Volume 8 • Issue 1 • 1000212 Global J Technol Optim, an open access journal ISSN: 2229-8711 Research Article Geetha et al., Global J Technol Optim 2017, 8:1 DOI: 10.4172/2229-8711.1000212 Review Article Open Access Global Journal of Technology & Optimization G l o b a l J o u r n a l o f T e c h n o lo g y a n d O p t i m i z a t i o n ISSN: 2229-8711 Predictor Analysis on Non-parametric Bulk Arrival Fuzzy Queueing System Sivaraman Geetha 1 , Bharathi Ramesh Kumar 2 * and Sankar Murugesan 3 1 National Engineering College, Kovilpatti, Tamil Nadu, India 2 Sree Sowdambika College of Engineering, Aruppukottai, Tamil Nadu, India 3 Sri.S.Ramasamy Naidu Memorial College, Sattur, Tamil Nadu, India 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 diffculty 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. *Corresponding author: Bharathi Ramesh Kumar, Sree Sowdambika College of Engineering, Aruppukottai, Tamil Nadu, India, Tel: 984-268-9899; E-mail: brkumarmath@gmail.com Received April 14, 2017; Accepted May 10, 2017; Published May 22, 2017 Citation: Geetha S, Kumar BR, Murugesan S (2017) Predictor Analysis on Non- parametric 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. 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. Te Researcher [1-3] has investigated the performance level. Te basic queue characters are involved the certain probability distribution. Te attractiveness is analyzing the observed data through statistical interference. Te 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. Te 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. Tey are investigated the queues parameters through statistical interfering the diferent 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. Te 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 diferent 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 ) ( , t p i n 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 frst phase of service and one is the last phase. We can derive the steady state balance equation is: Inter arrival time: 0 , ) ( ≥ = − t e t A t λ λ Service time: 1 ( ) () 0 ( 1)! k k t k k t e Bt t k µ µ µ − − = ≥ − , here µ / 1 ) ( = x E and 2 / 1 ) ( µ k x V = . Te k-phase queueing system shown in Figure 1. Defne 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). Ten ∑ = = k i i n P n P 1 ) , ( ) ( : at the 1st phase 1: at the last phase 0 : leaving the system or service completion i k i i =     =       =   (1) , , , 1 1, ( ) ( )(1 ) ( )( ) ( )( ) 2, 1 ni ni ni n i p t t p t t k t p t k t p t t n i k λ µ µ λ + − +∆ = − ∆− ∆ + ∆ + ∆ ≥ ≤ ≤ , , , 1 1, ( ) ( )(1 ) ()( ) ( )( ) 2, 1 nk nk nk n i p t t p t t k t p t k t p t t n i k λ µ µ λ + − +∆ = − ∆− ∆ + ∆ + ∆ ≥ ≤ ≤ (2)