CH 11 - Waiting Line Models. Questions and Answers

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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. a. True b. False 2. A waiting line situation with... a single server is referred to as an M/M/1 model with a finite calling population. a. True b. False 3. Use of the Poisson probability distribution assumes that arrivals are not random. a. True b. False 4. Queue discipline refers to the assumption that a customer has the patience to remain in a slow moving queue. a. True b. False 5. Before waiting lines can be analyzed economically, the arrivals' cost of waiting must be estimated. a. True b. False 6. A variation of the waiting line models involves a system in which no waiting is allowed, and arriving units are denied access to the system. a. True b. False 7. Little's flow equations indicate that the relationship of L to Lq is the same as that of W to Wq. a. True b. False 8. Kendall's notation is helpful when classifying the wide variety of different waiting line models and can indicate that the waiting line system is assumed to have infinite capacity. a. True b. False 9. When blocked customers are cleared, an important decision is how many servers to provide. a. True b. False 10. If service time follows an exponential probability distribution, approximately 63% of the service times are less than the mean service time. a. True b. False 11. Queue discipline refers to the manner in which waiting units are arranged for service. a. True b. False 12. Waiting line models describe the transient-period operating characteristics of a waiting line. a. True b. False 13. For a single-server waiting line, the utilization factor is the probability that an arriving unit must wait for service. a. True b. False 14. After the startup or transient period, a waiting system is in steady-state operation and considered to be the normal operation of the waiting line. a. True b. False 15. Adding more servers always improves the operating characteristics of the waiting line and reduces the waiting cost. a. True b. False 16. 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. a. True b. False 17. 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. a. True b. False 18. A multiple-server waiting line is one that has two or more parallel service facilities. a. True b. False 19. 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. a. True b. False 20. In waiting line applications, the exponential probability distribution indicates that approximately 63% of the service times are less than the mean service time. a. True b. False 21. Waiting line models consist of mathematical formulas and relationships that can be used to determine the operating characteristics (also known as performance measures) for a waiting line. a. True b. False 22. Little’s flow equations can apply to a single-server as well as multiple-server waiting line model. a. True b. False 23. The body of knowledge dealing with waiting lines is known as queueing theory. a. True b. False Multiple Choice 24. Decision makers in queuing situations attempt to balance a. operating characteristics against the arrival rate. b. service levels against service cost. c. the number of units in the system against the time in the system. d. the service rate against the arrival rate. 25. 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. 26. If arrivals occur according to the Poisson distribution every 20 minutes, then which of the following is NOT true? a. λ = 20 arrivals per hour b. λ = 3 arrivals per hour c. λ = 1/20 arrivals per minute d. λ = 72 arrivals per day 27. The manner in which units receive their service, such as FCFS, is the a. queue discipline. b. server. c. steady state. d. operating characteristic. 28. In a waiting line situation, arrivals occur, on average, every 10 minutes, and 10 units can be received every hour. Which of the following represents λ and μ in this situation? a. λ = 10, μ = 10 b. λ = 6, μ = 6 c. λ = 6, μ = 10 d. λ = 10, μ = 6 29. Operating characteristics formulas for the single-server queue do NOT require a. λ ≥ μ. b. Poisson distribution of arrivals. c. an exponential distribution of service times. d. an FCFS queue discipline. 30. The mean number of units that can be served per time period is called a. the service rate and is denoted by λ. b. the service rate and is denoted by μ. c. the steady state. d. None of these are correct. 31. 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. d. All of these are correct. 32. The total cost for a waiting line does NOT specifically depend on the a. cost of waiting. b. cost of service. c. number of units in the system. d. cost of a lost customer. 33. 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. c. have no limit placed on how many units may seek service. d. All of these are correct. 34. When no limit is placed on how many units may seek service, the waiting line model a. can assume that no units are in the system. b. is said to have an infinite calling population. c. maintains a constant arrival rate. d. None of these are correct. 35. The arrival rate in queuing formulas is expressed as the a. mean time between arrivals. b. minimum number of arrivals per time period. c. mean number of arrivals per server. d. mean number of arrivals per time period. 36. Which of the following queue disciplines 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 d. first-in, last-out 37. 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(n) a. normal probability distribution. b. exponential probability distribution. c. uniform probability distribution. d. Poisson probability distribution. 38. 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. 39. Single-booth ticket sales at a theater are an example of which of the following queuing models? a. single-server, Poisson service rate distribution, unlimited queue length b. single-server, Poisson service rate distribution, limited queue length c. single-server, constant service rate distribution, unlimited queue length d. single-server, normal service rate distribution, unlimited queue length 40. The equations provided in the textbook for computing operating characteristics apply to a waiting line operating a. at start-up. b. at steady state. c. at peak demand times. d. in transition. Subjective Short Answer 41. During summer weekdays, boats arrive at the inlet drawbridge according to the Poisson distribution at a rate of three per hour. In a two-hour period, a. what is the probability that no boats arrive? b. what is the probability that two boats arrive? c. what is the probability that eight boats arrive? 