Waiting Line Models

Waiting Line Models MULTIPLE CHOICE 1. Decision makers in queuing situations attempt to balance a. operating characteristics against the arrival rate. b. service levels against service cost. the number of units in the system against the time in the system. c. the service rate against the arrival rate. d. b ANSWER: 2. Performance measures dealing with the number of units in line and the time spent waiting are called a. queuing facts. b. performance queues. c. system measures. d. operating characteristics. ANSWER: d 3. If arrivals occur according to the Poisson distribution every 20 minutes, then which is NOT true? a. λ = 20 arrivals per hour b. λ = 3 arrivals per hour λ = 1/20 arrivals per minute c. λ = 72 arrivals per day d. a ANSWER: 4. The manner in which units receive their service, such as FCFS, is the a. queue discipline. b. channel. c. steady state. d. operating characteristic. a ANSWER: 2 5. In a waiting line situation, arrivals occur, on average, every 10 minutes, and 10 units can be received every hour. What are λ and µ? a. λ = 10, µ = 10 b. λ = 6, µ = 6 c. λ = 6, µ = 10 λ = 10, µ = 6 d. c ANSWER: 6. Operating characteristics formulas for the single-channel queue do NOT require a. λ > µ. b. Poisson distribution of arrivals. c. an exponential distribution of service times. an FCFS queue discipline. d. a ANSWER: 7. In a multiple channel system a. each server has its own queue. b. each server has the same service rate. c. µ>λ d. All of the alternatives are correct. ANSWER: b 8. Little's flow equations a. require Poisson and exponential assumptions. b. are applicable to any waiting line model. c. require independent calculation of W, L, Wq, and Lq. All of the alternatives are correct. d. b ANSWER: 9. The total cost for a waiting line does NOT specifically depend on a. the cost of waiting. b. the cost of service. the number of units in the system. c. d. the cost of a lost customer. ANSWER: d 10. Models with a finite calling population a. have an arrival rate independent of the number of units in the system. b. have a service rate dependent on the number of units in the system. use the size of the population as a parameter in the operating characteristics formulas. c. All of the alternatives are correct. d. c ANSWER: 11. Which of the following can NOT be found by the queuing formulas presented in the textbook? a. the probability that no units are in the system. b. the average number of units in the system. c. the maximum time a unit spends in the system. d. the average time a unit spends in the system. c ANSWER: 12. The arrival rate in queuing formulas is expressed as a. the mean time between arrivals. b. the minimum number of arrivals per time period. c. the mean number of arrivals per channel. d. the mean number of arrivals per time period. ANSWER: d 13. What queue discipline is assumed by the waiting line models presented in the textbook? a. first-come first-served. b. last-in first-out. c. shortest processing time first. No discipline is assumed. d. a ANSWER: 14. For many waiting line situations, the arrivals occur randomly and independently of other arrivals and it has been found that a good description of the arrival pattern is provided by a. a normal probability distribution. b. an exponential probability distribution. c. a uniform probability distribution. a Poisson probability distribution. d. d ANSWER: 15. The assumption of exponentially distributed service times indicates that a. 37% of the service times are less than the mean service time. b. 50% of the service times are less than the mean service time. c. 63% of the service times are less than the mean service time. d. service time increase at an exponential rate as the waiting line grows. c ANSWER: TRUE/FALSE 1. For an M/M/1 queuing system, if the service rate, µ, is doubled, the average wait in the system, W, is cut in half. ANSWER: False 2. A waiting line situation where every customer waits in the same line before being served by the same server is called a single server waiting line. ANSWER: False 3. Use of the Poisson probability distribution assumes that arrivals are not random. ANSWER: False 4. Queue discipline refers to the assumption that a customer has the patience to remain in a slow moving queue. ANSWER: False 5. For all waiting lines, P0 + Pw = 1. ANSWER: False 6. Before waiting lines can be analyzed economically, the arrivals’ cost of waiting must be estimated. ANSWER: True 7. In a multiple channel system it is more efficient to have a separate waiting line for each channel. ANSWER: False 8. Little's flow equations indicate that the relationship of L to Lq is the same as that of W to Wq. ANSWER: False 9. If some maximum number of customers is allowed in a queuing system at one time, the system has a