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Economic Analysis of the M/M/1/N Queuing System Cost Model in a Vague Environment

1Department of Industrial Engineering, School of Engineering, Damghan University, Damghan, Iran
2Department of Industrial Engineering, Mazandaran University of Science and Technology, Babol, Iran
Correspondence to: Hamed Fazlollahtabar (hfazl@du.ac.ir)
Received July 26, 2019; Revised September 3, 2019; Accepted September 15, 2019.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract

In the real world, values are not always accurate, since vague and uncertain data exist in many applications. In this paper, a single-server finite capacity Markovian queuing system being called M/M/1/N with encouraged arrivals is considered. The different arrival rates, service rates and processing times of labor/duties are usually supposed to be uncertain. Therefore, we describe the theory of queuing in a vague environment in which encourage arrivals rates and service rates are considered to be vague numbers. Then, economic analysis of the model is presented by developing a cost model proceeding to exert uncertainty of the primary information when some of the parameters of the models are vague.

Keywords : Encouraged arrivals rate, Queuing system, Vague environment, Vague numbers
1. Introduction

The study of waiting lines, called queuing theory, is one of the oldest and most widely used quantitative analysis techniques. Queuing theory is basically a mathematical approach, used for the analysis of expecting lines. One of the applications of queuing theory is to analyze the congeries and delay of waiting in line.

Queuing theory has a long background and has been handle to decide applied problems in manufacturing, communication, inventory control, etc. Queuing system is composed of three parts (1) the arrival or inputs to the system (sometimes referred to as the calling population), (2) the queue (or the waiting line itself), and (3) the service facility. In the queuing theory, it is usually supposed that the time between the two successively arrivals and the servicing time follow a particular probability distribution. However, in the real world, this kind of information is gained using qualitative data being expressed by words equal rapid, middle, and slow rather than the probability distribution. It can be said that fuzzy queuing models are more realistic and practical than the classical ones. Queuing theory has its beginning in the research work of a Danish engineer [1]. applied this theory extensively to study the manner of phone networks. Since in the actual world there is vague information about different applications, such as in sensor databases, we can formalize the measurements from different sensors to a vague set in the real world. The various arrival rates, service rates and processing times of jobs/tasks usually assumed to be uncertainty. Thus, fuzzy set theory has been suggested to batch such ambiguousness by distribute the concept of membership in a set. Fuzzy queuing models have been explained by several scholar, namely Li and Lee [2], Buckley [3], Negi and Lee [4], Kao et al. [5], Chen [6, 7] who dragged fuzzy queues using using Zadeh’s deployment origin [8]. Kao et al. [5] created the membership functions of the system characteristic for fuzzy queues using parametric linear programming. Basically, in a fuzzy set each base is related with a point value opted from the unit interval [0, 1], A vague set, as well as an Intuitionistic fuzzy set, is a further extension of a fuzzy set. Instead of using point-relying membership as in fuzzy sets, interval-based membership is used in a vague set [9]. Defined vague sets are used by Gonzalez-Lopez et al. [10] where they described the optimization of queuing theory based on vague environment.

Oz et al. [11] considered the Mn/Gn/1 queue with vacations and exhaustive service in which the server takes (repeated) vacations whenever it becomes idle, the service time distribution is queue length dependent, and the arrival rate varies both with the queue length and with the status of the server, being busy or on vacation. Dutkova et al. [12] displayed the usage of simulation method as a tool for optimizing costs of particular post office. Simulation methods make it possible to eliminate the most of the obstacles that arise from the analytical solution of queuing systems such as queuing system of post office.

Hanczewski et al. [13] presented a multi-service model of a queueing system with elastic and adaptive traffic. The presented model permits to determine the characteristics of a multi-service queueing system, such as blocking probability or the average queueing length.

Legros [14] analyzed the transient behavior of the M/M/1+D queue. Considering an Erlang distribution for customers’ waiting time, he approximated the real system by a Markov chain. Then, obtained the Laplace transform of the transient probabilities in the approximated model and the Laplace transform of the main performance measures for the real system.

