Question
Television sets for repair arrive at random at an average rate of 4 per day to a single repairman who takes an average of hours to carry out each repair. It being assumed that the repair times have an exponential distribution. What is the average number of television sets in the workshop? What is the probability that an arriving set will find at least 3 sets in front of it? The repairman works for eight hours a day.
Answer :
Word Count : 412
We can model this problem using the M/M/1 queue, which is a common model for systems with a single server (in this case, the repairman), exponential arrival rates, and exponential service times. ### Given: - Arrival rate (λ): 4 sets per day (since 4 televisions arrive each day). - Service rate (μ): 1.5 hours per set. The repairman works 8 hours a day, so he repairs \(\frac{8}{1.5} \approx 5.33\) sets per day. Now, let's calculate the average number of television sets in the workshop and the probability that an arriving set will find at least 3 sets in front of it. ### Step _____ ______ _______ ______ _____ ________ ________ ________ __________ __________ ______ _______.
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We can model this problem using the M/M/1 queue, which is a common model for systems with a single server (in this case, the repairman), exponential arrival rates, and exponential service times. ### Given: - Arrival rate (λ): 4 sets per day (since 4 televisions arrive each day). - Service rate (μ): 1.5 hours per set. The repairman works 8 hours a day, so he repairs \(\frac{8}{1.5} \approx 5.33\) sets per day. Now, let's calculate the average number of television sets in the workshop and the probability that an arriving set will find at least 3 sets in front of it. ### Step _____ ______ _______ ______ _____ ________ ________ ________ __________ __________ ______ _______.
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