Abstract: This paper considers the $M/G/1$ queueing system with single server vacation which can be interrupted immediately according to the Min($N,D,V$)-policy. By applying the total probability decomposition technique and the Laplace transformation, the transient and steady-state properties of the queue length from any initial state are discussed, and the Laplace transformation expression of the transient solution of queue length distribution is obtained. Moreover, we derive the recursive expressions of the equilibrium solution of queue length distribution for convenient calculation. Furthermore, we propose the stochastic decomposition structures of the steady-state queue length, the explicit expressions for the probability distribution of the additional queue length and the corresponding results for some special cases. Finally, by numerical examples, we discuss the sensitivity of the steady state queue length distribution towards system parameters and analyze the influence of different parameters on system performance.

Key words: single server vacation, Min($N,D,V$)-policy, queue length distribution, transient solution, equilibrium solution