finite calling population. ANSWER: False 10. When blocked customers are cleared, an important decision is how many channels to provide. ANSWER: True 11. If service time follows an exponential probability distribution, approximately 63% of the service times are less than the mean service time. ANSWER: True 12. Queue discipline refers to the manner in which waiting units are arranged for service. ANSWER: True 13. False Waiting line models describe the transient-period operating characteristics of a waiting line. ANSWER: 14. For a single-channel waiting line, the utilization factor is the probability that an arriving unit must wait for service. ANSWER: True 15. When a waiting system is in steady-state operation, the number of units in the system is not changing. ANSWER: False 16. Adding more channels always improves the operating characteristics of the waiting line and reduces the waiting cost. True ANSWER: 17. In developing the total cost for a waiting line, waiting cost takes into consideration both the time spent waiting in line and the time spent being served. True ANSWER: 18. In waiting line systems where the length of the waiting line is limited, the mean number of units entering the system might be less than the arrival rate. ANSWER: True 19. A multiple-channel system has more than one waiting line. ANSWER: False 20. For an M/M/k system, the average number of customers in the system equals the customer arrival rate times the average time a customer spends waiting in the system. ANSWER: True 21. For a single-server queuing system, the average number of customers in the waiting line is one less than the average number in the system. ANSWER: False SHORT ANSWER 1. Discuss the importance of the utilization factor in a queuing system and the assumptions made about its value. 2. How can a system be changed to improve the service rate? 3. Diagram the servers and arrivals in the single and multiple channel models. Designate the line and the system. 4. Explain what is meant by the following statement, "operating characteristics are non-optimizing." 5. Give examples of systems you have seen in which a) blocked arrivals are cleared, and b) there is a finite calling population. PROBLEMS 1. During summer weekdays, boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of 3 per hour. In a 2-hour period, a. what is the probability that no boats arrive? b. what is the probability that 2 boats arrive? what is the probability that 8 boats arrive? c. 2. The time to process a registration at the Sea View Resort follows the exponential distribution and has a mean of 6 minutes. a. What is the probability of a registration time shorter than 3 minutes? b. What is the probability of a registration time shorter than 6 minutes? c. What is the probability of a registration time between 3 and 6 minutes? 3. The Grand Movie Theater has one box office clerk. On average, each customer that comes to see a movie can be sold its ticket at the rate of 6 per minute. For the theater’s normal offerings of older movies, customers arrive at the rate of 3 per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution. a. b. c. d. e. What is the average number of customers waiting in line? What is the average time a customer spends in the waiting line? What is the average number of customers in the system? What is a customer’s average time in the system? What is the probability that someone will be buying tickets when an arrival occurs? The Grand has booked the Stars Wars Trilogy and expects more customers. From conversations with other theater owners, it estimates that the arrival rate will increase to 10 per minute. Output is supplied for a two- cashier and a three-cashier system. Number of Channels Arrival Rate Service Rate Probability of No Units in System Average Waiting Time Average Time in System Average Number Waiting Average Number in System Probability of Waiting Probability of 11 in System 4. 2 10 6 .0909 .3788 .5455 3.7879 5.4545 .7576 .0245 3 10 6 .1727 .0375 .2041 .3747 2.0414 .2998 less than .0088 f. The Grand has space for ten customers to wait indoors to buy tickets. Which system will be better? g. Do you think it is more sensible for them to continue the one-cashier system? The Arctic Flyers minor league hockey team has one box office clerk. On average, each customer that comes to see a game can be sold a ticket at the rate of 8 per minute. For normal games, customers arrive at the rate of 5 per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution. a. b. c. d. e. What is the average number of customers waiting in line? What is the average time a customer spends in the waiting line? What is the average number of customers in the system? What is a customer’s average time in the system? What is the probability that someone