Abstract:

This paper proposes an $M/G/1$ queueing model with a patient server and uninterrupted multiple vacations under the control of $(p,N)$-policy, in which the $(p,N)$-policy means that when the server returns from vacation and finds the number of customers waiting to be served in the system is greater than or equal a given threshold $N$, the server immediately serves the customers until the system becomes empty again. If there are less than $N$ customers but at least one customer in the system, the server begins its service with probability $p\left( {0 \le p \le 1} \right)$ or stays idle with probability $(1-p)$ until there are $N$ customers in the system and starts its service at once. We employ the total probability decomposition technology, renewal theory and Laplace transform tool to conduct a detailed analysis of the system's performance indicators. The expressions of the Laplace transform of the transient queue length distribution and the recursive expressions of the steady-state queue length distribution are obtained. Furthermore, the probability generating function of the steady-state queue length distribution and the display expression of the average queue length are presented. Finally, numerical examples are presented to discuss the system capacity optimization design and the sensitivity of system parameters on the system's idle rate and the additional average queue-length.

Key words: $(p,N)$-policy, patient server, uninterrupted multiple vacation, queue length distribution, system capacity optimization design