Queueing Theory Models Training Presentation

By: Seth Randall

Topics

• What is Queueing Theory?• How can your company benefit from it?• How to use Queueing Systems and Models?• Examples & Exercises• How can I learn more?

What is Queueing Theory?

• The study of waiting in lines (Queues)

• Uses mathematical models to describe the flow of objects through systems

Can queuing models help my firm?

• Increase customer satisfaction• Optimal service capacity and utilization

levels• Greater Productivity• Cost effective decisions

Examples

• How many workers should I employ?• Which equipment should we purchase?• How efficient do my workers need to be?• What is the probability of exceeding capacity

during peak times?

Brainstorm

• Can you identify areas in your firm where queues exist?

• What are the major problems and costs associated with these queues?

Queueing Systems and Models

Customer Exit

Servicing Systems

Customer Arrival and Distribution

Customer Arrivals

• Finite Population : Limited Size Customer Pool

• Infinite Population: Additions and Subtractions do not affect system probabilities.

Customer Arrivals

• Arrival Rate

λ = mean arrivals per time period

• Constant: e.g. 1 per minute• Variable: random arrival

2 ways to understand arrivals

• Time between arrivals– Exponential Distribution f(t) = λe- λt

• Number of arrivals per unit of time (T)– Poisson Distribution

!

)()(

n

eTnP

Tn

T

Time between arrivals

0 1 2 3 4 5 60.00

0.20

0.40

0.60

0.80

1.00

1.20

Exponential Distribution

Time Before Next Arrival

F(t

)

f(t) = λe- λt

f(t) = The probability that the next arrival will come in (t) minutes or more

Minutes (t) Probability that the next arrival will come in t minutes or more

Probability that the next arrival will come in t minutes or less

0 1.00 0.001 0.37 0.632 0.14 0.863 0.05 0.954 0.02 0.985 0.01 0.99

Time between arrivals

Number of arrivals per unit of time (T)

0 1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

Poisson Distribution

Number of arrivals (n)

Probability of n ar-rivals in time (T) !

)()(

n

eTnP

Tn

T

= The probability of exactly (n) arrivals during a time period (T))(nPT

Can arrival rates be controlled?

• Price adjustments• Sales• Posting business hours• Other?

Other Elements of Arrivals• Size of Arrivals

– Single Vs. Batch

• Degree of patience– Patient: Customers will stay in line– Impatient: Customers will leave

• Balking – arrive, view line, leave• Reneging – Arrive, join queue, then leave

Suggestions to Encourage Patience

• Segment customers• Train servers to be friendly• Inform customers of what to expect• Try to divert customer’s attention• Encourage customers to come during slack

periods

Types of Queues

• 3 Factors– Length– Number of lines

• Single Vs. Multiple

– Queue Discipline

• Infinite Potential– Length is not limited by any restrictions

• Limited Capacity– Length is limited by space or legal restriction

Length

Line Structures

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

Queue Discipline

• How to determine the order of service– First Come First Serve (FCFS)– Reservations– Emergencies – Priority Customers– Processing Time– Other?

Two Types of Customer Exit

• Customer does not likely return

• Customer returns to the source population

Notations for Queueing Concepts

λ = Arrival Rate

µ = Service Rate

1/µ = Average Service Time

1/λ = Average time between arrivals

р = Utilization rate: ratio of arrival

rate to service rate ( )

Lq = Average number waiting in line

Ls = Average number in system

Wq = Average time waiting in line

Ws = Average total time in system

n = number of units in system

S = number of identical service

channels

Pn = Probability of exactly n units in

system

Pw = Probability of waiting in line

Service Time Distribution

• Service Rate – Capacity of the server– Measured in units served per time period (µ)

Examples of Queueing Functions

)(

2

qL

sL

q

q

LW

s

s

LW

Exercise• Should we upgrade the copy machine?

– Our current copy machine can serve 25 employees per hour (µ)

– The new copy machine would be able to serve 30 employees per hour (µ)

– On average, 20 employees try to use the copy machine each hour (λ )

– Labor is valued at $8.00 per hour per worker

Current Copy Machine:

= 4 people in the system

hours waiting in the system

2025

20

sL

Exercise

2.020

4

s

s

LW

Upgraded Copy Machine:

people in system

hours

22030

20

sL

1.020

2

s

s

LW

Exercise

Current Machine: – Average number of workers in system = 4– Average time spent in system = 0.2 hours per worker– Cost of waiting = 4 * 0.2 * $8.00 = $6.40 per hour

New Machine: – Average number of workers in system = 2– Average time spent in system = 0.1 hours per worker– Cost of waiting = 2 * 0.1 * $8.00 = $1.60 per hour

Savings from upgrade = $4.80 per hour

Conclusion and Takeaways

• Queueing Theory uses mathematical models to observe the flow of objects through systems

• Each model depends on the characteristics of the queue

• Using these models can help managers make better decisions for their firm.

How Can I Learn More?• Fundamentals of Queueing Theory

– Donald Gross, John F. Shortle, James M. Thompson, and Carl M. Harris

• Applications of Queueing Theory– G. F. Newell

• Stochastic Models in Queueing Theory– Jyotiprasad Medhi

• Operations and Supply Management: The Core– F. Robert Jacobs and Richard B. Chase

References

• Jacobs, F. Robert, and Richard B. Chase. “Chapter 5." Operations and Supply Management The Core. 2nd Edition. New York: McGraw-Hill/Irwin, 2010. 100-131. Print.

• Newell, Gordon Frank. Applications of Queueuing Theory. 2nd Edition. London: Chapman and Hall, 1982.