Training Presentation
By: Seth Randall

• 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?

• The study of waiting in lines (Queues)
• Uses mathematical models to describe the flow of objects through systems

• Increase customer satisfaction
• Optimal service capacity and utilization levels
• Greater Productivity
• Cost effective decisions

• 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?

• Can you identify areas in your firm where queues exist?
• What are the major problems and costs associated with these queues?

Customer
Arrival and
Distribution
Servicing
Systems
Customer
Exit

• Finite Population : Limited Size Customer
Pool
• Infinite Population: Additions and
Subtractions do not affect system probabilities.

• Arrival Rate
λ = mean arrivals per time period
• Constant: e.g. 1 per minute
• Variable: random arrival

• Time between arrivals
– Exponential Distribution f(t) = λe - λt
• Number of arrivals per unit of time (T)
– Poisson Distribution
P
T
( n )

(

T ) n e
 
T n !

1,20
1,00
0,80
0,60
0,40
0,20
0,00
0
Exponential Distribution
1 2 3 4
Time Before Next Arrival
5 6 f(t) = λe - λt f(t) = The probability that the next arrival will come in (t) minutes or more

Minutes (t) Probability that come in t minutes or more
Probability that the next the next arrival will arrival will come in t minutes or less
2
3
0
1
4
5
1.00
0.37
0.14
0.05
0.02
0.01
0.00
0.63
0.86
0.95
0.98
0.99

0,25
Poisson Distribution
0,2
Probability of n arrivals in time (T)
0,15
0,1
0,05
0
-0,05
0 1 2 3 4 5 6 7 8 9 10
Number of arrivals (n
)
P
T
( n )

(

T ) n e
 
T n !
P
T
( n )
= The probability of exactly (n) arrivals during a time period (T)

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

• 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

• 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

• 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

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

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

• 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


L q
= Average number waiting in line
L s
= Average number in system
W q
= Average time waiting in line
W s
= Average total time in system n = number of units in system
S = number of identical service channels
P n
= Probability of exactly n units in system
P w
= Probability of waiting in line

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

L q


(


2
 
)
L s



 
W
 q
L q

W
 s
L s


• 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:
L s



 

20
25

20
= 4 people in the system
W s

L s


4
20

0 .
2 hours waiting in the system

Upgraded Copy Machine:
L s



 

20
30

20

2 people in system
W s

L s


2
20

0 .
1 hours

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

• 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.

• 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

• Jacobs, F. Robert, and Richard B. Chase. “Chapter 5." Operations and Supply Management The Core.
2 nd Edition. New York:
McGraw-Hill/Irwin, 2010. 100-131. Print.
• Newell, Gordon Frank. Applications of Queueuing Theory. 2 nd Edition.
London: Chapman and Hall, 1982.