Write down the generator matrix of this markov process

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Consider an exponential server system with service rate μ having no waiting space. Customers arrive according to a Poisson process with mean rate λ . As the server is failure-prone, the normal time of the server follows the exponential distribution with mean θ-1. When it breaks down, it will be repaired at once. The repair time of the server follows an exponential distribution with mean γ-1. The following are the four possible states of the system
{(N, 0), (D, 0) ,(N, 1), (D; 1)}

where N represents the server is normal, D represents the server is down, 0 represents the system is idle and 1 represents the system is busy.

(a) Write down the generator matrix of this Markov process.

(b) Obtain the steady-state probability distribution of the system.

(c) Assuming that the system is in steady-state, find the conditional probability that the server is under repair given that the system is idle.

Reference no: EM131162868

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