Article: Kernel estimation analysis of the finite source M/G /1 queue with retrial and vacation Journal: International Journal of Mathematics in Operational Research (IJMOR) 2026 Vol.33 No.2 pp.198 - 214 Abstract: 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. Inderscience Publishers - linking academia, business and industry through research

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

Authors: Lyes Ikhlef; Karim Labadi

Addresses: Department of Mathematics, University of Algiers, Benyoucef Benkhedda, 2 Rue Didouche Mourad, Algeria; Research Unit LaMOS (Modeling and Optimization of Systems) ECAM-EPMI-Quartz Lab, Cergy Pontoise, France ' Department of Mathematics, University of Algiers, Benyoucef Benkhedda, 2 Rue Didouche Mourad, Algeria; Research Unit LaMOS (Modeling and Optimization of Systems) ECAM-EPMI-Quartz Lab, Cergy Pontoise, France

Abstract: 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.

Keywords: retrial model; Markov regenerative process; gamma kernel; estimation; vacation policy.

DOI: 10.1504/IJMOR.2026.152314

International Journal of Mathematics in Operational Research, 2026 Vol.33 No.2, pp.198 - 214

Received: 27 Mar 2024
Accepted: 21 Apr 2024
Published online: 16 Mar 2026 *

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

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