I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = \lambda$ and $\mu_i = \mu \forall i$. In the current situation we're adding the non-trivial probability that a new client decides to leave depending on the number of people currently in the system.
For instance if at time $t$ we have $M_t=n$ then a client arriving in the system would leave with a probability $p_n=\frac{1}{1+n}$. This modifies the regular arrival rate (which would normally be $\lambda$) although the average service time remains the same ($\frac{1}{\mu}$) but I'm not sure how.
Any tips on how to tackle this situation? Thanks
What have you done so far, where are you stuck?
The transition rate matrix for the standard M/M/1 queue model is given by $$Q=\begin{pmatrix} -\lambda & \lambda \\ \mu & -(\mu+\lambda) & \lambda \\ &\mu & -(\mu+\lambda) & \lambda \\ &&\mu & -(\mu+\lambda) & \lambda &\\ &&&&\ddots \end{pmatrix}.$$ In your situation the arriving clients can immediately choose to leave, meaning that the actual arrival rate to the queue at state $n$ rather than being $\lambda$ is $\frac{n}{1+n}\lambda$. Therefore you have the following modified transition rate matrix $$Q=\begin{pmatrix} 0 & 0\\ \mu & -\left(\mu+\frac{1}{2}\lambda\right) & \frac{1}{2}\lambda \\ &\mu & -\left(\mu+\frac{2}{3}\lambda\right) & \frac{2}{3}\lambda \\ &&\mu & -\left(\mu+\frac{3}{4}\lambda\right) & \frac{3}{4}\lambda &\\ &&&&\ddots \end{pmatrix}.$$ Note that this means that $0$ is an absorbing state, so when the queue is empty any jobs which arrive depart immediately with probability $1$, therefore leaving the queue forever empty.
Edit: I see you also posted this question at mathoverflow, where you say you got the probability the wrong way around (it is the probability of a customer entering, rather than one leaving). In this case the transition rate matrix should be $$Q=\begin{pmatrix} -\lambda & \lambda \\ \mu & -\left(\mu+\frac{1}{2}\lambda\right) & \frac{1}{2}\lambda \\ &\mu & -\left(\mu+\frac{1}{3}\lambda\right) & \frac{1}{3}\lambda \\ &&\mu & -\left(\mu+\frac{1}{4}\lambda\right) & \frac{1}{4}\lambda &\\ &&&&\ddots \end{pmatrix}.$$