Kernel estimation analysis of the finite source M/G /1 queue with retrial and vacation

In this work, we study the finite source queueing system with retrial and vacation when the service distribution of customers is unknown. We compute the qualitative analysis of this queueing system by the use of the Markov regenerative process and the kernel method. The two main quantities, one step transition probability matrix P of the embedded Markov chain (EMC) defined at regenerative instants and the mean time matrix A, that the queue spends in any state between two successive regeneration instants, for the system considered are obtained. The matrix P (resp. A) depends on the probability density function (pdf) (resp. cumulative distribution function (cdf)) of the service time. These two functions (pdf and cdf) are unknown and are estimated by using the two asymmetric kernels gamma and Birnbaum-Saunders. Finally, we establish an algorithm which gives us the estimators of: the one step transition probability matrix P, the mean time matrix A, the probability distribution of the number of customers in the orbit and the state of the server and the characteristics of our queueing system considered.