M/M/c queue

Stationary solution

$\lambda _{k}=\lambda ,\forall k$

$\mu _{k}={\begin{cases}k\mu ,&{\mbox{if }}0\leq k\leq c\\c\mu ,&{\mbox{if }}c\leq k\end{cases}}$

$\rho ={\frac {\lambda }{c\mu }}$

Probability of state $k$:

$\pi _{k}={\begin{cases}\pi _{0}{\frac {(c\rho )^{k}}{k!}},&{\mbox{if }}0\leq k\leq c\\\pi _{0}{\frac {\rho ^{k}c^{c}}{c!}},&{\mbox{if }}c\leq k\end{cases}}$

Probability of an empty system:

$\pi _{o}={\frac {1}{\sum _{k=0}^{c-1}{\frac {(c\rho )^{k}}{k!}}+{\frac {(c\rho )^{c}}{c!}}{\frac {1}{1-\rho }}}}$

Probability that request has to wait in the queue (all servers all occupied):

$P[K\geq c]=\pi _{c+}=\sum _{k=c}^{\infty }={\frac {(c\rho )^{c}}{c!(1-\rho )}}\pi _{0}$

Average (expected) number of requests in the system:

${\overline {N}}=c\rho +{\frac {\rho }{1-\rho }}\pi _{c+}$

Average (expected) length of the queue:

${\overline {N}}_{Q}={\frac {\rho }{1-\rho }}\pi _{c+}$