Download - Queueing Theory Models
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Queueing Theory Models Training Presentation
By: Seth Randall
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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?
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What is Queueing Theory?
• The study of waiting in lines (Queues)
• Uses mathematical models to describe the flow of objects through systems
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Can queuing models help my firm?
• Increase customer satisfaction• Optimal service capacity and utilization
levels• Greater Productivity• Cost effective decisions
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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?
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Brainstorm• Can you identify areas in your firm where
queues exist?
• What are the major problems and costs associated with these queues?
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Queueing Systems and Models
Customer Exit
Servicing Systems
Customer Arrival and Distribution
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Customer Arrivals
• Finite Population : Limited Size Customer Pool
• Infinite Population: Additions and Subtractions do not affect system probabilities.
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Customer Arrivals• Arrival Rate
λ = mean arrivals per time period
• Constant: e.g. 1 per minute• Variable: random arrival
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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
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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
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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
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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
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Can arrival rates be controlled?
• Price adjustments• Sales• Posting business hours• Other?
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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
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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
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Types of Queues• 3 Factors
– Length– Number of lines
• Single Vs. Multiple– Queue Discipline
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• Infinite Potential– Length is not limited by any restrictions
• Limited Capacity– Length is limited by space or legal restriction
Length
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Line Structures• Single Channel, Single Phase• Single Channel, Multiphase• Multichannel, single phase• Multichannel, multiphase• Mixed
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Queue Discipline• How to determine the order of service
– First Come First Serve (FCFS)– Reservations– Emergencies – Priority Customers– Processing Time– Other?
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Two Types of Customer Exit
• Customer does not likely return
• Customer returns to the source population
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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
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Service Time Distribution• Service Rate
– Capacity of the server– Measured in units served per time period (µ)
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Examples of Queueing Functions
)(
2
qL
sL
q
q
LW
s
sLW
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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
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Current Copy Machine:
= 4 people in the system
hours waiting in the system
202520
sL
Exercise
2.0204
ss
LW
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Upgraded Copy Machine:
people in system
hours
22030
20
sL
1.0202
ss
LW
Exercise
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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
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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.
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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
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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.