An Interpretation of Non-Preemptive Priority Fuzzy Queuing Model with Asymmetrical Service Rates

This paper presents Non-Preemptive priority fuzzy queuing model with asymmetrical service rates. Arrival rate and service rate are taken to be hexagonal, heptagonal, and octagonal fuzzy numbers. Here an interpretation is given to determine the performance measures by applying a new ranking technique through which the fuzzy values are reduced to the crisp values. This ranking technique has the benefit of being precise and relevant compared to other methods such as alpha-cut method and LR method. The main intention is to evaluate the fuzziness before the performance measures are processed by utilizing the regular queueing hypothesis. Three numerical examples are exhibited to show the validity implementation of the methodology.


Introduction
These days, the idea of queuing hypothesis has numerous applications in the real time processes. On the whole, the priority queue has a wide scope of utilizations like communication networks, transport area, medical service executives, service industry, production etc. Meanwhile, the idea of fuzzy queues is broadly discussed by scientists such as (by Yager. (1981)), (by Li. and Lee.(1989)), ( There are two possible segments in priority situation, the Preemption and Non-Preemption. (by Yeo. (1963)) analysed Preemptive priority queues with K classes of customers with a Preemptive repeat and a Preemptive resume strategy. (by Chang. (1965)) proposed a single server queueing system with Non-Preemptive and Preemptive resume priorities. (by Miller.(1981)) obtained steady-state distributions of exponential single server (Preemptive and Non-Preemptive) priority queues with two classes of customers by using Transient analysis of a Non-Preemptive Neut's theory of matrix-geometric invariant probability vectors. (by Brandt. and Brandt. (2004)) contemplated a two-class M/M/1 queueing system under Preemptive resume and impatience of the prioritized customers. Non-Preemptive priority fuzzy queues have been studied (by Devaraj. and Jayalakshmi. (2012)) where fuzzy problem is reduced to crisp problem.
(by Palpandi. and Geetharamani. (2013)) computed performance measures of fuzzy Non-Preemptive priority queues by Robust ranking technique. The above overview shows that the analysis of Non-Preemptive priority fuzzy queueing systems has not been studied in many cases. Thus, in this paper, an approach to deal with an interpretation of a Non-Preemptive priority queueing system in fuzzy environment with asymmetrical service rates is given. This paper is coordinated as follows: Section 2 describes some basic definitions, Section 3 presents methodology, section 4 clarifies fuzzy queuing model, Section 5 depicts the plan of new ranking technique, Section 6 presents mathematical interpretation by taking numerical examples, Section 7 gives the results and discussion, section 8 finishes the article with future directions.

2.Essential ideas and definitions
The essential ideas and definitions are given as follows Fuzzy set (by Zadeh.1965)): Let X be a classical set or a universe. A fuzzy subset (or a fuzzy set ) in X is defined by the function µ , called membership function of , from X to the real unit interval [0,1]. µ (a) is called the grade or the membership degree of a, ∀ a ∈ . Zadeh.(1965)): A fuzzy set characterized on the set of real numbers R is said to be a fuzzy number, if has the accompanying qualities such as

Methodology
In this section we provide a solution methodology for the proposed model i.e. Non-Preemptive priority fuzzy queuing model. The crisp values of the fuzzy arrival rate and the fuzzy service rate were determined by new ranking method. The main purpose is to determine the performance measures by utilizing the regular queueing hypothesis . In this model the service rates are asymmetrical. Here fuzziness is evaluated before the performance measures are processed.

Non-Preemptive priority fuzzy queue with asymmetrical service rates
Let us consider a single server two-class Non-Preemptive priority queue with various service rates. The inter arrival rate of 1 and 2 are appropriated independently. The service rates 1 and 2 are additionally circulated independently. FCFS line discipline is followed, though the low priority customer gets prior service than the high priority customers. From the classical queueing hypothesis, The stability steady state is ≡ 1 + 2 < 1, essential, 0 < < 1. Where 1 = µ , ρ2 = µ Other exhibition estimations are characterized by: Ls (i) =λi ws (i) ; i=1, 2 (10)

Proposed New Ranking Technique
To change the fuzzy values into real crisp values, the accompanying new ranking technique mentioned below is utilized.
whereas d and d are the minimum and maximum values of the given fuzzy number.

Mathematical Interpretation
Let us now assume a critical situation that happens in a corporate hospital in Hyderabad during covid 19 time, where some corona patients with breathing difficulty have arrived in need of medical treatment. In such emergency, the doctor allows the patients to immediately receive the treatment (Non-Preemptive priority only). The average queue length and average waiting time of that two-class Non-Preemptive priority corona patients queue on this possibility is currently computed.

Results and Discussion
The obtained outcomes are given in Tables 1−3, which clarify various estimations of each priority class for a wide range of membership functions considered (hexagonal, heptagonal, and octagonal fuzzy numbers). It is likewise observed from Tables that all the performance measures of first priority class is less than the performance measures of second priority class in the framework for all the three sorts of above-mentioned fuzzy numbers.

Conclusions and future directions
This paper interprets the average queue length and average waiting time of two-class Non-Preemptive priority queue with asymmetric service rates. The fuzzy non-preemptive priority queue is represented more precisely and the scientific outcomes are derived by new ranking technique. Numerical examples for hexagonal, heptagonal and octagonal fuzzy numbers are disclosed viably to decide the validity of the proposed queuing model.The crisp values of the fuzzy arrival rate and the fuzzy service rate were determined by new ranking method. It is more efficient than other existing ranking method. The future work can be done in evaluating the potency of this new ranking technique to other queueing models and other types of linear membership functions. This paper can be extended by considering the probabilistic parameter in place of fuzzy numbers. Another possible area for future research work is to consider intuitionistic fuzzy numbers and neutrosophic sets. The authors are right now working on more complex examples of customer-server associations including multiple serving channels and/or stages, just as cases for which more than one server processes a customer all the while.