## A Survey on Queueing Systems with Mathematical Models and Applications

Read this paper on waiting line analysis and queues. It provides a good survey of the theory and uses of this type of analysis. Pay particular attention to sections 1 through 3. How might these models be used to balance firm costs with different levels of customer satisfaction?

### 4. Formulation of Queueing Models

#### 4.3. Queue (p/c < 1)

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

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