4.8 Exercises
Exercise 4.1 Using the supplied data set, draw the sample path for the state variable, \(Y(t)\). Assume that the value of \(Y(t)\) is the value of the state variable just after time \(t\). Compute the time average over the supplied time range.
| \(t\) | 0 | 1 | 6 | 10 | 15 | 18 | 20 | 25 | 30 | 34 | 39 | 42 |
| \(Y(t)\) | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 1 | 0 | 1 |
Exercise 4.2 Using the supplied data set, draw the sample path for the state variable, \(N(t)\). Give a formula for estimating the time average number in the system, \(N(t)\), and then use the data to compute the time average number in the system over the range from 0 to 25. Assume that the value of \(N(t\) is the value of the state variable just after time \(t\).
| \(t\) | 0 | 2 | 4 | 5 | 7 | 10 | 12 | 15 | 20 |
| \(N(t)\) | 0 | 1 | 0 | 1 | 2 | 3 | 2 | 1 | 0 |
Exercise 4.3 Consider the banking situation described within the chapter. A simulation analyst observed the operation of the bank and recorded the information given in the following table. The information was recorded right after the bank opened during a period of time for which there was only one teller working. From this information, you would like to re-create the operation of the system.
| Customer | Time of | Service |
| Number | Arrival | Time |
| 1 | 3 | 4 |
| 2 | 11 | 4 |
| 3 | 13 | 4 |
| 4 | 14 | 3 |
| 5 | 17 | 2 |
| 6 | 19 | 4 |
| 7 | 21 | 3 |
| 8 | 27 | 2 |
| 9 | 32 | 2 |
| 10 | 35 | 4 |
| 11 | 38 | 3 |
| 12 | 45 | 2 |
| 13 | 50 | 3 |
| 14 | 53 | 4 |
| 15 | 55 | 4 |
Complete a table similar to that used in the chapter and compute the average of the system times for the customers. What percentage of the total time was the teller idle? Compute the percentage of time that there were 0, 1, 2, and 3 customers in the queue.
Exercise 4.4 Consider the following inter-arrival and service times for the first 20 customers to a single server queuing system.
| Customer | Inter-Arrival | Service | Time of |
| Number | Time | Time | Arrival |
| 1 | 22 | 74 | 22 |
| 2 | 89 | 105 | 111 |
| 3 | 21 | 34 | 132 |
| 4 | 26 | 38 | 158 |
| 5 | 80 | 23 | |
| 6 | 81 | 26 | |
| 7 | 78 | 90 | |
| 8 | 20 | 26 | |
| 9 | 32 | 37 | |
| 10 | 13 | 88 | |
| 11 | 28 | 38 | |
| 12 | 18 | 73 | |
| 13 | 29 | 93 | |
| 14 | 19 | 25 | |
| 15 | 20 | 93 | |
| 16 | 23 | 5 | |
| 17 | 78 | 37 | |
| 18 | 20 | 51 | |
| 19 | 109 | 28 | |
| 20 | 78 | 85 |
We are given the inter-arrival times. Determine the time of arrival of each customer. Complete the event and state variable change table associated with this situation. Draw a sample path graph for the variable \(N(t)\) which represents the number of customers in the system at any time \(t\). Compute the average number of customers in the system over the time period from 0 to 700. Draw a sample path graph for the variable \(NQ(t)\) which represents the number of customers waiting for the server at any time \(t\). Compute the average number of customers in the queue over the time period from 0 to 700. Draw a sample path graph for the variable \(B(t)\) which represents the number of servers busy at any time t. Compute the average number of busy servers in the system over the time period from 0 to 700. Compute the average time spent in the system for the customers.
Exercise 4.5 Parts arrive at a station with a single machine according to a Poisson process with the rate of 1.5 per minute. The time it takes to process the part has an exponential distribution with a mean of 30 seconds. There is no upper limit on the number of parts that wait for process. Setup an model to estimate the expected number of parts waiting in the queue and the utilization of the machine. Run your model for 1,000,000 seconds for 30 replications and report the results. Use stream 1 for the time between arrivals and stream 2 for the service times. Use M/M/1 queueing results from the chapter to verify that your simulation is working as intended.