Here, the attracted customers are entitling as encourage arrivals by the obscure rate. The concept of customer mobilization was famous by [15]. They called it inverse balking. The phenomenon of encouraged arrivals can also be understood as opposite to disheartened arrivals bring up by [16]. Explain the M/M/1/N queuing system with encouraged arrivals [17].

Literature review shows that no researchers have discussed the problem of a single server with limited capacity in the M/M/1 N Markovian system with vague encouraged inputs. In real, the input data, arrival rate and service rate are uncertain. Uncertainty is resolved by using vague set theory. So, the classical queuing model will be more effective if it is proposed using vague models.

As a result, the major contribution of the work is to develop an M/M/1/N queue model using vague numbers and as a minor contribution, the corresponding economic analysis through a novel cost model is proposed.

The remainder of the paper is organized as follows: in Section 2, the M/M/1/N queue model is introduced. In Section 3, various concepts of the vague set theory are discussed. The vague queue model M/M/1/N is introduced in Section 4. In Section 5, the analysis of the model is expressed. In Section 6, a numerical example is illustrated to show the effectiveness of the proposed model. We conclude in Section 7.

2. M/M/1/N Queue Model

In this model, the customers arrive based on the Poisson distribution with rate (1+η) and receive service which is exponentially distributed with a mean service time of μ from a single server and the capacity of the system is finite say, N [17].

We first outline the assumptions of the problem:

1. Customers arrive based on the Poisson distribution with parameter (1+η), where ‘η’ displays the percent change in the number of customers computed from bygone or perceives data. That is λ, vague.

2. We show the service rate with μ that is vague.

3. The capacity of the system is finite say, N.

Due to the definition and assumption the following queue model computation are given.

The possibility of system being empty is computed as follows [17]:

$P0=Pr{system is empty}={1+∑n=1N[λ(1+η)μ]n}-1=1-(λ(1+η)μ)1-(λ(1+η)μ)N+1.$

And the probability that system is full is given by [17]:

$PN=Pr{system is full}=[λ(1+η)μ]N*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1).$

Expected number of customers in the system [12]:

$L=∑n=1Nn[λ(1+η)μ]n*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1).$

Expected number of the customers waiting on the queue [17]:

$Lq=∑n=1N(n-1) [λ(1+η)μ]n*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1).$

Expected waiting time of customers in the system [17]:

$W=Lλ, Wq=Lqλ.$

In the special case if η = 0 of that time [17]:

$Pn=Pr{system is full}=(λμ)N Pn=ρNP0,$$P0=Pr{system is empty}=1-(λμ)1-(λμ)N+1=1-ρ1-ρN+1,$

where $ρ=λμ<1$, is the traffic intensity.

3. Vague Environment

To conceptualize the vagueness, some preliminary definitions one presented below:

### Definition 1 (Fuzzy set) [10]

A fuzzy set Ã = {< u, μÃ(u) >| uU} in a universe of discourse U is characterized by a membership function, μÃ, as follows:

$μA˜:U→[0,1].$

### Definition 2 [10]

A vague set Ã in a universe of discourse U is characterized It is expressed by two membership functions, which is a true membership, TÃ, function and a false membership function, FÃ, as follows: TÃ : U → [0, 1], FÃ : U → [0, 1], and TÃ(u) + FÃ(u) ≤ 1, where TÃ(u) is a lower bound on the grade of membership of u derived from the evidence for u, FÃ(u) is a lower bound on the grade of membership of the negation of u derived from the evidence against u. Suppose U = {u1, u2, . . . , un}. A vague set Ã of the universe of discourse U can be represented by $A˜=∑i=1N[TA˜(ui),1-FA˜(ui)]ui$, where 0 ≤ T(ui) ≤ 1 – F(ui) ≤ 1 and 1 ≤ in. In this case, the degree of membership in ui will be limited to a period of time, [TÃ(ui), 1 – FÃ(ui)] Of [0, 1].

Therefore, generalized vague collections have fuzzy sets of membership degrees μÃ(u) of u in Definition 1 may be inexact in a vague set.

