Your team has been hired by a new pretzel shop, Twisted Pretzel. They are trying
ID: 3301894 • Letter: Y
Question
Your team has been hired by a new pretzel shop, Twisted Pretzel. They are trying to determine the number of employees they should schedule. Twisted Pretzel says each employee can handle 15 arrivals per hour. You decide the first step is to know the probability of the number of customers arriving at the store in a given hour. You conducted an experiment and recorded the number of arrivals per hour for 7 days. Assume the arrivals follow a Poisson Probability Distribution.
Deliverables:
Create a report to be submitted to Twisted Pretzel. It should include at least the following all presented in one document:
At least one page presenting your results.
The probability of each possible number of arrivals from 0 to 50 per hour.
The probability of more than 15 arrivals and the probability of more than 30 arrivals.
How many employees do you think you should have working?
Include any charts and tables needed to convey your results. Be sure to follow all of the data visualization rules.
Day
Time
Arrivals
1
10-11
24
1
11-12
18
1
12-1
22
1
1-2
27
1
2-3
28
1
3-4
20
1
4-5
13
1
5-6
25
1
6-7
20
1
7-8
19
2
10-11
16
2
11-12
21
2
12-1
33
2
1-2
23
2
2-3
21
2
3-4
20
2
4-5
18
2
5-6
21
2
6-7
23
2
7-8
27
3
10-11
19
3
11-12
14
3
12-1
21
3
1-2
23
3
2-3
17
3
3-4
23
3
4-5
20
3
5-6
31
3
6-7
23
3
7-8
26
4
10-11
22
4
11-12
26
4
12-1
37
4
1-2
29
4
2-3
27
4
3-4
19
4
4-5
24
4
5-6
19
4
6-7
20
4
7-8
28
5
10-11
24
5
11-12
18
5
12-1
27
5
1-2
33
5
2-3
20
5
3-4
24
5
4-5
18
5
5-6
21
5
6-7
21
5
7-8
23
6
10-11
27
6
11-12
19
6
12-1
29
6
1-2
19
6
2-3
24
6
3-4
22
6
4-5
32
6
5-6
23
6
6-7
22
6
7-8
21
7
10-11
20
7
11-12
28
7
12-1
17
7
1-2
20
7
2-3
29
7
3-4
20
7
4-5
18
7
5-6
23
7
6-7
22
7
7-8
25
Day
Time
Arrivals
1
10-11
24
1
11-12
18
1
12-1
22
1
1-2
27
1
2-3
28
1
3-4
20
1
4-5
13
1
5-6
25
1
6-7
20
1
7-8
19
2
10-11
16
2
11-12
21
2
12-1
33
2
1-2
23
2
2-3
21
2
3-4
20
2
4-5
18
2
5-6
21
2
6-7
23
2
7-8
27
3
10-11
19
3
11-12
14
3
12-1
21
3
1-2
23
3
2-3
17
3
3-4
23
3
4-5
20
3
5-6
31
3
6-7
23
3
7-8
26
4
10-11
22
4
11-12
26
4
12-1
37
4
1-2
29
4
2-3
27
4
3-4
19
4
4-5
24
4
5-6
19
4
6-7
20
4
7-8
28
5
10-11
24
5
11-12
18
5
12-1
27
5
1-2
33
5
2-3
20
5
3-4
24
5
4-5
18
5
5-6
21
5
6-7
21
5
7-8
23
6
10-11
27
6
11-12
19
6
12-1
29
6
1-2
19
6
2-3
24
6
3-4
22
6
4-5
32
6
5-6
23
6
6-7
22
6
7-8
21
7
10-11
20
7
11-12
28
7
12-1
17
7
1-2
20
7
2-3
29
7
3-4
20
7
4-5
18
7
5-6
23
7
6-7
22
7
7-8
25
Explanation / Answer
Solution
Let X = number of arrivals per hour. We are given that X ~ Poisson ().
