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Unformatted text preview: 1 ! PollaczekKhinchin (PK) Equation ! Steadystate probabilities ! π = 1 – ρ ! Finding L, W q , W M/ G /1/GD/ ∞ / ∞ System ) 1 ( 2 2 2 2 ρ ρ σ λ − + = 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 , , π ρ π ρ π 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 Jackson’s Theorem: If ! Interarrival times are exponential, rate λ ! Service time at each server is exponential ! Each stage has infinite capacity waiting room ⇒ Interarrival times for each stage exponential, rate λ ⇒ If λ < s j µ j , each stage is M/M/s j /GD/ ∞ / ∞ system. … … … … s 1 , µ 1 s 2 , µ 2 s k , µ 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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 Spring '08
 GeorgeWilson
 Operations Research, Exponential Function, Automobile, police car, Interarrival Times, Road Policing Unit

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