Abstract:

This paper considers an $M/G/1$ queueing system with multiple adaptive vacations for exhausted services under the modified dyadic Min($N, D$) in which the server who is on vacation resumes its service if either $N$ customers accumulate in the system or the total workload of the server for all the waiting customers is not less than a given threshold $D$. The essential meaning of the workload of the server for every customer refers to the quantity of events included in the completed service items required by the customer. The unit of measurement for the workload may be a counting unit, a weight unit, etc. According to the well-known stochastic decomposition property of the steady-state queue size, both the probability generating function of the steady-state queue length distribution and the expression of the expected queue length are obtained. Additionally, the mean server busy period and busy cycle period are discussed. Based on the analytical results, the explicit expressions of the expected queue length and the expected length of server busy cycle period for some special cases (e.g., the number of vacations is a fixed positive integer $J$) are derived. Finally, through the renewal theory, the explicit expression of the long-run expected cost per unit time is derived. Meanwhile, numerical examples are provided to determine the optimal joint control policy for economizing the system cost.

Key words: $M/G/1$ queueing system, Multiple adaptive vacations, Min($N, D$)-policy, Steady-state queue length distribution, Optimal joint control policy