Balance Equations for M/M/1/m Queue

Question

I think I found the solution for this problem, but they don't show all the steps. I was wondering if someone could explain to me how they get from the three balance equations to the solution.

Answer 1

Let $a = \lambda/\mu$. Then: $$\pi_1 = a\pi_0\\\pi_{n+1} = (1 + a)\pi_n - a\pi_{n-1}\\\pi_N = a\pi_{N-1}$$

A quick calculation shows $\pi_2 = a^2\pi_0$, which suggests that $\pi_n = a^n\pi_0$. This is certainly true for $n = 0, 1$. So suppose it is true for all values up to $n$. Then $$\pi_{n+1} = (1 + a)(a^n\pi_0) - a(a^{n-1}\pi_0) = (a^n\pi_0) + (a^{n+1}\pi_0) - (a^n\pi_0) = a^{n+1}\pi_0$$ And so the result hold for all $n$ by induction.

Now this result satisfies all three balance equations for any $\pi_0$. Therefore, they used some other condition not included in your problem statement to evaluate $\pi_0$ In particular, the condition that $$\sum_{n=0}^N \pi_n = 1$$

With the formula for $\pi_n$, this becomes for $a\ne 1$ $$1 = \sum_{n=0}^N a^n\pi_0 = \frac{1 - a^{N+1}}{1 - a}\pi_0$$

so $$\pi_0 = \frac{1 - a}{1 - a^{N+1}}$$ and $$p_n = \frac{a^n(1 - a)}{1 - a^{N+1}}$$which is the solution for $\lambda \ne \mu$.