42. The time to process a registration at the Sea View Resort follows the exponential distribution and has a mean of six minutes. a. What is the probability of a registration time shorter than three minutes? b. What is the probability of a registration time shorter than six minutes? c. What is the probability of a registration time between three and six minutes? 43. The Grand Movie Theater has one box office clerk. On average, each customer that comes to see a movie can be sold a ticket at the rate of six per minute. For the theater's normal offerings of older movies, customers arrive at the rate of three per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution. a. What is the average number of customers waiting in line? b. What is the average time a customer spends in the waiting line? c. What is the average number of customers in the system? d. What is a customer's average time in the system? e. 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 two- and three-cashier systems. Number of servers 2 3 Arrival rate 10 10 Service rate 6 6 Probability of no units in system 0.0909 0.1727 Average waiting time 0.3788 0.0375 Average time in system 0.5455 0.2041 Average number waiting 3.78790 0.3747 Average number in system 5.4545 2.0414 Probability of waiting 0.7576 0.2998 Probability of 11 in system 0.0245 less than 0.0088 f. The Grand has space for 10 customers to wait indoors to buy tickets. Which system will be better? g. Do you think it is more sensible for the Grand to continue the one-cashier system? 44. 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 eight per minute. For normal games, customers arrive at the rate of five per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution. a. What is the average number of customers waiting in line? b. What is the average time a customer spends in the waiting line? c. What is the average number of customers in the system? d. What is a customer's average time in the system? e. 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 two- and three-cashier systems. Number of servers 2 3 Arrival rate 12 12 Service rate 8 8 Probability of no units in system 0.1429 0.2105 Average waiting time 0.1607 0.0197 Average time in system 0.2857 0.1447 Average number waiting 1.9286 0.2368 Average number in system 3.4286 1.7368 Probability of waiting 0.6429 0.2368 Probability of 7 in system 0.0381 0.0074 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 the Flyers to continue the one-cashier system? 45. In a waiting line situation, arrivals occur at a rate of two 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 the probability of no units in the system. d. Find the average number of units in the system. e. Find the average time in the waiting line. f. Find the average time in the system. g. Find the probability that there is one person waiting. h. Find the probability that an arrival will have to wait. 46. 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 the probability of no units in the system. d. Find the average number of units in the system. e. Find the average time in the waiting line. f. Find the average time in the system. g. Find the probability that there is one person waiting. h. Find the probability that an arrival will have to wait. 47. 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 five per minute and costs $10 per hour to operate. The second system has an exponential service rate of eight per minute and costs $20 per hour to operate. Which system should be chosen? 48. 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, should the second counter be opened? 49. For an M/G/1 system with λ = 6, μ = 9, and σ = 0.03, find a. the probability the system is idle. b. the average length of the queue. c. the average number in the system. 50. For an M/G/1 system with λ = 20, μ = 35, and σ = 0.005, find a. the probability the system is idle. b. the average length of the queue. c. the average number in the system. 51. 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. 52. The eight students in a seminar class must come to the professor's office to turn in a paper and give a five-minute 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 five per hour. What is the probability there is no one in the office or waiting when you come? 53. Andy Archer, Ph.D., is a training consultant for six mid-sized manufacturing firms. On 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 the time between client calls follows an exponential distribution, determine a. the average number of clients Andy has on backlog. b. the average time a client must wait before Andy arrives. c. the proportion of time Andy is busy. 54. 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 a. the average number of customers waiting to use the photo machine. b. the average time a customer spends using the system. c. the probability an arriving customer must wait for service. 55. 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 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. How many operators should be hired to meet this objective? b. Given your answer to part (a), what is the probability that all the operators are idle? 56. 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 four minutes. John Cox is a veteran who expects $12 per hour. His average service time is two 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. What is the average waiting time a workman would spend in the system under each applicant? b. Which applicant should be hired? 57. The insurance department at Shear's has two agents, each working at a mean speed of eight 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 that leases for $18 per day and will increase an agent's efficiency by 25% or System B that leases for $23 per day and will increase an agent's efficiency by 50%. Agents work eight-hour days. If Shear's estimates its cost of having a customer in the system at $3 per hour, determine if it should install a new minicomputer system, and if so, which one? 58. The postmaster at the Oak Hill Post Office expects that the mean arrival rate of people to her customer counter will soon increase by 50% 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? [Show More]

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