"Consider a queueing process with two servers in which the inter-arrival times and service times are exponential with parameters $\lambda$ and $\mu$ respectively. Let $X_n$ be the queue length at the time of arrival of the $n^{\text{th}}$ customer. Compute $p_{11}$, $p_{32}$ and $p_{31}$ where $p_{ij}$ = $P(X_{n+1}=i|X_n = g)$."
Can someone please help?
A possible approach?
We see that the required probability has the following expression. $$p_{ij}=P(i+j-1 \text{services during the time in between one customer arrival})$$ Hence, $$p_{ij}= \int_0^{\infty}P(N_2(t)=i+j-1,N_1(t)=1) dt$$
Where $N_1(t)$ is the number of arrivals till time $t$ and $N_2(t)$ is the number of services till time $t$. So, Using the fact that these are exponentially distributed, and independently, we get that $$p_{ij}= \int_0^{\infty} \frac{e^{-(\lambda+\mu)t}~\lambda~\mu^{i+j}t^{i+j}}{(i+j-1)!} dt$$ Which is seen to be equal to $$p_{ij}=\frac{\lambda\mu^{i+j}}{(\lambda+\mu)^{i+j-1}(i+j-1)}$$
Is this right?