OPTIMIZATION OF A QUEUEING SYSTEM WITH INVERSE-GAUSSIAN SERVICE PATTERN

Copyright © 2021 the author(s).This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: In this paper parameters involvedin a single server waiting line system with poisson arrivals and Inversegaussian service times are estimated. Also, the same result has been obtained when it is assumed that the service time distribution is a finite range model namely, Mukheerji-Islam model, which is a well-known life testing model.


INTRODUCTION
The Inversegaussian family of distributions are often used in analyzing many of the realistic situations arising at life testing, economical analysis, insurance studies etc. The major advantage of this distribution is the interpretation of the inversegaussian random variable as the first passage time distribution of Brownian motion with positive drift.In textile industries the printing or bleaching processed are distribution approximately as Inversegaussian distribution . Here unit of cloth is to be taken as customer, the printing or bleaching is viewed as service. In spite of wide applicability of the inversegaussian distribution as approximate model of skewed data and having simple exact sampling theory. It has been not much utilized in analyzing waiting line systems.

QUEUEING MODEL WITH INVERSEGAUSSIAN SERVICE TIME DISTRIBUTION
Consider a single server queueing with infinite capacity having FCFS (First Come First Serve) queue discipline. we assume that the arrivals are Poission with arrival rate . The service time distribution of the process is aninversegaussian of the form.

2.1Maximum Likelihood Estimates
The and are parameters involved in the service time distribution given in eq n . (1).Consider a random sample T1 , T2,……Tn from the population with p.d.feq n . (3)The likelihood function is given as Taking logarithm both sides, we have Now, to obtain the maximum likelihood estimator of the parameter partially differentiating eq n .(3) with respect to and equating the resultant to zero, we get = 0 Similarly, to obtain the maximum likelihood estimator of the parameter partially differentiating eq n .(3) with respect to and equating the resultant to zero, we get = 0 Now, substituting the maximum likelihood estimate of in eq n .(4) , we get

2.2Analysis of the Model
To analyze the model we will obtain probability generating function of , the probability that there are arrivals during the service time of a customer.
Let be the probability that there are arrivals during the service time of a customer. Let ( ) denotes the probability generating function (p.d.f) of given as Following heuristic argument Kendall [12] and Gross and Hariss [9] , the probability that there are arrivals during the service time is given by

OPTIMIZATION OF A QUEUEING SYSTEM
Then the probability generating function of is On further simplification, we get The average number of arrivals during the service time is Let be the probability that are customers in the system that are steady state. Let ( ) be the probability generating function of .Therefore, By expanding ( ) and collecting the coefficients of , we get be the probability that are customers in the system.
The probability that the system is empty is The maximum likelihood estimate of the parameter is given in equation (5).
After substituting estimated value of in equation (12), we can obtain for various values of λ . we also observe that is independent of θ i.e. is not influenced by the variability of the service time.
The average number of customers in the system can be obtained as From the equation (13) it can be observe that the average number of customers in the system influenced by θ . The value of L can be computed by uing estimated values of μand θ from equation (5) and (6) and various values of λ .
The variability of the system size can be obtained by using the formula.
The above model is monotonic decreasing and highly skewed to the right. The graph is J-shaped thereby showing the unimodel feature. The distribution function of above model will be with = +1 . and = ( +1) 2 ( +2) . 2

MAXIMUM LIKELIHOOD ESTIMATES
Under the same consideration as in section 1.2.1 the likelihood function for the model (16) is given by Taking onboth the sides, we get Differentiating the equation (19) partially with respect to and equating it to zero, The

ANALYSIS OF THE MODEL
To analyze the model we will obtain probability generating function of , the probability that there are arrivals during the service time of a customer.
Let be the probability that there are arrivals during the service time of a customer. Let ( ) denotes the probability generating function (p.d.f) of given as Following heuristic argument Kendall [12] and Gross and Hariss [9] , the probability that there are arrivals during the service time is given by Then the probability generating function of is Furthermore, the analysis can be carried out in the same manner as in the section 1.2 for inversegaussian service time distribution system.

CONFLICT OF INTERESTS
The author declares that there is no conflict of interests.