5.10 Exercises
Exercise 5.1 Indicate whether or not the statistical quantity should be classified as tally-based or time-persistent.
| Classification | Description |
|---|---|
| The number of jobs waiting to be processed by a machine | |
| The number of jobs completed during a week | |
| The number in customers in queue: | |
| The time that the resource spends serving a customer | |
| The number of items in sitting on a shelf waiting to be sold | |
| The waiting time in a queue | |
| The number of patients processed during the first hour of the day | |
| The time that it takes a bus to complete its entire route | |
| The number of passengers departing the bus at Dickson street | |
| The amount of miles that a truck travel empty during a week |
Exercise 5.2 True or False A replication is the generation of one sample path which represents the evolution of the system from its initial conditions to its ending conditions.
Exercise 5.3 In a \(\underline{\hspace{3cm}}\) horizon simulation, a well defined ending time or ending condition can be specified which clearly demarks the end of the simulation.
Exercise 5.4 Which of the following are finite horizon situations? Select all that apply.
- Bank: bank doors open at 9 am and close at 5 pm
- Military battle: simulate until force strength reaches a critical value
- A factory where we are interested in measuring the steady state throughput
- A hospital emergency room which is open 24 hours a day, 7 days a week
Exercise 5.5 Consider a manufacturing system comprising two different machines and two operators. Each operator is assigned to run a single machine. Parts arrive with an exponentially distributed inter-arrival time with a mean of 4 minutes. The arriving parts are one of two types. Sixty percent of the arriving parts are Type 1 and are processed on Machine 1. These parts require the assigned operator for a one-minute setup operation. The remaining 40 percent of the parts are Type 2 parts and are processed on Machine 2. These parts require the assigned operator for a 1.5-minute (constant) setup operation. The service times (excluding the setup time) are lognormally distributed with a mean of 4.5 minutes and a variance of 1 minute for Type 1 parts and a mean of 7.5 minutes and a variance of \((1.5)^2\) minutes for Type 2 parts. The operator of the machine is required to both setup and operate the machine.
Run your model for 20000 minutes, with 10 replications. Use stream 1 for the arrival process, stream 2 for machine 1 processing, stream 3 for machine 2 processing, and stream 4 for the part type determination.
- Report the utilization of the operators. In addition, report the total time spent in the system for each type of part.
- Perform a warm up analysis on this system.
Exercise 5.6 YBox video game players arrive according to a Poisson process with rate 10 per hour to a two-person station for inspection. The inspection time per YBox set is exponentially distributed with a mean of 10 minutes. On the average 82% of the sets pass inspection. The remaining 18% are routed to an adjustment station with a single operator. Adjustment time per YBox is uniformly distributed between 7 and 14 minutes. After adjustments are made, the units are routed back to the inspection station to be retested. Assume that a part can be adjusted as many times as needed until it passes inspection. Build an simulation model of this system. Use a replication length of 30,000 minutes. Use stream 1 for the arrival process, stream 2 for the inspection time, stream 3 for the adjustment time, and stream 4 to determine if the part needs adjustment.
Perform a warm up analysis of the total time a set spends in the system and estimate the system time to within 2 minutes with 95% confidence.
Collect statistics to estimate the average number of times a given job is adjusted.
Suppose that any one job is not allowed more than two adjustments, after which time the job must be discarded. Modify your simulation model and estimate the number of discarded jobs.
Exercise 5.7 Create a model to simulate observations from a \(N(\mu, \sigma^2)\) random variable. Use your simulation to generate two independent samples of size \(n_1 = 20\) and \(n_2 = 30\) from normal distributions having \(\mu_1 = 2\), \(\sigma_1^2 = 0.64\) and \(\mu_2 = 2.2, \sigma_2^2 = 0.64\). Assume that you don’t know the true means and variances. Use the method of independent samples to test whether \(\mu_2 > \mu_1\).
