Analysis of the Performance Measures of a Non-Markovian Fuzzy Queue via Fuzzy Laplace Transforms Method

Laplace transforms play an essential role in the analysis of classical non-Markovian queueing systems. The problem addressed here is whether the Laplace transform approach is still valid for determining the characteristics of such a system in a fuzzy environment. In this paper, fuzzy Laplace transforms are applied to analyze the performance measures of a non-Markovian fuzzy queueing system FM/ FG/1. Starting from the fuzzy Laplace transform of the service time distribution, we define the fuzzy Laplace transform of the distribution of the dwell time of a customer in the system. By applying the properties of the moments of this distribution, the derivative of this fuzzy transform makes it possible to obtain a fuzzy expression of the average duration of stay of a customer in the system. This expression is the fuzzy formula of the same performance measure that can be obtained from its classical formula by the Zadeh extension principle. The fuzzy queue FM/ FE_k /1 is particularly treated in this text as a concrete case through its service time distribution. In addition to the fuzzy arithmetic of L-R type fuzzy numbers, based on the secant approximation, the properties of the moments of a random variable and Little's formula are used to compute the different performance measures of the system. A numerical example was successfully processed to validate this approach. The results obtained show that the modal values of the performance measures of a non-Markovian fuzzy queueing system are equal to the performance measures of the corresponding classical model computable by the Pollaczeck-Khintchine method. The fuzzy Laplace transforms approach is therefore applicable in the analysis of a fuzzy FM/FG/1 queueing system in the same way as the classical M/G/1 model.


INTRODUCTION
Today, no one is unaware of the crucial role played by the modelling of queueing systems in the implementation of computer systems, telecommunication systems, etc. A queuing system is often described as a circuit in which a client who, on arrival, finds the server(s) busy, decides: • Or wait for a random period until it is served to leave the circuit; • Or to go away and come back to request the service after a random time; • Or to leave once and for all.
When the descriptor parameters of the system are vague and imprecise, it is called a fuzzy queueing system, usually represented by the letter This is usually represented by the letter "Fuzzy".
Much work has already been done to analyse these types of queues based on fuzzy set theory. This is particularly the case for Markovian fuzzy queuing systems M/ M/c and the product-form fuzzy queueing network. The main results of this work can be found in (Fatoumata, Y., Adnane, A., & Ataoua, Z. (2021).). As for non-Markovian fuzzy models such as M/ G/1, G/ M/1 ..., the literature is not yet sufficiently extensive to our humble knowledge. Among these rare works, we quote for example of which would be the most recent to our humble knowledge (Al-Kridi, K., Anan, M. T., & Zeina, M. B. (2018).).
Most of these researchers have based their analyses on the method of mathematical optimization programs PNLP (Parametric Non-Linear Programming), combining both the Zadeh Extension Principle and the Arithmetic of -and intervals (Patel, K. R., & Desai, N. B. (2017).).
Others have applied and shown that the L-R method is the fastest and most flexible method to analyze the FM/FEk/1 model (Çitil, H. G. (2019).). In this article, we asked the question of how to calculate the performance measures of a fuzzy waiting system M/ G/1 using the fuzzy Laplace transform approach in steady state.
Our hypothesis is that the Laplace transform method would remain valid for analyzing both classical and fuzzy non-Markovian queue performance measures. Like the classical model, our methodology consists in calculating these measures from the fuzzy Laplace transforms of the distribution of the residence (waiting) times of a customer in the system (the queue) (Gong, Z., & Hao, Y. (2019).). This text is organized as follows: Section 2 covers the preliminaries, Section 3 deals with fuzzy Laplace transforms starting with the notion of fuzzy functions. Section 4 is devoted to the fuzzy model M/ G/1. The concrete case where = is discussed with a numerical example to validate the method. Finally, section 5 is reserved for the conclusion that ends the text (Chen, G., Liu, Z., & Zhang, J. (2020).).

Fuzzy arithmetic
The algebraic operations on fuzzy numbers, used in this text, are based on three main arithmetics: the Zadeh extension principle, the arithmetic of -cuts and intervals and the arithmetic of fuzzy numbers of the same type L-R (Gong, Z., & Hao, Y. (2019).).

The Zadeh extension principle
The arithmetic of the extension principle allows any classical binary operation to be extended * in to a fuzzy binary operation ⊛ in ( where −1 is the reciprocal of and ̃( ), ̃( ) are respectively the sets of all fuzzy subsets of and .

