Some New Results for Departure Process in the M^X/G/1 Queueing System with a Single Vacation and Exhaustive Service

Abstract

A batch arrival queueing system with a single vacation between two successive busy periods and with exhaustive service is considered.

The departure process h(t) is studied first on a single vacation cycle. The approach based on renewal theory is applied to obtain results in the general case. In particular, the explicit representation for the generating function of Laplace transform of the probability function of h(t) is derived. All formulae are written in terms of input parameters of the system and factors of a certain canonical factorization of Wiener–Hopf type. A numerical approach to results is discussed as well.

Keywords:

• Batch arrival queueing system
• Departure process
• Single vacation
• Wiener–Hopf factorization

AMS Subject Classification:

• 60K25
• 90B22

Acknowledgments

This article is dedicated to my father, Tadeusz.