Stochastic Mfg&Serv Sys
Stochastic Mfg&Serv Sys ISYE 3232
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This 0 page Class Notes was uploaded by Maryse Thiel on Monday November 2, 2015. The Class Notes belongs to ISYE 3232 at Georgia Institute of Technology - Main Campus taught by Jiheng Zhang in Fall. Since its upload, it has received 17 views. For similar materials see /class/234193/isye-3232-georgia-institute-of-technology-main-campus in Industrial Engineering at Georgia Institute of Technology - Main Campus.
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Date Created: 11/02/15
lSyE 3232 Stochastic Manufacturing and Service Systems Summer 2008 J Zhang Two Examples on Topic 7 July 24 2008 1 Consider a production system consisting of three singleserver stations in series Customer orders arrive at the system according to a Poisson process with rate 1 per hour Each customer order immediate triggers a job that is released to the production system to be processed at station 1 rst and then at station 2 After being processed at station 2 a job has p1 10 probability going back to station 1 for rework and 1 7 p1 probability continuing onto station 3 After being processed at station 3 a job has p2 5 probability going back to station 1 p3 10 probability going back to station 2 and 171927193 probability leaving the production system as a nished product Assume that the processing times of jobs at each station are iid having exponential distribution regardless of the history of the jobs The average processing times at stations 1 2 and 3 are m1 08 m2 070 and 7713 08 hours respectively a Find the long run fraction of time that there are 2 jobs at station 1 1 job at station 2 and 4 jobs at station 3 b Find the long run average system size at station 3 c Find the long run average time in system for each job d Reduce p1 to 5 Answer 10 again What story can you tell Solutions a Let A be the actual arrival rate to station 2 i 1 2 3 Then we have the following traf c equations 1 1012 0053 2 1 3 092 Solving the equations one has 1 11895 2 13072 and 3 11765 Thus p1 1m1 09516 p2 2m2 09150 p3 3m3 09412 Let Xit be the number ofjobs at station i at time t Then the long run fraction of time that there are 2 jobs at station 1 1 job at station 2 and 4 jobs at station 3 is equal to Pr X100 27X20017X300 4 Pr X1oo 2PrX2oo1PrX3oo 4 1 p1pi1 p2p21 psp x 15718 X1074 N b IDS1 p3160 c The total long run average system size is L 7 p1 7 p2 7 p3 jObS Using Little s law L W the average time in system per job is W 464449 hours since the arrival rate into the system is 1 job per hour d The average size of the system is L 21985 jobs Thus the average time in system per job is 21985 hours 5 of reduction of the rework cuts the average cycle time to less than half This is consistent with Goldratt theory in managing bottlenecks a small help to a bottleneck machine the great reduction to the cycle time CONWIP Consider the production system in Problem 1 with following modi cation p1 10 p2 p3 0 station 3 makes 100 reliable operations Furthermore the Management decides to adopt the make to stock policy using the CONWIP job release policy Recall that when the system is operating CONWIP policy only a job leaving station 3 triggers a new job to be released to station 1 Let N be the CONWIP level a For N 2 compute the throughput of the production system What is the average time in system per job b Is it possible to double the throughput If so what N is needed to achieve it What is the corresponding average time in system per job Solutions a Let y be the throughput rate going through station 2 Then we have traf c equations 71 01W 737 1 V2 V17 2 73 09W 3 Equations 2 and 3 imply equation Thus equation 1 is redundant In general in a closed network there is always a redundant traf c equation Thus one can not solve 39y s from 1 3 However if we can nd m by using some method then w and V3 follow readily For the moment we set 4 Setting V1 to any other positive number is OK Then 39yg 1 and V3 09 Remember these are not true throughput rates With 39y s set one can de ne V11 P1 39Ylml 087 P2 39Ylml 077 P3 39Ylml 072
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