***Performance Measures of a Non-Markovian Fuzzy Queue*** ***Using Fuzzy Laplace Transforms Method***

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**HIGHLIGHTS**

  - *Fuzzy Laplace Transforms applied to analyze performance measures*.

  - *A non-markovian fuzzy queuing system FM/FG/1.*

  - Queuing systems models play an important role in computers systems implementation

  - The **performance measures of a non-markovian queue in a fuzzy environment.**

  - **How to apply fuzzy transforms in the evaluation of the measures**

ABSTRACT

*Laplace Transforms play essential role in queuing systems investigation. In this paper, Fuzzy Laplace Transforms are applied to analyze performance measures of a non-markovian fuzzy queuing system FM/FG/1. The FM/FE-k/1 is particularly treated as concrete case. Outside of fuzzy arithmetic, random variable’s properties and Little formula are used and contribute to deduce expected performance measures. A numerical example is successfully treated to illustrate this study.*

*Keywords: Fuzzy Laplace Transforms, Erlang-k Service, Fuzzy Arithmetic, Performance Measures, Defuzzification*

# INTRODUCTION

*Actually,* queuing systems models play an important role in computers systems implementation, in telecommunication systems, in transport networks systems, etc. A queuing system is often described as a circuit in which, a customer arriving from outside, finds a server busy, and takes one of the three following options:

  - Wait with patience in the circuit until the moment where it will be served, and leave the circuit ;

  - Leave the circuit and after some random time, come-back and attempt to access the sever until the moment where it will be served, and leave the circuit;

  - Leave the circuit and give up coming back to the server again.

In queue systems literature, several researches have already analyzed successfully performance measures of different classical queues (Babu, P. S., Kumar, K. S., & Chandan, K. (2022).).

When the system parameters are vague and imprecise, the system is called fuzzy queue system, and represented generally by \(\mathbb{F}\), which derives to « ***Fuzzy*** ». An abundant theory of fuzzy set has already been developed to analyze the performance measures of such queues. This is particularly the case for markovian queues \(\mathbb{F}\) M/\(\mathbb{\text{\ F}}\) M/c and the product form queuing networks. Many results of this development can be found in (Fatoumata, Y., Adnane, A., & Ataoua, Z. (2021).).

Concerning the non-markovian fuzzy models such as \(\mathbb{F}\)M/\(\mathbb{\text{\ F}}\)G/1, \(\mathbb{F}\)G/\(\mathbb{\text{\ F}}\)M/1 …, the literature is not yet extensive to our knowledge. Among these rare works, we can cite for example among which is the most recent (Al-Kridi, K., Anan, M. T., & Zeina, M. B. (2018).).

Most of the researchers based their analysis on the optimization mathematical programs method called PNLP (*Parametric Non Linear Programming*), which mixes *Zadeh’s extension* principle and \(\alpha\)*-cuts and intervals arithmetic* (Patel, K. R., & Desai, N. B. (2017).). Some of them have applied and shown that the L-R method, is the fastest and the most flexible to analyze \(\mathbb{F}\)M/\(\mathbb{F}\)Ek/1 model (Çitil, H. G. (2019).).

In this article, we show how to calculate the performance measures of the fuzzy queue \(\mathbb{F}\)M/\(\mathbb{\text{\ F}}\)G/1 by the approach of fuzzy Laplace transforms at the steady state. *Our hypothesis is that Laplace’s method would remain valid for analyzing performance measures of both classical and fuzzy non-markovian queues. Like classical model (see (*Gong, Z., & Hao, Y. (2019).*), our methodology consists in calculating from fuzzy Laplace transforms the distribution of the customer sojourn time in the system.*

This article is organized as follows: Section 2 resumes the preliminaries. Section 3 deals with fuzzy Laplace transform starting with the notion of fuzzy functions. Section 4 is devoted to the fuzzy model \(\mathbb{F}\)M/\(\mathbb{\text{\ F}}\)G/1. The concrete case where \(G\  = \ E_{k}\) is approached with a numerical example to validate the method (Chen, G., Liu, Z., & Zhang, J. (2020).). Finally, the 5<sup>th</sup> section which concludes the expertise puts an end to our investigation.

**Preliminaries**

**Basic Notions**

**Definition 1** (Fazlollahtabar, H., & Gholizadeh, H. (2019).) : Let \(X\) be a classical set or a universe. A fuzzy subset \(\widetilde{A}\) of \(X\) is defined by a membership function \(\mu_{\widetilde{A}}\) from \(X\) to \[0,1\] such as :

\(\mu_{\widetilde{A}}\left( x \right) = \left\{ \begin{matrix}
\text{0\ \ \ \ \ \ \ \ if\ \ \ x\ belongs\ not\ to\ A\ \ \ \ \ \ \ \ \ } \\
\text{r}\text{∈}\left\rbrack \text{0,\ 1} \right\lbrack\text{\ \ \ \ if\ \ \ x\ partially\ belongs\ to\ A} \\
\text{\ \ \ 1\ \ \ \ \ \ \ \ if\ \ \ x\ belongs\ to\ A\ \ \ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (1)

The essential characteristics of a fuzzy subset \(\widetilde{A}\) are: the \(\alpha\)-cuts \({\widetilde{A}}_{\alpha}\) (\(\alpha\mathbb{\in R}\)), the support \(supp(\widetilde{A})\), the height \(h(\widetilde{A})\) and the kernel \(ker(\widetilde{A})\). Hanss \[17\] defines them as follows :

\({\widetilde{A}}_{\alpha} = \left\{ x \in X,\mu_{\widetilde{A}}\left( x \right) \geq \alpha\  \right\}\) (2)

\(\text{supp}\left( \widetilde{A} \right) = \left\{ x \in X,\mu_{\widetilde{A}}\left( x \right) > 0 \right\}\) (3)

\(h\left( \widetilde{A} \right) = \max\left\{ \mu_{\widetilde{A}}\left( x \right),\ x \in X \right\}\) (4)

\(\ker\left( \widetilde{A} \right) = \left\{ x \in X,\mu_{\widetilde{A}}\left( x \right) = 1 \right\}\) (5)

The cuts \({\widetilde{A}}_{\alpha}\) are also called *parametric* *representations of* \(\widetilde{A}\).

