# GE330_lect20 - Lecture 20 Queuing Systems Queuing Theory A...

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Lecture 20 Queuing Systems April 16, 2009

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Queuing Theory A framework to analyze the performance of waiting lines, such as average waiting time, average line length, etc. Not an optimization technique, but a modeling tool. Built on the underlying theory of stochastic process, espe- cially, Markov chains. Initiated by Agner Krarup Erlang (a Danish engineer who worked for the Copenhagen Telephone Exchange) in 1909. Recent research focus on various applications, approxima- tion methods and optimal control of queuing networks (Prof. Tolga Tezcan is an expert in this area). 2
The Scenario The principal actors in a queuing model are the customer and the server . Customers are generated from a source. On arrive at a ser- vice facility, they can start service if a server is available. Otherwise, they can wait in a queue . After getting service, they leave the system. Servers provide service to customers, when a server com- pletes a service, it automatically “pull” a waiting customer, if any, from the queue. If the queue is empty, the server becomes idle until the next customer arrives. server server …… customers queue 3

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Elements of a Queuing Model inter-arrival time: the random variable which represents the time between two successive customers. service time: the random variable which represents the time a service takes. queue size: inﬁnite or ﬁnite. queue discipline: ﬁrst come ﬁrst served (FCFS), last come ﬁrst served (LCFS), service in random order (SIRO), or ser- vice based on the order of priority. customer behavior: balking, jockeying, reneging customer source: inﬁnite or ﬁnite 4
Role of Exponential Distribution In a lot of queuing situations, we assume that the inter-arrival time and the service time obey exponential distribution. The PDF, mean, and CDF of the exponential distribution is given by: f ( t ) = λe - λt , t > 0 , E { t } = 1 λ , P { t T } = Z T 0 λe - λt dt = 1 - e - λT .

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## This note was uploaded on 08/31/2010 for the course IESE GE 330 taught by Professor Nedich during the Spring '09 term at University of Illinois at Urbana–Champaign.

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GE330_lect20 - Lecture 20 Queuing Systems Queuing Theory A...

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