Haven't see anything like it in queueing theory, the problem appears to be M/M/1 but with a twist?
Description: The server starts serving calls only when the number of customers in the system becomes 3. In all other cases, the system behaves like a typical M/M/1 queue.
When the server starts serving calls, it serves them normally, even if their number is less than 3. However, if all calls are served and the system becomes empty (n = 0), the server will again start serving calls only when their number in the system becomes 3.
I can't even create a state diagram?
To model this as a CTMC the simplest way would be to add states $1^*$ and $2^*$ as Henry mentioned in the comments. So the state space would be $S=\{0,1^*,2^*,1,2,3,\ldots\}$ and the transitions similar to a regular $M/M/1$ queue, with generator matrix as $$ Q=\begin{pmatrix} -\lambda & \lambda & 0&0&0&0&0\\ 0&-\lambda &\lambda & 0&0&0&0\\ 0&0&-\lambda&0&0&\lambda&0\\ \mu &0&0&-(\lambda+\mu) & \lambda & 0&0\\ 0&0&0&\mu&-(\lambda+\mu)&\lambda&0\\ 0&0&0&0&\mu&-(\lambda+\mu)&\lambda&\cdots \end{pmatrix}. $$ Solving the balance equations $$ \sum_{i\in A,j\in S\setminus A}q_{ij}\pi_{i} = \sum_{i\in S\setminus A,j\in S}q_{ij}\pi_i $$ for the sequence of subsets of the state space $A=\{0\}, \{0,1^*\}, \{0,1^*,2^*\},\ldots\ $ subject to the condition $\sum_{i\in S}\pi_i =1$ yields the stationary distribution \begin{align} \pi_0&=\frac{\mu (\mu -\lambda )}{2 \lambda ^2+\mu ^2}\\ \pi_{1^*} &= \frac{\lambda ^2 (\mu -\lambda )}{\mu(2 \lambda ^2 +\mu ^2)}\\ \pi_{2^*} &= \frac{\lambda ^2 (\mu -\lambda )}{\mu(2 \lambda ^2 +\mu ^2)}\\ \pi_1 &= \frac{\lambda (\mu -\lambda )}{2 \lambda ^2+\mu ^2}\\ \pi_2 &= \frac{\lambda ^2 (\mu -\lambda ) (\lambda +\mu )}{\mu ^2 \left(2 \lambda ^2+\mu ^2\right)}\\ \pi_n &= \frac{\lambda^n(\mu -\lambda ) (\lambda +2 \mu )}{\mu^n(2 \lambda ^2+\mu ^2)},\ n\geqslant 3. \end{align}