## 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

#### 4.4. M/M/c/k Queue

In this model, customers arrive according to a Poisson process describing independent and exponentially distributed inter-arrival times. The service times are also assumed to be independent and exponentially distributed. The difference of this model with the previous models is the only k customers who can get the service by fixed number of c servers. Followings are the formulae for some of the performance measures of this queueing system.

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

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

$\begin{array}{ll} \mathrm{P}_{\mathrm{N}}=\frac{1}{\mathrm{~N} !} \rho^{\mathrm{N}} \mathrm{P}_{0} & \text { for } 0 \leq \mathrm{N} \leq \mathrm{c} \\ \mathrm{P}_{\mathrm{N}}=\left(\frac{1}{\mathrm{c}^{\mathrm{N}-\mathrm{c}_{\mathrm{c}}}}\right) \rho^{\mathrm{N}} \mathrm{P}_{0} & \text { for } \mathrm{c} \leq \mathrm{N} \leq \mathrm{k} \end{array}$

iii. Average number of customers in the system

\begin{aligned} L_{s} &=\sum_{N=0}^{c-1} N \cdot P_{N}+\sum_{N=c}^{k} N \cdot P_{N} \\&=\frac{P_{0}}{c !}\left(\sum_{N=0}^{c-1} N \cdot \rho^{N}+\sum_{N=c}^{k} N \cdot \frac{\rho^{N}}{c^{N-c}}\right) \end{aligned}

iv. Average number of customers in queue

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

v. Average waiting time in the system

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

vi. Average waiting time in the queue

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