### Definition 3 [10]

The complement of a vague set Ã is denoted by Ã′ and is defined by

$TA˜′(u)=FA˜(u), 1-FA˜′(u)=1-FA˜(u).$

### Definition 4 [10]

Let ÃTÃFÃ be a vague set of U. Then, we define αT -cuts and αF -cuts of Ã as the crisp sets of U given by

$A˜αT={u:TA˜(u)≥αT}, αT∈[0,1],$$A˜αF={u:FA˜(u)≥αF}, αF∈[0,1].$

### Definition 5 [10]

A vague set ÃTÃFÃ of ℛ with continuous membership functions TÃ and FÃ is called a vague number if and only if ÃαT and ÃαF, for all αT, αF ∈ (0, 1], are bounded closed intervals; i.e.,

$A˜αT=[A˜αTL,A˜αTU], A˜αF=[A˜αFL,A˜αFU].$

We denote the class of all vague numbers by A(ℛ).

### Definition 6 [10]

The vague number ÃTÃFÃ is called a trapezoidal vague number, if

$TA˜(u)={u-a1w(a2-a1),a1≤u≤a2,1,a2≤u≤a3,a4-uw(a4-a3),a3≤u≤a4.0,otherwise,$$1-FA˜(u)={u-a1(a2-a1),a1≤u≤a2,1,a2≤u≤a3,a4-u(a4-a3),a3≤u≤a4,0,otherwise,$

where w ∈ [1,∞). We denote such a vague number by Ã = (a1, a2, a3, a4, w)T .

### 3.1 Vague Queues

This paper develops vague queuing M/M/1/N model in which the source encouraged arrivals rate and service rate and the percentage of change in the number of customers are uncertain parameters.

Since the arrival rate (λ̃), service rate (μ̃) and the percentage of change in number of customers (η̃) are not known precisely, therefore, we use trapezoidal vague numbers to represent them. Let λ̃ = (λ1, λ2, λ3, λ4), μ̃ = (μ1, μ2, μ3, μ4), η̃ = (η1, η2, η3, η4), where λ1λ2λ3λ4, μ1μ2μ3μ4, η1η2η3η4 based on subjective judgment. The membership of Tλ̃ (λ), 1 – Fλ̃(λ) and Tμ̃(μ), 1 – Fμ̃(μ) and Tη̃(η), 1 – Fη̃(η) are defined as follows:

$Tλ˜(λ)={λ-λ1w(λ2-λ1),λ1≤λ≤λ2,1,λ2≤λ≤λ3,λ4-λw(λ4-λ3),λ3≤λ≤λ4,0,otherwise,$$1-Fλ˜(λ)={λ-λ1(λ2-λ1),λ1≤λ≤λ2,1,λ2≤λ≤λ3,λ4-λ(λ4-λ3),λ3≤λ≤λ4,0,otherwise,$$Tμ˜(μ)={μ-μ1w(μ2-μ1),μ1≤μ≤μ2,1,μ2≤μ≤μ3,μ4-μw(μ4-μ3),μ3≤μ≤μ4,0,otherwise,$$1-Fμ˜(μ)={μ-μ1(μ2-μ1),μ1≤μ≤μ2,1,μ2≤μ≤μ3,μ4-μ(μ4-μ3),μ3≤μ≤μ4,0,otherwise,$$Tη˜(η)={η-η1w(η2-η1),η1≤η≤η2,1,η2≤η≤η3,η4-ηw(η4-η3),η3≤η≤η4,0,otherwise,$$1-Fη˜(η)={η-η1(η2-η1),η1≤η≤η2,1,η2≤η≤η3,η4-η(η4-η3),η3≤η≤η4,0,otherwise.$

