queueing theory models
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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) - PowerPoint PPT PresentationTRANSCRIPT
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.