I am not sure if the queue system I draw is correct.
Consider the M/G/1 queueing system, Customers arrive according to a Poisson process with rate $\lambda$ in batches of size 2. Customers are served individually and service times have a gamma distribution Gamma($\frac{1}{2}$, $\frac{3}{\lambda}$).
In an attempt to control the size of the queue the system implements the following policy: at most one batch of arrivals is allowed during a service time, i.e., if two separate batches of customers arrive during the service of a single customer the second batch is turned away and leaves the system. If we let $X_n$ be the number of customers in queue immediately following the nth service completion then $$X_{n+1} = X_n + Y_{n+ 1} - 1_{\{X_n > 0\}},$$ where $Y_n$ is the number of customers entering the system during the nth service time. The stochastic process $\{X_n\}_{n\geq 1}$ is a Markov chain. We want to find the limiting probability of this system.
Here is my solution. Let $\alpha_0$ denote the probability that no arrival during one service time then $\alpha_0 = \int_{0}^{\infty}e^{-\lambda x} \frac{x^{-0.5}}{\Gamma(0.5)(\frac{3}{\lambda})^{0.5}}e^{-\frac{\lambda x}{3}}dx = 0.5$.
Then I got the queue system as followings.
In this case, the limiting probabilities are all zeros, so I got a little confused about it and do not know if there is anything wrong.