queueing theory models

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

eTnPTn

T

Time between arrivals

0 1 2 3 4 5 60.000.200.400.600.801.001.20

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

eTnPTn

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

sLW

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

202520

sL

Exercise

2.0204

ss

LW

Upgraded Copy Machine:

people in system

hours

22030

20

sL

1.0202

ss

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.