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Unformatted text preview: 1 ! PollaczekKhinchin (PK) Equation ! Steadystate probabilities ! = 1 ! Finding L, W q , W M/ G /1/GD/ / System ) 1 ( 2 2 2 2 + = s q L 2 M/M/R/GD/ K / K System ! Draw the transition diagram ! Derive the steadystate probabilities: ! Find L,L q ,W,W q R repair people K machines + + = = = K R R j R R j j K R j j K j R j j j ,..., 2 , 1 , ! ! ,..., 1 , , 3 Example 1 ! The Gotham Township Police Department has 5 patrol cars. A patrol car breaks down and requires service once every 30 days. The police department has two repair workers,each of whom takes an average of 3 days to repair a car. Breakdown times and repair times are exponential. ! E(# police cars in good condition) ! E(down time for car needing repairs) ! Fraction of time a particular repair worker is idle 4 Queues in Series Jacksons Theorem: If ! Interarrival times are exponential, rate ! Service time at each server is exponential ! Each stage has infinite capacity waiting room Interarrival times for each stage exponential, rate If < s j j , each stage is M/M/s j /GD/ / system. s 1 , 1 s 2 , 2 s k , k 5 Example 2 ! The last two things that are done to a car before its manufacture is complete are installing the engine and putting on the tires. An average of 54 cars per hour arrive requiring these two tasks. One worker is available to install the engine and can service 60 cars per hour. the engine and can service 60 cars per hour....
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This note was uploaded on 05/05/2008 for the course IE 221 taught by Professor Georgewilson during the Spring '08 term at Lehigh University .
 Spring '08
 GeorgeWilson
 Operations Research

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