**Definition 2** (Fazlollahtabar, H., & Gholizadeh, H. (2019).) : A fuzzy subset \(\widetilde{A}\) is said :

  - *normal* if \(h\left( \widetilde{A} \right) = 1\) ;

  - *convex* if \(\forall x,y \in X,\ \forall\lambda \in \left\lbrack 0,\ 1 \right\rbrack\), \(\mu_{\widetilde{A}}\left( \lambda x + (1 - \lambda y) \right) \geq \min\left\{ \mu_{\widetilde{A}}\left( x \right),\ \mu_{\widetilde{A}}\left( y \right) \right\}\) ;

  - *fuzzy number* if \(\widetilde{A}\) is a fuzzy subset of \(\mathbb{R}\) such as :

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  - \(\text{noy}\left( \widetilde{A} \right) \neq \varnothing\),

  - \(\text{supp}\left( \widetilde{A} \right)\) is bounded,

  - \({\widetilde{A}}_{\alpha}\) are bounded intervals of \(\mathbb{R}\).

In this work, the set of fuzzy numbers will be noted by \(\mathbb{F(R)}\).

**Definition 3** : All real number \(x\) such as \(\mu_{\widetilde{A}}\left( x \right) = 1\) is said *modal value, mode* or *mean value* of \(\widetilde{A}\).

A fuzzy number \(\widetilde{A}\) is *strictly positive* if \(\forall x < 0\), \(\mu_{\widetilde{A}}\left( x \right) = 0\); and strictly negative if \(\forall x > 0\), \(\mu_{\widetilde{A}}\left( x \right) = 0\).

**Definition 4** (\[28\]) : A fuzzy number \(\widetilde{A}\) is said *triangular* if there exists three real numbers \(a < b < c\) such as 

\(\mu_{\widetilde{A}}\left( x \right) = \left\{ \begin{matrix}
\ \frac{x - a}{b - a}\ \ \ \ \ \ if\ \ a \leq x \leq b\ \ \ \ \  \\
\frac{c - x}{c - b}\ \ \ \ \ \ \ if\ \ \ \ \ b < x \leq c \\
0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise \\
\end{matrix} \right.\ \) (6)

Notation : \(\widetilde{A} = (a,\ b,\ c)\) or \(\widetilde{A} = (a/b/c)\).

**Definition 5** (Gong, Z., & Hao, Y. (2019).) : A fuzzy number \(\widetilde{A}\) is said *L-R type fuzzy number if* there exists three real numbers \(m,\ \ a > 0,\ \ b > 0\) and two positive functions, continuous and decreasing L and R, from \(\mathbb{R}\) to \(\left\lbrack 0,\ 1 \right\rbrack\) such as :

\(L(0) = R(0) = 1\ \); \(L(1) = 0\) or \(L\left( x \right) > 0\) \(\forall x\mathbb{\in R}\) with \(\lim_{x \rightarrow \infty}{L\left( x \right) = 0}\ \);

\(R(1) = 0\) or \(R\left( x \right) > 0\) \(\forall x\mathbb{\in R}\) with \(\lim_{x \rightarrow \infty}{R\left( x \right) = 0}\ \);

\(\mu_{\widetilde{A}}\left( x \right) = \left\{ \begin{matrix}
\text{\ L}\left( \frac{m - x}{a} \right)\ \ \ \ \ \ if\ \ x \in \left\lbrack m - a,\ m \right\rbrack\text{\ \ \ \ \ } \\
R\left( \frac{x - m}{b} \right)\ \ \ \ \ \ \ if\ \ \ \ \ x \in \left\lbrack m,\ m + b \right\rbrack \\
0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise \\
\end{matrix} \right.\ \) (7)

Notation : \(\widetilde{A} = \left\langle m,\ a,\ b \right\rangle_{\text{LR}}\) or simply \(\widetilde{A} = \left( m,\ a,\ b \right)_{\text{LR}}.\)

Moreover, any triangular fuzzy number \(\widetilde{A} = (a,\ b,\ c)\) is an L-R fuzzy number. It can be written :

\(\widetilde{A} = (a,\ b,\ c)\  = \left\langle b,\ b - a,\ c - b \right\rangle_{\text{LR}}\).

The family of L-R fuzzy numbers is noted \(\mathbb{F}_{\text{LR}}\mathbb{(R)}\). All real number \(r\) (called also in this case *fuzzy singleton*) is by convention an L-R fuzzy number noted \(r\) = \(\widetilde{r}\) = \({(r,0,0)}_{\text{LR}}\).

**Fuzzy arithmetic**

Algebraic operations used on fuzzy numbers so far are based on three kinds of fuzzy arithmetic: *fuzzy arithmetic using Zadeh’s extension principle, fuzzy arithmetic of alpha-cuts and intervals, and L-R fuzzy arithmetic* (Gong, Z., & Hao, Y. (2019).).

**Arithmetic based on extension principle**

According to (Chen, G., Liu, Z., & Zhang, J. (2020).), arithmetic of extension principle makes it possible to extend any classical operation \(\mathbf{*}\) \(\mathbb{R}\) to a fuzzy binary operation \(\circledast\) in \(\mathbb{F(R)}\) defined \(\forall\widetilde{A},\ \widetilde{B} \in \mathbb{F(R)}\), \(\forall z \in \mathbb{R}\) by :

\(\mu_{\widetilde{A}\  \circledast \widetilde{B}}\left( z \right) = \sup\left\{ \min\left\{ \mu_{\widetilde{A}}\left( x \right),\ \mu_{\widetilde{B}}\left( y \right) \right\}\ :\ x,y\mathbb{\in R,\ \ }x*y = z \right\}\) (8)

**Définition 6** (Fazlollahtabar, H., & Gholizadeh, H. (2019).) : Let *E<sub>1</sub> x E<sub>2</sub> x … x E<sub>n</sub>* and *F* be two crisp sets. Suppose that *f* is a map from *E* to *F*. The extension principle is another map \(\widetilde{f}\) from \(\widetilde{\mathcal{P}}(E)\) to \(\widetilde{\mathcal{P}}(F)\) such as \(\forall\widetilde{A} \in \widetilde{\mathcal{P}}(E)\), \(\exists\widetilde{B} \in \widetilde{\mathcal{P}}(F)\) : \(\widetilde{f}\left( \widetilde{A} \right) = \widetilde{B}\ ;\) and \(\forall y \in F\), we have : 

\(\left\{ \begin{matrix}
\mu_{\widetilde{B}}\left( y \right) = \sup_{x \in \frac{E}{f\left( x \right)} = y}\left\{ \min\left\{ \mu_{{\widetilde{A}}_{1}}\left( x_{1} \right),\ldots,\ \mu_{{\widetilde{A}}_{n}}\left( x_{n} \right) \right\} \right\}\text{\ if\ }f^{- 1}(y) \neq \varnothing\  \\
\mu_{\widetilde{B}}\left( y \right) = 0\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (9)

where \(f^{- 1}\) is the reciprocal map of \(f\), and \(\widetilde{\mathcal{P}}(E)\), \(\widetilde{\mathcal{P}}(F)\) are respectively the sets of all fuzzy subsets of *E* and *F.*

