1. Lindsay Mullen Seminar Presentation #2 November 4, 2013
Queueing Theory
Lindsay Mullen
Seminar Presentation #2
November 4, 2013
Computer Science Applied Mathematics Discrete Mathematics
Operation Research
Queueing Theory

2. 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.

3. Successful Areas of Operation Research
Airline Industry
Telecommunications
Manufacturing Industry
Healthcare
Transportation

4. 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)

5. 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

6. 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,...)

7. 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.

8. 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

9. 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

10. A/S/C

11. A/S/C

12. Service Disciplines
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

13. Models of Queueing Nodes

14. Models continued…

15. 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

16. 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,…

17. Disney
Build the story
Split lines
Turns
Holding areas
Virtual spots

18. 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

19. 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

20. References