**7. Conclusion**

At the end of this chapter related to "the calculation of the performance parameters of the fuzzy Markovian queueing system FM/FM/1 in transient state by the L–R method," it was a question of evaluating these performance parameters of the queueing system considered in transient regime by means of a method called the L–R

#### *Advances in Fuzzy Logic Systems*

method based primarily on the arithmetic of the fuzzy numbers of type L–R restricted to the secant approximations.

The L–R method facilitated us to find, in transient state, the L–R representation of the performance parameters, their support and mode as well as their membership functions which allowed us to represent graphically the performance parameters of the queueing model in study in the space in three dimensions. This is the originality of this scientific work.

In this chapter, a fuzzy queue (or waiting system) analysis method called L–R method, essentially based on the fuzzy L–R arithmetic deduced from secant approximations has been studied. The L–R representation was used to find the performance measures of the studied model. Using this method, the fuzzy functions were computed for the fuzzy expectation model FM/FM/1 and the results are found in L–R representation. Under this representation, the fuzzy results give much more reliable information than the relaxed alpha-cuts method which will be the subject of a future research study. The solutions obtained show that this method has three major advantages: it is short, convenient and flexible compared to other methods used in this field.

We are sure that the L–R method can still help, to obtain results of other similar problems posed in this field in the framework of evaluating the performance parameters of fuzzy Markovian expectation models in transient regime.

An illustrative example was given in the medical field to show the relevance of this study in operational research and in particular in queueing.