**Apha-cuts and intervals arithmetic**

This arithmetic is based on intervals arithmetic whose the basic formula is defined in equation (10) below :

**Definition 7 :** Let \(\lbrack a,b\rbrack\), \(\lbrack c,d\rbrack\) be two real intervals bounded. The arithmetical operation \(*\) is defined between these intervals as follows :

\(\lbrack a,b\rbrack*\lbrack c,d\rbrack\) = \(\lbrack\alpha,\beta\rbrack\) (10)

where \(\lbrack\alpha,\beta\rbrack\) \(=\) \(\{ x*y\ |\ a \leq \ x\  \leq \ b\ ;\ c\  \leq \ y\  \leq \ d\}\), with 0∉\(\lbrack c,d\rbrack\ \)in

the division case.

Concretely, we have the following:

\(\left\lbrack a,b \right\rbrack + \left\lbrack c,\ d \right\rbrack = \left\lbrack a + c,\ b + d \right\rbrack\) (11)

\(\left\lbrack a,b \right\rbrack - \left\lbrack c,\ d \right\rbrack = \left\lbrack a - d,\ b - c \right\rbrack\) (12)

\(\left\lbrack a,b \right\rbrack \times \left\lbrack c,\ d \right\rbrack = \left\lbrack \min\left\{ ac,\ ad,\ bc,\ bd \right\},\max\left\{ ac,\ ad,\ bc,\ bd \right\} \right\rbrack\) (13)

\(\left\lbrack a,b \right\rbrack \div \left\lbrack c,\ d \right\rbrack = \left\lbrack \min\left\{ \frac{a}{c},\ \frac{a}{d},\ \frac{b}{c},\ \frac{b}{d} \right\},\max\left\{ \frac{a}{c},\ \frac{a}{d},\ \frac{b}{c},\ \frac{b}{d} \right\} \right\rbrack\) (14)

**Definition 8** : Let \(\widetilde{A},\ \widetilde{B}\) be two fuzzy numbers with respective α-cuts \({\widetilde{A}}_{\alpha} = \left\lbrack A^{L}\left( \alpha \right),\ A^{U}(\alpha) \right\rbrack\) and \({\widetilde{B}}_{\alpha} = \left\lbrack B^{L}\left( \alpha \right),\ B^{U}(\alpha) \right\rbrack\) (\(0 \leq \alpha \leq 1\)). The fuzzy arithmetical operations on \(\widetilde{A}\) and\(\ \widetilde{B}\) are defined using their cuts as follows

\(\left\lbrack \widetilde{A}\  \oplus \widetilde{B} \right\rbrack_{\alpha} = {\widetilde{A}}_{\alpha} + {\widetilde{B}}_{\alpha} = \left\lbrack A^{L}\left( \alpha \right),\ A^{U}(\alpha) \right\rbrack + \left\lbrack B^{L}\left( \alpha \right),\ B^{U}(\alpha) \right\rbrack\) (15)

\(\left\lbrack \widetilde{A}\  \circleddash \widetilde{B} \right\rbrack_{\alpha} = {\widetilde{A}}_{\alpha} - {\widetilde{B}}_{\alpha} = \left\lbrack A^{L}\left( \alpha \right),\ A^{U}(\alpha) \right\rbrack - \left\lbrack B^{L}\left( \alpha \right),\ B^{U}(\alpha) \right\rbrack\) (16)

\(\left\lbrack \widetilde{A}\  \otimes \widetilde{B} \right\rbrack_{\alpha} = {\widetilde{A}}_{\alpha} \times {\widetilde{B}}_{\alpha} = \left\lbrack A^{L}\left( \alpha \right),\ A^{U}(\alpha) \right\rbrack \times \left\lbrack B^{L}\left( \alpha \right),\ B^{U}(\alpha) \right\rbrack\) (17)

\(\left\lbrack \widetilde{A}\  \oslash \widetilde{B} \right\rbrack_{\alpha} = {\widetilde{A}}_{\alpha} \div {\widetilde{B}}_{\alpha} = \left\lbrack A^{L}\left( \alpha \right),\ A^{U}(\alpha) \right\rbrack \div \left\lbrack B^{L}\left( \alpha \right),\ B^{U}(\alpha) \right\rbrack\) (18)

These last operations on intervals are calculated by the formulas from Equation (11) to Equation (14) above (Sanga, S. S., & Jain, M. (2019).).

**Arithmetic of L-R fuzzy numbers**

Let \(\widetilde{M} = \left\langle m,\ a,\ b \right\rangle_{\text{LR}}\) and \(\widetilde{N} = \left\langle n,\ c,\ d \right\rangle_{\text{LR}}\) be two L-R fuzzy numbers having the same L-R type. *Hanss* defines arithmetic operations on \(\widetilde{M}\) and \(\widetilde{N}\) as follows (Sanga, S. S., & Jain, M. (2019).\()\):

\(\widetilde{M} \oplus \widetilde{N} = \left\langle m + n,\ a + c,\ b + d \right\rangle_{\text{LR}}\) (19)

\(\widetilde{M}\  \circleddash \widetilde{N} = \left\langle m - n,\ a + d,\ b + c \right\rangle_{\text{LR}}\) (20)

\(\widetilde{M}\bigodot\widetilde{N} \approx \left\{ \begin{matrix}
\left\langle mn,\ mc + na - ac,md + nb + bd \right\rangle_{\text{LR}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ si\ }\widetilde{M},\widetilde{N} > 0 \\
\left\langle mn, - md - nb - bd,\  - mc - na + ac \right\rangle_{\text{LR}}\text{\ \ \ \ \ \ \ \ \ \ si\ }\widetilde{M},\widetilde{N} < 0 \\
\left\langle mn,\  - md + na + ad,\  - mc + nb - bc \right\rangle_{\text{LR}}\text{\ \ si\ }\widetilde{M} < 0,\ \widetilde{N} > 0 \\
\end{matrix} \right.\ \) (21)

\(\frac{1}{\widetilde{N}} \approx \left\langle \frac{1}{n},\ \frac{d}{n(n + d)},\frac{c}{n(n - c)} \right\rangle_{\text{LR}}\), \(\widetilde{N} > 0\) (22)

\(\widetilde{M} \oslash \widetilde{N} \approx \left\langle \frac{m}{n},\ \frac{\text{md}}{n(n + d)} + \frac{a}{n} - \frac{\text{ad}}{n(n + d)},\frac{\text{mc}}{n(n - c)} + \frac{b}{n} + \frac{\text{bc}}{n(n - c)} \right\rangle_{\text{LR}}\), \(\widetilde{M},\widetilde{N} > 0\) (23)

Here, the product and the quotient are obtained through secant approximations mentioned by *Hanss in \[17\] and developed by* *Mukeba in (*Panta, A. P., Ghimire, R. P., Panthi, D., & Pant, S. R. (2021).).

