Computing Fuzzy Queueing Performance Measures by L-R Method

This article shows that the L-R method introduced in this work is one of valid methods for computing performance measures of fuzzy queues. Using this calculation technique, we find the number of customers and the waiting time of a simple queue M/M/1 in fuzzy environment. L-R method has the advantage of being short, convenient and flexible compared to the well-known and called alpha-cuts method.


Introduction
In fuzzy logic literature, fuzzy queues are largely studied : see e.g Prade [15], Li and Lee ( [7]; [8]), Negi and Lee [12], Kao et al [6], Ning and Zhao [13], Ritha and Robert [16], Ritha and Menon [17], Palpandi and Geetharamani [14], Srinivasan [18], Wang et al [19], Mary et al [9], etc.Most of these papers are devoted to find system performance measures using alpha-cuts method.Here we compute the number of customers and the waiting time in fuzzy queue by a new technique called L-R method, essentially based on L-R fuzzy arithmetic.The paper is organized as follows: Second section recalls the number of customers and the waiting time formulas of a classic simple queue M/M/1.Third section recalls fuzzy set, fuzzy number, alpha-cuts and interval arithmetic, L-R fuzzy number, arithmetic of L-R fuzzy numbers and triangular fuzzy number notions; which are basic ingredients of this analysis.Fourth section computes customers number and waiting time in the queue using L-R method.Fifth section treats a numerical example in two times separately, using successively alpha-cuts method and L-R method, in idea to show L-R method advantages.Sixth and last section concludes this work.
2 Number of customers and waiting time in a simple traditional queue M/M/1 Suppose a classic simple queue with one server whose characteristics are described in the following manner.A single class of customers have come to the queue according to a Poisson process with parameter λ .If the server is busy, they wait in queue, otherwise, they enter one after another into the server to receive an exponential service with rate µ.The calling source is set to be infinite and the service discipline is FIFO (first in first out).At the end of service, each customer leaves the system.Under these assumptions, it is shown in the open literature (see e.g [2]) that a such queue is stable if only if: λ < µ This condition allows to determine the number of customers N and the average waiting time T at the steady state as follows (see [1], pp.168-169 ): and where ρ = λ µ .
3 Fuzzy set, Fuzzy number, Alpha-cuts and interval arithmetic, L-R fuzzy number, Arithmetic of L-R fuzzy numbers, Triangular fuzzy number η Ã(a) is called the grade or the membership degree of a, ∀ a ∈ Ã (see [20]; [11]).
Definition 3.2.Let Ã be a subset in the universe E. The α − cut of Ã noted Ãα is a classical set defined as follows: Let Ã be a subset in the universe E. The support supp( Ã) and the height hgt( Ã) of Ã are the crisp sets defined respectively as follows: 3.2 Fuzzy number Definition 3.5.A fuzzy set Ã in the universe E is a fuzzy number if only if it satisfies the following conditions: 1. E = R; 4. The membership function η Ã is piecewise continous; 5. There exists one and only one x ∈ R such that η Ã(x) = 1.
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Alpha-cuts and interval arithmetic
Alpha-cuts and interval arithmetic is one of arithmetic methods most used and developped in fuzzy mathematics.It operates with the help of the α − cuts notion defined above and interval arithmetic formulas defined as follows (see e.g [3]): Let [a 1 , b 1 ] and [a 2 , b 2 ] be two closed, bounded, intervals of reals numbers.If * denotes addition, substraction, multiplication, or division, then If * is division, we must assume that zero does not belong to [a 2 , b 2 ].We may simplify the last equation as follows: The L-R representation of the fuzzy number M is M = ⟨m, a, b⟩ LR .m is called the mean value (the mode or the modal value) of M , a and b are called respectively the left spread and right spread of M. Conventionally, ⟨m, 0, 0⟩ LR is the ordinary real number m; also called fuzzy singleton.According to (3.4), the support of M is the following open interval: Remark 3.1.From Hanss [5], p. 54, we can state that: - -If M = ⟨m, a, b⟩ LR is semi-symmetric and a = b; then M = ⟨m, a, b⟩ LR is said symmetric.
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Arithmetic of L-R fuzzy numbers
Regarding the arithmetic of L-R fuzzy numbers, some results exist in the literature.
• [4] shows that if there are two L-R fuzzy numbers of the same type M = ⟨m, a, b⟩ LR and Ñ = ⟨n, c, d⟩ LR ; then their sum and their difference are also L-R fuzzy numbers of the same type defined by the following formulas: • [10] shows that if there are two L-R fuzzy numbers of the same type M = ⟨m, a, b⟩ LR and Ñ = ⟨n, c, d⟩ LR ; then their product does not be always an L-R fuzzy number.However, this reference confirms the following result, found before in [5] on secant approximation for two positive L-R fuzzy numbers of the same type M = ⟨m, a, b⟩ LR and Ñ = ⟨n, c, d⟩ LR : Definition 3.7.The inverse secant approximation of an L-R fuzzy number defined in [5], allows to determine the quotient secant approximation of two positive L-R fuzzy numbers M = ⟨m, a, b⟩ LR and Ñ = ⟨n, c, d⟩ LR as follows: Interested reader must refer for details on L-R fuzzy numbers to [4], [20], [5] and [10].
4 Expected number of customers and expected waiting time in a fuzzy queue

