Waiting Line and Queuing Theory Solutions Essay Example

Waiting Line and Queuing Theory Solutions Paper Students should realize that different organizations place different values on customer waiting time. Ask students to consider different scenarios, from a drive-through restaurant to a doctors office to a registration line in their college or motor vehicle office. It becomes clear that organizations place different values on their customers time (with most colleges and Dams unfortunately placing minimal cost on waiting time). Teaching Suggestion 14. 3: use of Poisson and Exponential Probability Distributions to Describe Arrival and Service Rates. These two distributions are very common in basic models, but students should not take their appropriateness for granted. As a project, ask students to visit a bank or drive-through restaurant ND time arrivals to see if they indeed are Poisson distributed. Note that other distributions (such as exponential, normal, or Erelong) are often more valid. Teaching Suggestion 14. 4: Balking and Reneging Assumptions, Note that most queuing models assume that balking and reneging are not permitted. Since we know they do occur in supermarkets, what can be done? This is one of many places to prepare students for the need for simulation, the topic of the next chapter. Teaching Suggestion 14. 5: use of Queuing Software The Excel KM and KM for Windows queuing software modules are among the easiest models n the program to use since there are so few inputs. Yet students should be reminded of how long it would take to produce the programs in Chapter 14 by hand. Teaching Suggestion 14. 6: Importance Of LLC and Was in Economic Analysis. Although many parameters are computed for a queuing study, the two most important ones are LLC and Was when it comes to an actual cost analysis. We will write a custom essay sample on Waiting Line and Queuing Theory Solutions specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Waiting Line and Queuing Theory Solutions specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Waiting Line and Queuing Theory Solutions specifically for you FOR ONLY $16.38 $13.9/page Hire Writer Teaching Suggestion 14. 7: Teaching the New England Foundry Case. Here is a tip for this very teachable case. About half the students who tackle the case forget that time walking to the counter must be noted and that the return time also needs to be added. ALTERNATIVE EXAMPLES Alternative Example 14. 1: A new shopping mall is considering setting up an information desk manned by one employee. Based on information obtained from similar information desks, it is believed that people Will arrive at the desk at the rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed. A. Find the probability that the employee is idle. B. Find the proportion of the time that the employee is busy. C. Find the average number of people receiving and waiting to receive information. D. Find the average number f people waiting in line to get information. E. Find the average time a person seeking information spends at the desk. F. Find the expected time a person spends just waiting in line to have a question answered. ANSWER: a. B. C. L 20/hour 1 0. 6 20 30 20 1 2030 30/hour . 33 33% 2 people q (1 (20)2 1. 33 people ) 3030 20) 1 0. 1 hour 30 20 Wq 20 30130 20) 0. 0667 hours Alternative Example 14. 2: In Alternative Example 14. 1 the information desk employee earns $5/hour. The cost of waiting time, in terms of customer unhappiness with the mall, is 512/hour of time spent waiting in line. Vind the total expected costs over an 8hour day. . The average person waits 0. 0667 hour and there are 160 arrivals per day. So total waiting time (1 10. 67 hours @ $12/hour, implying a waiting cost of $128/day. b. The salary cost is $40/day. C. Total costs are $128 $40 $168/day. 5/12/08 1:01 PM Page 218 CHAPTER 14 WAITING LINE AND QUEUING THEORY MODELS Alternative Example 14. 3: A new shopping mall is considering setting up an information desk manned by two employees. Based on information obtained from similar information desks, it is believed that people will arrive at the desk distributed. A. Find the proportion of the time that the employees are idle. B. Find he average number Of people waiting in the system. C. Pin the expected time 3 person spends waiting in the system. ANSWER: (servers). A. P 20/hour, 30/hour, M 2 open channels SOLUTIONS TO DISCUSSION QUESTIONS AND PROBLEMS 14-1. The waiting line problem concerns the question of finding the ideal level of service that an organization should provide. The three components of a queuing system are arrivals, waiting line, and service facility. 