Using the concept of α-cut method, we have

$λ˜-λ1w(λ2-λ1)=αT⇒λ˜=wαT(λ2-λ1)+λ1,λ4-λ˜w(λ4-λ3)=αT⇒λ˜=λ4-wαT(λ4-λ3),λ˜αT=[wαT(λ2-λ1)+λ1,λ4-wαT(λ4-λ3)],$$λ˜-λ1(λ2-λ1)=αF⇒λ˜=αF(λ2-λ1)+λ1,λ4-λ˜(λ4-λ3)=αF⇒λ˜=λ4-αF(λ4-λ3),λ˜αF=[αF(λ2-λ1)+λ1,λ4-αF(λ4-λ3)],$$μ˜-μ1w(μ2-μ1)=αT⇒μ˜=wαT(μ2-μ1)+μ1,μ4-μ˜w(μ4-μ3)=αT⇒μ˜=μ4-wαT(μ4-μ3),μ˜αT=[wαT(μ2-μ1)+μ1,μ4-wαT(μ4-μ3)],$$μ˜-μ1(μ2-μ1)=αF⇒μ˜=αF(μ2-μ1)+μ1,μ4-μ˜(μ4-μ1)=αF⇒μ˜=μ4-αF(μ4-μ3),μ˜αF=[αF(μ2-μ1)+μ1,μ4-αF(μ4-μ3)],$$η˜-η1w(η2-η1)=αT⇒η˜=wαT(η2-η1)+η1,η4-η˜w(η4-η3)=αT⇒η˜=η4-wαT(η4-η3),η˜αT=[wαT(η2-η1)+η1,η4-wαT(η4-η3)],$$η˜-η1(η2-η1)=αF⇒η˜=αF(η2-η1)+η1,η4-η˜(η4-η3)=αF⇒η˜=η4-αF(η4-η3),η˜αF=[αF(η2-η1)+η1,η4-αF(η4-η3)].$

Then, λ̃αT, λ̃αF and μ̃αT, μ̃αF and η̃αT, η̃αF are vague numbers.

$λ˜αT=[λ˜αTL, λ˜αTU], λ˜αF=[λ˜αFL, λ˜αFU],$$μ˜αT=[μ˜αTL, μ˜αTU], μ˜αF=[μ˜αFL, μ˜αFU],$$η˜αT=[η˜αTL, η˜αTU], η˜αF=[η˜αFL, η˜αFU].$
4. Vague M/M/1/N Model

Given what have explained above, we’ll analyze M/M/1/N model under vague environment.

Vague empty system:

$(P˜0)αTL=1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1=1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))N+1,$$(P˜0)αTU=1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1=1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))N+1,$$(P˜0)αFL=1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1=1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))N+1,$$(P˜0)αFU=1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1=1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))N+1.$

Vague probability that system is full is given by:

$(P˜N)αTL=[λαTL(1+ηαTL)μαTL]N*(1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1)=[((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))]N*(1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))N+1),$$(P˜N)αTU=[λαTU(1+ηαTU)μαTU]N*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1)=[((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))]N*(1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))N+1),$$(P˜N)αFL=[λαFL(1+ηαFL)μαFL]N*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1)=[((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))]N*(1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))N+1),$$(P˜N)αFU=[λαFU(1+ηαFU)μαFU]N*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1)=((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))N*(1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))N+1).$

Vague expected system size:

$(L˜)αTL=∑n=1Nn[λαTL(1+ηαTL)μαTL]n*(1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1)=∑n=1Nn[((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))]n*(1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))N+1),$$(L˜)αTU=∑n=1Nn[λαTU(1+ηαTU)μαTU]n*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1)=∑n=1Nn[((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))]n*(1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))N+1),$$(L˜)αFL=∑n=1Nn[λαFL(1+ηαFL)μαFL]n*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1)=∑n=1Nn[((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))]n*(1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))N+1),$$(L˜)αFU=∑n=1Nn[λαFU(1+ηαFU)μαFU]n*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1)=∑n=1Nn[((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))]n*(1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))N+1),$

Vague expected number of the customers waiting in the queue is computed by:

$(L˜q)αTL=∑n=1N(n-1)[λαTL(1+ηαTL)μαTL]n*(1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1)=∑n=1N(n-1)*[((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))]n*(1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))1-((wαT(λ2-λ1)+λ1) (1+(wαT(η2-η1)+η1))(wαT(μ2-μ1)+μ1))N+1),$$(L˜q)αTU=∑n=1N(n-1)[λαTU(1+ηαTU)μαTU]n*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1)=∑n=1N(n-1)*[((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))]n*(1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))1-((λ4-wαT(λ4-λ3)) (1+(η4-wαT(η4-η3)))(μ4-wαT(μ4-μ3)))N+1),$$(L˜q)αFL=∑n=1N(n-1)[λαFL(1+ηαFL)μαFL]n*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1)=∑n=1N(n-1)*[((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))]n*(1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))1-((αF(λ2-λ1)+λ1) (1+(αF(η2-η1)+η1))(αF(μ2-μ1)+μ1))N+1),$$(L˜q)αFU=∑n=1N(n-1)[λαFU(1+ηαFU)μαFU]n*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1)=∑n=1N(n-1)*[((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))]n*(1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))1-((λ4-αF(λ4-λ3)) (1+(η4-αF(η4-η3)))(μ4-αF(μ4-μ3)))N+1).$

