‎In this article‎, ‎the method of determining the optimal sample size is based on Linex asymmetric loss function and has been expressed through Bayesian method for normal‎, ‎Poisson and exponential distributions‎. ‎The desirable sample size has been calculated through numerical method‎. ‎In numerical method‎, ‎the average posterior risk is calculated and then it is added to the Lindley linear cost function to achieve the average of the total cost‎. ‎Then‎, ‎the diagram of sample size is drawn in comparison to the average of total cost and eventually‎, ‎the optimal sample size that minimizes the cost has been achieved.

‎In this paper‎, ‎a new distribution of the three-parameter lifetime model called the Marshall-Olkin Gompertz is proposed on the basis of the Gompertz distribution‎. ‎It is a generalization of the Gompertz distribution having decreasing failure rate and can also be increasing and bathtub-shaped depending on its parameters‎. ‎The probability density function‎, ‎cumulative distribution function‎, ‎hazard rate function and some mathematical properties of this model such as‎, ‎central moments‎, ‎moments of order statistics‎, ‎Renyi and Shannon entropies and quantile function are derived‎. ‎In addition‎, ‎the maximum likelihood of its parameters method is estimated and this new distribution compared with some Gompertz distribution generalizations by means of a set of real data‎. 

‎In this paper we propose a utility function and obtain the Bayese stimate and the optimum sample size under this utility function‎. ‎This utility function is designed especially to obtain the Bayes estimate when the posterior follows a gamma distribution‎. ‎We consider a Normal with known mean‎, ‎a Pareto‎, ‎an Exponential and a Poisson distribution for an optimum sample size under the proposed utility function‎, ‎so that minimizes the cost of sampling‎. ‎In this process‎, ‎we use Lindley cost function in order to minimize the cost‎. ‎Here‎, ‎because of the complicated form of computation‎, ‎we are unable to solve it analytically and use the mumerical methids to get the optimum sample size.

In this article, the optimal model is determined for the family of models ‎$‎‎‎{E_r/M/1, rin N}‎$‎ with interarrival times with Erlang distribution‎‏ ‎and service times with exponential distribution‎. The method of choosing the optimal model is that first, a cost function is introduced, and then a new index is introduced according to the cost function and the ‎stationary‎ probability of the system called ‎‏‎‎$‎‎‎SER‎$‎.‎‏ A model with a larger ‎$‎SER‎$‎ index is ‎optimal.‎ Numerical analysis is also used to describe the method of determining optimal ‎model.

‎The cost function and the system ‎s‎tationary‏‎ probability are two key criteria in the design of queuing systems. In this paper, the aim is to design a single server queuing models with infinite capacity, where the service times in the first model and the interarrival times in the second model are assumed to have an Erlang distribution. For this purpose, a new index based on the cost function and the system reliability probability is introduced, the larger of which indicates the optimality of the model. Several numerical examples and an applied example are presented to explain the computational details of the proposed method.