Sankhya: The Indian Journal of Statistics

2006, Volume 68, Pt. 2, 198--226

A Single Vacation Model G/M/1/K with $N$ Threshold Policy

Jau-Chuan Ke, National Taichung Institute of Technology, Taichung, Taiwan, R.O.C.
Kuo-Hsiung Wang, National Chung-Hsing University, Taichung, Taiwan, R.O.C.
Cheng-Hwai Liou, National Taichung Institute of Technology, Taichung, Taiwan, R.O.C.

SUMMARY. This paper studies a G/M/1/K queueing system, where the server applies an $N$ policy and takes a single vacation when the system is empty. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining inter-arrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential inter-arrival time distribution. Hereby, we establish the distributions of the number of customers in the queue at pre-arrival epochs and at arbitrary epochs as well as the distributions of the waiting time and the busy period.

AMS (2000) subject classification. 60K25, 90B22.

Key words and phrases. Single vacation, steady-state probability, supplementary variable.

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