Suppose there are 4 machines and 2 technicians. The machines are assumed to be identical, and the life of the machines h

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Suppose there are 4 machines and 2 technicians. The machines are
assumed to be identical, and the life of the machines has an
exponential distribution with the rate 𝜇 . If there is a
faulty machine, the machine is repaired in order by an available
technician. Each technician who works is assumed to be identical
with the work time has an exponential distribution with the
rate 𝜆 . After repair, the machine can function as before.
Suppose {𝑋(𝑡), 𝑡 ≥ 0} is a birth and death process that represents
the number of machines that function up to time .
a. Determine the rate diagram for the above process.
b. Suppose we know the birth rate 𝜆1 = 1,
𝜆2 = 1, 𝜆3 = 1, 𝜆4 = ½ and
the death rate 𝜇1= 1/72, 𝜇2= 2/72,
𝜇3= 3/72, 𝜇4= 4/72. What is the long-run
fraction of the time that the two technicians are not working? And
what is the long-run fraction of the time that all the machines are
down?