	2.26	16.26
176	10	4	0.73	14.73
177	10	4	3.24	17.24
178	10	4	0.95	14.95
179	10	4	3.08	17.08
180	10	4	0.63	14.63
181	10	4	1.29	15.29
182	10	4	2.23	16.23
183	10	4	1.65	15.65
184	10	4	0.26	14.26
185	10	4	1.71	15.71
186	10	4	0.66	14.66
187	10	4	3.71	17.71
188	10	4	1.50	15.50
189	10	4	1.14	15.14
190	10	4	0.37	14.37
191	10	4	3.58	17.58
192	10	4	2.30	16.30
193	10	4	1.38	15.38
194	10	4	2.96	16.96
195	10	4	-0.34	13.66
196	10	4	1.18	15.18
197	10	4	5.76	19.76
198	10	4	-0.21	13.79
199	10	4	3.63	17.63
200	10	4	0.85	14.85
201	10	4	2.54	16.54
202	10	4	1.50	15.50
203	10	4	2.88	16.88
204	10	4	2.23	16.23
205	10	4	1.08	15.08
206	10	4	2.56	16.56
207	10	4	0.25	14.25
208	10	4	1.13	15.13
209	10	4	1.59	15.59
210	10	4	1.31	15.31
211	10	4	4.90	18.90
212	10	4	2.04	16.04
213	10	4	1.15	15.15
214	10	4	1.55	15.55
215	10	4	1.88	15.88
216	10	4	2.21	16.21
217	10	4	0.65	14.65
218	10	4	3.61	17.61
219	10	4	2.65	16.65
220	10	4	1.02	15.02
221	10	4	3.13	17.13
222	10	4	1.95	15.95
223	10	4	4.51	18.51
224	10	4	1.43	15.43
225	10	4	2.79	16.79
226	10	4	1.50	15.50
227	10	4	3.02	17.02
228	10	4	3.25	17.25
229	10	4	0.49	14.49
230	10	4	1.85	15.85
231	10	4	-0.83	13.17
232	10	4	0.84	14.84
233	10	4	2.64	16.64
234	10	4	-0.32	13.68
235	10	4	1.14	15.14
236	10	4	5.91	19.91
237	10	4	0.67	14.67
238	10	4	2.30	16.30
239	10	4	1.40	15.40
240	10	4	1.98	15.98
241	10	4	2.81	16.81
242	10	4	1.79	15.79
243	10	4	1.17	15.17
244	10	4	4.71	18.71
245	10	4	1.40	15.40
246	10	4	-0.59	13.41
247	10	4	2.45	16.45
248	10	4	2.96	16.96
249	10	4	3.21	17.21
250	10	4	1.10	15.10
251	10	4	0.40	14.40
252	10	4	0.90	14.90
253	10	4	1.65	15.65
254	10	4	1.13	15.13
255	10	4	2.83	16.83
256	10	4	0.74	14.74
257	10	4	2.63	16.63
258	10	4	2.23	16.23
259	10	4	2.75	16.75
260	10	4	4.41	18.41
261	10	4	0.11	14.11
262	10	4	3.04	17.04
263	10	4	0.46	14.46
264	10	4	1.68	15.68
265	10	4	3.22	17.22
266	10	4	2.54	16.54
267	10	4	1.42	15.42
268	10	4	1.23	15.23
269	10	4	3.53	17.53
270	10	4	0.61	14.61
271	10	4	2.99	16.99
272	10	4	2.28	16.28
273	10	4	2.55	16.55
274	10	4	2.64	16.64
275	10	4	1.51	15.51
276	10	4	2.36	16.36
277	10	4	0.32	14.32
278	10	4	2.60	16.60
279	10	4	0.34	14.34
280	10	4	3.59	17.59
281	10	4	0.09	14.09
282	10	4	1.84	15.84
283	10	4	1.21	15.21
284	10	4	3.70	17.70
285	10	4	2.15	16.15
286	10	4	2.44	16.44
287	10	4	1.83	15.83
288	10	4	-1.31	12.69
289	10	4	1.76	15.76
290	10	4	2.46	16.46
291	10	4	2.81	16.81
292	10	4	3.54	17.54
293	10	4	1.47	15.47
294	10	4	2.44	16.44
295	10	4	4.82	18.82
296	10	4	2.12	16.12
297	10	4	0.24	14.24
298	10	4	5.56	19.56
299	10	4	3.37	17.37
300	10	4	3.31	17.31
301	10	4	2.89	16.89
302	10	4	1.82	15.82
303	10	4	2.51	16.51
304	10	4	4.26	18.26
305	10	4	3.83	17.83
306	10	4	3.61	17.61
307	10	4	0.11	14.11
308	10	4	1.44	15.44
309	10	4	3.07	17.07
310	10	4	-2.07	11.93
311	10	4	1.97	15.97
312	10	4	1.70	15.70
313	10	4	0.83	14.83
314	10	4	1.03	15.03
315	10	4	0.94	14.94
316	10	4	3.26	17.26
317	10	4	1.28	15.28
318	10	4	3.87	17.87
319	10	4	3.32	17.32
320	10	4	1.19	15.19
321	10	4	-0.32	13.68
322	10	4	2.76	16.76
323	10	4	3.48	17.48
324	10	4	1.68	15.68
325	10	4	1.62	15.62
326	10	4	2.80	16.80
327	10	4	1.03	15.03
328	10	4	3.15	17.15
329	10	4	4.76	18.76
330	10	4	0.13	14.13
331	10	4	1.59	15.59
332	10	4	-0.70	13.30
333	10	4	1.34	15.34
334	10	4	2.69	16.69
335	10	4	1.50	15.50
336	10	4	1.61	15.61
337	10	4	2.69	16.69
