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.