17 Customers arrive in a queue according to a Poisson process with rate , and their service times have the exponential distribution with parameter Show that, if there is only one customer in the queue, then the probability that the next customer arrives within time t and has to wait for service is (1 e( )t )

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Question: 17. Customers arrive in a queue according to a Poisson process with rate , and their service times have the exponential distribution with parameter .

17. Customers arrive in a queue according to a Poisson process with rate λ, and their service times have the exponential distribution with parameter μ. Show that, if there is only one customer in the queue, then the probability that the next customer arrives within time t and has to wait for service is

λ + μ

(1 − e−(λ+μ)t ).

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