reid & sanders, operations management © wiley 2002 waiting line models a supplement

Reid & Sanders, Operations Management© Wiley 2002

Waiting Line Models ASUPPLEMENT


Learning Objectives

• Describe the elements of a waiting line problem

• Use waiting line models to estimate system performance

• Use waiting line models to make managerial decisions


Waiting Line System

• A waiting line system consists of two components:– The customer population (people or

objects to be processed)– The process or service system

• Whenever demand exceeds available capacity, a waiting line or queue forms


Terminology

• Finite versus Infinite populations:– Is the number of potential new customers affected by the

number of customers already in queue?

• Balking– When an arriving customer chooses not to enter a queue

because it’s already too long

• Reneging– When a customer already in queue gives up and exits

without being serviced

• Jockeying– When a customer switches back and forth between alternate

queues in an effort to reduce waiting time


Service System

• The service system is defined by:– The number of waiting lines– The number of servers– The arrangement of servers– The arrival and service patterns– The service priority rules


Number of Lines

• Waiting lines systems can have single or multiple queues.– Single queues avoid jockeying behavior & all

customers are served on a first-come, first-served fashion (perceived fairness is high)

– Multiple queues are often used when arriving customers have differing characteristics (e.g.: paying with cash, less than 10 items, etc.) and can be readily segmented


Servers

• Single servers or multiple, parallel servers providing multiple channels

• Arrangement of servers (phases)– Multiple phase systems require customers to visit

more than one server– Example of a multi-phase, multi-server system:

C C C CC Depart

Arrivals

1

2

3 6

5

4

Phase 1 Phase 2


Arrival & Service Patterns

• Arrival rate:– The average number of customers arriving

per time period– Modeled using the Poisson distribution

• Service rate:– The average number of customers that can

be serviced during the same period of time– Modeled using the exponential distribution


Priority Rules

• First come, first served

• Best customers first (reward loyalty)

• Highest profit customers first

• Quickest service requirements first

• Largest service requirements first

• Earliest reservation first

• Emergencies first


Common Performance Measures

• The average number of customers waiting in queue

• The average number of customers in the system (multiphase systems)

• The average waiting time in queue• The average time in the system• The system utilization rate (% of time

servers are busy)


Infinite Population, Single-Server, Single Line, Single Phase Formulae

systemincustomersofnumberaverageL

nutilizatiosystemaverage

rateservicemeanmu

ratearrivalmeanlambda


Infinite Population, Single-Server, Single Line, Single Phase Formulae

systemtheincustomersnofyprobabilitP

waitingspenttimeaverageWW

serviceincludingsystemintimeaverageW

lineincustomersofnumberaverageLL

nn

Q

Q

1

1


Example

• A help desk in the computer lab serves students on a first-come, first served basis. On average, 15 students need help every hour. The help desk can serve an average of 20 students per hour.

• Based on this description, we know:– Mu = 20 (exponential distribution)– Lambda = 15 (Poisson distribution)


Average Utilization

%7575.020

15or


Average Number of Studentsin the System

studentsL 31520

15


Average Number of StudentsWaiting in Line

studentsLLQ 25.2375.0


Average Time a Student Spends in the System


Average Time a StudentSpends Waiting (Before Service)

minutes9

hours15.02.075.0

or

WWQ


Probability of nStudents in the Line

079.075.075.011

105.075.075.011

141.075.075.011

188.075.075.011

25.0175.011

444

333

222

11

00

P

P

P

P

P


Changing System Performance

• Demand management:– change customer arrival rates through non-peak

discounts or price promotions

• Modify the number of servers• Division of labor:

– change the number of phases in the system– change the number of workers at each station

(e.g.: add a bagger to assist each cashier)


Changing System Performance

• Apply technology to improve efficiency – e.g.: price scanners

• Change priority rules – e.g.: implement a reservation protocol

• Change the number of lines:– Reduce multiple lines to single queue to avoid

jockeying– Dedicate specific servers to specific transactions


The End

Copyright © 2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.

reid & sanders, operations management © wiley 2002 waiting line models a supplement

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