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- 1. 1 Waiting Line Management
- 2. 2 OBJECTIVES Waiting Line Characteristics Suggestions for Managing Queues Examples (Models 1, 2, 3, and 4)
- 3. Components of the Queuing System Servicing System Servers Queue or Customer Arrivals Waiting Line Exit 3
- 4. Customer Service Population Sources Population Source Finite Infinite Example: Number of Example: Number of machines needing machines needing repair when a repair when a company only has company only has three machines. three machines. Example: The Example: The number of people number of people who could wait in who could wait in a line for a line for gasoline. gasoline. 4
- 5. Service Pattern Service Pattern Constant Example: Items Example: Items coming down an coming down an automated automated assembly line. assembly line. Variable Example: People Example: People spending time spending time shopping. shopping. 5
- 6. 6 The Queuing System Length Queue Discipline Queuing System Service Time Distribution Number of Lines & Line Structures
- 7. Examples of Line Structures Single Phase Single Channel Multichannel Multiphase 7
- 8. Suggestions for Managing Queues 1. Determine an acceptable waiting time for your customers 2. Try to divert your customer’s attention when waiting 3. Inform your customers of what to expect 4. Keep employees not serving the customers out of sight 5. Segment customers 8
- 9. 9 Suggestions for Managing Queues (Continued) 6. Train your servers to be friendly 7. Encourage customers to come during the slack periods 8. Take a long-term perspective toward getting rid of the queues
- 10. 10 Waiting Line Models Model Layout 1 Single channel Source Population Infinite Service Pattern Exponential 2 Single channel Infinite Constant 3 Multichannel Infinite Exponential 4 Single or Multi Finite Exponential These four models share the following characteristics: • Single phase • Poisson arrival • FCFS • Unlimited queue length
- 11. 11 Notation: Infinite Queuing: Model 1 λ = Arrival rate µ = Service rate 1 = Average service time µ 1 = Average time between arrivals λ λ ρ = = Ratio of total arrival rate to sevice rate µ for a single server Lq = Average number waiting in line
- 12. Infinite Queuing Model 1 (Continued) Ls = Average number in system (including those being served) Wq = Average time waiting in line Ws = Average total time in system (including time to be served) n = Number of units in the system S = Number of identical service channels Pn = Probability of exactly n units in system Pw = Probability of waiting in line 12
- 13. Example: Model 1 Assume a drive-up window at a fast food restaurant. Customers arrive at the rate of 25 per hour. The employee can serve one customer every two minutes. Assume Poisson arrival and exponential service rates. Determine: Determine: A) What is the average utilization of the employee? A) What is the average utilization of the employee? B) What is the average number of customers in line? B) What is the average number of customers in line? C) What is the average number of customers in the C) What is the average number of customers in the system? system? D) What is the average waiting time in line? D) What is the average waiting time in line? E) What is the average waiting time in the system? E) What is the average waiting time in the system? F) What is the probability that exactly two cars will be F) What is the probability that exactly two cars will be in the system? in the system? 13
- 14. 14 Example: Model 1 A) What is the average utilization of the employee? λ = 25 cust / hr 1 customer µ = = 30 cust / hr 2 mins (1hr / 60 mins) λ 25 cust / hr ρ = = = .8333 µ 30 cust / hr
- 15. 15 Example: Model 1 B) What is the average number of customers in line? λ (25) Lq = = = 4.167 µ ( µ - λ ) 30(30 - 25) 2 2 C) What is the average number of customers in the system? λ 25 Ls = = =5 µ - λ (30 - 25)
- 16. 16 Example: Model 1 D) What is the average waiting time in line? Lq Wq = = .1667 hrs = 10 mins λ E) What is the average waiting time in the system? Ls Ws = = .2 hrs = 12 mins λ
- 17. Example: Model 1 F) What is the probability that exactly two cars will be in the system (one being served and the other waiting in line)? pn λ λ = (1- )( ) µ µ n 25 25 2 p 2 = (1- )( ) = .1157 30 30 17
- 18. 18 Question Bowl The central problem for virtually all queuing problems is which of the following? a. Balancing labor costs and equipment costs b. Balancing costs of providing service with the costs of waiting c. Minimizing all service costs in the use of equipment d. All of the above e. None of the above Answer: b. Balancing costs of providing service with the costs of waiting
- 19. 19 Question Bowl Customer Arrival “populations” in a queuing system can be characterized by which of the following? a. Poisson b. Finite c. Patient d. FCFS e. None of the above Answer: b. Finite
- 20. 20 Question Bowl Customer Arrival “rates” in a queuing system can be characterized by which of the following? a. Constant b. Infinite c. Finite d. All of the above e. None of the above Answer: a. Constant
- 21. 21 Question Bowl An example of a “queue discipline” in a queuing system is which of the following? a. Single channel, multiphase b. Single channel, single phase c. Multichannel, single phase d. Multichannel, multiphase None of the above Answer: e. None of the above (These are the rules for determining the order of service to customers, which include FCFS, reservation first, highest-profit customer first, etc.) e.
- 22. 22 Question Bowl Withdrawing funds from an automated teller machine is an example in a queuing system of which of the following “line structures”? a. Single channel, multiphase b. Single channel, single phase c. Multichannel, single phase d. Multichannel, multiphase e. None of the above Answer: b. Single channel, single phase
- 23. 23 Question Bowl Refer to Model 1 in the textbook. If the service rate is 15 per hour, what is the “average service time” for this queuing situation? a. 16.00 minutes b. 0.6667 hours c. 0.0667 hours d. 16% of an hour e. Can not be computed from data above Answer: c. 0.0667 hours (1/15=0.0667)

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