Operations Research: Queuing Theory Before completing production a product must

Question

Operations Research: Queuing Theory Before completing production a product must pass through three stages of production. On average, a new product begins at stage 1 every 6 minutes. The average time it takes to process the product at each stage is as follows: stage 1, 3 minutes; stage 2, 2 minutes; stage 3, 1 minute. After finishing at stage 3, the product is inspected (assume this takes no time). Ten percent of the final products are found to have a defective part and must return to stage 1 and go through the entire system again. After completing stage 3, 20% of the final products are found to be defective. They must return to stage 2 and pass through 2 and 3 again. On average, how many jobs are in the system? Assume that all inter-arrival times and service times are exponential and that each stage consists of a single server.

Explanation / Answer

jobs per minute Fresh Products Arrival rate at stage 1 1/6 0.167 Defective products arrival rate for rework at stage 1 1/6*10% 0.017 defective products arrival rate at stage 2 1/6*20% 0.033 Arrival Rate at first stage 1/6 + 1/6*10% 0.183 Arrival Rate at second stage 1/6 + 1/6*10% + 1/6*20% 0.217 Arrival Rate at third stage 1/6 + 1/6*10% + 1/6*20% 0.217 Service Rate at first stage 1/3 0.333 Service Rate at second stage 1/2 0.500 Service Rate at third stage 1/1 1.000 Stage 1 Stage 2 Stage 3 Arrival Rate () 0.183 0.217 0.217 Service Rate () 0.333 0.500 1.000 Average number of jobs in system at each stage = /(-) 1.22 0.76 0.28 Average number of Jobs in the system 2.26

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