ANALYSIS OF PRIORITY QUEUES WITH PENTAGON FUZZY NUMBER OF PRIORITY QUEUES WITH PENTAGON FUZZY NUMBER.”

: Fuzziness is a sort of recent incoherence. Fuzzy set theory is asserted to depict vagueness. This study explores the queuing model of priority classes adopting pentagon fuzzy number with the inclusions of fuzzy set operations. A mathematical programming method is designed to establish the membership function of the system performance, in which the arrival rate and service rate of the system performance of two priority classes are utilized as fuzzy numbers. Based on  -cut approach and Zadeh’s extension principle, the fuzzy queues are scaled down to a family of ordinary queues. The potency of the performance measures of the characteristics of the queuing model is ensured with an illustration and its graph .


Introduction
Queues (or) waiting lines are universal. A queuing model is formulated so that queue lengths and waiting time can be envisaged. Queuing theory examines every component of waiting lists to be served, including the arrival and service process, and the number of customers and system places. The prime vitality is to find the most desirable and supreme level of service.
The most common queue discipline is the "first come, first served" (FCFS), or the "first in, first out" (FIFO) rule under which the customers are serviced in the strict order of their arrivals. Other queue disciplines are "last in, first out" (LIFO) rule according to which the last arrival in the system is serviced first, "selection service in random order" (SIRO) rule according to which the arrivals are serviced randomly irrespective of their arrivals in the system and a variety of priority schemes according to which a customer's service is done in preference over some other customer.
In priority discipline, the service is of two types preemptive and non-preemptive. In pre-emptive priority, the customers of high priority are given service over the low priority customers. In nonpremption, the highest priority customer goes ahead in the queue, but his service is started only after the completion of the service of the currently being served customer. Fuzzy queuing models have been described by such researchers like Li and Lee [4], Kaufmann [3], Negi and Lee [5], kao et al [1]. Chen [10,11] has analyzed fuzzy queues using Zadeh's extension principle and has developed (FM/FM/1): (/FCFS) and (FM/FM k/1) : (/FCFS) where FM denotes fuzzified exponential time based of queuing theory. Kao et al constructed the membership functions of the system characteristic for fuzzy queues using parametric linear programming. S. Thamotharan [9] studied multiserver queuing model in triangular and trapezoidal fuzzy numbers using  cuts in 2016. In recent times, Usha Madhuri and Chandan [8] designed FM/FM/1 queuing model with Pentagon fuzzy numbers using -cuts and Ashok Kumar. V [12] using DSW algorithm.
In this paper, fuzzy set theory is employed to formulate the membership function of fuzzy priority queues in which two arrival rates and single service rate are Pentagon fuzzy numbers. -cut and fuzzy arithmetic operations are used to derive system characteristics.

1) Fuzzy set: A fuzzy set Ã is defined by
, 2)  -Cut: Given a fuzzy set A in X and any real number   [0,1], then the  -level set of A denoted by A a is the ordinary set 3) Support: The support of fuzzy set Ã is the set of all points x in X such that MÃ (x) > 0.

5) Trapezoidal Fuzzy Number
The trapezoidal fuzzy number is defined as Ã = (a, b, c, d) where a < b < c < d with its membership function as

8) Zadeh's Extension Principle
The membership function of performance measures of the queuing model is examined in accordance to Zadeh's principle.
Let P(x, y) denote the system performance measure of interest where the arrival rate  % and service rate , ( , ) P    % %% are all fuzzy numbers. ( , ) PM  % % is formalized based on Zadeh's principle as

Fuzzy Priority Queues
Queue discipline is based on a priority system. We elaborate a fuzzy queuing model FM/FM/1 with a priority. Each arrival unit is framed as a member of one of two priority classes. Assume the arrivals of the first or higher priority with mean rate 1  % and the second or lower priority with mean rate 2  % such that 12 .  =  +  % % % The first or higher priority units have the right to be served ahead of the others without preemption. It is ascertained that the capacity of the system and the calling source population are interminable.
In this model, 1 L % and 2 L % denote the average system length of first and second priorities.

Parametric Programming Problem for Fuzzy Priority Queuing Model
Consider a queuing system with single server FM/FM/I model. The time between successive arrivals is taken as , 1, 2 i Ai = % of units in the first and second priority and service time S % are represented by the following fuzzy sets.
Where X and Y are crisp universal sets of the inter arrival times and service times and

Expected waiting time and expected number of customers in the queue for FM/FM/1 queue with two priority classes
Suppose that the arrival rates of first and second priority with the same service rate are fuzzy numbers represented by    (6) and (7), the parametric programming problems are formulated to derive the membership functions of (i) °1 q L = average queue length of first priority.
 The optimal solution is ( )

Conclusion
Fuzzy queuing models mark its richness for analysis of vague ideas. The fuzzy set theory has been applied to explore the queuing model of two priority classes using pentagon fuzzy numbers. Based on Zadeh's extension principle, system performance measures are structured. In forthcoming research, we shall deal with generalizing other methods for supporting decision-makers to clinch the direction for modification in priority queues.