will be buying tickets when an arrival occurs? The Flyers are playing in the league playoffs and anticipate more fans, estimating that the arrival rate will increase to 12 per minute. Output is supplied for a two-cashier and a three-cashier system. Number of Channels Arrival Rate Service Rate Probability of No Units in System Average Waiting Time Average Time in System Average Number Waiting Average Number in System Probability of Waiting 2 12 8 .1429 .1607 .2857 1.9286 3.4286 .6429 3 12 8 .2105 .0197 .1447 .2368 1.7368 .2368 Probability of 7 in System .0381 .0074 f. The rink has space for six customers to wait indoors to buy tickets. Which system will be better? g. Do you think it is more sensible for them to continue the one cashier system? 5. In a waiting line situation, arrivals occur at a rate of 2 per minute, and the service times average 18 seconds. Assume the Poisson and exponential distributions. a. What is λ? b. What is µ? c. Find probability of no units in the system. d. Find average number of units in the system. e. Find average time in the waiting line. f. Find average time in the system. g. Find probability that there is one person waiting. h. Find probability an arrival will have to wait. 6. In a waiting line situation, arrivals occur around the clock at a rate of six per day, and the service occurs at one every three hours. Assume the Poisson and exponential distributions. a. What is λ? b. What is µ? c. Find probability of no units in the system. d. Find average number of units in the system. e. Find average time in the waiting line. f. Find average time in the system. Find probability that there is one person waiting. g. Find probability an arrival will have to wait. h. 7. The Sea View Resort uses a multiple-channel queue registration system. If the average service time is 8 minutes, there are three registration clerks, and guests arrive at the rate of one every 5 minutes, find a. λ and µ. b. the probability all three clerks are idle. c. the probability a guest will have to wait. d. the average time a customer is in line. e. the average number of customers in line. 8. The post office uses a multiple channel queue, where customers wait in a single line for the first available window. If the average service time is 1 minute and the arrival rate is 7 customers every five minutes, find, when two service windows are open, a. the probability both windows are idle. b. the probability a customer will have to wait. c. the average time a customer is in line. d. the average time a customer is in the post office. 9. Two new checkout scanning systems are under consideration by a retail store. Arrivals to the checkout stand follow the Poisson distribution with λ = 2 per minute. The cost for waiting is $18 per hour. The first system has an exponential service rate of 5 per minute and costs $10 per hour to operate. The second system has an exponential service rate of 8 per minute and costs $20 per hour to operate. Which system should be chosen? 10. Circle Electric Supply is considering opening a second service counter to better serve the electrical contractor customers. The arrival rate is 10 per hour. The service rate is 14 per hour. If the cost of waiting is $30 and the cost of each service counter is $22 per hour, then should the second counter be opened? 11. For an M/G/1 system with λ = 6 and µ = 9, with σ = .03, find a. the probability the system is idle. the average length of the queue. b. the average number in the system. c. 12. For an M/G/1 system with λ = 20 and µ = 35, with σ = .005, find a. the probability the system is idle. b. the average length of the queue. the average number in the system. c. 13. Arrivals at a box office in the hour before the show follow the Poisson distribution with λ = 7 per minute. Service times are constant at 7.5 seconds. Find the average length of the waiting line. 14. The 8 students in a seminar class must come to the professor's office to turn in a paper and give a 5minute oral summary. Assume there is a service rate of 10 per hour and adequate time is available for all. The arrival rate for each unit is 5 per hour. What is the probability there is no one in the office or waiting when you come? 15. Andy Archer, Ph.D., is a training consultant for six mid-sized manufacturing firms. On the average, each of his six clients calls him for consulting assistance once every 25 days. Andy typically spends an average of five days at the client's firm during each consultation. Assuming that the time between client calls follows an exponential distribution, determine the following: a. b. c. 16. The Quick Snap photo machine at the Lemon County bus station takes four snapshots in exactly 75 seconds. Customers arrive at the machine according to a Poisson distribution at the mean