L-R method description
Let us reconsider the queue defined in the section 2, and suppose that the arrival rate and service rate are all triangular fuzzy numbers noted respectively λ = (λ 1 /λ 2 /λ 3 ) and μ = (µ 1 /µ 2 /µ 3 ).The fuzzy information transmitted through λ and μ in the system affect N and T, which become fuzzy numbers Ñ and T .The model becomes in this case a fuzzy queue FM/FM/1, where FM denotes a fuzzified exponential distribution.To determine its fuzzy performance measures, L-R method puts first the fuzzy rates λ and μ in L-R representation, and replaces them in equations (2.1) and (2.2) to obtain two L-R fuzzy numbers Ñ and T .It finds finally L-R representations of fuzzy numbers Ñ and T using suitable formulas among (3.16), (3.17), (3.18) and (3.19).For multiplication and division, it is required to use secant approximations, which give the exact support as shown in [10].

4.2
Expected number of customers formulas Theorem 4.1.If FM/FM/1 is a simple fuzzy queue whose arrival and service rate are respectively positive triangular fuzzy numbers λ = (λ 1 /λ 2 /λ 3 ) and μ = (µ 1 /µ 2 /µ 3 ), with λ 3 < µ 1 ; then the expected number of customers in the system at stationary state is approximately the fuzzy number Ñ whose Proof.In classical queueing theory, parameters such as number of customers, waiting time, arrival rate, service rate, ... are always positive.It is the same in fuzzy environment where these numbers are imprecise and given in fuzzy information.Let Ñ be the number of customers of this given queue at the steady state.By L-R method described in the preceding subsection, Ñ can be computed by equation (2.1) after replacing λ and µ by their suitable L-R fuzzy values.Positive real numbers k 1 = λ 2 − λ 1 , k 2 = λ 3 − λ 2 being spreads of λ ; and l 1 = µ 2 − µ 1 , l 2 = µ 3 − µ 2 being spreads of μ; L-R representation of λ and μ are respectively: From equation (2.1), Ñ can be computed using formulas in equations (3.17) and (3.19) as follows: with and  (ii) Support is the real closed interval [u, v]; where ) where N 1 , N 2 and N 3 are defined above in equations ( 4.27), (4.28) and (4.29).
Proof.The proof is similar to previous theorem.

Numerical example
In order to show the benefits of this new method, this section compares it through the following problem to the alpha-cuts method.

Problem
After finding that about 45 people per hour seek photocopies in his neighborhood, Mr. Mukeba has gone to provide a high-speed copier that can serve about 48 people per hour.For his business, Mr. Mukeba wants to build cheaply a comfortable waiting room, so that none of his customers wait up.Following comments are requested 1) What should be the room capacity ?
2) What is the expected number of customers in the room after a long time of service?
3) What is the average waiting time of a customer in the room?

