Find the probability in the long run at a random time

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Question: In Springfield, Massachusetts, people drive their cars to a state inspection center for annual safety and emission certification at a Poisson rate of λ. For n ≥ 0, if there are n cars at the center either being inspected or waiting to be inspected, the probability is 1/(n + 1) that an additional driver will join the queue. Hence the probability is n/(n + 1) that he or she will not join the queue and will leave. Suppose that inspection times are independent and identically distributed exponential random variables with mean 1/µ and they are independent of the arrival process. For n ≥ 0, find the probability that in the long run, at a random time, there are n cars at the center either being inspected or waiting to be inspected.

Reference no: EM131812242

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