The membership functions for , q:

$TL˜(L)={L-L1w(L2-L1),L1≤L≤L2,1,L2≤L≤L3,L4-Lw(L4-L3),L3≤L≤L4,0,otherwise,$$1-FL˜(L)={L-L1(L2-L1),L1≤L≤L2,1,L2≤L≤L3,L4-L(L4-L3),L3≤L≤L4,0,otherwise,$$TL˜q(Lq)={Lq-Lq1w(Lq2-Lq1),Lq1≤Lq≤Lq2,1,Lq2≤Lq≤Lq3,Lq4-Lqw(Lq4-Lq3),Lq3≤Lq≤Lq4,0,otherwise,$$1-FL˜q(Lq)={Lq-Lq1(Lq2-Lq1),Lq1≤Lq≤Lq2,1,Lq2≤Lq≤Lq3,Lq4-Lq(Lq4-Lq3),Lq3≤Lq≤Lq4,0,otherwise,$$(W˜)αTL=(L˜)αTLλ˜αTL, (W˜)αTU=(L˜)αTUλ˜αTU,$$(W˜)αFL=(L˜)αFLλ˜αFL, (W˜)αFU=(L˜)αFUλ˜αFU,$$(W˜q)αTL=(L˜q)αTLλ˜αTL, (W˜q)αTU=(L˜q)αTUλ˜αTU,$$(W˜q)αFL=(L˜q)αFLλ˜αFL, (W˜q)αFU=(L˜q)αFUλ˜αFU.$
5. Economic Analysis

Economic analysis of the proposed model leads to strategically determine the role of decision making for organizations and its ability and capacity. It is required to optimize the queuing system. In other words, for an optimal system, it is needed to explore a system in which the total cost is minimized.

Here, we present the cost model:

$Total Cost=CS*C+Ch*L+ClλPn+CW*Lq+CI*(m-L+Lq),$$Total expected cost of the system=CS*μ+Ch∑n=1Nn[λ(1+η)μ]n*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1)+Clλ[λ(1+η)μ]N*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1)+Cw∑n=1N(n-1)[(λ(1+η)μ)n]*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1)+CI*(m-(∑n=1Nn[λ(1+η)μ]n*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1))+(∑n=1N(n-1) [λ(1+η)μ]n*(1-(λ(1+η)μ)1-(λ(1+η)μ)N+1))).$

Let:

1. CW : Cost of one customer waiting in queue,

2. CS : Cost per service per unit time,

3. Ch : Holding cost per unit per unit time,

4. Cl : Cost of cancellation per customer per unit time in queue,

5. CI : Cost of keeping an idle server in a unit of time,

6. M: Number of the servers.

Vague total expected cost of the system:

