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ISSN: 2229-8711
Global Journal of Technology and Optimization
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Predictor Analysis on Non-parametric Bulk Arrival Fuzzy Queuing System

Sivaraman Geetha1, Bharathi Ramesh Kumar2* and Sankar Murugesan3

1National Engineering College, Kovilpatti, Tamil Nadu, India

2Sree Sowdambika College of Engineering, Aruppukottai, Tamil Nadu, India

3Sri.S.Ramasamy Naidu Memorial College, Sattur, Tamil Nadu, India

Corresponding Author:
Bharathi Ramesh Kumar
Sree Sowdambika College of Engineering
Aruppukottai, Tamil Nadu, India
Tel: 984-268-9899
E-mail: [email protected]

Received Date: April 14, 2017; Accepted Date: May 10, 2017; Published Date: May 22, 2017

Citation: Geetha S, Kumar BR, Murugesan S (2017) Predictor Analysis on Nonparametric Bulk Arrival Fuzzy Queueing System. Global J Technol Optim 8:212. doi: 10.4172/2229-8711.1000212

Copyright: © 2017 Geetha S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

In general a management does not like the arriving customer wait for service in a system. It is not possible for all the times because the situation. In this case parameter estimation is helpful to rectify this difficulty and to analyze the modeling of system performance. Practically the queue parameters are not deterministic. So in this paper we estimate the queue parameter. Initially we construct the inverse membership function of the k-phase fuzzy queueing system and proposed an algorithm of performing the system. Finally, obtained the level of uncertainty range in the system and analyze the interval optimality level of k-phase fuzzy queueing system. The idea is extended to the work. A numerical example is included.

Keywords

Fuzzy sets; Mixed integer nonlinear programming; K-phase Erlang distribution; α- cut Membership function

Introduction

In general, any management system did not like that the arriving customer waiting in the service stage on long time. Sometime it is not possible because of the situation due natural calamities, the server providing the worst service, service time factor, etc., In this case our system fully block to the service. So, management likes to avoid this kind of manners, in this connection we estimated the queues parameters. In this situation bulk arriving queueing model is useful for recovery the problem in this case service may talented in many phases. The Researcher [1-3] has investigated the performance level. The basic queue characters are involved the certain probability distribution. The attractiveness is analyzing the observed data through statistical interference. The observable data’s unquestionable on the queueing system actually are. It is important to utilize the data of extend possible. Many algebraic problems are connected with the simulation modeling in queueing analysis. A statistical formula can support the best use of remaining data should be taken its important of the queueing studies. The initial works on the measurements of queue was totally observed a period of time and complete information was available in the form of the arrival moments and service of each customer. In general, model of queue liable on the markov process. Clarke and Benes are assumed the processing time consider as a special distribution. They are investigated the queues parameters through statistical interfering the different models (M/M/1) and (M/M/∞). In general depends on the situation queueing parameters are uncertainty. For this case fuzzy set theory is most helpful to analyze the optimality level of the system performance. In [4] classical queueing models are extended in fuzzy model with more applications. The fuzzy queuing models are more truthful for the classical ones [5-11] have analyzed and proved important results on fuzzy applications using α-level membership function, [12-14] analyze the nonlinear programming for single phase fuzzy queues in general discipline [15] Provided the overview on the conceptual aspects for the phase service in different queueing model. Clearly, many researchers are analyzing the queueing system modeling. In this paper, we analyze the interval optimality level of k-phase fuzzy queueing system; the above work extended in [14,16] and derived the uncertainty range k-phase fuzzy queueing system with the help of inverse membership function.

Generalized Erlang k-phase service distribution

In this model service time consider as an Erlang distribution. More specially, the overall rate of each service phase is kμ. Even though the service may not actually contain in k phases, Let pn,i(t) be the steady state probability, here “n, i” denotes customers in the system and service in k-phase. Here, we considered the number of phases in backward, so k is the first phase of service and one is the last phase. We can derive the steady state balance equation is:

Inter arrival time: A(t) =λe−λt, t ≥ 0

Service time: image t≥0, here E(x) =1/μ and V(x) =1/ kμ2.

The k-phase queueing system shown in Figure 1.

global-journal-technology-optimization-queueing-system

Figure 1: K-Phase queueing system.

Define the 2-dim state variable (n, i) to be the total number of customers n in the system and the customer being served is at i-stage (phase). Then image

image

image

image

At steady state for n>0, 1≤ i ≤ k

image

If n=1, then

image

image

We obtained the total phase service is:

image

image

image

If we let j = (n −1)k + i , n ≥ 0, 1≤ i ≤ k be the total number of phases for (n,i) state in the system with problem image

image

image The prob of j in the bulk-input system. Then we rewrite

image

The Erlangian type k service model is equivalent to Bulk input model where ck = 1, ck =0, x ≠ k.

image

Using partial fraction expansion may yield

image

The performance measures as follows:

image

image

image

image

image

image

image

image - Denotes the average total phase in the system.

Fuzzy queues with k-phase infinite capacity

Consider the arrival and service rates of k-phase queueing systems in triangular fuzzy number and defined as imageimage being the membership function of the arrival and service rate. Let P denote measuring the queueing system performance and defined the fuzzy system is

image

Equation [9,10] formulated the fuzzy membership function as follows:

image

image

Now, the idea is establish the mathematical programming technique a pair of nonlinear programs is developed and the different possibility levels are calculated. So, we estimated the system performance through the statistical interference.

