**1. Introduction**

Nowadays, in many areas of life, whether it is computer systems, communication systems, production systems, medical or health systems or any other system of daily life, the world is looking for the best quality of service and performance of systems.

Thus, Baynat [1] points out that "it is becoming inconceivable to build any system without first doing a performance analysis. This analysis is related to the prior knowledge of the performance parameters of the queueiting systems such as the average number of customers in the queue and in the system; as well as the average waiting time of customers in the queue and in the system."

The fundamental question posed in this chapter is: "Would the L–R method be able to compute the performance parameters of the fuzzy Markovian queueing system FM/FM/1 in transient regime?"

In the literature browsed in fuzzy mathematics, it is well shown that fuzzy queues are widely studied in steady state by Ning and Zhao [2], Ritha and Robert [3], Ritha and Menon [4].

Many researches on Markovian fuzzy queueing systems and scientific papers have been based on computing the performance parameters of Markovian fuzzy queueing systems in steady state by the method of sluggish alpha-cuts and the L–R method, see for example Li and Lee [5, 6]; Kao et al. [7]; Palpandi and Geetharamani [8]; Wang et al. [9]. But, the calculation of these parameters of the system under study in transient regime is a major preoccupation of operational researchers nowadays.

In this chapter, the novelty of our study is due to the fact that we have computed the performance parameters of the queueing system, in the transient regime and in a fuzzy environment where these parameters are time-dependent fuzzy functions, whereas for all the authors presented above, the performance parameters have been analyzed in steady state where the results obtained are real numbers.

L–R Fuzzy Mathematics plays an important role if it is widened to secant approximation. This arithmetic conducts to the same results as those obtained by the most used and well-known alpha-cut and interval arithmetic (see Mukeba; Dubois D. and Prade [10–12]).

To achieve this, our approach is broken down as follows: The second section will present the classical M/M/1 queueing model and give the performance parameters of the model in a transient state The third section will recall the notions of fuzzy set, fuzzy numbers, fuzzy number of L–R type, arithmetic of fuzzy numbers of L–R type and triangular fuzzy number. The fourth section will be devoted to fuzzy functions. The fifth section will give a description of the L–R method and the calculation procedure. The sixth section will deal with a numerical example that uses the L–R method in transient state. The seventh section will give the conclusion of this chapter.
