Optimal Control Policy of M/G/1 Queueing System with Delayed Randomized Multiple Vacations Under the Modified Min(N, D)-Policy Control

Based on the number of customers and the server’s workload, this paper proposes a modified Min(N, D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy. Applying the well-known stochastic decomposition property of the steady-state queue size, the probability generating function of the steady-state queue length distribution is obtained. Moreover, the explicit expressions of the expected queue length and the additional queue length distribution are derived by some algebraic manipulations. Finally, employing the renewal reward theorem, the explicit expression of the long-run expected cost per unit time is given. Furthermore, we analyze the optimal policy for economizing the expected cost and compare the optimal Min(N, D)-policy with the optimal N-policy and the optimal D-policy by using numerical examples.