**2. Presentation of the queueing model M/M/1 et some performance parameters in transient state**

### **2.1 Classical M/M/1 queueing system**

**Definition 1**: A queueing system or M/M/1 queue is a Markovian process unfolding in L.D.P.(Life and Death Process) with birth and death rates defined respectively by:

$$\begin{cases} \lambda\_n = \lambda, \lambda > 0 \\ \mu\_n = \begin{cases} \mu & \text{if } n > 0 \\ 0 & \text{if } n = 0 \end{cases} \end{cases} \tag{1}$$

### *2.1.1 Assumptions (or characteristics) of the model*


*Computing the Performance Parameters of the Markovian Queueing System FM/FM/1… DOI: http://dx.doi.org/10.5772/intechopen.110388*


#### **2.2 Performance parameters of the M/M/1 queue in transient state**

In the literature, it is well-known that a queue is stable if and only if (cfr. [13]): *λ*< *μ*

• This condition makes it possible to determine the following performance parameters in transient state:

$$\tilde{N}\_S(t) = \frac{\rho}{1 - \rho} \left( \mathbf{1} - e^{-(\mu - \lambda)t} \right) \tag{2}$$

$$\tilde{T}\_S(t) = \frac{\rho\left(\mathbf{1} - e^{-(\mu-\lambda)t}\right)}{\mu(\mathbf{1}-\rho)[\rho + (\mathbf{1}-\rho)e^{-(\mu-\lambda)t}]} \tag{3}$$

where *<sup>N</sup>*<sup>~</sup> *<sup>S</sup>*ð Þ*<sup>t</sup>* and *<sup>T</sup>*<sup>~</sup> *<sup>S</sup>*ð Þ*<sup>t</sup>* are, respectively, the average number of customers in the system and the average waiting time in the system at time *t t*ð Þ ≥ 0 *:*.