338	10	4	0.43	14.43
339	10	4	2.46	16.46
340	10	4	2.67	16.67
341	10	4	1.92	15.92
342	10	4	2.59	16.59
343	10	4	2.53	16.53
344	10	4	2.40	16.40
345	10	4	3.27	17.27
346	10	4	2.95	16.95
347	10	4	2.10	16.10
348	10	4	0.72	14.72
349	10	4	3.12	17.12
350	10	4	1.82	15.82
351	10	4	1.25	15.25
352	10	4	0.92	14.92
353	10	4	0.11	14.11
354	10	4	1.65	15.65
355	10	4	2.53	16.53
356	10	4	5.03	19.03
357	10	4	-0.63	13.37
358	10	4	3.76	17.76
359	10	4	3.11	17.11
360	10	4	5.32	19.32
361	10	4	3.04	17.04
362	10	4	1.84	15.84
363	10	4	4.29	18.29
364	10	4	2.25	16.25
365	10	4	1.39	15.39
366	10	4	4.98	18.98
367	10	4	2.90	16.90
368	10	4	0.96	14.96
369	10	4	1.69	15.69
370	10	4	2.95	16.95
371	10	4	0.14	14.14
372	10	4	1.43	15.43
373	10	4	3.60	17.60
374	10	4	0.18	14.18
375	10	4	1.30	15.30
376	10	4	1.51	15.51
377	10	4	1.39	15.39
378	10	4	0.69	14.69
379	10	4	1.24	15.24
380	10	4	1.25	15.25
381	10	4	-0.92	13.08
382	10	4	4.41	18.41
383	10	4	1.24	15.24
384	10	4	-1.28	12.72
385	10	4	0.86	14.86
386	10	4	3.35	17.35
387	10	4	1.67	15.67
388	10	4	0.26	14.26
389	10	4	-0.80	13.20
390	10	4	2.06	16.06
391	10	4	2.41	16.41
392	10	4	0.30	14.30
393	10	4	4.36	18.36
394	10	4	2.72	16.72
395	10	4	0.70	14.70
396	10	4	2.67	16.67
397	10	4	-0.89	13.11
398	10	4	3.54	17.54
399	10	4	3.93	17.93
400	10	4	-0.32	13.68
401	10	4	2.95	16.95
402	10	4	4.64	18.64
403	10	4	3.11	17.11
404	10	4	3.63	17.63
405	10	4	1.87	15.87
406	10	4	-1.58	12.42
407	10	4	1.92	15.92
408	10	4	1.98	15.98
409	10	4	0.32	14.32
410	10	4	1.45	15.45
411	10	4	2.33	16.33
412	10	4	-0.17	13.83
413	10	4	0.35	14.35
414	10	4	2.79	16.79
415	10	4	0.53	14.53
416	10	4	1.12	15.12
417	10	4	0.86	14.86
418	10	4	2.42	16.42
419	10	4	3.41	17.41
420	10	4	2.02	16.02
421	10	4	1.43	15.43
422	10	4	1.31	15.31
423	10	4	3.44	17.44
424	10	4	1.90	15.90
425	10	4	2.21	16.21
426	10	4	0.18	14.18
427	10	4	-0.26	13.74
428	10	4	2.10	16.10
429	10	4	3.00	17.00
430	10	4	2.10	16.10
431	10	4	0.00	14.00
432	10	4	-0.40	13.60
433	10	4	2.26	16.26
434	10	4	3.35	17.35
435	10	4	2.60	16.60
436	10	4	1.02	15.02
437	10	4	4.17	18.17
438	10	4	2.27	16.27
439	10	4	3.37	17.37
440	10	4	2.67	16.67
441	10	4	2.44	16.44
442	10	4	3.14	17.14
443	10	4	1.14	15.14
444	10	4	1.63	15.63
445	10	4	1.99	15.99
446	10	4	1.80	15.80
447	10	4	0.58	14.58
448	10	4	2.47	16.47
449	10	4	-0.94	13.06
450	10	4	1.01	15.01
451	10	4	2.62	16.62
452	10	4	3.57	17.57
453	10	4	0.60	14.60
454	10	4	-2.85	11.15
455	10	4	1.42	15.42
456	10	4	1.72	15.72
457	10	4	-0.83	13.17
458	10	4	-0.67	13.33
459	10	4	3.98	17.98
460	10	4	0.36	14.36
461	10	4	0.08	14.08
462	10	4	3.34	17.34
463	10	4	1.93	15.93
464	10	4	1.10	15.10
465	10	4	1.78	15.78
466	10	4	1.19	15.19
467	10	4	1.39	15.39
468	10	4	1.34	15.34
469	10	4	1.55	15.55
470	10	4	0.99	14.99
471	10	4	2.40	16.40
472	10	4	2.16	16.16
473	10	4	1.97	15.97
474	10	4	3.50	17.50
475	10	4	1.93	15.93
476	10	4	-0.98	13.02
477	10	4	1.28	15.28
478	10	4	4.85	18.85
479	10	4	1.69	15.69
480	10	4	3.71	17.71
481	10	4	0.72	14.72
482	10	4	2.75	16.75
483	10	4	1.33	15.33
484	10	4	1.98	15.98
485	10	4	1.97	15.97
486	10	4	1.35	15.35
487	10	4	-0.49	13.51
488	10	4	0.75	14.75
489	10	4	2.33	16.33
490	10	4	2.52	16.52
491	10	4	0.50	14.50
492	10	4	3.66	17.66
493	10	4	4.22	18.22
494	10	4	2.42	16.42
495	10	4	0.67	14.67
496	10	4	1.82	15.82
497	10	4	2.35	16.35
498	10	4	1.83	15.83
499	10	4	1.98	15.98
500	10	4	2.65	16.65