Classical Fuzzy Retrial Queue with Working Vacation using Hexagonal Fuzzy Numbers

Using fuzzy techniques “Classical Fuzzy Retrial Queue with Working Vacation(WV) using Hexagonal Fuzzy Numbers” is discussed in this paper. We acquire model in fuzzy environment as the average orbit length, Probability(Pr) that the server busy, and Pr(the server is in a WV period), the sojourn time of a customer in the queue. Finally numerical results are presented.


Introduction
Recently, the retrial queueing systems with working vacations have been investigated extensively Do [1] first studied an M/M/1 retrial queue with working vacations. Seyed Behrouz et al. [2] analyzed A fuzzy based threshold policy for a single server retrial queue with vacations. Following zadeh, many researchers and considered the problem of fuzzy queueing systems. On the basis of zadeh's extension principle [4]. Dhurai.K, Karpagam.A [3] investigated A new membership function of hexagonal fuzzy numbers. In this paper the Part 2 describe fuzzy queue model. In Part 3 we discuss the average orbit length, Pr (the server is busy) and Pr (the server is in a WV period), the sojourn time of a customer in the queue are studied in fuzzy model. In Part 4 gives the numerical study of this model. Finally, conclusions are gained in part 5.

The fuzzy queue model
The model of this paper as M/M/1 retrial queue. we consider fuzzy arrival rate ̅ then the fuzzy service rate is , fuzzy retrial rate , fuzzy vacation time θ, customer served fuzzy rate are assume to be fuzzy numbers respectively.  Where ( ̅ ), ( ̅ ) , ( ̅ ), ( ̅ ), ( ̅ ) are the universal sets of the arrival rate, service rate, retrial rate, vacation time, customer served rate respectively. It defines f (g, h, i, j, l) as the system performance measure related to the above defined fuzzy queuing model, which depends on the fuzzy membership functions ( ̅ ), ( ̅ ) , ( ̅ ), ( ̅ ), ( ̅ ). Applying Zadeh's extension principle (1978), the membership functions of the measure of efficiency f ( ̅ , ̅ , ̅ , ̅ , ̅ )can be given as.
If the -cuts of f ( ̅ , ̅ , ̅ , ̅ , ̅ ) degenerate to some fixed value, then the system showing is a crisp number, otherwise it is a fuzzy number.
We acquire the membership function of some measure of efficiency, namely the average orbit length, Pr(the server is busy), Pr(the server is in a WV period), the sojourn time of a customer in the queue for the system in terms of these membership functions are , as follows ( )

Performance of measure
We calculate the performance measures for this model in fuzzy model.

The average orbit length
We have the following five types: ---------(6) Using the above method we get the upcoming results.

Numerical Study
The average orbit length: Opine the arrival rate ̅ , service rate ̅ , retrial rate ̅ , vacation time ̅ and customer served rate ̅ are assumed to be hexagonal fuzzy numbers expressed by: The upcoming four graphs are constitute the measure of efficiency

Conclusion
Here, we discuss Classical fuzzy retrial queue with WV using hexagonal fuzzy numbers. Also, we obtained the Pr (the server is busy) and Pr (server is in a WV period), the sojourn time of a customer in the queue. The numerical results show the effective of performance measure. Example for this fuzzy queue model is telephone psychological counselling and computer systems.