The Copy Shop is open 5 days per week for copying materials that are brought to the shop. It has three identical copying machines that are run by employees of the shop. Only two operators are kept on

  

The Copy Shop is open 5 days per week for copying materials that are brought to the shop. It has three identical copying machines that are run by employees of the shop. Only two operators are kept on duty to run the machines, so the third machine is a spare that is used only when one of the other machines breaks down. When a machine is being used, the time until it breaks down has an exponential distribution with a mean of 2 weeks. If one machine breaks down while the other two are operational, a service representative is called in to repair it, in which case the total time from the breakdown until the repair is completed has an exponential distribution with a mean of 0.2 week. However, if a second machine breaks down before the first one has been repaired, the third machine is shut off while the two operators work together to repair this second machine quickly, in which case its repair time has an exponential distribution with a mean of only 1/15 week. If the service representative finishes repairing the first machine before the two operators complete the repair of the second, the operators go back to running the two operational machines while the representative finishes the second repair, in which case the remaining repair time has an exponential distribution with a mean of 0.2 week. (a) Letting the state of the system be the number of machines not working, construct the rate diagram for this queueing system. (b) Use the balance equations to find the steady-state distribution of the number of machines not working. (c) What is the expected number of operators available for copying?

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