Queuing Theory Models By Nancy Hutchins. Agenda What is queuing Why is queuing important How can...
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Transcript of Queuing Theory Models By Nancy Hutchins. Agenda What is queuing Why is queuing important How can...
Agenda• What is queuing• Why is queuing important• How can this help our company• Explanation• How it works• Summary• Reading list
What is Queuing?
• A queue is a line of waiting people, vehicles, products, etc.
• Queuing theory models use a mathematical approach to study queues and make them as efficient as possible
Why is this important?
Inadequate queue management may lead to• Customers leaving before completing their
transaction• Decrease in customer satisfaction• Reduction in number of return customers
Why is this important?
• Retaining customers much more cost effective than finding new customers
• Many businesses depend on revenue from repeat customers
How can this help your company?
• Decrease average customer wait time• Increase customer satisfaction• Increase number of return customers• Increase revenue• Increase positive word-of-mouth customer
advertising
Basic Ways to Manage Queues
• Train employees to be friendly• Segment customers by needs• Ensure customers know what to expect• Divert the customer’s attention during wait
times• Encourage customers to come during slack
times**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin Professional Pub, 2006. 112. Print.
Source Population
Finite• Limited size• Probabilities affected by an
increase/decrease in the population
Infinite• Large size• Probabilities not affected by
an increase/decrease in the population
Exponential Distribution
t (minutes)
Probability that the next arrival will occur in t
minutes or more
Probability that the next arrival will occur in t
minutes or less
0 1.00 0
0.5 0.61 0.39
1.0 0.37 0.63
1.5 0.22 0.78
2.0 0.14 0.86 λe ^ (-λt)**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin
Professional Pub, 2006. 114. Print.
Customer Arrivals in Queues
• Arrival Characteristics– Distribution– Pattern– Size of Arrival– Degree of Patience
Poisson Distribution
1 2 3 4 5 6 7 8 9 10 11
0.149
0.224 0.224
0.16
0.102
0.05
Number of Arrivals (n)
Pro
babi
lity
of
n A
rriv
als
in
Tim
e T
PT(n) =
Mean = = 3
Variance = = 3
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin
Professional Pub, 2006. 115. Print.
Degree of Patience
Degree of Patience
Patient (in line and stay)
Impatient
Arrive, View, and Leave
Arrive, Wait Awhile, then
Leave
Queuing System Factors
• Length– Infinite potential length– Limited capacity
• Number of Lines– Single – Multiple
• Queue Discipline
Queue Discipline
First Come, First Served (FCFS)Shortest Processing
TimeReservations First
Emergencies First Limited Needs Other
Service Time Distribution
• Service rate: the capacity of the server in number of units per time period and not as service time.
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The
Core. New York: Irwin Professional Pub, 2006. 118. Print.
Line Structures
• Single Channel, Single Phase• Single Channel, Multiphase• Multichannel, Single phase• Multichannel, multiphase• Mixed
Properties of Some Specific Line ModelsModel
Layout ServicePhase
SourcePopulation
Arrival Pattern
QueueDiscipline
Service Pattern
Permissible Queue Length
Example
1 SingleChannel
Single Infinite Poisson FCFS Exponential Unlimited One-lane toll bridge
2 SingleChannel
Single Infinite Poisson FCFS Constant Unlimited Roller coaster rides in amusement park
3 Multi-channel
Single Infinite Poisson FCFS Exponential Unlimited Parts counter in auto agency
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin
Professional Pub, 2006. 121. Print.
Infinite Queuing Notation: Models 1-3
• λ = arrival rate• µ = service rate• 1/µ = average service time• 1/λ = average time between arrivals• ρ = ratio of total arrival rate to service rate for a single server
(λ/µ)• Lq = average number waiting in line
• Ls = average number in system (including and being served)
**Jacobs, F. Robert, and Richard B. Chase. "5." Operations and Supply Management The Core. New York: Irwin
Professional Pub, 2006. 121. Print.
Infinite Queuing Notation: Models 1-3
• Wq = average time waiting in line
• Ws = average total time in system (including time to be served)
• n = number of units in the system• S = number of identical service channels• Pn = Probability of exactly n units in system
• Pw = Probability of waiting in line
Equations for Model 2 and 3
Model 2
• Lq =
• Ls = Lq
• Wq =
• Ws =
Model 3
• Ls = Lq
• Wq =
• Ws =
• Pw = Lq
Summary
• Effective queue management may lead to improved customer satisfaction and increased revenue
• Many queue management methods require little money to implement
• Software is available to help with queue analysis
Reading List• An Introduction to Queuing Theory: Modeling and
Analysis in Applications (Statistics for Industry and Technology) by U. Narayan Bhat
• Introduction to Queuing Networks by Erol Gelenbe and Guy Pujolle
• Optimal Design of Queuing Systems by Shaler Stidham• Fundamentals of Queuing Theory by Donald Gross and
Carl M. Harris• Operations and Supply Management The Core by
Jacobs, F. Robert, and Richard B. Chase