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## Queueing Theory

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**Queueing Theory**Lindsay Mullen Seminar Presentation #2 November 4, 2013 Computer Science Applied Mathematics Discrete Mathematics Operation Research Queueing Theory**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**• 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) • Queueingtheory 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 AgnerK. Erlangof the Copenhagen Telephone Company • Erlangderived several important formulas for teletrafficengineering • 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 AleksandrKhinchin and now known as the Pollaczek–Khinchinformula. • 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 queueingnode. (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**• Queueingmodel 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 formA/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**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**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**• 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 kservers • 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-and-operations-research-in-industry • http://www.cse.fau.edu/~bob/publications/encyclopedia.pdf • http://en.wikipedia.org/wiki/Queue_theory • http://www4.ncsu.edu/~hp/SSME_QueueingTheory.pdf • http://www.factoryphysics.com/principle/littleslaw.htm • http://weakonomics.com/2012/10/24/queuing-theory-for-amusement-parks/