$(TEC)αTL=(CS*μαTL)+Ch*(∑n=1Nn[λαTL(1+ηαTL)μαTL]n*(1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1))+(Cl*λαTL)*([λαTL(1+ηαTL)μαTL]N*(1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1))+CW*(∑n=1N(n-1)[λαTL(1+ηαTL)μαTL]n*(1-(λαTL(1+ηαTL))μαTL1-(λαTL(1+ηαTL)μαTL)N+1))+CI*(m-(∑n=1Nn[λαTL(1+ηαTL)μαTL]n*(1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1))+(∑n=1N(n-1) [λαTL(1+ηαTL)μαTL]n*(1-(λαTL(1+ηαTL)μαTL)1-(λαTL(1+ηαTL)μαTL)N+1))),$$(TEC)αTU=(CS*μαTU)+Ch*(∑n=1N[λαTU(1+ηαTU)μαTU]n*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1))+(Cl=λαTU)*([λαTU(1+ηαTU)μαTU]n*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1))+CW*(∑n=1N(n-1)[λαTU(1+ηαTU)μαTU]n*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1))+CI*(m-(∑n=1N(n)[λαTU(1+ηαTU)μαTU]n*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1))+(∑n=1N(n-1)[λαTU(1+ηαTU)μαTU]n*(1-(λαTU(1+ηαTU)μαTU)1-(λαTU(1+ηαTU)μαTU)N+1))),$$(TEC)αFL=(CS*μαFL)+Ch*(∑n=1Nn[λαFL(1+ηαFL)μαFL]N*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1))+(Cl*λαFL)*([λαFL(1+ηαFL)μαFL]n*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1))+CW*(∑n=1N(n-1)[λαFL(1+ηαFL)μαFL]n*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1))+CI*(m-(∑n=1Nn[λαFL(1+ηαFL)μαFL]n*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1))+(∑n=1N(n-1) [λαFL(1+ηαFL)μαFL]n*(1-(λαFL(1+ηαFL)μαFL)1-(λαFL(1+ηαFL)μαFL)N+1))),$$(TEC)αFU=(CS*μαFU)+Ch*(∑n=1Nn[λαFU(1+ηαFU)μαFU]n*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1))+(Cl*λαFU)*([λαFU(1+ηαFU)μαFU]N*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1))+CW*(∑n=1N(n-1)[λαFU(1+ηαFU)μαFU]n*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1))+CI*(m-(∑n=1Nn[λαFU(1+ηαFU)μαFU]n*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1))+(∑n=1N(n-1) [λαFU(1+ηαFU)μαFU]n*(1-(λαFU(1+ηαFU)μαFU)1-(λαFU(1+ηαFU)μαFU)N+1))).$
6. Numerical Example

Consider the carwash station as an example (Figure 1). Assume that due to the spatial constraints on the road, carwash only has a capacity of 6 cars for waiting in the system. The driver of the car takes an average of 2 minutes to move the car. Consider now the arrival speed of 9 cars per hour and service time of 6 minutes. Poisson distribution entry rate and distribution service rates are exponential. Also, the percentage of customer change over the past years is equal to 0.75.

Then, we can extract the following input data (Tables 1 and 2): λ̃ = 9̃, $μ˜=7.5˜,η˜=0.75˜$, N = 7, CS = 15, Cw = 13, Ch = 2, Cl = 10, CI = 5, W = 2.

Tables 3 to 6 show that in the sensitivity analysis for different αT and αF, there is an increase in the average rate of entry and service rates, which is expected to rise by increasing system costs and reach the maximum at a certain level.

As the service rate increases, the cost increases quickly by reducing the number of lines increased customer waiting time. Also, by increasing the percentage of change in the number of customers, the length of the waiting line increases. According to the sensitivity analysis, by the increase in entry rates and service rates in αT and αF, the potential for emptying or filling up the system is reduced.

7. Conclusions

In this paper, we considered the vague set to study a single server queuing model with encouraged arrivals. The arrival rate and service rate were vague numbers. conducted action to exert uncertainty of the primary information when some of the parameters of the models are vague. The system can also be studied for non-heterogeneous service. The yield of the paper is of huge use for any organization exposed the phenomenon of encouraged customers augment numbers of lines and service station number inducement the time of customer queuing is cutback, and the customer satisfaction is increased load on service. By pursue and enforcement this model and useful strategy can be plotted the economic analysis can be computed and cost aspect of the business can also be beholden. To study the model validity, the mentioned model is analyzed in different scenarios of costs by the sensitivity analysis on the models’ fuzzy vague costs. Regarding the conditions of the system, the obtained results can help the decision maker according to the costs of the system.

Figures
Fig. 1.

A carwash station.

