There are three machines, two working and one that is used as a spare. A working machine will function for an exponential time with rate λ=.1 and will then fail. Upon failure, it is immediately replaced by another machine if that one is in working order, and it goes to the repair facility. The repair facility consists of a single person who takes an exponential time with rate μ=3 to repair a failed machine. At the repair facility, the newly failed machine enters service if the repair person is free. If the repair person is busy, it waits until the other machine is fixed; at that time, the newly repaired machine is put in service and repair begins on another one. Starting with all machines in working condition, find, in the long run, the proportion of time there are three working machines (2 actually working, the other not being used). π_0= ___ (round answer to 3 decimals)

There are three machines, two working and one that is used as a spare. A working machine will function for an exponential time with rate λ=.1 and will then fail. Upon failure, it is immediately replaced by another machine if that one is in working order, and it goes to the repair facility. The repair facility consists of a single person who takes an exponential time with rate μ=3 to repair a failed machine. At the repair facility, the newly failed machine enters service if the repair person is free. If the repair person is busy, it waits until the other machine is fixed; at that time, the newly repaired machine is put in service and repair begins on  another one. Starting with all machines in working condition, find, in the long run, the proportion of time there are three working machines (2 actually working, the other not being used). π_0= ___  (round answer to 3 decimals)