Consider a single server queuing system where customers arrive according to a Poisson process with rate $\lambda$, service times are exponential with rate $\mu$, and customers are served in the order of their arrival. Suppose that a customer arrives and finds $n-1$ others in the system. Let $X$ denote the number in the system at the moment that customer departs. Find the probability mass function of $X$. Hint: Relate this to a negative binomial random variable.
Added by Kathy V.
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Step 1: Define the random variable $X$ as the number of customers in the system at the moment the customer departs. Show more…
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