### Queuing Theory Models

```Queueing Theory Models
Training Presentation
By: Seth Randall
Topics
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What is Queueing Theory?
How can your company benefit from it?
How to use Queueing Systems and Models?
Examples & Exercises
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
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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
Arrival and
Distribution
Servicing
Systems
Customer
Exit
Customer Arrivals
• Finite Population : Limited Size Customer
Pool
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
( T ) n e  T
PT (n) 
n!
Time between arrivals
Exponential Distribution
F(t)
1.20
1.00
f(t) = λe- λt
0.80
0.60
0.40
0.20
0.00
0
1
2
3
4
5
6
Time Before Next Arrival
f(t) = The probability that the next arrival will come in (t) minutes or more
Time between arrivals
Minutes (t)
Probability that
Probability that the next
the next arrival will arrival will come in t
come in t minutes minutes or less
or more
0
1
2
3
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
Number of arrivals per unit of time (T)
Poisson Distribution
0.25
0.2
Probability
of n
arrivals in
time (T)
0.15
( T ) n e  T
PT (n) 
n!
0.1
0.05
0
0
-0.05
PT (n)
1
2
3
4
5
6
7
8
9
10
Number of arrivals (n)
= The probability of exactly (n) arrivals during a time period (T)
Can arrival rates be controlled?
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Sales
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
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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
Length
• Infinite Potential
– Length is not limited by any restrictions
• Limited Capacity
– Length is limited by space or legal restriction
Line Structures
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Single Channel, Single Phase
Single Channel, Multiphase
Multichannel, single phase
Multichannel, multiphase
Mixed
Queue Discipline
• How to determine the order of service
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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
Wq = Average time waiting in line
µ = Service Rate
Ws = Average total time in system
1/µ = Average Service Time
n = number of units in system
1/λ = Average time between arrivals
S = number of identical service
р = Utilization rate: ratio of arrival

)

Lq = Average number waiting in line
rate to service rate (
Ls = Average number in system
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
Lq 
 (   )
Ls 

 
Wq 
Lq
Ws 
Ls


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
Exercise
Current Copy Machine:

20
Ls 

 
25  20
Ls
= 4 people in the system
4
Ws 

 0.2 hours waiting in the system
 20
Exercise

20
Ls 

 2 people in system
   30  20
Ls
2
Ws 

 0.1 hours
 20
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.
• 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