I have a balance equation, representing a Markov Chain, which yields $$ (K - z) \pi(Z_c = z) = (\lambda_c + (z+1)x) \pi(Z_c = z+1) $$ where K is the maximum state of the server. The term $\lambda_c$ and $x$ are the constants.
From this, the following steady-state probability is obtained: $$ \pi(Z_c = z) = \pi(Z_c = K) x^{(K-z)} * C $$ where C is given by $$ {K+\frac{\lambda_c}{x}}\choose{K-z} $$
The approximation of $\pi(Z_c = K)$ is given by: $$ \pi(Z_c = K) = (\sum_{y=0}^{K} x^y {K+ \frac{\lambda_c}{y}\choose{y} })^{-1} $$
Can anybody help me in approximation $\pi(Z_c = K)$? How is $\pi(Z_c = K)$ approximated?