I have this interesting M/M/1 queue question which I need help in order to accurately solved it.
$\textbf{Question:}$
Peter is doing grilling which can be modeled as an M/M/1 queue with a rate $\lambda$ and service rate $\mu.$ Peter is inexperienced and as a result his clients may be dissatisfied with his service and ask for re-service. Peter can also work one item one at a time.
Assume that upon receiving an order, there is a probability $1-\alpha$, $0 < \alpha <1$ that a customer is dissatisfied, independent of whether that customer has been dissatisfied one or more times previously. Subsequent service time are also independent and have the same exponential distribution with parameter $\mu$.
(a) Check that the length of time it takes for a customer to be satisfied with his/her order is exponentially distributed. What is the parameter?
(b) Suppose dissatisfied customers are served again immediately until they are satisfied. What are the conditions for the queue to be stable?
(c) Suppose some of his customers object to a dissatisfied customer being treated with priority, so a new rule is initiated requiring a dissatisfied customer to go to the end of the line. How does this affect the birth and death rates? (Recall a CTMC cannot jump from a state to itself.)
$\textbf{My solution:}$ I am not sure of what I did. Any help is welcome.
(a) We apply birth and death process to the question which are respectively given as: $$Q_{i, i+1} = p_i = \frac{\lambda_i}{\lambda_i + \mu_i} \text{ and } Q_{i, i-1} = 1- p_i = \frac{\mu_i}{\lambda_i + \mu_i}, \quad i \geq 1 $$ Let $D(i)$ is the time until a death (no customers) when population size is $i$ and $B(i)$ is the time until a birth (arrival of customers) when population size is $i$ and the birth and death are $$ B(i) \sim \text {Exp}(\lambda_i) = \lambda \text{ and } D(i) = \mu_i = \mu $$
(b) It has to be stable if we have $P(B(i) \leq D(i)) = Q_{i, i+1}$ and $0 < \lambda (i) < \infty$.
(c) For c, I have no idea.