Principles of operations management 9th by heizer and render module d

D

MODULE

Waiting-Line Models

PowerPoint presentation to accompany
Heizer and Render
Operations Management, Eleventh Edition
Principles of Operations Management, Ninth Edition
PowerPoint slides by Jeff Heyl
© 2014
© 2014
Pearson
Pearson
Education,
Education,
Inc.Inc.

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Outline
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Queuing Theory

Characteristics of a Waiting-Line
System
Queuing Costs
The Variety of Queuing Models
Other Queuing Approaches

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Learning Objectives
When you complete this chapter you
should be able to:
1. Describe the characteristics of arrivals,
waiting lines, and service systems
2. Apply the single-server queuing model
equations
3. Conduct a cost analysis for a waiting line

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Learning Objectives
When you complete this chapter you
should be able to:
4. Apply the multiple-server queuing
model formulas

5. Apply the constant-service-time
model equations
6. Perform a limited-population model
analysis
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Queuing Theory
▶ The study of waiting lines
▶ Waiting lines are common situations
▶ Useful in both
manufacturing
and service
areas

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Common Queuing Situations
TABLE D.1

Common Queuing Situations

SITUATION

ARRIVALS IN QUEUE

SERVICE PROCESS

Supermarket

Grocery shoppers

Checkout clerks at cash register

Highway toll booth

Automobiles

Collection of tolls at booth

Doctor’s office

Patients

Treatment by doctors and nurses

Computer system

Programs to be run

Computer processes jobs

Telephone company

Callers

Switching equipment to forward calls

Bank

Customer

Transactions handled by teller

Machine maintenance

Broken machines

Repair people fix machines

Harbor

Ships and barges

Dock workers load and unload

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Characteristics of Waiting-Line
Systems
1. Arrivals or inputs to the system
▶ Population size, behavior, statistical

distribution

2. Queue discipline, or the waiting line itself
▶ Limited or unlimited in length, discipline of
people or items in it

3. The service facility
▶ Design, statistical distribution of service times
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Parts of a Waiting Line
Population of
dirty cars

Arrivals
from the
general
population …

Queue
(waiting line)

Service
facility
Dave’s
Car Wash

Enter

Arrivals to the system

Arrival Characteristics
► Size of the population
► Behavior of arrivals
► Statistical distribution of
arrivals
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Exit the system

In the system

Waiting-Line
Characteristics
► Limited vs.
unlimited
► Queue discipline

Exit

Exit the system

Service Characteristics
► Service design
► Statistical distribution of
service
Figure D.1

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Arrival Characteristics
1. Size of the arrival population
▶ Unlimited (infinite) or limited (finite)

2. Pattern of arrivals
▶ Scheduled or random, often a Poisson
distribution

3. Behavior of arrivals
▶ Wait in the queue and do not switch lines
▶ No balking or reneging
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Poisson Distribution
e-λλx
P(x) =
x!
where

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for x = 0, 1, 2, 3, 4, …

P(x)= probability of x arrivals

x = number of arrivals per unit of
time
λ = average arrival rate
e = 2.7183 (which is the base of
the natural logarithms)
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Poisson Distribution
Figure D.2

e-λλx
x!

0.25 –

0.25 –

0.02 –

0.02 –
Probability

Probability

Probability = P(x) =

0.15 –
0.10 –

0.15 –
0.10 –

0.05 –

0.05 –

–

–
0 1 2 3 4 5 6 7 8 9

Distribution for λ = 2
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x

0 1 2 3 4 5 6 7 8 9 10 11 x

Distribution for λ = 4
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Waiting-Line Characteristics
▶ Limited or unlimited queue length
▶ Queue discipline - first-in, first-out
(FIFO) is most common
▶ Other priority rules may be used in
special circumstances

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Service Characteristics
1. Queuing system designs
▶ Single-server system, multiple-server
system
▶ Single-phase system, multiphase system

2. Service time distribution
▶ Constant service time
▶ Random service times, usually a negative
exponential distribution
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Queuing System Designs
A family dentist’s office
Queue
Service
facility

Arrivals

Departures
after service

Single-server, single-phase system
A McDonald’s dual-window drive-through
Queue
Arrivals