Problem data
The rates being given in fuzzy information (linguistic terms), it is not possible to analyze this problem with traditional queueing theory.The room that Mr. Mukeba tries to build is a simple queue FM/FM/1 having fuzzy rates .Let us suppose that these rates are triangular fuzzy numbers given by λ = (44/45/46) and μ = (47/48/49).

Alpha-cuts method solution
To solve this problem, alpha-cuts method uses following steps: (5.36) We define membership functions of Ñ and T in equations (5.33) and (5.34).Those membership functions are not in usual forms; it is very difficult to imagine their shapes.For this, alpha-cuts method appeals to the problem using mathematical programming techniques called Parametric NLP (Parametric Non Linear Programming).This problem requires to determine α − cuts of Ñ and T .Those α − cuts are computed using alpha-cuts definition in equation (3.3) and intervals arithmetic formulas in (3.8) and (3.10) as follows: For Ñα , we have: where Using Parametric NLP, we finally find For Tα , we have: (5.38) 2 The waiting time in the queue at the steady state is approximately between 0.191 and 1.045.

L-R method solution
These are L-R method steps: (i) We put in (2.1) and (2.2) λ = λ and µ = μ to get respectively following fuzzy numbers

Responses
Results obtained in subsubsections 5.2.2 and 5.2.3 allow to make answers to three questions put above in Mr. Mukeba problem.
1) Room capacity corresponds to "queue length" or "customers number" in terminology of queueing theory.This measure depends to mean values and supports of fuzzy rates λ and μ.But in the problem, mean values and supports of λ and μ are not specified.Consequently, we do not specify the room capacity.However, if the problem formulation had specified for example that, these fuzzy rates were those considered in subsubsection 5.2.1, then , the room capacity was going to be an integer c such as ∀x ∈ supp( Ñ); x ≤ c.That is c = 46.
2) According to supp( Ñ) in figure 1, the expected number of customers in the room after a long time of service is approximately between 8 and 46.
3) According to supp( T ) in figure 2, the expected waiting time of a customer in the room is approximately between 11 and 63 minutes.

Conclusion
In this paper, a new analysis method of fuzzy queues called L-R method, essentially based on L-R fuzzy arithmetic has been studied.The L-R representation has been used to derive performance measures of the model.With the help of this method, the expected number of customers and the average waiting time of FM/FM/1 model are successfully computed and results are found in L-R representation.Under this representation, fuzzy results give many informations than the crisp ones.Mean value of the fuzzy measure corresponds to the average measure in traditional queueing theory.The two spreads help to get lower and upper bound of the fuzzy measure.The problem solved in section 5 shows that the proposed approach is more suitable for designer and practitioner since it deals with imprecise information.Solutions made in subsubsections 5.2.2 and 5.2.3 show that this new method presents three major advantages: It is short, convenient and flexible compared to alpha-cuts method.We are confident that L-R method could help in further reseachs to get results for some open problems in this field as evaluation of fuzzy queueing networks performance measures .

3. 1
Fuzzy set Definition 3.1.Let E be a classical set or a universe.A fuzzy subset Ã (or a fuzzy set Ã) in E is defined by the function η Ã , called membership function of Ã, from E to the real unit interval [0,1].

20 ) 3 . 2 .
Remark Such triangular fuzzy number is often noted Ã = (a, b, c) or Ã = (a/b/c).Remark 3.3.According to definitions 3.6 and 3.8, a triangular fuzzy number Ã = (a/b/c) is always an L-R fuzzy number.In L-R representation, it can be writen .29) These positive real numbers N 1 , N 2 and N 3 are respectively approximate values of the mode, the left spread and the right spread of Ñ.According to equation (3.15), lower and upper bound of Ñ are respectively N 1 − N 2 and N 1 − N 3 .Noting m = N 1 , u = N 1 − N 2 and v = N 1 − N 3 ; equations (4.21), (4.22) and (4.23) are established.

(
iv) To deduce fuzzy values for Ñ et T , we need only their modal values and their supports as shown respectively on figure1and figure 2 represented below.Their membership functions shapes do not be important in this case.

Figure 1 :
Figure 1: Membership function of fuzzy number of customers in the room.

Figure 2 :Figure 1
Figure 2: Membership function of fuzzy waiting time in the room.