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.

  1. Bank: bank doors open at 9 am and close at 5 pm
  2. Military battle: simulate until force strength reaches a critical value
  3. A factory where we are interested in measuring the steady state throughput
  4. 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.

  1. Report the utilization of the operators. In addition, report the total time spent in the system for each type of part.
  2. 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.

  1. 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.

  2. Collect statistics to estimate the average number of times a given job is adjusted.

  3. 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:

  1. 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?

  2. How does your running average track the theoretical value? What would happen if you increased the number of customers?

  3. 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?

  4. 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)?

  5. 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.

  6. 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?

Exercise 5.13 You have been hired to analyze the needs for loading dock facilities at a trucking terminal. The present terminal has 4 docks on the main building. Any trucks that arrive when all docks are full are assigned to a secondary terminal, which is a short distance away from the main terminal. Assume that the arrival process is Poisson with a rate of 5 trucks each hour. There is no available space at the main terminal for trucks to wait for a dock. At the present time nearly 50% of the arriving trucks are diverted to the secondary terminal. The average service time per truck is two hours on the main terminal and 3 hours on the secondary terminal, both exponentially distributed. Two proposals are being considered. The first proposal is to expand the main terminal by adding docks so that at least 80% of the arriving trucks can be served there with the remainder being diverted to the secondary terminal. The second proposal is to expand the space that can accommodate up to 8 trucks. Then, only when the holding area is full will the trucks be diverted to secondary terminal.

Build KSL models to analyze this situation. Run the models for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, answer the following questions.

  1. For the first proposal, what is the required number of docks so that at least 80% of the arriving trucks can be served?

  2. For the second proposal, will the specified number of spaces result in at least 80% of the arriving trucks being served at the main facility?

Use stream 1 for the time between arrivals and stream 2 for the service times.

Exercise 5.14 Two machines are being considered for processing a job within a factory. The first machine has an exponentially distributed processing time with a mean of 10 minutes. For the second machine the vendor has indicated that the mean processing time is 10 minutes but with a standard deviation of 6 minutes with a lognormal distribution.

Build KSL models to analyze this situation. Run the models for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, which machine is better in terms of the average waiting time of the jobs?

Use stream 1 for the time between arrivals and stream 2 for the service times.

Exercise 5.15 SuperFastCopy wants to install self-service copiers, but cannot decide whether to put in one or two machines. They predict that arrivals will be Poisson with a rate of 30 per hour, and the time spent copying is exponentially distributed with a mean of 1.75 minutes. Because the shop is small they want the probability of 5 or more customers in the shop to be small, say less than 7%.

Build a KSL model to analyze this situation. Run the model for 20,000 minutes with a warm up period of 5,000 minutes Based on 30 replications of your simulation, make a statistically valid recommendation concerning the number of machines needed to meet the customer service requirement.

Use stream 1 for the time between arrivals and stream 2 for the service times.