Continued questions for this one:
- Explain why, for some $\theta$ in $(0, 1]$, and $k = 0, 1, 2, ..., $
$$P(X \ \text{hits} \ A_0 \mid X_0 = A_k) = \theta ^k $$
- Show that $(\theta - 1)f(\theta) = 0, $ where
$$f(\theta) = \lambda^2 \theta^2 - \lambda(\lambda+\mu+\alpha+\beta)\theta + (\lambda + \beta)\mu.$$
- By considering $f(1)$ or otherwise, prove that $X$ is transient if $\mu \beta < \lambda(\alpha + \beta)$, explain why this is intuitively obvious.
For 1. I thought that all I need to show is that
$$P(X \ \text{hits} \ A_{k-1} \mid X_0 = A_k) = \theta$$
and $\theta$ independent of $k$. The later part is easy from Strong Markov property by using the hitting time.
- Show that when $\theta \in (0,1), f(\theta) = 0.$
But I have no idea how function $f()$ comes into being.
It seems that it can be separated into
$$f(\theta) = \lambda(\theta - 1)(\lambda\theta - \mu - \alpha - \beta) + \mu \beta - \lambda(\alpha + \beta)$$
- Intuitive explanation: The inequality is equivalent to $$\frac{1}{\lambda} \le \frac{1/\alpha + 1/\beta}{1/\alpha} \frac{1}{\mu},$$ which means that the expected time takes for a customer to arrive is less than the expected time takes for one to be served. Then there will be more and more customers staying in the shop.
Don't know how to prove it strictly.