Recall that an M/M/1 queueing system is a GI/G/1 system in which there is one server, customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/µ. For...

Recall that an M/M/1 queueing system is a GI/G/1 system in which there is one server, customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/µ. For an M/M/1 queueing system,

(a) show that the number of arrivals during a period in which a customer is being served is geometric with parameter µ/(λ + µ);

(b) suppose that there are n customers waiting in the queue, and a customer is being served. Find the probability mass function of the number of new customers arriving by the time that all of these n + 1 customers are served.