# How to determine the rate of input in a queue M/M/c?

I know the exit rate ($$\mu$$) and the average waiting time in the queue ($$W_q$$). I need solve to rate of input ($$\lambda$$) in a queue.

$$\rho = \frac{\lambda}{c\mu} < 1$$

$$\pi_0 = \left[\left(\sum_{k=0}^{c-1}\frac{(c\rho)^k}{k!} \right) + \frac{(c\rho)^c}{c!}\frac{1}{1-\rho}\right]^{-1}$$

$$W_q=\frac{1}{(c-1)!}\left(\frac{\lambda}{\mu}\right)^c\pi_0\frac{\mu}{(c\mu-\lambda)^2}$$

But I do not know how to solve this sum. Is there a simple analytical solution?

• Why not just add up the $c$ values $(c\rho)^k/k!$? – jbowman Jan 29 at 17:47