**Fuzzy transforms**

**Fuzzy function**

There are several types of fuzzy functions: fuzzy functions by constraint, fuzzy functions by propagation of a fuzzy variable and fuzzy functions themselves (Ritha, W., & Rajeswari, N. (2021).*).*

**Definition 9** (Panta, A. P., Ghimire, R. P., Panthi, D., & Pant, S. R. (2021).) : Let \(X\), \(Y\) be two universes and \(\widetilde{\mathcal{P}}(Y)\) the set of all subsets of \(Y\). The map \(\widetilde{f}\ :\ X\  \longrightarrow \ \widetilde{\mathcal{P}}\left( Y \right),\ \ x \mapsto \widetilde{B} = \widetilde{f}\ (x)\) is a fuzzy function if

\(\mu_{\widetilde{B}}\left( y \right) = \mu_{\widetilde{R}}(x,\ y)\), ( \(\forall\ (x,\ y) \in X\)×\(Y\) ) (24)

where \(\widetilde{R}\) is a fuzzy relation between memberships of \(X\)*×*\(Y\).

When \(X = \lbrack a,b\rbrack\) and \(Y\mathbb{= R}\), then \(\widetilde{\mathcal{P}}(Y)\ \mathbb{= F(R)}\) and \(\widetilde{f}\) is a fuzzy real valued function. This is the case for expressions : \(t\  \rightarrow \ \widetilde{A}.t + \widetilde{B}\), \(t\  \rightarrow e^{\widetilde{a}t}\).

**Definition 9** (Wang, F. F. (2022).) : Let *f* be a classical map of variable \(t\). A fuzzy function, extension of *f* is a map noted \(\widetilde{f}\), from \(\mathbb{R}\) to \(\mathbb{F(R)}\), such as \(\widetilde{f}\left( t \right) = \widetilde{Z}\) has the following \(\alpha\)-cuts :

\({\widetilde{Z}}_{\alpha} = \left\lbrack Z^{L}\left( \alpha \right),\ Z^{U}(\alpha) \right\rbrack\) (25)

**Definition 10** (Wang, F. F. (2022).)\(\ \) : Let \(f\left( x_{1},\ldots,x_{n} \right)\) be real valued function from \(\mathbb{R}^{n}\) to \(\mathbb{R}\) and let \({\widetilde{A}}_{1},\ldots,{\widetilde{A}}_{n}\) be \(n\) subsets of \(\mathbb{R}\). The extension principle allows to induce from \(f\left( x_{1},\ldots,x_{n} \right)\) a fuzzy function \(\widetilde{f}\ :\ \mathbb{F}^{n}\mathbb{(R)\  \longrightarrow \ F}\left( \mathbb{R} \right)\) such as \(\widetilde{f}\ \left( {\widetilde{A}}_{1},\ldots,{\widetilde{A}}_{n} \right)\) is a fuzzy subset \(\widetilde{B}\) of \(\mathbb{R}\), whose membership function is defined for all \(y\) of \(\mathbb{R}\) by :

\(\mu_{\widetilde{B}}\left( y \right) = \left\{ \begin{matrix}
\sup_{\left( x_{1},\ldots,x_{n} \right)}\left\{ \min\left\{ \mu_{{\widetilde{A}}_{1}}\left( x_{1} \right),\ldots,\mu_{{\widetilde{A}}_{n}}\left( x_{n} \right) \right\} \right\}\text{\ \ if\ }f^{- 1}\left( y \right) \neq \varnothing \\
0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{if}\ f^{- 1}\left( y \right) = \varnothing \\
\end{matrix} \right.\ \) (26)

Its parametric representation or α-cut is given for all \(\alpha \in \left\lbrack 0,\ 1 \right\rbrack\) by :

> \({\widetilde{B}}_{\alpha} = \left( \widetilde{f}\ \left( {\widetilde{A}}_{1},\ldots,{\widetilde{A}}_{n} \right) \right)_{\alpha} = f\ \left( {{\widetilde{A}}_{1}}_{\alpha},\ldots,{{\widetilde{A}}_{n}}_{\alpha} \right)\) (27)

This definition establishes the compatibility between the approach of the Zadeh’s extension principle and the arithmetic of alpha-cuts and intervals (Zhang, Q., Sun, H., Gao, X., Wang, X., & Feng, Z. (2022).*)*.

For example, the expressions \(\widetilde{z} = \frac{2\widetilde{x} + 10}{3\widetilde{x} + 4}\) and \(\widetilde{z} = \frac{\widetilde{A}\widetilde{x} + \widetilde{B}}{\widetilde{C}\widetilde{x} + \widetilde{D}}\) are fuzzy functions, respective extensions of classical functions \(h(x) = \frac{2x + 10}{3x + 4}\) and \(g\left( x_{1},x_{2},x_{3},x_{4},x\  \right) = \frac{x_{1}x + x_{2}}{x_{3}x + x_{4}}\) .

**Fuzzy Laplace Transforms**

**Definition 11** (Gong, Z., & Hao, Y. (2019).): Let \(\widetilde{f}\left( t \right)\) be a fuzzy function and \(s\) a real parameter. The fuzzy Laplace transform of \(\widetilde{f}\left( t \right)\) is a fuzzy function \(\widehat{F}\) defined by :

\(\widehat{F}\left( s \right)\mathcal{= L}\left\lbrack \widetilde{f}\left( t \right) \right\rbrack = \int_{0}^{\infty}{e^{- st}\widetilde{f}\left( t \right)\text{dt}} = \lim_{\tau \rightarrow \infty}{\int_{0}^{\tau}{e^{- st}\widetilde{f}\left( t \right)\text{dt}}}\), (28)

(Provided that this limit exists).

