student slides chapter 18

7/30/2019 Student Slides Chapter 18

1/15

McGraw-Hill/Irwin Copyrig ht 2009 by The McGraw -Hill Com panies, Inc. All Rights Reserved.

Chapter 18

Waiting Lines


2/15

18-2

Chapter 18: Learning Objectives

You should be able to: 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


3/15

18-3

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


4/15

18-4

Simple Queuing System

Calling population

Arrivals Waitingline

ExitService

System

Processing Order


5/15

18-5

Queuing Models: Infinite Source

Four basic infinite source models All assume a Poisson arrival rate

1. Single server, exponential service time

2. Single server, constant service time3. Multiple servers, exponential service time4. Multiple priority service, exponential service time


6/15

18-6

Infinite-Source Symbols

lineintingnumber waiexpectedmaximumThe

(channels)serversof number The

systemin theunitsof y probabilitThe

systemin theunitszeroof y probabilitThe timeService1

systemin thespendcustomerstimeaverageThe

lineinwaitcustomerstimeaverageThe

nutilizatiosystemTheserved beingcustomersof number averageThe

systemin thecustomer of number averageThe

servicefor waitingcustomersof number averageThe

server per rateServiceratearrivalCustomer

max

0

L

M

n P

P

W

W

r

L

L

n

s

q

s

q


7/1518-7

System Utilization

Average number of customers being served

Basic Relationships

M

r


8/1518-8

Basic Relationships

Littles Law For a stable system the average number of customers

in line or in the system is equal to the averagecustomers arrival rate multiplied by the average timein the line or system

qq

s s

W L

W L


9/1518-9

Basic Relationships

The average number of customers Waiting in line for service:

In the system:

The average t ime customers are Waiting in line for service

In the system

q L

r L L q s

q

q LW

s

q s

LW W

1


10/1518-10

Single Server, Exponential Service Time

M/M/1

n

n

n

n

q

P

P P

P

L

1

1

0

0

2

2


11/1518-11

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


12/1518-12

Multiple Servers (M/M/S)

Assumptions: A Poisson arrival rate and exponential service time Servers all work at the same average rate

Customers form a single waiting line (in order tomaintain FCFS processing)


13/1518-13

M/M/S

s

qW

s

M

n

M n

M

q

W

W P

M W

M M

n P

P M M

L

1

1!!

!11

1

00

02


14/15

18-14

Maximum Line Length

An issue that often arises in service system design ishow much space should be allocated for waiting lines

The approximate line length, n , that will not be exceededa specified percentage of the time can be determinedusing the following:

1 percentage

specified1

where

lnln

or loglog

q L K

K K n


15/15

18-15

Operations Strategy Managers must carefully weigh the costs and benefits of

service system capacity alternatives Options for reducing wait times:

Work to increase processing rates, instead of increasing thenumber of servers

Use new processing equipment and/or methods Reduce processing time variability through standardization Shift demand

student slides chapter 18

Documents