14-2, The seven underlying assumptions are: 1. Arrivals are FIFO. 2. There is no balking or reneging. 3. Arrivals are independent. 4 Arrivals are Poisson. 5. Service times are independent. . Service times are negative exponential. 7. Average service rate exceeds average arrival rate, 14-3, The seven operating characteristics are: 1. Average number to customers in the system (L) 2. Average time spent in the system (W) 3, Average number in the queue (LLC) 4. Average time in the queue (Was) 5. Utilization factor ( ) 6. Percent idle time (Pop) 7. Probability there are more than K customers in the system 1 (20 10! Icily 123 911 J 1312 II (6020) J b. / 3012 (1 2012 12 J 20 30 (J 1, 600121 L 3/ 420 23 112 812 912 3 people p 4 144. If the service rate is not greater than the arrival rate, an infinite queue will eventually build up. 4-5. First-in, first-out (FIFO) is often not applicable. Some examples are (I) hospital emergency rooms, (2) an elevator, (3) an airplane trip, (4) a small store where the shopkeeper serves whoever can get his or her attention first, (5) a computer system set to accept priority runs, (6) a college registration system that allows juniors and seniors to register ahead of freshmen and sophomores, (7) a restaurant that may seat a party Of 2 before a party Of 4 even though the latter group arrived earlier, (8) a garage that repairs cars with minor problems before it works on major overhauls. 4-6. Examples Of finite ensuing situations include (1) a firm that has only 3 or 4 machines that need servicing, (2) a small airport at which only 10 or 15 flights land each day, (3) a classroom that seats only 30 students for class, (4) a physician who has a limited number of patients, and (5) a hospital ward with only 20 patients who need care. 14-7. A. Barbershop: usually a single-channel, multiplicities system (if there is more than one barber). Arrivals Waiting line Service customers wanting haircuts seated customers who informally recognize who arrived first among them haircut, style, shampoo, and so forth: it service involves barber, then shampooing, hen manicurist, it becomes a multiphase system 3 her. 80 0. 0375 Alternative Example 14. 4: Three students arrive per minute at a coffee machine that dispenses exactly 4 cups/minute at a constant rate. Describe the operating system parameters. ANSWER: 3/millet 2 4/minute 9 4 3) was 1. 125 people in queue on average 3 0. 375 minutes in the queue waiting alls 1. 254 1. 87 people in the system 1 1 WA _375 4 0. 625 minutes in the system 5/12/08 1:01 PM Page 219 b. Car wash: usually either a single-channel, single-server system, or else a system with each service bay having its own queue. Arrivals Waiting time Service arty cars or trucks cars in one line (or more lines if there are service parallel wash systems); always FIFO either multiphase (if car first vacuumed, then soaped, then sent through automatic cleaner, then dried by hand) or single-phase if all automatic or performed by one person 14-8. The vitiating time cost should be based on time in the queue in situations where the customer does not mind how long it takes to complete service once the service starts. The classic example of this is waiting in line for an amusement park ride. Waiting time cost should be based on the time in the system when the entire time is important to the customer. When a computer or an automobile is taken into the shop to be repaired, the customer is Without use Of the item until the service is finished. In such a situation, the time in the system is the relevant time. 4-9. The use Of Poisson to describe arrivals: a. Cafeteria: probably not. Most people arrive in groups and eat at the same time. B. Barbershop: probably acceptable, especially on a weekend, in which case people arrive at the same rate all day long. C. Hardware store: okay. D. Dentists office: usually not. Patients are most likely scheduled at IS. To 30. Minute intervals and do not arrive randomly. , College class: number of students come in groups at the beginning of class period; very few arrive during the class or very early before class. F. Movie theater: probably not if only one movie is shown (if there are four or more auditoriums each playing a different movie simultaneously, it may be okay). Patrons all tend to arrive in batches S to 20 minutes before a show, c. Laundromat: basically a single-channel, multiplexer, two-phase system. Arrivals Waiting line Service customers with dirty clothes usually first-come, first-served in terms of selecting an available machine first phase consists of washing clothes n washing machines; second-phase is again queuing for the first available drying machine d.