Arithmetic of -cuts and intervals
The arithmetic of -The arithmetic of the cuts is based on the interval arithmetic as defined below:

FUZZY WAITING SYSTEM / /1 Classical M/G/1 system and Laplace Transforms
The Laplace transform plays a crucial role in the calculation of performance measures of a classical non-Markovian system M/G/1; this is done through the Laplace transform of the general service law G, defined by : where and = 1 are respectively the average arrival rate of customers in the system and the traffic rate in the system ( 1 being the first order moment of the G law).
On the other hand, * ( ) intervenes in the analysis of the variable distribution of the residence (and waiting) times of a of a customer in the system (the queue) via its Laplace transform defined by : This allows us to derive the average length of stay and waiting time of a customer as a first order moment of the variable (variable ) : = (−1) * ( ) (0) and = (−1) * ( ) (0) (34) Other performance measures such as the average number of customers in the system or in the queue are derived by applying Little's law: = . and = . (35)

Fuzzy system / /1 and Fuzzy Laplace Transforms
A fuzzy waiting system is defined as one with vague and imprecise descriptor parameters. In this case, the probability density of the general service law is a fuzzy function of a real variable ̃( ). Hence the opportunity of its analysis by the fuzzy Laplace transforms, object of this article.
Being then in a fuzzy environment where the system's descriptor parameters are fuzzy, the probability density of the general service law ̃( ) admits a fuzzy Laplace transform of expression (Gong, Z., & Hao, Y. (2019).). : These are nothing more than fuzzy extensions of the Laplace transforms of the variables and of the residence and waiting times of a customer in the classical M/G/1 system (in the queue).
On the other hand, although fuzzy, these expressions of relation (37) are functions of a real variable . Hence the opportunity to exploit the properties of the moments of the variables ̃ and ̃ to define the average stay and waiting time of a customer in the system (the queue) by the formulas : It will therefore be sufficient to apply Little's Law to obtain the other performance measures of the FM/FG/1 system, including : ̃=̃⨀̃ and ̃=̃⨀̃ (39) FM/F case study 2 /1

Position of the problem
As announced above, this case had just been successfully analyzed by Merlyn Margaret and her friends (Kannadasan, G., & Sathiyamoorthi, N. (2018).). Using the NLP approach, these authors showed that the problem of analysing a performance measure of a fuzzy expectation system M/ G/1 can be reduced to the solution of a pair of non-linear parametric programs. After solving these PNLPs, they used the mean degree integral scheme defined by the relation below to defuzzify the obtained fuzzy features: It is a question of dealing with this system M/ G/1 by the fuzzy Laplace transform method as described in the previous subsection.
The results obtained are defuzzified by the "integral centroid" approach named "COA method", i.e. "Center of Area" defined by (Fazlollahtabar, H., & Gholizadeh, H. (2019).). : where * is the classical defuzzified value corresponding to the fuzzy result ̃ and ̃( ) is the membership function.

Resolution
In the classical model M/ 2 /1model, the density of the service law 2 and its Laplace transform are given (cf. In the FM/ /1 the service density is a fuzzy extension of the function ( ) given by : By a simple integration calculation, we can establish that : And according to relation (38) above, we have :

Numerical example
Statement Consider a waiting system in which customers arrive according to a Poisson process with an imprecise average rate of about 1 customer per minute. Let us also assume that the service time is distributed according to a 2 process with an imprecise mean of about 1/3. Let us determine the average time a customer stays in the system and waits in the queue respectively.

Resolution procedure
To say that the average service time of 2 is about 1/3 means that the service rate is about 3. Since these two descriptors are vague and imprecise, we will proceed as follows: 1. Represent these parameters by two triangular fuzzy numbers of modes 1 and 3 respectively: ̃= ( 1 2 , 1, 3 2 ) and ̃= (2, 3, 4) for example, then write them in L-R form; 2. Apply the formulas in relation (48), according to the fuzzy Laplace transform approach; 3. Use L-R fuzzy number arithmetic (see relations (19) to (23)) to obtain the expected fuzzy results; 4. Defuzzify these results by relation (41) as announced in 4.3.1 above.

Results obtained
1. Fuzzy numbers ̃= ( ℎ if the minute is the chosen unit of time.

Discussion
We find that all modal values of the fuzzy results correspond exactly to the performance measures (average waiting time, average dwell time, average number of customers in the system and in the queue) of the classical model M/ 2 /1 which can be obtained by the Pollaczeck-Khintchine formula mentioned above (Babu, P. S., Kumar, K. S., & Chandan, K. (2022).).
As for the defuzzified values, they are all slightly higher than these modes which are performance measures of the classical model. Shouldn't we see the effects of a fuzzy environment on the performance measures of a waiting system?

CONCLUSION
In this paper, we have sought both to answer the question of what happens to the performance measures of a non-Markovian system in a fuzzy environment and to apply fuzzy transforms in the evaluation of these measures.
To achieve this, we used both the L-R arithmetic of triangular fuzzy numbers and especially the Zadeh extension principle to obtain the fuzzy transform of the distribution of the dwell times of a customer in the system (the queue).
The numerical example treated revealed that, when the descriptor parameters of a system are vague and uncertain, the performance measures, which are fuzzy numbers, have as modal values the performance measures of the corresponding classical model.
The Laplace transform method is therefore still applicable in the evaluation of the performance measures of a fuzzy FM/FG/1 queueing system. Will this be the case for a fuzzy FG/ FM/1 queue? This is a question worth considering.