Abstract:

This paper studies an $M/G/1$ queueing system with Bernoulli feedback and uninterrupted multiple vacations under $(m,N)$-policy, in which the server immediately activates the system upon returning from vacation if the number of customers reaches or exceeds a predefined lower threshold $m(\geq 1)$. After the system startup is completed, the server begins service only if the number of customers is not less than a predefined upper threshold $N(\geq m)$, continuing until the system becomes empty again. Utilizing the stochastic decomposition theorem for the steady-state queue length distribution, explicit expressions for the probability generating function (PGF) and the average queue size are derived. Furthermore, other important queueing performance indicators of the system are also discussed, such as the probabilities of the server busy and non-busy period in the steady state, as well as the number of consecutive vacations taken by the server. Through numerical examples, sensitivity analyses are conducted to examine the effects of the parameters $m$, $N$, the startup time, and the server vacation time on the mean additional queue length and the system idle probability. Finally, the objective function of the long-run expected cost per unit time of the system is constructed by applying the renewal reward theorem. Numerical examples are presented to discuss the bi-level optimal control strategy $(m^*,N^*)$ that minimizes the long-run expected cost per unit time of the system, and the impact of system activation time, server vacation time and customer feedback rate on the expected cost and optimal control strategy are also analyzed.

Key words: $M/G/1$ queue, bi-level threshold $(m,N)$-policy, uninterrupted multiple vacation, Bernoulli feedback, optimal control policy