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

[A-fil]

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 β₀ p₀ = μ p₁
λ β_n p_n + μ p_n = μ p_n+1 + λ β<sub>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.</sub>

 

[mighty2000]

Thank you for your interesting question. I can provide some insights and guidance on how to approach this problem.

Firstly, your approach in part (1) is correct. The key here is to understand that the arrival of a new customer and the decision to join the line or not are two separate events, and thus their probabilities can be multiplied together.

In terms of solving the equation, there are a few ways to go about it. One approach would be to use the fact that the balance equation expresses the steady-state probabilities, and thus the equation can be solved by setting the left and right side equal to each other and solving for pn. Another approach would be to use generating functions to solve the equation. I would recommend consulting with your textbook or professor for more guidance on the specific method to use for solving this type of equation.

I hope this helps and good luck with your problem-solving!