Operations management by stevenson 9th student slides chapter 18

Chapter 18: Learning Objectives – Explain why waiting lines form in systems that are underloaded – Identify the goal of queuing management – List the measures of system performance that

Chapter 18

Waiting Lines

Chapter 18: Learning Objectives

– Explain why waiting lines form in systems that are underloaded – Identify the goal of queuing management

– List the measures of system performance that are used in

queuing

– Discuss the assumptions of the basic queuing models presented – Solve typical problems

Queuing Theory

• Queuing theory

– Mathematical approach to the analysis of waiting lines

– Applicable to many environments

• Call centers

• Banks

• Post offices

• Restaurants

• Theme parks

• Telecommunications systems

• Traffic management

Simple Queuing System

Calling population Arrivals Waiting

line

Exit Service

System

Processing Order

Queuing Models: Infinite Source

• Four basic infinite source models

1 Single server, exponential service time

2 Single server, constant service time

3 Multiple servers, exponential service time

4 Multiple priority service, exponential service time

Infinite-Source Symbols

line

in ting number wai expected

maximum The

(channels) servers

of number The

system

in the units

of

y probabilit The

system

in the units

zero of

y probabilit The

time Service

1

system

in the spend

customers time

average The

line

in wait customers

time average

The

n utilizatio system

The

served being

customers of

number average

The

system

in the customer

of number average

The

service for

waiting customers

of number average

The

server per

rate Service

rate arrival

Customer

max

0



























L M

n P

P

W W

r L L

n

s q

s q









System Utilization

Average number of customers being served

Basic Relationships







M









r

Basic Relationships

• Little’s Law

in line or in the system is equal to the average customers arrival rate multiplied by the average time

in the line or system

q q

s

s

W L

W

L









Basic Relationships

• The average number of customers

– Waiting in line for service:

– In the system:

• The average time customers are

– Waiting in line for service

– In the system

q L

r L

L s  q 

q q

L

W 

Single Server, Exponential Service Time

• M/M/1

n n

n n

q

P

P P

P L





























































1

1

0 0

2 2

Single Server, Constant Service Time

• M/D/1

– If a system can reduce variability, it can shorten waiting lines

noticeably

– For, example, by making service time constant, the average number of

customers waiting in line can be cut in half

– Average time customers spend waiting in line is also cut by half.

– Similar improvements can be made by smoothing arrival rates (such

as by use of appointments)

) (

2

2













q

L

Multiple Servers (M/M/S)

• Assumptions:

maintain FCFS processing)

   

M

n

M n

M

q

W

M M

n P

P M

M L























































































































1

1

!

!

!

1

1

1

0 0

0 2

Maximum Line Length

how much space should be allocated for waiting lines

a specified percentage of the time can be determined

using the following:

  













1 percentage

specified 1

where

ln

ln

or log

log

q

L K

K K

n

Operations Strategy

service system capacity alternatives

– Work to increase processing rates, instead of increasing the

number of servers

– Use new processing equipment and/or methods

– Reduce processing time variability through standardization

– Shift demand