Mean Waiting time Analysis on Bukl Arrival Fuzzy Queueing System Using Maximum Likelihood Estimation

: Consider the Erlang k-phase queueing system on fuzzy environment and to obtain the optimal value of mean waiting time for the concern system. In this regard, here we proposed new algorithm, using alpha - cut method to construct the different value in the range of uncertain and analyze the system performance level by the MINLP method and discussed a numerical example.


Introduction
The present world is very highly competitive.Due to the situation decision maker decisions are uncertain.In particular, consider industrial people they would expect to deliver the product as on time.But some time not possible to do, it's based on the delay parameters (Service, whether, Cost etc.).Consider a general queueing system, we can provide the service on time the level of queue cannot be increased otherwise not possible, because service time taken in more than the expected service time the entire system will collapse.So, we rectify the above problem by fuzzy environment and the decision parameters are considered TFN's.In this case, we estimate the queue parameter and using a statistical procedure can help to overcome the above behavior or obtained the error of the system performance based on the existing data.In recent years, some authors did the work in statistical manner and also stated the results but did not occur the optimum level (system Performance).Generally, the basic queueing system has used for Markov processes.In the year 1957, Clarke has published in some sequence of papers on MLE for general queueing system in addition to the covariance matrixvariance in the statistics.Benes(1957) followed by the Clarke work and stated the results in special distribution on a queueing model in fuzzy environment.Li and Lee [5] have to analyze the principle of fuzzy queue infinite capacity queueing system. .Negi and Lee [6] simulated the two variable fuzzy queueing systems using an alpha cut method for FM/FE k /1/∞.Kao et al [4] constructed the membership functions and analyzed the system performance of the special methods as follows (i)  [8,3,5] analyzed the queueing parameter via possibility theory and obtained the optimization levels in fuzzy numbers.Chen.S.P [1,2] analyzed the system performance level Fuzzy environment.Nakamura [7]] has developed nonlinear programs to calculate the range of the system and measured the possibility level of the system performance.In this paper, we proposed an algorithm and obtained system concert level with the help of using statistical approach.

Generalized Erlang K-phase service distribution
The entire system consists as on kphase Erlang k phase distribution.We know that the phase of the service time is  k .Here ) ( .t P i n is probability of n th customer in the i th phase service channel system.The numbers of phases are considering the process in backward substitution method.Here, we provided the first service in the 'k th ' node.So the phase "1" is leaves last channel.The parameters are considered as  The probability of "n" service customer being served at the 'i th ' phase stage is Initially the k th service channel customer has been removed in the initial stage and the first service channel customer leaves the system at the end of the stage. ) The relation about the nth customer service channel is and obtain the cumulative value of the total phase is The probability on n th customer in the i th phase of the system is

Fuzzy Queues with k-phase infinite capacity
Here we consider two cases, First one is time axis it may be decaying in two sequences because the system is busy n (t)>0 or its free n(t)=0.Consider the interval time 21 tt  and system parameter is independent.Now, we discussed V.Ramalakshmi a , Bharathi Ramesh Kumar b , D.Kanagajothi c 571 only in busy periods.So, we fixed the some pre-assigned value, the initial queue size is (0) n  = , 'm' denotes the total number of servers, T be period of server, n be the arrival of T. The given system is balanced ( 1   ) and  taken to the Erlang distribution with the ratio value of , & 0,1, 2.... ( ) , where  not depends for the basic queue parameters.
(vi) Obtained the MLE of  and  is and eliminating the service parameter we obtain the MLE of  in the quadratic equation. because it provides a better approximation value and 'm' get larger than the.

Numerical Example
A three engine airline maintenance base schedules the service providing one by one.Hence, to return the airplanes into use at the earliest, the policy is to stumble the overhaul.Under this policy, arrival and service parameter consider in Poisson and Erlang distribution.The parameter  = [1,4,5] and  = [2,5,7] per hour, respectively.Find all the waiting time character.

 
Table : 1 Range of Uncertainty Value:

Figure 2 . 1
Figure 2.1 Flow Diagram here & ii xz is the transition time of queue parameter, Using likelihood function L as follows; (i) The index of the maximum value consider as m z   , here 'm' is the function of z and  is the independent of 'm'.The frequency function considers Erlang distribution as follows:

iz
is the conditional distribution of given 'm'.The random subdivision of the fixed interval length  into m+1 parts and it's fixed.(iv) Depends on time the number of arrival is () nm −+ and the frequency domain of Erlang distribution is We estimate the unique solution in the interval ' (0,1) Obtained two roots, One root is unit and the other root is we get the bounded value of 1 (x) Using back substation method we obtained the MLE value of queue parameters.
Using Origin software from table (i), we occurred the error value by statistical approach and also obtained the curve linear value of Ls and Wq in Fig(i),Fig(ii)and Table(ii) The system considered as bulk arrival queueing system on fuzzy environment and system performance level can be obtained in different level.Finally, we concluded that the optimum queue length are exist: (i) If  =0 then q L =[ 0.045,6.238]and  =1 then q L 2.134 and (ii) If  =0 then q W =[ 0.0474, 1.458] and  =1 the range of  is 0.354.The curve linear value is 0.6238 & 0.7196.In our proposed method easily obtained the error value and it's helpful to simulate the system performance level.
q W