Chapter 7A Waiting Line Management. OBJECTIVES Waiting Line Characteristics Suggestions for Managing Queues Examples (Models 1, 2, 3, and 4)

Transcription

1 Chapter 7A Waiting Line Management 1 OBJECTIVES Waiting Line Characteristics Suggestions for Managing Queues Examples (Models 1, 2, 3, and 4)

2 Components of the Queuing System 2 Customer Arrivals Servicing System Queue or Waiting Line Servers Exit

3 Customer Service Population Sources 3 Population Source Example: Number of machines needing repair when a company only has three machines. Example: The number of people who could wait in a line for gasoline.

4 The Queuing System 4 Length Queue Queuing System Number of Lines & Line Structures Service Distribution

5 5 Examples of Line Structures Single Phase Multiphase Single Channel Multichannel

6 6

7 7 Degree of Patience No Way! No Way!

8 Suggestions for Managing Queues 8 1. Determine an waiting time for your customers 2. Try to divert your customer s when waiting 3. Inform your customers of 4. Keep employees not serving the customers 5. Segment customers

9 9 Suggestions for Managing Queues (Continued) 6. Train your servers to be friendly 7. Encourage customers to come during the 8. Take a long-term perspective toward getting rid of the queues

10 Waiting Line Models Source Model Layout Population Service Pattern 1 Single channel Infinite 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 10

11 Notation: Infinite Queuing: Models = Arrival rate 1 Average time between arrivals = Service rate 1 Average service time = = Utilization, (single server) L q Average number waiting in line

12 Example: Model 1 12 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: A) What is the average utilization of the employee? B) What is the average number of customers in line? C) What is the average number of customers in the system? D) What is the average waiting time in line? E) What is the average waiting time in the system? F) What is the probability that exactly two cars will be in the system?

13 Example: Model 1 13 A) What is the average utilization of the employee?

14 Example: Model 1 14 B) What is the average number of customers in line? 2 2 (25) L q = = = ( - ) 30(30-25) C) What is the average number of customers in the system? L = = - 25 (30-25) = 5

15 15 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)? p n = 1 n

16 Example: Model 2 16 An automated pizza vending machine heats and dispenses a slice of pizza in 4 minutes. Customers arrive at a rate of one every 6 minutes with the arrival rate exhibiting a Poisson distribution. Determine: A) The average number of customers in line. B) The average total waiting time in the system.

17 Example: Model 2 17 A) The average number of customers in line. 2 2 (10) L q = = = ( - ) (2)(15)(15-10) B) The average total waiting time in the system. L W = L W L q 10 L customers =.1333 hrs 10 8 min

18 Example: Model 3 18 Recall the Model 1 example: 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. If an identical window (and an identically trained server) were added, what would the effects be on the average number of cars in the system and the total time customers wait before being served? Number of Service Channels =

19 Example: Model 3 Average number of cars in the system 19 L q L = = Lq + = = Total time customers wait before being served W = q L q λ = Interpolation: Lookup and interpolate Exhibit 8A.9 Or use the Excel function =FORECAST(25/30,{0.1523,0.1873},{0.8,0.85}). 176 customers 25 customers/ min =.007 min (No Waiting!)

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