rate of 20 per hour. On the basis of this information, determine the following: a. b. c. 17. the average number of clients Andy has on backlog the average time a client must wait before Andy arrives to it the proportion of the time Andy is busy the average number of customers waiting to use the photo machine the average time a customer spends in the system the probability an arriving customer must wait for service. Quick Clean Rooter cleans out clogged drains. Due to the competitive nature of the drain cleaning business, if a customer calls Quick Clean and finds the line busy, they immediately try another company and Quick Clean loses the business. Quick Clean management estimates that on the average, a customer tries to call Quick Clean every three minutes and the average time to take a service order is 200 seconds. The company wishes to hire enough operators so that at most 4% of its potential customers get the busy signal. a. b. How many operators should be hired to meet this objective? Given your answer to a), what is the probability that all the operators are idle? 18. A company has tool cribs where workmen draw parts. Two men have applied for the position of distributing parts to the workmen. George Fuller is fresh out of trade school and expects a $6 per hour salary. His average service time is 4 minutes. John Cox is a veteran who expects $12 per hour. His average service time is 2 minutes. A workman's time is figured at $10 per hour. Workmen arrive to draw parts at an average rate of 12 per hour. a. b. What is the average waiting time a workman would spend in the system under each applicant? Which applicant should be hired? 19. The insurance department at Shear's has two agents, each working at a mean speed of 8 customers per hour. Customers arrive at the insurance desk at a mean rate of one every six minutes and form a single queue. Management feels that some customers are going to find the wait at the desk too long and take their business to Word's, Shear's competitor. In order to reduce the time required by an agent to serve a customer Shear's is contemplating installing one of two minicomputer systems: System A which leases for $18 per day and will increase an agent's efficiency by 25%; or, System B which leases for $23 per day and will increase an agent's efficiency by 50%. Agents work 8-hour days. If Shear's estimates its cost of having a customer in the system at $3 per hour, determine if Shear's should install a new minicomputer system, and if so, which one. 20. The postmaster at the Oak Hill Post Office expects the mean arrival rate of people to her customer counter will soon increase by fifty percent due to a large apartment complex being built. Currently, the mean arrival rate is 15 people per hour. The postmaster can serve an average of 25 people per hour. By what percentage must the postmaster's mean service rate increase when the apartment complex is completed in order that the average time spent at the post office remains at its current value? SOLUTIONS TO PROBLEMS 1. a. b. c. P(0) = .0025 P(2) = .0446 P(8) = .1033 2. a. b. c. P(time < 3) = 1 - .6065 = .3935 P(time < 6) = 1 - .3679 = .6321 P(3 < time < 6) = .6321 - .3935 = .2386 3. a. b. c. d. e. f. g. Lq = .5 Wq = .1667 L =1 W = .3333 Pw = .5 The three-cashier system is probably too good and not cost effective. The one cashier system won’t work because now the arrival rate is faster than the service rate. 4. a. b. c. d. e. f. Lq = 1.04 Wq = .2083 L = 1.665 W = .3333 Pw = .625 The three-cashier system is probably too good and not cost effective. g. The one cashier system won’t work because now the arrival rate is faster than the service rate. 5. a. b. c. d. e. f. g. h. λ = 3/min. µ = 4/min. P0 = .25 L =3 Wq = .75 min. W = 1.00 min. P(2) = .140625 Pw = .75 6. a. b. c. d. e. f. g. h. λ=6 µ=8 P0 = .25 L =3 Wq = .375 W = .5 P(2) = .1406 Pw = .75 7. a. b. c. d. e. λ = 12 per hour, µ = 7.5 per hour P0 = .1872 Pw = .2738 Wq = .0261 Lq = .3129 8. a. b. c. d. P0 = .1765 Pw = .5765 Wq = .1922 (of a five-minute period) W = .3922 (of a five-minute period) 9. First system costs .37 per minute, second system costs .43 per minute. Choose first system. 10. With one window the cost per hour is 97. With two windows the cost per hour is 68.56. Choose two windows. 11. a. b. c. P0 = .67 Lq = .7153 L = 1.3819 12. a. b. c. P0 = .4286 Lq = .3926 L = .964 13. Lq = 3.0625 14. λ = 5, µ = 10, N = 8 P0 = .0009 15. a. b. c. Lq = .7094 Wq = 4.96 days 1 – P0 = .7151 16. a. b. c. Lq = .15 W = .028 hour = 1.7 minutes PW = .417 17. a. b. k = 4; PW = .021 P0 = .331 18. a. b. George: W = 20 minutes; John: W = 3 1/3 minutes Total Cost: George = $46.00, John = $18.67. Hire John 19. No system: $49.20; System A: $50.00; System B: $47.50. Install System B 20. 30 percent