Phase 1
service
facility

Phase 2
service
facility

Departures
after service

Single-server, multiphase system
Figure D.3
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Queuing System Designs
Most bank and post office service windows

Service
facility
Channel 1

Queue
Service
facility
Channel 2

Arrivals

Departures
after service

Service
facility
Channel 3

Multi-server, single-phase system
Figure D.3
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Queuing System Designs
Some college registrations

Queue
Arrivals

Phase 1
service

facility
Channel 1

Phase 2
service
facility
Channel 1

Phase 1
service
facility
Channel 2

Phase 2
service
facility
Channel 2

Departures
after service

Multi-server, multiphase system
Figure D.3
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Probability that service time ≥ 1

Figure D.4

1.0 –

Negative Exponential
Distribution
Probability that service time is greater than t = e-µt for t ≥ 1
µ = Average service rate
e = 2.7183

0.9 –
0.8 –
0.7 –

Average service rate (µ) = 3 customers per hour
⇒ Average service time = 20 minutes (or 1/3 hour)
per customer

0.6 –
0.5 –
0.4 –
0.3 –

Average service rate (µ) =
1 customer per hour

0.2 –
0.1 –
|
|

|
|
|
|
|
|
|
|
|
|
0.0 |–
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

Time t (hours)
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Measuring Queue Performance
1. Average time that each customer or object spends
in the queue
2. Average queue length
3. Average time each customer spends in the system
4. Average number of customers in the system
5. Probability that the service facility will be idle
6. Utilization factor for the system
7. Probability of a specific number of customers in the
system

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Queuing Costs
Cost

Figure D.5

Minimum
Total
cost

Total expected cost
Cost of providing service
Cost of waiting time
Low level
of service

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Optimal
service level

High level
of service
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Queuing Models
The four queuing models here all assume:
1. Poisson distribution arrivals
2. FIFO discipline
3. A single-service phase

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Queuing Models
TABLE D.2

Queuing Models Described in This Chapter

MODEL
A

NAME

Single-server
system (M/M/1)

EXAMPLE
Information counter at
department store

NUMBER OF
SERVERS

(CHANNELS)

NUMBER
OF
PHASES

ARRIVAL
RATE
PATTERN

SERVICE
TIME
PATTERN

POPULATION QUEUE
SIZE
DISCIPLINE

Single

Single

Poisson

Exponential

Unlimited

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FIFO

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Queuing Models
TABLE D.2

Queuing Models Described in This Chapter

MODEL
B

NAME

Multiple-server
(M/M/S)

EXAMPLE
Airline ticket counter

NUMBER OF
SERVERS
(CHANNELS)

NUMBER
OF
PHASES

ARRIVAL

RATE
PATTERN

SERVICE
TIME
PATTERN

POPULATION QUEUE
SIZE
DISCIPLINE

Multi-server

Single

Poisson

Exponential

Unlimited

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FIFO

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Queuing Models
TABLE D.2

Queuing Models Described in This Chapter

MODEL
C

NAME

EXAMPLE

Constant-service
(M/D/1)

Automated car wash

NUMBER OF
SERVERS
(CHANNELS)

NUMBER
OF
PHASES

ARRIVAL
RATE
PATTERN

SERVICE
TIME
PATTERN

POPULATION QUEUE
SIZE
DISCIPLINE

Single

Single

Poisson

Constant

Unlimited

© 2014 Pearson Education, Inc.

FIFO

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Queuing Models
TABLE D.2

Queuing Models Described in This Chapter

MODEL
D

NAME

EXAMPLE

Limited population
(finite population)

Shop with only a dozen
machines that might break

NUMBER OF
SERVERS
(CHANNELS)

NUMBER
OF
PHASES

ARRIVAL
RATE
PATTERN

SERVICE
TIME
PATTERN

POPULATION QUEUE
SIZE
DISCIPLINE

Single

Single

Poisson

Exponential

Limited

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FIFO

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Model A – Single-Server
1. Arrivals are served on a FIFO basis and
every arrival waits to be served regardless
of the length of the queue
2. Arrivals are independent of preceding
arrivals but the average number of arrivals
does not change over time
3. Arrivals are described by a Poisson
probability distribution and come from an
infinite population
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