Question

In a small hospital, there are four operating rooms used for surgeries. After each surgery, the operating room needs to be cleaned and sanitized before it can be used again. The hospital employs two cleaning staff members responsible for this task. The time an operating room remains in use for the surgery (before requiring cleaning) is exponentially distributed with mean 1/2 hours. The time it takes for a cleaning staff member to sanitize an operating room is exponentially distributed with mean 1/4 hours. Only one cleaning staff member can clean an operating room at a time. Whenever the cleaning is over, the operating room is assigned to another surgery, and the operations start right away. Find: a) What is the average number of operating rooms out of service (awaiting cleaning) in the long run? b) What proportion of time are both cleaning staff members busy cleaning operating rooms? Could you solve this question with Markov chain process and could you show the transition matrix, thank you!

          In a small hospital, there are four operating rooms used for surgeries. After each surgery, the operating room needs to be cleaned and sanitized before it can be used again. The hospital employs two cleaning staff members responsible for this task. The time an operating room remains in use for the surgery (before requiring cleaning) is exponentially distributed with mean 1/2 hours. The time it takes for a cleaning staff member to sanitize an operating room is exponentially distributed with mean 1/4 hours. Only one cleaning staff member can clean an operating room at a time. Whenever the cleaning is over, the operating room is assigned to another surgery, and the operations start right away. Find: a) What is the average number of operating rooms out of service (awaiting cleaning) in the long run? b) What proportion of time are both cleaning staff members busy cleaning operating rooms?
Could you solve this question with Markov chain process and could you show the transition matrix, thank you!

Added by Donald L.

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