Concerning its parametric representations (α-cuts), the fuzzy Laplace transform of \(\widetilde{f}\left( t \right)\) is written :

\(\widehat{F}\left( s,\alpha \right)\mathcal{= L}\left\lbrack \widetilde{f}\left( t,\alpha \right) \right\rbrack = \left\lbrack \mathcal{L}\left\lbrack f^{L}(t,\alpha) \right\rbrack\mathcal{,\ L}\left\lbrack f^{U}(t,\alpha) \right\rbrack \right\rbrack\), \(0 \leq \alpha \leq 1\) (29)

where \(\left\{ \begin{matrix}
\mathcal{L}\left\lbrack f^{L}(t,\alpha) \right\rbrack = \int_{0}^{\infty}{e^{- st}f^{L}(t,\alpha)dt} \\
\mathcal{L}\left\lbrack f^{U}(t,\alpha) \right\rbrack = \int_{0}^{\infty}{e^{- st}f^{U}(t,\alpha)dt} \\
\end{matrix} \right.\ \) (30)

**Example :** Let \(\widetilde{a} = (3,\ 4,\ 5)\) be a triangular fuzzy number and \(\widetilde{f}\left( t \right) = e^{\widetilde{a}t}\) a fuzzy function. Following the definition,

\(\widehat{F}\left( s \right) = \int_{0}^{\infty}{e^{- st}e^{\widetilde{a}t}\text{dt}} = \frac{1}{s - \widetilde{a}}\).

As,

\({\widetilde{a}}_{\alpha} = \ \left\lbrack 3 + \alpha,\ 5 - \alpha \right\rbrack\) and \(\widetilde{f}\left( t,\ \alpha \right) = \left\lbrack e^{\left( 3 + \alpha \right)t},\ e^{\left( 5 - \alpha \right)t} \right\rbrack\).

Then, the \(\alpha\)-cuts of the fuzzy Laplace transform of \(\widetilde{f}\left( t \right) = e^{\widetilde{a}t}\) equals :

\(\widehat{F}\left( s,\ \alpha \right) = \left\lbrack \int_{0}^{\infty}{e^{- st}e^{\left( 3 + \alpha \right)t}\text{dt}},\ \int_{0}^{\infty}{e^{- st}e^{\left( 5 - \alpha \right)t}\text{dt}} \right\rbrack = \left\lbrack \frac{1}{s - \alpha - 3},\ \frac{1}{s + \alpha - 5} \right\rbrack\).

**Properties :** Like classical transforms, fuzzy transforms also possess properties such as linearity, translation theorems, derivation theorems, whose eloquent proofs can be consulted in (Gong, Z., & Hao, Y. (2019).).

**Fuzzy Queuing** \(\mathbb{F}\) \(\mathbf{M}\)**/**\(\mathbb{\text{\ F}}\) \(\mathbf{G}\)**/1**

**Crisp queue M/G/1 and Laplace transform**

The Laplace transform plays a capital role in the calculation of performance measures of a classical non-markovian queue M/G/1, through the Laplace transform of service general law G, defined by :

\(B^{*}\left( s \right) = \ \int_{0}^{\infty}{e^{- st}b\left( t \right)\text{dt}}\), (31)

where \(b(t)\) denotes the probability density of this law.

On the one hand, this transform \(B^{*}\left( s \right)\) intervenes in the definition of the generating function of the stationary probabilities of the system given (Gong, Z., & Hao, Y. (2019).) by :

\(G\left( z \right) = (1 - \rho)\frac{(1 - z)B^{*}\left( \lambda - \lambda z \right)}{B^{*}\left( \lambda - \lambda z \right) - z}\) (32)

where \(\lambda\) and \(\rho = \lambda m_{1}\) are respectively the mean arrival rate of the customers at the system and the trafic rate in the system (\(m_{1}\) is the moment of order 1 of the law G).

On the orther hand, \(B^{*}\left( s \right)\) intervenes in the analysis of the variable \(W\), which is the distribution of customer sojourn time (\(W_{q}\) : waiting time) in the queue, through its Laplace transform defined by :

\(W^{*}\left( s \right) = (1 - \rho)\frac{sB^{*}\left( s \right)}{\text{λB}^{*}\left( s \right) - \lambda + s}\) and \({W_{q}}^{*}\left( s \right) = (1 - \rho)\frac{s}{\text{λB}^{*}\left( s \right) - \lambda + s}\) (33)

This allows to deduce the mean sojourn time and the mean waiting time of a customer, as the moment of order 1 of \(W\):

\(\tau_{s} = ( - 1)\frac{dW^{*}\left( s \right)}{\text{ds}}(0)\) and \(\tau_{q} = ( - 1)\frac{d{W_{q}}^{*}\left( s \right)}{\text{ds}}(0)\) (34)

The other measures such as the mean customers number in the system or in the queue are deduced applying the Little law :

\(N_{s} = \lambda.\tau_{s}\) and \(N_{q} = \lambda.\tau_{q}\) (35)

**Fuzzy queue** \(\mathbb{F}\) \(\mathbf{M}\)**/**\(\mathbb{\text{\ F}}\) \(\mathbf{G}\)**/1 and fuzzy Laplace transforms**

A queuing system is said fuzzy when its descriptors parameters are vague and imprecise. In this case, the probability density of the general service law is a fuzzy function of real variable \(\widetilde{b}\left( t \right)\). Hence, the opportunity to analyze it using fuzzy Laplace transforms, which is the subject of this article (Gong, Z., & Hao, Y. (2019).).

Being given the fuzzy environment where system parameters are vague and imprecise, the probability density of the service law \(\widetilde{b}\left( t \right)\) admits a following fuzzy Laplace transform :

\({\widetilde{B}}^{*}\left( s \right) = \int_{0}^{\infty}{e^{- st}\widetilde{b}\left( t \right)\text{dt}}\) (36)

On the one hand, by Zadeh’s extension principle, the formulas in Eq. (33) above become :

\({\widetilde{W}}^{*}\left( s \right) = (1 - \widetilde{\rho})\frac{s{\widetilde{B}}^{*}(s)}{s - \widetilde{\lambda} + \widetilde{\lambda}{\widetilde{B}}^{*}(s)}\) and \({{\widetilde{W}}_{q}}^{*}\left( s \right) = (1 - \widetilde{\rho})\frac{s}{s - \widetilde{\lambda} + \widetilde{\lambda}{\widetilde{B}}^{*}(s)}\) (37)

These are fuzzy extensions of Laplace transforms of variables *W* and *W<sub>q</sub>*, which designate customer sojourn and customer waiting time in the classical queuing system M/G/1.

