There is a selective group that people can join. Arrival of people who want to join the group is Poisson Process with parameter $\lambda$. The person who arrive at the group can join the group only if he/she gets approval from all existing members. Each member will say 'yes' with probability 1/2 and 'no' with probability 1/2. Also, each member of the group leaves the group independently at a Exponential distribution of time $\mu$. The number of members of this group defines a birth and death process. What is the limiting probability?
My attempt:
I came up with a very nasty answer.
$\alpha_i = \frac{1}{i!}(\frac{1}{2})^{\sum_0^{i-1}i}(\frac{\lambda}{\mu})^i\alpha_0 \ $ for $i = 1,2,3,...$
$\alpha_0 = (1 + \sum_{1}^{\infty}\frac{1}{i!}(\frac{1}{2})^{\sum_0^{i-1}i}(\frac{\lambda}{\mu})^i\alpha_0 )^{-1}$
Does anyone has any idea if this is correct?