Queueing Theory a. Parts arrive for processing at a bottleneck tool at an averag
ID: 670563 • Letter: Q
Question
Queueing Theory
a. Parts arrive for processing at a bottleneck tool at an average rate of 15 per hour. The tool processes a part in 3.75 minutes, on average, with a standard deviation of 1.3 minutes.
What is the average number of parts waiting for processing at this bottleneck station? Show all your work.
b. A tool crib that serves a large group of mechanics has exponential inter-arrival and service times. The mean time between arrivals is 3.5 minutes, and it takes 3 minutes on the average for a single tool-crib attendant to provide service to each mechanic. The tool crib attendant is paid $15/hour, and each mechanic is paid $25/hour. Turn in answers to the following questions (show all your work):
i. What is the utilization of the tool crib attendant?
ii. What is the hourly cost at the tool crib?
Explanation / Answer
Data Tabulated first:
Hours
Arrival of parts per hour = 15
tool capacity in minutes
standard deviation = 1.3 minutes
1
15
50.55
34.82501
2
30
101.1
70.56926
3
45
151.65
106.3135
4
60
202.2
142.0578
5
75
252.75
177.802
6
90
303.3
213.5462
7
105
353.85
249.2905
8
120
404.4
285.0347
9
135
454.95
320.779
10
150
505.5
356.5232
11
165
556.05
392.2675
12
180
606.6
428.0117
13
195
657.15
463.756
14
210
707.7
499.5002
15
225
758.25
535.2445
16
240
808.8
570.9887
17
255
859.35
606.733
18
270
909.9
642.4772
19
285
960.45
678.2215
20
300
1011
713.9657
Now to find the answer:
(a) The average number of parts waiting for processing:
Hour
Average Number of parts waiting for processing
Tool Capacity in Hours
1
-187.2
0.8425
2
-172.2
1.685
3
-157.2
2.5275
4
-142.2
3.37
5
-127.2
4.2125
6
-112.2
5.055
7
-97.2
5.8975
8
-82.2
6.74
9
-67.2
7.5825
10
-52.2
8.425
11
-37.2
9.2675
12
-22.2
10.11
13
-7.2
10.9525
14
7.8
11.795
15
22.8
12.6375
16
37.8
13.48
17
52.8
14.3225
18
67.8
15.165
19
82.8
16.0075
20
97.8
16.85
Tool Crib: ( A simulation of 20 arrivals in the Queue):
Arrivals
Arrival Times in minutes
1
0
2
3.5
3
7
4
10.5
5
14
6
17.5
7
21
8
24.5
9
28
10
31.5
11
35
12
38.5
13
42
14
45.5
15
49
16
52.5
17
56
18
59.5
19
63
20
66.5
Time taken by Attendant to delegate(Provide service to) Mechanic (in minutes)
3
6
9
12
15
18
21
24
27
30
33
36
39
42
45
48
51
54
57
60
Hours
Arrival of parts per hour = 15
tool capacity in minutes
standard deviation = 1.3 minutes
1
15
50.55
34.82501
2
30
101.1
70.56926
3
45
151.65
106.3135
4
60
202.2
142.0578
5
75
252.75
177.802
6
90
303.3
213.5462
7
105
353.85
249.2905
8
120
404.4
285.0347
9
135
454.95
320.779
10
150
505.5
356.5232
11
165
556.05
392.2675
12
180
606.6
428.0117
13
195
657.15
463.756
14
210
707.7
499.5002
15
225
758.25
535.2445
16
240
808.8
570.9887
17
255
859.35
606.733
18
270
909.9
642.4772
19
285
960.45
678.2215
20
300
1011
713.9657
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