I am reading over the stationary distribution section Durrett's textbook Probability: Theory and Example, and I got stuck by a statement in an example he gave about the M/G/1 queue.
Here's the problem: Let $\mu = \Sigma k a_k$ be the mean number of customers that arrive during one service time where $a_k$ is the probability of the event that k customers arrive. We would like to show that the chain is positive recurrent if and only if $\mu < 1$. (The full example and proof are given in the attached screenshot)
Summary: In the forward direction, Durrett first proved that $E_x N \leq \frac{x}{1-\mu}$. He then showed that this $\leq$ is actually $=$.
To prove the equality, Durrett brought up several observations including $X_n$ decreases by at most one each time and for $x \geq 1, E_x T_{x-1} = E_1 T_0$. These are fine.
My question is: Durrett then claimed that based on the given observations, $E_x N = cx$, where $N = \{n \geq 0: X_n = 0\}$. This is not very obvious to me. May you show me how he got this result?
Thank you very much in advance.
The attached screenshot is the full example and proof, and my question is highlighted in blue.
We have $$ E_x N = E_x T_{x-1} + E_{x-1} T_{x-2} + E_{x-2} T_{x-3} + \dots + E_1 T_0 $$ since in order to get to $0$, we must first get to $x-1$, then to $x-2$, then to $x-3$, and so on. This simplifies to $x E_1 T_0$.