Queueing Theory
Lindsay Mullen
Seminar Presentation #2
November 4, 2013
Computer Science
Applied Mathematics
Operation Research
Queueing Theory
Discrete Mathematics
Operation Research
• Science of decision-making
• Combines mathematics, statistics, computer science,
physics, engineering, economics, and social sciences to
solve real-world business problems
• Established discipline during World War II when the British
government recruited scientists to solve problems in critical
military operations
• There are now many Operation Research departments in
industry, government, and academia throughout the world.
Successful Areas of Operation
Research
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Airline Industry
Telecommunications
Manufacturing Industry
Healthcare
Transportation
What is Queueing Theory?
• A queue is a waiting line (like customers waiting
at a supermarket checkout counter)
• Queueing theory is the mathematical theory of
waiting lines
• Concerned with the mathematical modeling and
analysis of systems that provide service to
random demands (i.e. business decisions
providing a service)
History of Queueing Theory
• Queueing theory was born in the early 1900s
with the work of Agner K. Erlang of the
Copenhagen Telephone Company
• Erlang derived several important formulas for
teletraffic engineering
• Erlang published the first paper on what would
now be called queueing theory in 1909
History continued…
• He modeled the number of telephone calls
arriving at an exchange by a Poisson process and
solved the M/D/1 queue in 1917 and M/D/k
queueing model in 1920
– M (Markov chain or memoryless) - arrivals occur
according to a Poisson process
– D (deterministic) - jobs arriving at the queue require
a fixed amount of service
– k - number of servers at the queueing node (k = 1,
2,...)
History continued…
• The M/G/1 model was solved by Felix Pollaczek in 1930, a solution
later recast in probabilistic terms by Aleksandr Khinchin and now
known as the Pollaczek–Khinchin formula.
• After World War II queueing theory became an area of research
interest to mathematicians.
• In 1953, David George Kendall created the Kendall’s notation which
is the standard system used to describe and classify a queueing
node. (A/S/C)
• Work on queueing theory used in modern packet
switching networks was performed in the early 1960s by Leonard
Kleinrock.
• It was in this period that John Little gave a proof of the formula
which now bears his name: Little's law
– Inventory =Throughput (number of customers served per unit time)×
Flow Time
• In 1961 John Kingman gave a formula for the mean waiting time in
a G/G/1 queue: Kingman's formula.
Queueing Model
• Queueing model represents
(1) the system's physical configuration,
by specifying the number and arrangement of the
servers, which provide service to the customers
AND
(2) the stochastic (that is, probabilistic or
statistical) nature of the demands, by specifying the
variability in the arrival process and in the service
process
Queue Nodes and Networks
• Single queueing nodes are usually described
using Kendall's notation in the
form A/S/C where A describes the time between
arrivals to the queue, S the size of jobs and C the
number of servers at the node.
• Many theorems in queue theory can be proven by
reducing queues to mathematical systems known
as Markov chains (It is a random process usually
characterized as memoryless)
• Networks of queues are systems in which a number
of queues are connected by customer routing
A/S/C
A/S/C
Service Disciplines
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First in first out This principle states that customers are served one at a time
and that the customer that has been waiting the longest is served first.
Last in first out This principle also serves customers one at a time, however
the customer with the shortest waiting time will be served first. Also known as
a stack.
Processor sharing Service capacity is shared equally between customers.
Priority Customers with high priority are served first.
Shortest job first The next job to be served is the one with the smallest size
Preemptive shortest job first The next job to be served is the one with the
original smallest size
Shortest remaining processing time The next job to serve is the one with the
smallest remaining processing requirement
Models of Queueing Nodes
Models continued…
Examples of How This Theory is Used
Now
•Queues are visibly found in everyday life!
– Supermarket checkout
– Traffic lights
– Waiting for the elevator
– Waiting at a gas station
– Waiting at passport control
– Waiting at a doctor’s office
– Paperwork waiting at somebody’s office to be
processed
Measures of Interest
• Mean waiting time
• Percentile of the waiting time, i.e. what percent
of the waiting customers wait more than x
amount of time.
• Utilization of the server
• Throughput, i.e. number of customers served per
unit time.
• Average number of customers waiting
• Distribution of the number of waiting customers,
i.e. Probability [n customers wait], n=01,1,2,…
Disney
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Build the story
Split lines
Turns
Holding areas
Virtual spots
Queues that Cannot Be Seen
• There are also queues that we cannot see
(unless we use a software/hardware system),
such as:
– Streaming a video
– Web services
– On hold at a call center
Current Research
• Problems such as performance metrics for
the M/G/k queue remain an open problem
• M/G/k queue is a queue model where arrivals
are Markovian (modulated by a Poisson
process), service times have
a General distribution and there are k servers
• An extension of the M/M/c queue, where
service times must be exponentially
distributed and of the M/G/1 queue with a
single server
References
• http://www.maa.org/mathematics-andoperations-research-in-industry
• http://www.cse.fau.edu/~bob/publications/ency
clopedia.pdf
• http://en.wikipedia.org/wiki/Queue_theory
• http://www4.ncsu.edu/~hp/SSME_QueueingThe
ory.pdf
• http://www.factoryphysics.com/principle/littlesl
aw.htm
• http://weakonomics.com/2012/10/24/queuingtheory-for-amusement-parks/