On the other hand, although these expressions in Eq. (37) are fuzzy, they are defined by the real variable *s*. Hence the opportunity to exploit the properties of the moments of the variables \(\widetilde{W}\) and \({\widetilde{W}}_{q}\) to define the sojourn and the waiting time of a customer in the system by the formulas :

\({\widetilde{\tau}}_{s} = ( - 1)\frac{d{\widetilde{W}}^{*}\left( s \right)}{\text{ds}}(0)\) and \({\widetilde{\tau}}_{q} = ( - 1)\frac{d{{\widetilde{W}}_{q}}^{*}\left( s \right)}{\text{ds}}(0)\) (38)

It will therefore suffice to apply Little’s law to obtain the other performance measures of the system \(\mathbb{F}\) M/\(\mathbb{\text{\ F}}\) G/1, in particular :

\({\widetilde{N}}_{s} = \widetilde{\lambda}\bigodot{\widetilde{\tau}}_{s}\) and \({\widetilde{N}}_{q} = \widetilde{\lambda}\bigodot{\widetilde{\tau}}_{q}\) (39)

**Concrete case** \(\mathbb{F}\) **M/**\(\mathbb{F}\mathbf{E}_{\mathbf{2}}\)**/1**

**Problem position**

As announced above, this case had just been successfully analyzed by *Merlyn Margaret* and her friends in (Kannadasan, G., & Sathiyamoorthi, N. (2018).). Having already used the PNLP approach, these authors have shown that the problem of analyzing a performance measure of a fuzzy queue \(\mathbb{F}\)M/\(\mathbb{F}\)G/1 is reduced to the resolution of a pair of parametric non-linear programs. After solving these PNLPs, they used the mean degree integral scheme defined by the relation below to defuzzify the obtained fuzzy features :

\(\Phi\left( \widetilde{z} \right) = \frac{\int_{0}^{1}{\frac{\alpha}{2}\left( Z_{\alpha}^{L} + Z_{\alpha}^{U} \right)\text{dα}}}{\int_{0}^{1}\text{αdα}} = \int_{0}^{1}{\alpha\left( Z_{\alpha}^{L} + Z_{\alpha}^{U} \right)\text{dα}}\) (40)

For us, in this article, we study this fuzzy queue \(\mathbb{F}\)M/\(\mathbb{F}\)G/1 by fuzzy Laplace transforms as described above.

As for results obtained, we will defuzzify them by the integral centroid approach named *COA Method* or *Center of Area* defined by (Fazlollahtabar, H., & Gholizadeh, H. (2019).) :

\(\left\lbrack \int_{supp(\widetilde{Z})}^{}{\text{x.}\mu_{\widetilde{Z}}\left( x \right)\text{dx}} \right\rbrack \div \left\lbrack \int_{supp(\widetilde{Z})}^{}{\mu_{\widetilde{Z}}\left( x \right)\text{dx}} \right\rbrack\) (41)

**Resolution**

For crisp model M/\(E_{2}/1\), the density of Erlang<sub>2</sub> and its Laplace transform are given by :

\(b(t) = \left( 2\mu \right)^{2}te^{- 2\mu t}\) and \(B^{*}\left( s \right) = \left( \frac{2\mu}{s + 2\mu} \right)^{2}\) (pour \(k = 2\)) (42)

This permits to write Eq. (33) as follows :

\(W^{*}\left( s \right) = \frac{4\mu(\mu - \lambda)}{s^{2} + \left( 4\mu - \lambda \right)s + 4\mu(\mu - \lambda)}\) and \({W_{q}}^{*}\left( s \right) = \frac{\mu - \lambda}{\mu}\frac{{(s + 2\mu)}^{2}}{s^{2} + \left( 4\mu - \lambda \right)s + 4\mu(\mu - \lambda)}\) (43)

In the classical model \(\mathbb{F}\)M/\({\mathbb{F}E}_{k}\)/1, the density of the service is a fuzzy extension function of the function \(b(t)\) given by :

\(\widetilde{b}(t) = \frac{{(k\widetilde{\mu})}^{k}}{\left( k - 1 \right)!}t^{k - 1}e^{- k\widetilde{\mu}t}\) (44)

By integration, we obtain :

\({\widetilde{B}}^{*}\left( s \right) = \int_{0}^{\infty}{e^{- st}\widetilde{b}\left( t \right)\text{dt}} = \left( \frac{k\widetilde{\mu}}{s + k\widetilde{\mu}} \right)^{k}\) (45)

or

\({\widetilde{B}}^{*}\left( s \right) = \left( \frac{2\widetilde{\mu}}{s + 2\widetilde{\mu}} \right)^{2}\) for \(k = 2\) (46)

Therefore,

\({\widetilde{W}}^{*}\left( s \right) = \frac{4\widetilde{\mu}(\widetilde{\mu} - \widetilde{\lambda})}{s^{2} + \left( 4\widetilde{\mu} - \widetilde{\lambda} \right)s + 4\widetilde{\mu}(\widetilde{\mu} - \widetilde{\lambda})}\) and \({W_{q}}^{*}\left( s \right) = \frac{\widetilde{\mu} - \widetilde{\lambda}}{\widetilde{\mu}}\frac{{(s + 2\widetilde{\mu})}^{2}}{s^{2} + \left( 4\widetilde{\mu} - \widetilde{\lambda} \right)s + 4\widetilde{\mu}(\widetilde{\mu} - \widetilde{\lambda})}\) (47)

And according to Eq. (38) above, we have : 

\({\widetilde{\tau}}_{s} = \left( - 1 \right)\frac{d{\widetilde{W}}^{*}\left( s \right)}{\text{ds}}\left( 0 \right) = \frac{3\widetilde{\lambda}}{4\widetilde{\mu}(\widetilde{\mu} - \widetilde{\lambda})} + \frac{1}{\widetilde{\mu}}\) and \({\widetilde{\tau}}_{q} = \left( - 1 \right)\frac{d{{\widetilde{W}}_{q}}^{*}\left( s \right)}{\text{ds}}\left( 0 \right) = \frac{3\widetilde{\lambda}}{4\widetilde{\mu}(\widetilde{\mu} - \widetilde{\lambda})}\) (48)

**Numerical Example**

**Consider a queuing system in which customers arrive according to a Poisson process, with an imprecise mean rate of about 1 customer per minute. Suppose also that the service time is distributed according to an Erlang<sub>2</sub> process with an imprecise mean of approximately 1/3. Let us determine the mean sojourn time in the system and the mean waiting time in the queue respectively.**

**Resolution process**

Saying the service time average of Erlang2 is about 1/3 means the service service rate is about 3. Since these two descriptor parameters are vague and imprecise, we will proceed as follows :

> 1\. Represent these parameters by two triangular fuzzy numbers of modes 1 and 3 respectively. For example, \(\widetilde{\lambda} = \left( \frac{1}{2},\ 1,\ \frac{3}{2} \right)\) and \(\widetilde{\mu} = \left( 2,\ 3,\ 4 \right)\), then write them in L-R form ;
> 
> 2\. Use formulas the formulas of Eq. (48), in accordance with the approach of fuzzy Laplace Transform;
> 
> 3\. Use L-R fuzzy arithmetic (cf. relations from (19) to (23)) to obtain the results;
> 
> 4\. Defuzzify these results through Eq. (41) as announced at the subsection 4.3.1 above.

