M/G/1 Multiple Vacation Model with Balking for a Class of Disciplines

Abstract

In this paper an M/G/1 multiple vacation model with balking is considered. In this model the probability of joining to the system upon arrival depends on the number of customers at previous (customer or vacation) start epoch. The distribution of the joining probabilities is general, i.e. it is not limited to any class of distributions. The model enables a wide range of service disciplines including the exhaustive, the non-exhaustive, the semi-exhaustive, the gated, the G-limited and the E-limited ones. We establish stationary relationships between the number of customers in the system at different characteristic epochs. This leads to a system of linear equations for the stationary number of customers in terms of unknown probabilities. We provide the solution for these unknown probabilities on discipline specific way for all the above listed disciplines. Additionally the stability, the special case of state independent joining probabilities and the numerical solution are also discussed.

Key Words:

• Balking
• M/G/1
• queueing theory
• vacation model.

Additional information

Notes on contributors

Zsolt Saffer

Zsolt Saffer received the Msc. degree in electrical engineering from Technical University of Budapest, in 1989. He received a special engineering postgraduate degree and the PhD degree in informatics from the Budapest University of Technology and Economics (BUTE) in 2005 and in 2010, respectively. From 1992 to 1997 he worked at R&D of Alcatel Austria, Vienna, on software development of Private Automatic Branch Exchange (PABX) systems. In 1997, he joined to the technology group of Philips Speech Recognition Systems, Vienna, where he took part in development of Philips speech recognition technology. Since 2007 he takes part in research activity of Department of Networked Systems and Services, BUTE, Budapest. Moreover he is teacher at University of Applied Sciences, Vienna from 2012. His current research interests includes queueing theory, stochastic processes, optimization methods and their application to performance modelling, analysis and evaluation and optimal resource allocation of modern wireless/mobile telecommunication networks and manufacturing systems as well as reliability modelling and system engineering.

Wuyi Yue

Wuyi Yue received the B.Eng. degree in electronic engineering from Tsinghua University, China, and the M.Eng. and Dr.Eng. degrees in applied mathematics and physics from Kyoto University, Japan. She was a researcher and a chief researcher of ASTEM RI, an associate Professor of Wakayama University, an associate Professor and a Professor at the Department of Applied Mathematics, a Professor at Department of Information Science and Systems Engineering, Konan University, Japan. She is currently a Professor at Department of Intelligence and Informatics, Konan University, Japan. She is also the dean of the faculty of Intelligence and Informatics, Konan University, Japan, now. She is a fellow of the Operations Research Society of Japan, a senior member of IEICE of Japan, a member of the IEEE, the System Engineers Society of China and the Operations Research Society of China. Her research interests include queueing theory, stochastic processes and optimal methods as applied to system modeling, performance analysis and evaluation, and optimal resource allocation of wired and wireless/mobile communication networks (including mobile cellular, multi-hop, multi-traffic mobile communication networks), multimedia communication networks, traffic systems, stochastic systems, information systems, systems engineering and operations research.