TABLES

### Table 1

Vague inputs of the problem for αF

αF λ̃ μ̃ η̃
$(λ˜)αFL$ $(λ˜)αFU$ $(μ˜)αFL$ $(μ˜)αFU$ $(η˜)αFL$ $(η˜)αFU$
0.55 6.75 16.25 5.25 14.75 0.525 1.475
0.95 8.75 14.25 7.25 12.75 0.725 1.275

### Table 2

Vague inputs of the problem for αT

αT λ̃ μ̃ η̃
$(λ˜)αTL$ $(λ˜)αTU$ $(μ˜)αTL$ $(μ˜)αTU$ $(η˜)αTL$ $(η˜)αTU$
0.15 5.5 17.5 4 16 0.4 1.6
0.45 8.5 14.5 7 13 0.7 1.3

### Table 3

αF n q q TEC
$(L˜)αFL$ $(L˜)αFU$ $(L˜q)αFL$ $(L˜q)αFU$ $(W˜)αFL$ $(W˜)αFU$ $(W˜q)αFL$ $(W˜q)αFU$ $(TEC˜)αFL$ $(TEC˜)αFU$
0.55 1 0.6620 0.7313 0 0 0.0980 0.0450 0 0 126.449 342.1611
2 1.1294 1.3316 0.5647 0.6658 0.1673 0.0819 0.0684 0.0446 128.6437 341.1005
3 1.5744 1.9325 1.0496 1.2883 0.2332 0.1189 0.1282 0.0861 133.3436 346.3757
4 2.0289 2.5401 1.5185 1.9051 0.3005 0.1563 0.1863 0.1271 139.2323 353.5698
5 2.4919 3.1611 1.9935 2.5289 0.3691 0.1945 0.2450 0.1684 145.7926 365.0194
6 2.9594 3.6931 2.4661 3.0776 0.4384 0.2272 0.3047 0.2062 152.5526 370.5862
7 3.4224 3.9151 2.9335 3.3558 0.5070 0.2409 0.3604 0.2418 155.8642 365.7954

0.95 1 0.6752 0.7695 0 0 0.0771 0.0540 0 0 170.8044 296.1995
2 1.1683 1.4847 0.5841 0.6455 0.1335 0.1041 0.0667 0.0452 171.8677 295.3987
3 1.6437 2.2971 1.0972 1.2417 0.1878 0.1612 0.1253 0.0871 176.5098 300.1733
4 2.1304 3.2249 1.5978 1.8288 0.2434 0.2263 0.1826 0.1283 182.7217 306.3617
5 2.6147 4.2901 2.0918 2.4199 0.2988 0.3010 0.2390 0.1698 189.3124 313.1362
6 3.1186 5.4166 2.5988 2.9678 0.3564 0.3801 0.2970 0.2082 196.6464 317.9944
7 3.5491 6.2642 3.0421 3.2920 0.4056 0.4395 0.3476 0.2310 202.2230 314.8889

### Table 4

αT n q q TEC
$(L˜)αTL$ $(L˜)αTU$ $(L˜q)αTL$ $(L˜q)αTU$ $(W˜)αTL$ $(W˜)αTU$ $(W˜q)αTL$ $(W˜q)αTU$ $(TEC˜)αTL$ $(TEC˜)αTU$
0.15 1 0.6579 0.7394 0 0 0.1196 0.0422 0 0 99.2108 372.1768
2 1.1176 1.3546 0.5588 0.6773 0.2032 0.0774 0.1016 0.0387 102.4396 371.6551
3 1.5536 1.9715 1.0357 1.3143 0.2824 0.1126 0.1883 0.0751 107.4608 377.7354
4 1.9938 2.5870 1.4953 1.9402 0.3625 0.1478 0.2718 0.1108 113.3460 385.3351
5 2.4450 3.1574 1.9560 2.5259 0.4445 0.1804 0.3556 0.1443 119.7680 391.4890
6 2.9004 3.8018 2.4170 3.1681 0.5273 0.2172 0.4394 0.1810 126.3918 401.5004
7 3.3597 4.2033 2.8798 3.6028 0.6108 0.2401 0.5236 0.2058 133.1518 402.3105

