Queuing Theory Models

By

Nancy Hutchins

Agenda• What is queuing• Why is queuing important• How can this help our company• Explanation• How it works• Summary• Reading list

What is Queuing?

• A queue is a line of waiting people, vehicles, products, etc.

• Queuing theory models use a mathematical approach to study queues and make them as efficient as possible

Video Clip

Office Space Grid Lock

Why is this important?

Inadequate queue management may lead to• Customers leaving before completing their

transaction• Decrease in customer satisfaction• Reduction in number of return customers

Why is this important?

• Retaining customers much more cost effective than finding new customers

• Many businesses depend on revenue from repeat customers

How can this help your company?

• Decrease average customer wait time• Increase customer satisfaction• Increase number of return customers• Increase revenue• Increase positive word-of-mouth customer

advertising

Basic Ways to Manage Queues

• Train employees to be friendly• Segment customers by needs• Ensure customers know what to expect• Divert the customer’s attention during wait

times• Encourage customers to come during slack

times**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, 2006. 112. Print.

The Queuing System

Source Population &

Arrival RateServicing System

Condition of Exiting

Customers

Source Population

Finite• Limited size• Probabilities affected by an

increase/decrease in the population

Infinite• Large size• Probabilities not affected by

an increase/decrease in the population

Distribution of Arrivals

• Arrival Rate: is the number of units per period– Constant– Variable

Exponential Distribution

t (minutes)

Probability that the next arrival will occur in t

minutes or more

Probability that the next arrival will occur in t

minutes or less

0 1.00 0

0.5 0.61 0.39

1.0 0.37 0.63

1.5 0.22 0.78

2.0 0.14 0.86 λe ^ (-λt)**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin

Professional Pub, 2006. 114. Print.

Customer Arrivals in Queues

• Arrival Characteristics– Distribution– Pattern– Size of Arrival– Degree of Patience

Poisson Distribution

1 2 3 4 5 6 7 8 9 10 11

0.149

0.224 0.224

0.16

0.102

0.05

Number of Arrivals (n)

Pro

babi

lity

of

n A

rriv

als

in

Tim

e T

PT(n) =

Mean = = 3

Variance = = 3

**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin

Professional Pub, 2006. 115. Print.

Pattern of Arrivals

Pattern

Controllable

Uncontrollable

Size of Arrival Units

Size of Arrival Units

Single

Batch

Degree of Patience

Degree of Patience

Patient (in line and stay)

Impatient

Arrive, View, and Leave

Arrive, Wait Awhile, then

Leave

Queuing System Factors

• Length– Infinite potential length– Limited capacity

• Number of Lines– Single – Multiple

• Queue Discipline

Queue Discipline

First Come, First Served (FCFS)Shortest Processing

TimeReservations First

Emergencies First Limited Needs Other

Service Time Distribution

• Service rate: the capacity of the server in number of units per time period and not as service time.

**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The

Core. New York: Irwin Professional Pub, 2006. 118. Print.

Line Structures

• Single Channel, Single Phase• Single Channel, Multiphase• Multichannel, Single phase• Multichannel, multiphase• Mixed

Exiting the Queuing System

Exit

Low Probability of Re-

service

Return to Source

Population

Properties of Some Specific Line ModelsModel

Layout ServicePhase

SourcePopulation

Arrival Pattern

QueueDiscipline

Service Pattern

Permissible Queue Length

Example

1 SingleChannel

Single Infinite Poisson FCFS Exponential Unlimited One-lane toll bridge

2 SingleChannel

Single Infinite Poisson FCFS Constant Unlimited Roller coaster rides in amusement park

3 Multi-channel

Single Infinite Poisson FCFS Exponential Unlimited Parts counter in auto agency

**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin

Professional Pub, 2006. 121. Print.

Infinite Queuing Notation: Models 1-3

• λ = arrival rate• µ = service rate• 1/µ = average service time• 1/λ = average time between arrivals• ρ = ratio of total arrival rate to service rate for a single server

(λ/µ)• Lq = average number waiting in line

• Ls = average number in system (including and being served)

**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin

Professional Pub, 2006. 121. Print.

Infinite Queuing Notation: Models 1-3

• Wq = average time waiting in line

• Ws = average total time in system (including time to be served)

• n = number of units in the system• S = number of identical service channels• Pn = Probability of exactly n units in system

• Pw = Probability of waiting in line

Equations for Model 1

• Lq =

• Ls =

• Wq =

• Ws =

• Pn =

• ρ = • Po =

Equations for Model 2 and 3

Model 2

• Lq =

• Ls = Lq

• Wq =

• Ws =

Model 3

• Ls = Lq

• Wq =

• Ws =

• Pw = Lq

Brainstorming Exercise

What are inexpensive ways

our company can reduce

customer wait times?

Summary

• Effective queue management may lead to improved customer satisfaction and increased revenue

• Many queue management methods require little money to implement

• Software is available to help with queue analysis

Reading List• An Introduction to Queuing Theory: Modeling and

Analysis in Applications (Statistics for Industry and Technology) by U. Narayan Bhat

• Introduction to Queuing Networks by Erol Gelenbe and Guy Pujolle

• Optimal Design of Queuing Systems by Shaler Stidham• Fundamentals of Queuing Theory by Donald Gross and

Carl M. Harris• Operations and Supply Management The Core by

Jacobs, F. Robert, and Richard B. Chase

Reference

** Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, 2006. 111-127. Print.