Chap008A-Waiting Line Analysis

7/31/2019 Chap008A-Waiting Line Analysis

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8A-1

McGraw-Hill/Irwin Copyright 2009 by The McGraw-Hill Companie

s, Inc. All rights re


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Chapter 8A

Waiting Line Analysis


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Waiting Line Characteristics

Suggestions for ManagingQueues

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

OBJECTIVES


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Components of the Queuing System

Customer

Arrivals

Servers

Waiting Line

Servicing System

Exit

Queue or


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Customer Service Population Sources

Population Source

Finite Infinite

Example: Number of

machines needingrepair when a

company only has

three machines.

Example: Number of

machines needingrepair when a

company only has

three machines.

Example: The

number of peoplewho could wait in

a line for

gasoline.

Example: The

number of peoplewho could wait in

a line for

gasoline.


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Service Pattern

Service

Pattern

Constant Variable

Example: Items

coming down an

automated

assembly line.

Example: Items

coming down anautomated

assembly line.

Example: People

spending time

shopping.

Example: People

spending timeshopping.


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The Queuing System

Queue Discipline

Length

Number of Lines &

Line Structures

Service Time

Distribution

Queuing

System


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Examples of Line Structures

Single Channel

Multichannel

Single

PhaseMultiphase

One-personbarber shop

Car wash

Hospital

admissions

Bank tellers

windows


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Degree of Patience

No Way!

BALK

No Way!

RENEG


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Suggestions for Managing Queues

1. Determine an acceptable waitingtime for your customers

2. Try to divert your customers

attention when waiting

3. Inform your customers of what to

expect

4. Keep employees not serving the

customers out of sight

5. Segment customers


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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


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Waiting Line Models

Model Layout

Source

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

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Notation: Infinite Queuing: Models 1-3

lineintingnumber waiAverage

serversingleafor

ratesevicetoratearrivaltotalofRatio==

arrivalsbetweentimeAverage

timeserviceAverage

rateService=

rateArrival=

1

1

=

=

=

Lq

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Infinite Queuing Models 1-3 (Continued)

lineinwaitingofyProbabilit

systeminunitsexactlyofyProbabilit

channelsserviceidenticalofNumber=

systemin theunitsofNumber

served)betotime(includingsystemintimetotalAverage

lineinwaitingtimeAverage=

served)beingthose(including

systeminnumberAverage=s

=

=

=

=

Pw

nPn

S

n

Ws

Wq

L

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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?


F) What is the probability that exactly two cars will be

in the system?

Example: Model 1

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= 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?

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Example: Model 1

B) What is the average number of customers in

line?

4.167=25)-30(30

(25)=)-(

=

22

Lq

C) What is the average number of customers in

the system?

5=

25)-(30

25=

-

=

Ls

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Example: Model 1

D) What is the average waiting time in line?

mins10=hrs.1667==

LqWq


mins12=hrs.2==

LsWs

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Example: Model 1

F) What is the probability that exactly two cars willbe in the system (one being served and the other

waiting in line)?

p = (1 -n

n

)( )

p = (1- =2

225

30

25

30)( ) .1157

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Example: Model 2

An automated pizza vending machine

heats anddispenses 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.

Determine:



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Example: Model 2


.6667=10)-(2)(15)(15

(10)=

)-(2=

22

Lq


mins4=hrs.06667=

10

6667.=

=

LqWq

mins8=hrs.1333=15/hr

1+hrs.06667=

1+=

WqWs

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Example: Model 3

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 trainedserver) 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?

If an identical window (and an identically trainedserver) 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?

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Example: Model 3

Average number of cars in the system

ion)interpolatlinear-using-TN7.11(Exhibit

1760= .Lq

1.009=30

25+.176=+=

LqLs

Total time customers wait before being served

)(=mincustomers/25

customers.176== Wait!No

LqWq mins.007

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Notation: Finite Queuing: Model 4

channelsserviceofNumber

lineinunitsofnumberAverage)(system

queuinginthoselesssourcePopulation=

servedbeingunitsofnumberAverage

lineinwaittohaving

ofeffecttheofmeasureafactor,Efficiency

lineinmust waitarrivalany thatProbabilit=

=

=

=

=

S

Ln-N

J

H

F

D

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Finite Queuing: Model 4 (Continued)

requiredtimeserviceofproportionorfactor,ServicelineintimewaitingAverage

tsrequiremenservicecustomerbetweentimeAverage

servicetheperformtotimeAverage=

systemqueuinginunitsexactlyofyProbabilit

sourcepopulationinunitsofNumber

served)beingonethe(including

systemqueuinginunitsofnumberAverage=

=

=

=

=

=

XW

U

T

nPn

N

n

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Example: Model 4

The copy center of an electronics firm has four copy

machines that are all serviced by a single technician.