0.45 1 0.6734 0.7194 0 0 0.0792 0.0496 0 0 165.2188 302.1548
2 1.1630 1.2961 0.5815 0.6480 0.1368 0.0893 0.0684 0.0446 166.4055 301.7357
3 1.635 1.8737 1.0903 1.2491 0.1924 0.1292 0.1282 0.0861 171.0527 307.4152
4 2.1124 2.4586 1.5843 1.8439 0.2485 0.1695 0.1863 0.1271 177.0687 314.9314
5 2.6033 3.0533 2.0826 2.4426 0.3062 0.2105 0.2450 0.1684 183.9279 323.3439
6 3.1080 3.5883 2.5900 2.9902 0.3656 0.2474 0.3047 0.2062 191.3260 329.7687
7 3.5745 4.0907 3.0638 3.5063 0.4205 0.2821 0.3604 0.2418 197.9759 335.5648

### Table 5

Problem answers for aT according to the system possibility

αT n 0 N
$(P˜0)αTL$ $(P˜0)αTU$ $(P˜N)αTL$ $(P˜N)αTU$
0.15 1 0.3418 0.2601 0.6579 0.7394
2 0.1508 0.0838 0.5588 0.6773
3 0.0726 0.0286 0.5178 0.6571
4 0.0363 0.0099 0.4984 0.6467
5 0.0185 0.0034 0.4890 0.6314
6 0.0095 0.0012 0.4834 0.6336
7 0.0049 0.0004 0.4799 0.6004

0.45 1 0.3263 0.2805 0.6734 0.7194
2 0.1365 0.0985 0.5815 0.6480
3 0.0620 0.0370 0.5451 0.6245
4 0.0291 0.0142 0.5281 0.6146
5 0.0139 0.0055 0.5206 0.6106
6 0.0067 0.0021 0.5180 0.5980
7 0.0032 0.0008 0.5106 0.5843

### Table 6

Problem answers for aF according to the system possibility

αF n 0 N
$(P˜0)αFL$ $(P˜0)αFU$ $(P˜N)αFL$ $(P˜N)αFU$
0.55 1 0.3378 0.2683 0.6620 0.7313
2 0.1470 0.0896 0.5647 0.6658
3 0.0697 0.0318 0.5248 0.6441
4 0.0343 0.0115 0.5072 0.6350
5 0.0172 0.0042 0.4983 0.6322
6 0.0087 0.0015 0.4932 0.6155
7 0.0044 0.0005 0.4889 0.5593

0955 1 0.3245 0.2823 0.6752 0.7176
2 0.1349 0.0999 0.5841 0.6455
3 0.0608 0.0378 0.5479 0.6208
4 0.0284 0.0146 0.5326 0.6096
5 0.0134 0.0057 0.5229 0.6049
6 0.0064 0.0022 0.5197 0.5935
7 0.0030 0.0008 0.5070 0.5486

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Biographies

Hamed Fazlollahtabar earned a BSc and an MSc in Industrial Engineering from Mazandaran University of Science and Technology, Iran, in 2008 and 2010, respectively. He received his PhD in Industrial and Systems Engineering from Iran University of Science and Technology in 2015, and completed a postdoctoral research fellowship at Sharif University of Technology, Iran, in the area of reliability engineering for complex systems in 2017. He currently works in the Department of Industrial Engineering at Damghan University, Iran, and is on the editorial boards of several journals and on the technical committees of several conferences. His research interests are advanced robotic production systems, reliability engineering, supply chain planning, and business intelligence and analytics. He has published more than 280 research papers and eight books.

E-mail: hfazl@du.ac.ir

Hadi Gholizadeh was born in Amol in Mazandaran province of Iran. He earned his BSc from Shomal University, Amol, Iran and MSc from Mazandaran University of Science and Technology, Babol, Iran (MUST).His researches are mainly about Metaheuristics and Optimization techniques to solve the single and multi-objective optimization problems and also industrial engineering issues including Closed-loop Supply Chain Network Design, Sustainable developments for these supply chain systems and health care networks and also related multilevel programming models in all mentioned contents and Maintenance Engineering, Queuing systems and Machin learning and Fuzzy Computing and MCDM. He has also published more than 15 papers in leading scientific and engineering conferences e.g., IEEE and journals e.g., ASOC, IJAMT, NCCA, IJFS.

September 2019, 19 (3)