093009Introduction to Queuing Theory

093009Introduction to Queuing Theory - Introduction to...

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Unformatted text preview: Introduction to Queuing Theory Introduction to Queuing Theory and Its Use in Manufacturing and Its Use in Manufacturing Rob Leachman 10, IEOR Module E10, IEOR Module September 30, 2009 Sept. 30, 2009 Intro to Queueing Theory Prof. Leachman 1 urpose urpose Purpose Purpose • In most service and production systems, the time required to provide the service or to complete the product is important. – We may want to design and operate the system to achieve certain service standards. • Generally, the time required includes “hands-on” time (actually processing) plus time waiting. • Queuing theory is about the estimation of waiting times. Sept. 30, 2009 Intro to Queueing Theory Prof. Leachman 2 erminology and Framework erminology and Framework Terminology and Framework Terminology and Framework • ustomers rrive randomly for service and await Customers arrive randomly for service and await availability of a server – When the server(s) has (have) finished servicing previous customers, the new customer can begin service ime between arrival of customer and start of • Time between arrival of customer and start of service is called the queue time ustomer departs the system after completion of • Customer departs the system after completion of the service time otal time in system = queue time + service time Sept. 30, 2009 Intro to Queueing Theory Prof. Leachman 3 • Total time in system = queue time + service time Analytical Approximation Analytical Approximation • The mathematics of queuing theory is much easier if we assume the customer inter-arrival time has an exponential distribution, and if we assume the service time also has an exponential istribution The exponential distribution has the distribution. The exponential distribution has the memoryless property : – Suppose the average inter-arrival time is t . Given it pp g a has been t since the last customer arrival, what is the expected time until the next customer arrival? nswer: Still t ! a – Suppose the average service time is t s . Given it has been t time units since service started, what is the xpected time until service ends? Answer: Still Sept. 30, 2009 Intro to Queueing Theory Prof. Leachman 4 expected time until service ends? Answer: Still t s ! The M/M/1 Queue The M/M/1 Queue • Queuing notation: A / B / n means inter-arrival times have distribution A , service times have distribution B , n means there are n servers • M means Markovian (memoryless), 1 means one erver server • In a Markovian queuing system, the only information we need to characterize the state of the system is the number of customers n in the system • We write = 1/ t a as the arrival rate and = 1/ t s as the service rate . The utilization of the server is u = t / t = / Note that we must have u < 1 for the Sept. 30, 2009 Intro to Queueing Theory Prof. Leachman 5 s a queue to be stable. he M/M/1 Queue (cont ) he M/M/1 Queue (cont ) The M/M/1 Queue (cont.) The M/M/1 Queue (cont.)The M/M/1 Queue (cont....
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093009Introduction to Queuing Theory - Introduction to...

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