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## Homework Statement

Customers arrive to a register as a poisson process with arrival rate λ and are serviced with an exponential distribution with service rate μ. When a customer arrives he'll decide to join the line or not depending on how many people are currently in the system (not including himself). In other words, if at a given time n people are in the system then a new customer will join the line with probability β

_{n}, and not join the line and not take the service with probability 1-β

_{n}. However, 0 ≤ β

_{n}≤ 1 (0 ≤ n ≤ N), β

_{n}= 0 (n > N).

(1) Find the balance equation expressed with the balance probability state p

_{n}(where n is the number of customers in the system).

(2) Solve the equation in (1).

## Homework Equations

## The Attempt at a Solution

(1) So, my problem here is where the β will enter the problem. I'm thinking that the events "Customer stays" and "Customer arrives" are independent, resulting in the following:

With this I would get:

λ β

_{0}p

_{0}= μ p

_{1}

λ β

_{n}p

_{n}+ μ p

_{n}= μ p

_{n+1}+ λ β

_{n-1/SUB] pn-1 for 1 ≤ n ≤ N-1 λ βN-1 pN-1 = μ pN Is this correct? If not, any hint? (2) If the above is correct, how to solve it? Any hint is appreciated.}