A Survey on Queueing Systems with Mathematical Models and Applications

The queuing system $\mathrm{M} / \mathrm{M} / \mathrm{c}$ is the queueing discipline where c service channels are ready for the arriving customers following Poisson process. $\lambda$ and $\mu$ have the usual meanings with all the random variables independent as described in the subsection 4.2. Followings are some of the formulae to for the performance measures of this model.

i. The probability of having zero customers in the system

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

ii. Probability of having $N$ customers in the system

$\begin{array}{cc}\mathrm{P}_{\mathrm{N}}=\mathrm{P}_{0} \frac{\rho^{N}}{\mathrm{~N} !} & \text { for } \mathrm{N} < \mathrm{c} \\ \mathrm{P}_{\mathrm{N}}=\mathrm{P}_{0} \frac{\rho^{N}}{\mathrm{c}^{N-\mathrm{c}} \mathrm{c} !} & \text { for } \mathrm{N} > \mathrm{c}\end{array}$

iii. Average number of customers in the queue

$L_{q}=P_{0} \frac{\rho^{c+1}}{c . c !} \frac{1}{\left(1-\frac{\rho}{c}\right)^{2}}$

iv. Average number of customers in system

$L_{s}=L_{q}+\rho$

v. Average waiting time in the system

$\mathrm{W}_{\mathrm{s}}=\frac{\mathrm{L}_{\mathrm{s}}}{\lambda}$

vi. Average waiting time in the queue

$\mathrm{W}_{\mathrm{q}}=\frac{\mathrm{L}_{\mathrm{q}}}{\lambda}$