Exercise 5.8 Create a model to simulate observations from a \(N(\mu, \sigma^2)\) random variable. Use your simulation to generate two independent samples of size \(n_1 = 20\) and \(n_2=30\) from normal distributions having \(\mu_1 = 2\), \(\sigma_1^2 = 0.64\), and \(\mu_2 = 2.2\), \(\sigma_2^2 = 0.36\). Assume that you don’t know the true means and variances. Use the method of independent samples to test whether \(\mu_2 > \mu_1\).
Exercise 5.9 Create a model to simulate observations from a \(N(\mu, \sigma^2)\) random variable. Use your simulation to generate two independent samples of size \(n_1 = 30\) and \(n_2 = 30\) from normal distributions having \(\mu_1 = 2\), \(\sigma_1^2 = 0.64\) and \(\mu_2 = 2.2\), \(\sigma_2^2 = 0.36\). Assume that you don’t know the true means and variances. Use the paired-t method to test whether \(\mu_2 > \mu_1\).
Exercise 5.10 Create a model to simulate observations from a \(N(\mu, \sigma^2)\) random variable. Use your simulation to generate two dependent samples of size \(n_1 = 30\) and \(n_2 = 30\) from normal distributions having \(\mu_1 = 2\), \(\sigma_1^2 = 0.64\) and \(\mu_2 = 2.2\), \(\sigma_2^2 = 0.36\). Use the method of common random number. Assume that you don’t know the true means and variances. Use the paired-t method to test whether \(\mu_2 > \mu_1\).
Exercise 5.11 Consider a M/M/1 queueing system with arrival rate of 1 customer per minute and a service rate of 1.2 customers per minute. Use stream 1 for the arrival process and stream 2 for the service process and perform the following:
Develop a 95% confidence interval for your estimate of the mean waiting time based on the data from 1 replication of 1000 customers. Discuss why this is inappropriate. How does your simulation estimate compare to the theoretical value?
How does your running average track the theoretical value? What would happen if you increased the number of customers?
Construct a Welch plot using 5 replications of the 1000 customers. Determine a warm up point for this simulation. Do you think that 1000 customers are enough?
Make an autocorrelation plot of the first 1000 customer wait times. What are the assumptions for forming the confidence interval in part (a). Is this data independent and identically distributed? What is the implication of your answer for your confidence interval in part (a)?
Use your warm up period from part (c) and generate an addition 1000 customers after the warm up point. Use the method of batch means to batch the 1000 observations into 40 batches of size 25. Make an autocorrelation plot of the 40 batch means. Compute a 95% confidence interval for the mean waiting time using the 40 batches.
Use the method of replication deletion to develop a 95% confidence interval for the mean waiting time. Use your warm period from part (c). Compare the result with that of (a) and (e) and discuss. What do you recommend for the warmup period?
Exercise 5.12 Reconsider Exercise 4.10 of Chapter 4, with stream 1 for the arrival process, stream 2 for the service process, and stream 3 for the inspection process. Suppose that after the passenger identification inspection, we now need to model a simplified version of the baggage screening process.
Part (a): Once passengers clear the identification check, they proceed to the X-ray baggage screening. Those that were denied, exit the system. For those proceeding to X-ray screening, at a minimum, it takes 1.5 minutes per passenger for the X-ray process to complete. Typically, this process takes 2.5 minutes. At the most, this process can last 7 minutes. There are two X-ray machines. Use stream 4 for the X-ray screening time.
- What is the maximum time that a passenger had to wait in line for the X-ray machine?
- What is the utilization of the X-ray machines?
Part (b): To increase security measures, a more extensive security check of passengers is performed after the baggage scan. Every 15th passenger will go through a full-body scan and manual baggage review. This inspection usually takes 5 minutes. At the least, it will take 3 minutes and at the most 10 minutes. Assume there is unlimited availability of resources to model this extra security check. Use stream 5 for the full-body scan and manual baggage review inspection.
- What is the overall average cycle time of passengers (from the time they enter the system until they are through all security points) who are selected for this check?