Question: 18. Customers arrive at a two-server system at a Poisson rate . An arrival finding the system empty is equally likely to enter service with

18. Customers arrive at a two-server system at a Poisson rate λ. An arrival finding the system empty is equally likely to enter service with either server. An arrival finding one customer in the system will enter service with the idle server. An arrival finding two others in the system will wait in line for the first free server. An arrival finding three in the system will not enter. All service times are exponential with rate μ, and once a customer is served (by either server), he departs the system.

(a) Define the states.

(b) Find the long-run probabilities.

(c) Suppose a customer arrives and finds two others in the system. What is the expected times he spends in the system?

(d) What proportion of customers enter the system?

(e) What is the average time an entering customer spends in the system?

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