Every two hours, on average, the machines requireadjustment. The technician spends an average of 10

minutes per machine when adjustment is required.

Assuming Poisson arrivals and exponential service,how many machines are down (on average)?

The copy center of an electronics firm has four copy

machines that are all serviced by a single technician.

Every two hours, on average, the machines requireadjustment. The technician spends an average of 10

minutes per machine when adjustment is required.

Assuming Poisson arrivals and exponential service,how many machines are down (on average)?

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Example: Model 4

N, the number of machines in the population = 4

M, the number of repair people = 1T, the time required to service a machine = 10 minutes

U, the average time between service = 2 hours

X =T

T + U

10 min

10 min +120 min

= .077=

From Table TN7.11, F = .980 (Interpolation)From Table TN7.11, F = .980 (Interpolation)

L, the number of machines waiting to be

serviced = N(1-F) = 4(1-.980) = .08 machines

L, the number of machines waiting to be

serviced = N(1-F) = 4(1-.980) = .08 machines

H, the number of machines being

serviced = FNX = .980(4)(.077) = .302 machines

H, the number of machines being

serviced = FNX = .980(4)(.077) = .302 machines

Number of machines down = L + H = .382 machinesNumber of machines down = L + H = .382 machines

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Queuing Approximation

This approximation is quick way to analyze a queuing

situation. Now, both interarrival time and service time

distributions are allowed to be general.

In general, average performance measures (waiting

time in queue, number in queue, etc) can be very well

approximated by mean and variance of the distribution

(distribution shape not very important). This is very good news for managers: all you need is

mean and standard deviation, to compute average

waiting time

( )

=

= =22

2

Define:

Standard deviation of Xcoefficient of variation for r.v. X =Mean of X

Variancesquared coefficient of variation (scv) =

mean

x

x x

C

C C

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Queue Approximation

2( 1) 2 2

1 2

S

a s

q

C C

L

+

+=

= +s qL L S

ComputeS

=

2 2,a sC CInputs: S, , ,

(Alternatively: S, , , variances of interarrival and service time distributions)

as before, , andq sq sL L

W W

= =

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Approximation Example

Consider a manufacturing process (for example making

plastic parts) consisting of a single stage with fivemachines. Processing times have a mean of 5.4 days

and standard deviation of 4 days. The firm operates

make-to-order. Management has collected date on

customer orders, and verified that the time between

orders has a mean of 1.2 days and variance of 0.72days. What is the average time that an order waits

before being worked on?

Using our Waiting Line Approximation spreadsheet we

get:Lq = 3.154 Expected number of orders waiting to be

completed.

Wq = 3.78 Expected number of days order waits.

= 0.9 Expected machine utilization.

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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 ofequipment

d. All of the above

e. None of the above

Answer: b. Balancing costs of providing

service with the costs of waiting

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Question Bowl

Customer Arrival populations in aqueuing system can be

characterized by which of the

following?

a. Poissonb. Finite

c. Patient

d. FCFSe. None of the above

Answer: b. Finite

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Question Bowl

Customer Arrival rates in a queuingsystem can be characterized by which

of the following?

a. Constant

b. Infinitec. Finite

d. All of the above


Answer: a. Constant

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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 phasec. Multichannel, single phase

d. Multichannel, multiphase


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.)

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Question Bowl

Withdrawing funds from an automatedteller machine is an example in aqueuing system of which of thefollowing line structures?

a. Single channel, multiphaseb. Single channel, single phasec. Multichannel, single phased. Multichannel, multiphase


Answer: b. Single channel, single phase

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Question Bowl

Refer to Model 1 in the textbook. If

the service rate is 15 per hour,what is the average service timefor this queuing situation?

a. 16.00 minutesb. 0.6667 hoursc. 0.0667 hoursd. 16% of an hour

e. Can not be computed from dataabove

Answer: c. 0.0667 hours (1/15=0.0667)

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Question Bowl

Refer to Model 1 in the textbook. If the

arrival rate is 15 per hour, what is theaverage time between arrivals 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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Question Bowl

Refer to Model 4 in the textbook. If the

average time to perform a service is 10minutes and the average time betweencustomer service requirements is 2minutes, which of the following is theservice factor for this queuing

situation?a. 0.833b. 0.800c. 0.750d. 0.500


Answer: a. 0.833 (10/(10+2)=0.833)

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End of Chapter 8A

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Chap008A-Waiting Line Analysis