**Results**

**1. The L-R type of** \(\widetilde{\lambda} = \left( \frac{1}{2},\ 1,\ \frac{3}{2} \right)\) and \(\widetilde{\mu} = \left( 2,\ 3,\ 4 \right)\) are respectively:

\(\widetilde{\lambda} = \left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}}\) and \(\widetilde{\mu} = \left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}}\)

2\. According to Eq. (48),

\({\widetilde{\tau}}_{s} = \frac{3\widetilde{\lambda}}{4\widetilde{\mu}(\widetilde{\mu} - \widetilde{\lambda})} + \frac{1}{\widetilde{\mu}} = \frac{3\left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}}}{4\left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}}\left( \left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}} - \left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}} \right)} + \frac{1}{\left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}}}\) (49)

\({\widetilde{\tau}}_{q} = \frac{3\widetilde{\lambda}}{4\widetilde{\mu}(\widetilde{\mu} - \widetilde{\lambda})} = \frac{3\left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}}}{4\left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}}\left( \left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}} - \left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}} \right)}\) (50)

However,

\(3\left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}} = \left\langle 3,\ \frac{3}{2},\ \frac{3}{2} \right\rangle_{\text{LR}}\) ; \(\frac{1}{\left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}}} \simeq \left\langle \frac{1}{3},\ \frac{1}{12},\frac{1}{6}\  \right\rangle_{\text{LR}}\) ;

\(4\left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}} = \left\langle 12,\ 4,\ 4 \right\rangle_{\text{LR}}\)  and \(\left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}} - \left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}} = \left\langle 2,\ \frac{3}{2},\ \frac{3}{2} \right\rangle_{\text{LR}}\).

> So, from the secant approximation formula of relations from Eq. (21) to Eq. (23), we get :

\[4\left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}}\left( \left\langle 3,\ 1,\ 1 \right\rangle_{\text{LR}} - \left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}} \right) = \left\langle 12,\ 4,\ 4 \right\rangle_{\text{LR}} \otimes \left\langle 2,\ \frac{3}{2},\ \frac{3}{2} \right\rangle_{\text{LR}}\]

\[\simeq \left\langle 24,\ 20,\ 32 \right\rangle_{\text{LR}}\]

> **From where,**

\({\widetilde{\tau}}_{s} \simeq \left\langle \frac{1}{8},\ \frac{9}{56},\ 1 \right\rangle_{\text{LR}} + \left\langle \frac{1}{3},\ \frac{1}{12},\frac{1}{6}\  \right\rangle_{\text{LR}} \simeq \left\langle \frac{11}{24},\ \frac{41}{168},\ \frac{7}{6} \right\rangle_{\text{LR}}\) (51)

\({\widetilde{\tau}}_{q} = \frac{\left\langle 3,\ \frac{3}{2},\ \frac{3}{2} \right\rangle_{\text{LR}}\ }{\left\langle 24,\ 20,\ 32 \right\rangle_{\text{LR}}} \simeq \left\langle \frac{1}{8},\ \frac{9}{56},\ 1 \right\rangle_{\text{LR}}\) (52)

> **These approximative results are none other than triangular fuzzy numbers**

\({\widetilde{\tau}}_{s} \simeq \left( \frac{3}{14},\ \frac{11}{24},\ \frac{13}{8} \right)\) \(\ \) and \({\widetilde{\tau}}_{q} \simeq \left( \frac{- 1}{28},\ \frac{1}{8},\ \frac{9}{8} \right)\) (53)

**with modal values** \(\frac{11}{24}\) and \(\frac{1}{8}\) (*unit of time*) respectively.

3\. The use of Little’s formula and that of the secant approximation of the product two L-R fuzzy numbers allows to us to obtain the other performance measures ; notably the customers number in the system and in the queue, noted \({\widetilde{N}}_{s} = \ \widetilde{\lambda}\bigodot{\widetilde{\tau}}_{s}\) and \({\widetilde{N}}_{q} = \ \widetilde{\lambda}\bigodot{\widetilde{\tau}}_{q}\ \):

\({\widetilde{N}}_{s} = \ \widetilde{\lambda}\bigodot{\widetilde{\tau}}_{s} = \left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}}\bigodot\left\langle \frac{11}{24},\ \frac{41}{168},\ \frac{7}{6} \right\rangle_{\text{LR}}\) \(\simeq \left\langle \frac{11}{24},\ \frac{59}{168},\ \frac{95}{48} \right\rangle_{\text{LR}}\) (54)

\({\widetilde{N}}_{q} = \ \widetilde{\lambda}\bigodot{\widetilde{\tau}}_{q} = \left\langle 1,\ \frac{1}{2},\ \frac{1}{2} \right\rangle_{\text{LR}}\bigodot\left\langle \frac{1}{8},\ \frac{9}{56},\ 1 \right\rangle_{\text{LR}}\) \(\simeq \left\langle \frac{1}{8},\ \frac{1}{7},\ \frac{25}{16} \right\rangle_{\text{LR}}\) (55)

In triangular form, we have :

\({\widetilde{N}}_{s} \approx \left( \frac{3}{28},\ \frac{11}{24},\ \frac{39}{16} \right)\) and \({\widetilde{N}}_{q} \approx \left( \frac{- 1}{56},\ \frac{1}{8},\ \frac{27}{16} \right)\) (56)

> where modal values \(\ \frac{11}{24}\) and \(\ \frac{1}{8}\) indicate the customers numbers per unit of time respectively.