Exercise 4.6 A large car dealer has a policy of providing cars for its customers that have car problems. When a customer brings the car in for repair, that customer has use of a dealer’s car. The dealer estimates that the dealer cost for providing the service is $10 per day for as long as the customer’s car is in the shop. Thus, if the customer’s car was in the shop for 1.5 days, the dealer’s cost would be $15. Arrivals to the shop of customers with car problems form a Poisson process with a mean rate of one every other day. There is one mechanic dedicated to the customer’s car. The time that the mechanic spends on a car can be described by an exponential distribution with a mean of 1.6 days. Setup a model to estimate the expected time within the shop for the cars and the utilization of the mechanic. Run your model for 10000 days for 30 replications and report the results. Estimate the total cost per day to the dealer for this policy. Use the M/M/1 queueing results from the chapter to verify that your simulation is working as intended. Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.7 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 depart the system. The company is interested in the total time spent in the system. Run your model for 10000 minutes for 30 replications and report the results. Use stream 1 for the time between arrivals, stream 2 for inspection times, stream 3 for inspections, and stream 4 for adjustments.
Exercise 4.8 Referring to the pharmacy model discussed in Section 4.4.4, suppose that the customers arriving to the drive through pharmacy can decide to enter the store instead of entering the drive through lane. Assume a 90% chance that the arriving customer decides to use the drive through pharmacy and a 10% chance that the customer decides to use the store. Model this situation with and discuss the effect on the performance of the drive through lane. Use stream 3 for the decision process. Run your model for 30 replications of length 20000 minutes and a warm up period of 5000 minutes.
Exercise 4.9 SQL queries arrive to a database server according to a Poisson process with a rate of 1 query every minute. The time that it takes to execute the query on the server is typically between 0.6 and 0.8 minutes uniformly distributed. The server can only execute 1 query at a time. Develop a simulation model to estimate the average delay time for a query. Use stream 1 for the time between query arrivals and stream 2 for the query execution time. Run your model for 30 replications having a length of 100,000 minutes. Use stream 1 for the query arrival process and stream 2 for the time to execute the query.
Exercise 4.10 Passengers arrive to an airport security check point at a small airport for identification inspection according to a Poisson process with a rate of 30 per hour. Assume that the check point is staffed by a single security officer. The officer can check the passenger’s identification with a minimum time of .75 minute, most likely value of 1.5 minutes, and a maximum of 3 minutes, triangularly distributed. Past data has shown that 93% of the passengers immediately pass the identification inspection. Those passengers that immediately pass move on to the security area. Those that do not pass are sent to a separate area for further investigation. For the purposes of this exercise, the activities after the determination of passing identification inspection are outside the scope of this modeling. Develop a simulation model that can estimate the following quantities:
- utilization of the security officer
- average number of passengers waiting for identification inspection
- average time spent by passengers waiting for identification inspection
- the probability that a passenger must wait longer than 5 minutes for identification inspection
- the average number of passengers cleared per day
- the average number of passengers denied per day
Assume that the check point operates for 10 hours per day, starting at 8 am. Also, assume that when the check point opens, there are no passengers waiting. Finally, for simplicity, assume that we are only interested in the statistics collected during the 10-hour time span. Analyze this situation for 20 independent days of operation. Use stream 1 for the arrival process, stream 2 for the service process, and stream 3 for the inspection process.
Exercise 4.11 Consider a single pump gas station where the arrival process is Poisson with a mean time between arrivals of 10 minutes. The service time is exponentially distributed with a mean of 6 minutes.
Build a KSL model to simulate 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, estimate the following performance measures.
What is the probability that you have to wait for service?
What is the expected number of customers at the station?
What is the expected time waiting in the line to get a pump?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.12 Suppose an operator has been assigned to the responsibility of maintaining 3 machines. For each machine the probability distribution of the running time before a breakdown is exponentially distributed with a mean of 9 hours. The repair time also has an exponential distribution with a mean of 2 hours.
Build a KSL model to simulate this situation. Run the model for 20,000 hours with a warm up period of 5,000 hours Based on 30 replications of your simulation, estimate the following performance measures.
What is the probability that the operator is idle?
What is the expected number of machines that are running?
What is the expected number of machines that are not running?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.13 Each airline passenger and his or her carry-on baggage must be checked at the security checkpoint. Suppose XNA averages 10 passengers per minute with exponential inter-arrival times. To screen passengers, the airport must have a metal detector and baggage X-ray machines. Whenever a checkpoint is in operation, two employees are required (one operates the metal detector, one operates the X-ray machine). The passenger goes through the metal detector and simultaneously their bag goes through the X-ray machine. A checkpoint can check an average of 12 passengers per minute according to an exponential distribution.
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, estimate the following performance measures.
What is the probability that a passenger will have to wait before being screened?
On average, how many passengers are waiting in line to enter the checkpoint?
On average, how long will a passenger spend at the checkpoint?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.14 Customers arrive at a one-window drive in bank according to a Poisson distribution with a mean of 10 per hour. The service time for each customer is exponentially distributed with a mean of 5 minutes. There are 3 spaces in front of the window including that for the car being served. Other arriving cars can wait outside these 3 spaces.
