(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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).

2. Relevant equations

3. 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.}

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# Queuing Theory problem, M/M/1/K queue with twist

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