4\. Finally, the defuzzification of results obtained through Eq. (41) requires us to define first their membership functions by Eq (6), and then, to carry out the various calculations required :

1° For \({\widetilde{\tau}}_{s} \simeq \left( \frac{3}{14},\ \frac{11}{24},\ \frac{13}{8} \right)\), the membership function is given by :

\(\mu_{{\widetilde{\tau}}_{s}}\left( x \right) = \left\{ \begin{matrix}
\frac{168x - 36}{41}\text{\ \ \ \ if\ }\frac{3}{14} \leq x \leq \ \frac{11}{24}\  \\
\frac{39 - 24x}{28}\text{\ \ \ \ \ if\ }\frac{11}{24} \leq x \leq \ \frac{13}{8} \\
0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ot}h\text{erwise}\text{\ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (57)

\({\tau_{s}}^{*} = \left\lbrack \int_{\frac{3}{14}}^{\frac{11}{24}}{\text{x.}\frac{168x - 36}{41}\text{dx}} + \int_{\frac{11}{24}}^{\frac{13}{8}}{\text{x.}\frac{39 - 24x}{28}\text{dx}} \right\rbrack \div \left\lbrack \int_{\frac{3}{14}}^{\frac{11}{24}}{\frac{168x - 36}{41}\text{dx}} + \int_{\frac{11}{24}}^{\frac{13}{8}}{\frac{39 - 24x}{28}\text{dx}} \right\rbrack\)

> \(= 0.766\ unit\ of\ time\ \);

2° For \({\widetilde{\tau}}_{q} \simeq \left( \frac{- 1}{28},\ \frac{1}{8},\ \frac{9}{8} \right)\), we have :

\(\mu_{{\widetilde{\tau}}_{q}}\left( x \right) = \left\{ \begin{matrix}
\frac{56x + 2}{9}\text{\ \ \ if\ }\frac{- 1}{28} \leq x \leq \ \frac{1}{8} \\
\frac{9 - 8x}{8}\text{\ \ \ \ \ \ if\ }\ \frac{1}{8} \leq x \leq \ \frac{9}{8} \\
0\ \ \ \ \ \ \ \ \ \ \text{ot}h\text{erwise}\text{\ \ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (58)

\({\tau_{q}}^{*} = \left\lbrack \int_{\frac{- 1}{28}}^{\frac{1}{8}}{\text{x.}\frac{56x + 2}{9}\text{dx}} + \int_{\frac{1}{8}}^{\frac{9}{8}}{\text{x.}\frac{9 - 8x}{8}\text{dx}} \right\rbrack \div \left\lbrack \int_{\frac{- 1}{28}}^{\frac{1}{8}}{\frac{56x + 2}{9}\text{dx}} + \int_{\frac{1}{8}}^{\frac{9}{8}}{\frac{9 - 8x}{8}\text{dx}} \right\rbrack\)

> \(= 0.405\ \ unit\ of\ time\ \);

3° For \({\widetilde{N}}_{s} \approx \left( \frac{3}{28},\ \frac{11}{24},\ \frac{39}{16} \right)\), the membership function is given by :

\(\mu_{{\widetilde{N}}_{s}}\left( x \right) = \left\{ \begin{matrix}
\frac{168x - 18}{59}\text{\ \ \ if\ \ }\frac{3}{28} \leq x \leq \ \frac{11}{24} \\
\frac{117 - 48x}{95}\text{\ \ \ if\ \ \ }\frac{11}{24} \leq x \leq \ \frac{39}{16} \\
0\ \ \ \ \ \ \ \ \ \ \text{ot}h\text{erwise}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ } \\
\end{matrix} \right.\ \) (59)

\({N_{s}}^{*} = \left\lbrack \int_{\frac{3}{28}}^{\frac{11}{24}}{\text{x.}\frac{168x - 18}{59}\text{dx}} + \int_{\frac{11}{24}}^{\frac{39}{16}}{\text{x.}\frac{117 - 48x}{95}\text{dx}} \right\rbrack \div \left\lbrack \int_{\frac{3}{28}}^{\frac{11}{24}}{\frac{168x - 18}{59}\text{dx}} + \int_{\frac{11}{24}}^{\frac{39}{16}}{\frac{117 - 48x}{95}\text{dx}} \right\rbrack = 1.000\ \ \text{customers\ per\ unit\ of\ time}\ \);

> or \(60\ customers\ per\ hour\ if\) the unit of time is the \(\text{minute}\).

4° Pour \({\widetilde{N}}_{q} \approx \left( \frac{- 1}{56},\ \frac{1}{8},\ \frac{27}{16} \right)\), we have :

\(\mu_{{\widetilde{N}}_{q}}\left( x \right) = \left\{ \begin{matrix}
\frac{56x + 1}{8}\text{\ \ \ if\ }\frac{- 1}{56} \leq x \leq \ \frac{1}{8}\  \\
\frac{27 - 16x}{25}\text{\ \ \ if\ \ }\frac{1}{8} \leq x \leq \ \frac{27}{16}\  \\
0\ \ \ \ \ \ \ \ \ \text{ot}h\text{erwise}\text{\ \ \ \ \ } \\
\end{matrix} \right.\ \) (60)

\[{N_{q}}^{*} = \left\lbrack \int_{\frac{- 1}{56}}^{\frac{1}{8}}{\text{x.}\frac{56x + 1}{8}\text{dx}} + \int_{\frac{1}{8}}^{\frac{27}{16}}{\text{x.}\frac{27 - 16x}{25}\text{dx}} \right\rbrack \div \left\lbrack \int_{\frac{- 1}{56}}^{\frac{1}{8}}{\frac{56x + 1}{8}\text{dx}} + \int_{\frac{1}{8}}^{\frac{27}{16}}{\frac{27 - 16x}{25}\text{dx}} \right\rbrack\]

> \(= 0.598\ \ \text{customer\ per\ unit\ of\ time}\ \),
> 
> or \(36\ \ \text{customers\ per\ hour\ if}\) the unit of time is the \(\text{minute}\).

**Discussion**

We find that all the modal values of the fuzzy results correspond exactly to the performance measures of the classical model M/E<sub>2</sub>/1 (average waiting time in the queue and in the system, average customers number in the queue and in the system) which can be obtained from the Pollaczeck-Khintchine formula (Babu, P. S., Kumar, K. S., & Chandan, K. (2022).).

As for the defuzified values, they are all slightly higher than these modes which are perfomance measures of the classic model. Shouldn’t we see the effects of a fuzzy environment on the performance measures of a queing system ?

**Conclusion**

**In this work, we sought to know what happens to the performance measures of a non-markovian queue in a fuzzy environment and how to apply fuzzy transforms in the evaluation of these measures.**

**To achieve this, we used both the L-R fuzzy arithmetic and especially the Zadeh’s extension principle to obtain the fuzzy transform of the customers sojourn time distribution in the system.**

**The numerical exercise treated, revealed that, when descriptor parameters of a system are vague and uncertain, the performance measures are not the same as those of the corresponding classical model, except for the modal values of the fuzzy results.**

**  
**

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