Solution Procedure

To derive the inverse membership function of image on the basis of α-cuts. The fuzzy arrival and service are defined as follows:

image

image

In [16], represented the queue parameter in different possibility levels of intervals, therefore; FM/FEk/1 can be summarized the family of crisp M/Ek/1 queues in the α-level sets. The Appearance of above two sets is relation between ordinary and fuzzy sets [16]. The bounds of α cut fuzzy interval obtained as imageimage Clearly, defined the membership function of image is also parameterized by α. Therefore, rendering to (i), image is the minimum ofimage We need to eitherimage and Wq(z) to satisfy that μLq(z) =α and μWq(z) =α. From (i), y ∈μ (α ) and x∈λ (α ) improved by image respectively, so, μLq(z) and μWq(z) can be written as

image

image

Many researchers are proposed that the different methods for solving these problems. If Lq(z) and Wq(z) are invertible with respect to α, then the shape of the function is

image obtained from membership function μWq(z) = μLq(z) andconstructed:

image

Otherwise, the values of image cannot obtained analytically, the numerical solutions for image at different possibility level of α can be collected to approximate the shapes of L(z) and R(z). The fuzziness values are converted to crisp value using Robust Ranking Technique and we estimated the queues parameters used in statistical interference.

Numerical Example

Consider the centralize three parallel processing system in which the arrival at different level of phases. The arrival and service rate are triangular fuzzy number and the service distribution follows an Erlang distribution. The rates of arrival and service λ=[1,5,7] & μ=[9,11] per minute respectively, the system manager wants to evaluate the performance measures of the system such as the expected number of customers in the queue and waiting in the queue and to analyze optimality level of the system. It is clear the system consisting three phases and the steady state condition is image our proposed methodologyimage can be formulated as follows:

image

The inverse functions of image exist which give the membership function image as

image

Similarly we derived,

image

The Inverse function of is image exist, which give the membership function

image

We analyze system performance optimality (XL-Stat 2016) relation between the average number of customer waiting in the queue and customer has to spend the time in queue shown in Figure 2 and Table 3. The classification system performance in tree structure

global-journal-technology-optimization-tree-structure

Figure 2: Performance classification of tree structure.

Conclusion

In this study, we considered an optimal K-Policy in the FM/FE(k)/1 queue with general server setup time. Especially, the entire queueing system performances are based in the server providing the service status. The Interval Optimization status if α=0 then image is [0.0061 1.8148] and [0.0061, 0.2593] respectively as shown in Table 1. If, the degree of certainty α=1 then image is 0.3333 & 0.0667 respectively. The fuzziness values are converted to crisp value using robust ranking function the value are image is 0.1327 & 0.91945 shown in Table 2. So, performance of the system is not optimized, because the frequency of the system is 0.954 and the estimation of the sample percentage level image is not good because level is reached only 18.18% (based on the estimation Table 4 degree of certainty not good). Our proposed methodology is most helpful for the system operation (simulation) studies and then we quickly identified blocking to the system.

α image image image image image image image image
0 1 7 9 11 0.0061 1.8148 0.0061 0.2593
0.1 1.4 6.8 9.1 10.9 0.0126 1.4728 0.0091 0.2166
0.2 1.8 6.6 9.2 10.8 0.0222 1.2140 0.0123 0.1839
0.3 2.2 6.4 9.3 10.7 0.0355 1.0125 0.0161 0.1582
0.4 2.6 6.2 9.4 10.6 0.0531 0.8520 0.0204 0.1374
0.5 3 6 9.5 10.5 0.0762 0.7218 0.0254 0.1203
0.6 3.4 5.8 9.6 10.4 0.1059 0.6148 0.0311 0.1060
0.7 3.8 5.6 9.7 10.3 0.1438 0.5254 0.0378 0.0939
0.8 4.2 5.4 9.8 10.2 0.1922 0.4508 0.0458 0.0835
0.9 4.6 5.2 9.9 10.1 0.2539 0.3874 0.0552 0.0745
1 5 5 10 10 0.3333 0.3333 0.0667 0.0667

Table 1: The α cuts of the performance measures image

α image image α image image
0 0.1327 0.91945 0.6 0.06855 0.3603
0.1 0.11285 0.7427 0.7 0.06585 0.3346
0.2 0.0981 0.6181 0.8 0.06465 0.3215
0.3 0.0871 0.524 0.9 0.06485 0.32065
0.4 0.0789 0.45255 1 0.0667 0.3333
0.5 0.07285 0.399      

Table 2: Uncertainty performance measures.

Node Object Percentage Split Variable Values Purity%
1 11 100 Lq   9.09
2 1 9.09 Wq 0.3 100
3 10 90.01 Wq 0.1,0.2 & 0.4 to 10 10

Table 3: Purity level of interval [Lq & Wq].

  0.064 0.064 0.065 0.06 0.068 0.072 0.078 0.087 0.098 0.112 0.13 % Correct
0.064 1 0 0 0 0 0 0 0 0 0 0 100
0.064 1 0 0 0 0 0 0 0 0 0 0 0
0.065 1 0 0 0 0 0 0 0 0 0 0 0
0.066 1 0 0 0 0 0 0 0 0 0 0 0
0.068 1 0 0 0 0 0 0 0 0 0 0 0
0.072 1 0 0 0 0 0 0 0 0 0 0 0
0.078 1 0 0 0 0 0 0 0 0 0 0 0
0.087 0 0 0 0 0 0 0 1 0 0 0 100
0.098 1 0 0 0 0 0 0 0 0 0 0 0
0.112 1 0 0 0 0 0 0 0 0 0 0 0
0.132 1 0 0 0 0 0 0 0 0 0 0 0
Total 10 0 0 0 0 0 0 1 0 0 0 18.18

Table 4: Interval optimal value.

References

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