A facility of 3 identical machines is sharing a single repair person. The time to repair a failed machine is exponentially distributed with mean 1/3. A machine once operational, fails after a time that is exponentially distributed with mean 1/4. All failure and repair times are independent. (1). Find the steady-state probability that there is no operational machine. (2). Find the expected number of operational machines, in steady-state.

Step-by-step explanation:

hello,

this question is drawn from queuing theory where the servers is the single repair person and the customer is the machine, thus we have 3 customers and a single server.

we shall obtain a Markov chain of the form M/M/1/3 queue where 1/3 and 1/4 are the arrival rate and service rate respectively.

the steady-state is given as ;

\\where\ \ \beta = \frac{3}{4}" alt="p_{3} =\frac{\beta ^{3}(1-\beta )}{1-\beta ^{4} }" /> \\where\ \ \beta = \frac{3}{4}" />

⇒ = " alt="\frac{\frac{27}{64}(1-\frac{3}{4})}{1-\frac{81}{256}}" />" /> =

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step-by-step explanation: