# M/M/1 queue with probability of new client leaving

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

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 $$Q=\begin{pmatrix} -\lambda & \lambda \\ \mu & -\left(\mu+\frac{1}{2}\lambda\right) & \frac{1}{2}\lambda \\ &\mu & -\left(\mu+\frac{1}{3}\lambda\right) & \frac{1}{3}\lambda \\ &&\mu & -\left(\mu+\frac{1}{4}\lambda\right) & \frac{1}{4}\lambda &\\ &&&&\ddots \end{pmatrix}.$$