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, estimate the following performance measures.
What is the probability that an arriving customer can enter one of the 3 spaces in front of the window?
What is the probability that an arriving customer will have to wait outside the 3 spaces?
How long is an arriving customer expected to wait before starting service?
How many spaces should be provided in front of the window so that an arriving customer can wait in front of the window at least 20% of the time? In other words, the probability of at least one open space must be greater than 20%.
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.15 Joe Rose is a student at Big State U. He does odd jobs to supplement his income. Job requests come every 5 days on the average, but the time between requests is exponentially distributed. The time for completing a job is also exponentially distributed with a mean of 4 days.
Build a KSL model to analyze this situation. Run the model for 20,000 days with a warm up period of 5,000 days Based on 30 replications of your simulation, estimate the following performance measures.
What is the chance that Joe will not have any jobs to work on?
What is the average value of the waiting jobs if Joe gets about $25 per job?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.16 The manager of a bank must determine how many tellers should be available. For every minute a customer stands in line, the manager believes that a delay cost of 5 cents is incurred. An average of 15 customers per hour arrive at the bank. On the average, it takes a teller 6 minutes to complete the customer’s transaction. It costs the bank $9 per hour to have a teller available. Inter-arrival and service times can be assumed to be exponentially distributed.
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, answer the following questions.
What is the minimum number of tellers that should be available in order for the system to be stable (i.e. not have an infinite queue)?
If the system has 3 tellers, what is the probability that there will be no one in the bank?
What is the expected total cost of the system per hour, when there are 2 tellers?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.17 Sly’s convenience store operates a two-pump gas station. The lane leading to the pumps can house at most five cars, including those being serviced. Arriving cars go elsewhere if the lane is full. The distribution of the arriving cars is Poisson with a mean of 20 per hour. The time to fill up and pay for the purchase is exponentially distributed with a mean of 6 minutes.
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, answer the following questions.
What is the percentage of cars that will seek business elsewhere?
What is the utilization of the pumps?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.18 An airline ticket office has two ticket agents answering incoming phone calls for flight reservations. In addition, two callers can be put on hold until one of the agents is available to take the call. If all four phone lines (both agent lines and the hold lines) are busy, a potential customer gets a busy signal, and it is assumed that the call goes to another ticket office and that the business is lost. The calls and attempted calls occur randomly (i.e. according to Poisson process) at a mean rate of 15 per hour. The length of a telephone conversation has an exponential distribution with a mean of 4 minutes.
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, answer the following questions.
What is probability of losing a potential customer?
What is the probability that an arriving phone call will not start service immediately but will be able to wait on a hold line?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.19 SuperFastCopy has three identical copying machines. When a machine is being used, the time until it breaks down has an exponential distribution with a mean of 2 weeks. A repair person is kept on call to repair the machines. The repair time for a machine has an exponential distribution with a mean of 0.5 week. The downtime cost for each copying machine is $100 per week.
Build a KSL model to analyze this situation. Run the model for 20,000 weeks with a warm up period of 5,000 weeks Based on 30 replications of your simulation, what is the expected downtime cost per week.
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.20 NWH Cardiac Care Unit (CCU) has 5 beds, which are virtually always occupied by patients who have just undergone major heart surgery. Two registered nurses (RNs) are on duty in the CCU in each of the three 8 hour shifts. About every two hours following an exponential distribution, one of the patients requires a nurse’s attention. The RN will then spend an average of 30 minutes (exponentially distributed) assisting the patient and updating medical records regarding the problem and care provided.
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, answer the following questions.
What is the average number of patients being attended by the nurses?
What is the average time that a patient spends waiting for one of the nurses to arrive?
Use stream 1 for the time between arrivals and stream 2 for the service times.
Exercise 4.21 HJ Bunt, Transport Company maintains a large fleet of refrigerated trailers. For the purposes of this problem assume that the number of refrigerated trailers is conceptually infinite. The trailers require service on an irregular basis in the company owned and operated service shop. Assume that the arrival of trailers to the shop is approximated by a Poisson distribution with a mean rate of 3 per week. The length of time needed for servicing a trailer varies according to an exponential distribution with a mean service time of one-half week per trailer. The current policy is to utilize a centralized contracted outsourced service center whenever more than two trailers are in the company shop, so that, at most one trailer is allowed to wait. Assume that there is currently one 1 mechanic in the company shop.
Build a KSL model to analyze this situation. Run the model for 20,000 weeks with a warm up period of 5,000 weeks Based on 30 replications of your simulation, what is the expected number of repairs that are outsourced per week?
Use stream 1 for the time between arrivals and stream 2 for the service times.