**5. Description of the L: R method and procedure of computing**

#### **5.1 Description of L: R method**

Let us consider the classical Markovian queue M/M/1 defined in Section (2.2) and assume that the arrival rate *λ* and service rate *μ* are triangular fuzzy numbers denoted *Computing the Performance Parameters of the Markovian Queueing System FM/FM/1… DOI: http://dx.doi.org/10.5772/intechopen.110388*

<sup>~</sup>*<sup>λ</sup>* <sup>¼</sup> ð Þ *<sup>λ</sup>*1j*λ*2j*λ*<sup>3</sup> and *<sup>μ</sup>*<sup>~</sup> <sup>¼</sup> *<sup>μ</sup>*1j*μ*2j*μ*<sup>3</sup> ð Þ, respectively. In this case, these rates are imprecise (or fuzzy) and also make the performance measures of the transient fuzzy functions and, we note:

$$
\tilde{\boldsymbol{\varphi}}(\mathbf{t}) = \tilde{\boldsymbol{f}}\left(\mathbf{t}, \tilde{\boldsymbol{\lambda}}, \tilde{\boldsymbol{\mu}}\right) \tag{28}
$$

Where *t* is a real variable called time and ~*λ* and *μ*~ are fuzzy variables. In this case, the queueing model becomes a fuzzy Markovian queue FM/FM/1, where FM is a fuzzy exponential distribution.

To determine the fuzzy performance measure *ψ*~ð Þ*t* , the L–R method proceeds as follows:

#### **5.2 Procedure**

Determine the L–R expressions of the fuzzy rates ~*λ* and *μ*~ and substitute them:

$$
\tilde{\boldsymbol{\varphi}}(\mathbf{t}) = \tilde{\boldsymbol{f}}\left(\mathbf{t}, \tilde{\boldsymbol{\lambda}}, \tilde{\boldsymbol{\mu}}\right) \tag{29}
$$

Apply the arithmetic of fuzzy numbers of (14)–(17) in (28) and we find:

$$
\bar{\psi}(t) = \langle m(t), \rho(t), \alpha(t) \rangle\_{L-R} \tag{30}
$$

Where *m t*ð Þ is the modal function of *ψ*~ð Þ*t* (or the kernel of *ψ*~ð Þ*t* ) and where *φ*ð Þ*t* and *<sup>ω</sup>*ð Þ*<sup>t</sup>* represent respectively the *left spread* and *right spread* of *<sup>A</sup>*~*:*.

The support of *ψ*~ð Þ*t* is:

$$\text{supp}(\ddot{\boldsymbol{\varphi}}(t)) = \left[ m(t) - \boldsymbol{\varrho}(t), m(t) + o(t) \right] \tag{31}$$

And its kernel (or modal) is:

$$\ker(\tilde{\boldsymbol{\mu}}(t)) = \boldsymbol{m}(t) \tag{32}$$
