1. Consider the following downtimes (in minutes) in the painting department of a manufacturing plant.

37.2 10.9 69.8 2.7 73.7 14.6 14.8 29.1 10.1 45.4 99.5 16.0 30.2 41.3 61.0 2.6 26.1 20.1 71.2 7.7

3.9 1.6 17.3 30.8 24.3 61.0 24.7 15.4 26.1

6.9 7.4 71.2 54.7 9.7 99.5 4.5 20.8 30.2 21.2 31.3 2.6 70.4 7.5 20.1 6.9 32.6 7.4 39.0 43.3 9.7

a. Use the chi-square test to see if the exponential distribution is a good fit to this sample data at the significance level a 1⁄4 0:05.

b. Using Arena’s Input Analyzer, find the best fit to the data. For what range of significance levels a can the fit be accepted? (Hint: revisit the concept of p-value in Section 3.11.)

2. Consider the following data for the monthly number of stoppages (due to failures or any other reason) in the assembly line of an automotive assembly plant.

12 14 26 13 19 15 18 17 14 10 11 11 14 18 13

9 20 19 9 12 17 20 25 18 12 16 20 20 14 11

Apply the chi-square test to the sample data of stoppages to test the hypothesis that the underlying distribution is Poisson at significance level a 1⁄4 0:05.

3. Let {xi} be a data set of observations (sample).

- Show that for the distribution Unif(a, b), the maximal likelihood estimates of

the parameters a and b are ^a 1⁄4 minfxig and ^b 1⁄4 maxfxig.

- What would be the estimates of a and b obtained using the method of

moments?

- Demonstrate the difference between the two sets of estimates for a and b using

a data set of 500 observations from a uniform distribution generated by the Arena Input Analyzer.

Input Analysis 139

4. The Revised New Brunswick Savings bank has three tellers serving customers in their Highland Park branch. Customers queue up FIFO in a single combined queue for all three tellers. The following data represent observed iid interarrival times (in minutes).

2.3 4.7 1.8 4.9 2.7 1.0 0.1 0.7 5.5 6.6 3.0 2.0 1.8 4.1 0.1 1.7 4.7 4.7 2.5 0.7

0.2 2.1 1.6 1.0 1.6 1.0 1.9 0.5 0.5 3.0 3.6 0.6 1.1 0.3 1.4 2.8 1.4 2.1 0.2 4.7 0.5 1.3 0.5 2.2 0.5 2.6 2.9 0.3 0.1 1.2

The service times are iid exponentially distributed with a mean of 5 minutes.

- Fit an exponential distribution to the interarrival time data in the previous table. Then, construct an Arena model for the teller queue at the bank. Run the model for an 8-hour day and estimate the expected customer delay in the

teller’s queue.

- Fit a uniform distribution to the interarrival time data above. Repeat the

simulation run with the new interarrival time distribution.

- Discuss the differences between the results of the scenarios of 4.a and 4.b. What do you think is the reason for the difference between the corresponding

mean delay estimates?

- Rerun the scenario of 4.a with uniformly distributed service times, and com-

pare to the results of 4.a and 4.b. Use MLE to obtain the parameters of the

service time distribution.

- Try to deduce general rules from the results of 4.a, 4.b, and 4.d on how

variability in the arrival process or the service process affects queue performance measures.

,

1. Consider the following downtimes (in minutes) in the painting department of a manufacturing plant.

37.2 10.9 69.8 2.7 73.7 14.6 14.8 29.1 10.1 45.4 99.5 16.0 30.2 41.3 61.0 2.6 26.1 20.1 71.2 7.7

3.9 1.6 17.3 30.8 24.3 61.0 24.7 15.4 26.1

6.9 7.4 71.2 54.7 9.7 99.5 4.5 20.8 30.2 21.2 31.3 2.6 70.4 7.5 20.1 6.9 32.6 7.4 39.0 43.3 9.7

a. Use the chi-square test to see if the exponential distribution is a good fit to this sample data at the significance level a 1⁄4 0:05.

b. Using Arena’s Input Analyzer, find the best fit to the data. For what range of significance levels a can the fit be accepted? (Hint: revisit the concept of p-value in Section 3.11.)

2. Consider the following data for the monthly number of stoppages (due to failures or any other reason) in the assembly line of an automotive assembly plant.

12 14 26 13 19 15 18 17 14 10 11 11 14 18 13

9 20 19 9 12 17 20 25 18 12 16 20 20 14 11

Apply the chi-square test to the sample data of stoppages to test the hypothesis that the underlying distribution is Poisson at significance level a 1⁄4 0:05.

3. Let {xi} be a data set of observations (sample).

- Show that for the distribution Unif(a, b), the maximal likelihood estimates of

the parameters a and b are ^a 1⁄4 minfxig and ^b 1⁄4 maxfxig.

- What would be the estimates of a and b obtained using the method of

moments?

- Demonstrate the difference between the two sets of estimates for a and b using

a data set of 500 observations from a uniform distribution generated by the Arena Input Analyzer.

Input Analysis 139

4. The Revised New Brunswick Savings bank has three tellers serving customers in their Highland Park branch. Customers queue up FIFO in a single combined queue for all three tellers. The following data represent observed iid interarrival times (in minutes).

2.3 4.7 1.8 4.9 2.7 1.0 0.1 0.7 5.5 6.6 3.0 2.0 1.8 4.1 0.1 1.7 4.7 4.7 2.5 0.7

0.2 2.1 1.6 1.0 1.6 1.0 1.9 0.5 0.5 3.0 3.6 0.6 1.1 0.3 1.4 2.8 1.4 2.1 0.2 4.7 0.5 1.3 0.5 2.2 0.5 2.6 2.9 0.3 0.1 1.2

The service times are iid exponentially distributed with a mean of 5 minutes.

- Fit an exponential distribution to the interarrival time data in the previous table. Then, construct an Arena model for the teller queue at the bank. Run the model for an 8-hour day and estimate the expected customer delay in the

teller’s queue.

- Fit a uniform distribution to the interarrival time data above. Repeat the

simulation run with the new interarrival time distribution.

- Discuss the differences between the results of the scenarios of 4.a and 4.b. What do you think is the reason for the difference between the corresponding

mean delay estimates?

- Rerun the scenario of 4.a with uniformly distributed service times, and com-

pare to the results of 4.a and 4.b. Use MLE to obtain the parameters of the

service time distribution.

- Try to deduce general rules from the results of 4.a, 4.b, and 4.d on how

variability in the arrival process or the service process affects queue performance measures.

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