1. Ch 6 Tech Notes OBJECTIVES
Waiting Line Characteristics
Suggestions for Managing Queues
Examples (Models 1, 2, 3, and 4) 2

2. Components of a Queuing System
Exhibit TN 6.3
Components of a Queuing System

3. Arrival and Service Profiles
Exhibit TN 6.2
Arrival and Service Profiles

4. Exponential Distribution
Exhibit TN 6.4
Exponential Distribution

5. Exhibit TN 6.5
Poisson Distribution

6. Customer Arrivals in Queues
Exhibit TN 6.6
Customer Arrivals in Queues

7. Exhibit TN 6.7
Line Structures

8. Notation: Infinite Queuing: Models 1-3

9. Infinite Queuing Models 1-3 (Continued)

10. 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:
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? 11

11. Example: Model 1 A) What is the average utilization of the employee?12

12. Example: Model 1 B) What is the average number of customers in line?
C) What is the average number of customers in the system? 13

13. Example: Model 1 D) What is the average waiting time in line?
E) What is the average waiting time in the system? 14

14. 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)? 15

15. Little’s Law
For nearly all queuing systems, there is a relationship between the expected time a unit spends in the system or queue and the expected number of units in the system or queue. Known as Little's flow equations, they are:
Ls = Ws and Lq = Wq
(Anderson, Sweeney, & Williams)

16. Economic Analysis
Total Cost = (cost of providing service) + (cost of waiting)
TC = csk + cq(/)Wq = csk + cqWs
= css + cqLs, by Little’s Law, where
cs is the cost of service per channel
cq Is the cost for one customer to wait one unit of time

17. Economic Analysis, suggested problem (Anderson, Sweeney & Williams)
A fast food franchise is considering operating a drive-up window food-service operation. Assume that customer arrivals follow a Poisson probability distribution, with a mean arrival rate of 24 cars per hour, and that service times follow an exponential probability distribution. Arriving customers place orders at an intercom station at the back of the parking lot and then drive to the service window to pay for and receive their orders. The following three service alternatives are being considered:

18. Economic Analysis, suggested problem
A single-channel operation in which one employee fills the order and takes the money from the customer. The average service time for this alternative is 2 minutes.
A single-channel operation in which one employee fills the order while a second employee takes the money from the customer. The average service time for this alternative is 1.25 minutes.
A two-channel operation with two service windows and two employees. The employee stationed at each window fills the order and takes the money for customers ar­riving at the window. The average service time for this alternative is 2 minutes for each channel.

19. What is the lowest-cost design for the fast food business?
For the suggested problem, the following cost information is available for the fast food franchise
Customer waiting time is valued at $25 per hour to reflect the fact that waiting time is costly to the fast food business.
The cost of each employee is $6.50 per hour.
To account for equipment and space, an additional cost of $20 per hour is attributable to each channel.
What is the lowest-cost design for the fast food business?