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 decide to stay 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



M/M/1 Queue with probability of new customer leaving
You are interested in a model with "balking". See e.g. 



> 
> János Sztrik, Basic Queueing Theory, Chapter 2.2., p. 25-29, [link](http://irh.inf.unideb.hu/~jsztrik/education/16/SOR_Main_Angol.pdf)
> 
> 
> 


I assume that the probability you gave was not meant to be a probability of leaving but rather of staying. 


