1

Waiting Line Management

2

Waiting Line Characteristics

Suggestions for Managing Queues

Examples (Models 1, 2, 3, and 4)

OBJECTIVES

3Components of the Queuing System

CustomerArrivals

Servers

Waiting Line

Servicing System

Exit

Queue or

4

Customer Service Population Sources

Population Source

Finite Infinite

Example: Number of machines needing repair when a company only has three machines.

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.

Example: The number of people who could wait in a line for gasoline.

5

Service Pattern

ServicePattern

Constant Variable

Example: Items coming down an automated assembly line.

Example: Items coming down an automated assembly line.

Example: People spending time shopping.

Example: People spending time shopping.

6

The Queuing System

Queue Discipline

Length

Number of Lines &Line Structures

Service Time Distribution

Queuing System

7

Examples of Line Structures

Single Channel

Multichannel

SinglePhase Multiphase

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

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

Waiting Line Models

Model LayoutSourcePopulation 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

11

Notation: Infinite Queuing: Model 1

linein tingnumber wai Average

server single afor

rate sevice torate arrival totalof Ratio = =

arrivalsbetween timeAverage

timeservice Average

rate Service =

rate Arrival =

1

1

Lq

linein tingnumber wai Average

server single afor

rate sevice torate arrival totalof Ratio = =

arrivalsbetween timeAverage

timeservice Average

rate Service =

rate Arrival =

1

1

Lq

12

Infinite Queuing Model 1 (Continued)

linein waitingofy Probabilit

systemin units exactly ofy Probabilit

channels service identical ofNumber =

system in the units ofNumber

served) be to time(including

systemin time totalAverage

linein waiting timeAverage =

served) being those(including

systemin number Average = s

Pw

nPn

S

n

Ws

Wq

L

linein waitingofy Probabilit

systemin units exactly ofy Probabilit

channels service identical ofNumber =

system in the units ofNumber

served) be to time(including

systemin time totalAverage

linein waiting timeAverage =

served) being those(including

systemin number Average = s

Pw

nPn

S

n

Ws

Wq

L

13

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?

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?

Example: Model 1

14

= 25 cust / hr

= 1 customer

2 mins (1hr / 60 mins) = 30 cust / hr

= = 25 cust / hr

30 cust / hr = .8333

= 25 cust / hr

= 1 customer

2 mins (1hr / 60 mins) = 30 cust / hr

= = 25 cust / hr

30 cust / hr = .8333

Example: Model 1

A) What is the average utilization of the employee?

15

Example: Model 1

B) What is the average number of customers in line?

4.167 = 25)-30(30

(25) =

) - ( =

22

Lq 4.167 = 25)-30(30

(25) =

) - ( =

22

Lq

C) What is the average number of customers in the system?

5 = 25)-(30

25 =

- =

Ls 5 = 25)-(30

25 =

- =

Ls

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Example: Model 1

D) What is the average waiting time in line?

mins 10 = hrs .1667 =

=Lq

Wq mins 10 = hrs .1667 =

=Lq

Wq

E) What is the average waiting time in the system?

mins 12 = hrs .2 = =Ls

Ws mins 12 = hrs .2 = =Ls

Ws

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)?

p = (1-n

n

)( )p = (1-n

n

)( )

p = (1- = 2

225

30

25

30)( ) .1157p = (1- =

2

225

30

25

30)( ) .1157

18

Question Bowl

The central problem for virtually all queuing problems is which of the following?

a. Balancing labor costs and equipment costsb. Balancing costs of providing service with the

costs of waitingc. Minimizing all service costs in the use of

equipmentd. All of the abovee. None of the above Answer: b. Balancing

costs of providing service with the costs of waiting

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

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

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

e. 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.)

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

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)