From the given study data on arrivals for 7 days, is estimated by Xbar which is found to be
= 22.8
With this value of , using Excel Function on Poisson Distribution, probabilities of arrivals 0 (1) 50 are tabulated below:
=
22.8
x
P(X = x)
0
1.2534E-10
1
2.8577E-09
2
3.2578E-08
3
2.4759E-07
4
1.4113E-06
5
6.4354E-06
6
2.4455E-05
7
7.9652E-05
8
0.00022701
9
0.00057509
10
0.00131121
11
0.00271777
12
0.00516377
13
0.00905646
14
0.01474909
15
0.02241862
16
0.03194654
17
0.04284594
18
0.05427153
19
0.06512583
20
0.07424345
21
0.08060717
22
0.08353834
23
0.08281192
24
0.07867133
25
0.07174825
26
0.0629177
27
0.0531305
28
0.04326341
29
0.03401399
30
0.02585063
31
0.01901272
32
0.01354656
33
0.00935944
34
0.00627633
35
0.00408858
36
0.00258944
37
0.00159565
38
0.00095739
39
0.00055971
40
0.00031903
41
0.00017741
42
9.631E-05
43
5.1067E-05
44
2.6462E-05
45
1.3407E-05
46
6.6454E-06
47
3.2237E-06
48
1.5313E-06
49
7.1251E-07
50
3.249E-07
Total
0.99999974
P(X < 16)
0.05633127
P(X < 31)
0.94131778
P(X > 15)
0.94366873
P(X > 30)
0.05868222
Probability of more than 15 arrivals = P(X > 15) = 0.9437
Probability of more than 30 arrivals = P(X > 30) = 0.0587
Number of employees to be working
Let this number be c. The desirable value of c would depend on the managerial stipulations on queue characteristics like average queue length, average waiting time, average time spent in the system etc.
Since these stipulations are not specified in the question, a definite answer cannot be derived. However, a general methodology is presented here.
We have average arrival rate, = 22.8[computed from the given data] and average service rate, µ = 15 [given].
Then, with number of employees working = c,
Average queue length = E(m) = P0{(µ)(/µ)c}/{(c - 1)!(cµ - )2}…………………………..(1)
Average number of customers in the system = E(n) = E(m) + (/µ)………………………..(2)
Average waiting time = E(w) = E(m)/()…… ……………………………………………..(3)
Average time spent in the system = E(v) = E(w) + (1/µ)..…………………………………..(4)
Percentage idle time of service channel = P0 ………. …………………………………….(5)
P0 = 1/(S1 + S2), where S1 = [0,c - 1](n/n!) and S2 = (c)/[c!{1 - (/c)}]……………….(6)
Using these formulae, and the given stipulations on queue characteristics, the desirable value of c can be found.
=
22.8
x
P(X = x)
0
1.2534E-10
1
2.8577E-09
2
3.2578E-08
3
2.4759E-07
4
1.4113E-06
5
6.4354E-06
6
2.4455E-05
7
7.9652E-05
8
0.00022701
9
0.00057509
10
0.00131121
11
0.00271777
12
0.00516377
13
0.00905646
14
0.01474909
15
0.02241862
16
0.03194654
17
0.04284594
18
0.05427153
19
0.06512583
20
0.07424345
21
0.08060717
22
0.08353834
23
0.08281192
24
0.07867133
25
0.07174825
26
0.0629177
27
0.0531305
28
0.04326341
29
0.03401399
30
0.02585063
31
0.01901272
32
0.01354656
33
0.00935944
34
0.00627633
35
0.00408858
36
0.00258944
37
0.00159565
38
0.00095739
39
0.00055971
40
0.00031903
41
0.00017741
42
9.631E-05
43
5.1067E-05
44
2.6462E-05
45
1.3407E-05
46
6.6454E-06
47
3.2237E-06
48
1.5313E-06
49
7.1251E-07
50
3.249E-07
Total
0.99999974
P(X < 16)
0.05633127
P(X < 31)
0.94131778
P(X > 15)
0.94366873
P(X > 30)
0.05868222
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