Abstract:

This paper considers a discrete-time Geo/G/1 queueing system in which the server takes multiple adaptive vacations and the system adopts modified Min$(N, D)$-control policy. By using the renewal process theory、total probability decomposition technique and $z$-transform tool, we study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain the expressions of the $z$-transformation of the transient queue length distribution at arbitrary time epoch $n^+$. Then, the recursive expressions of the steady-state queue length distribution are obtained by using L'Hospital's rule. Meanwhile, both the probability generating function of the stochastic decomposition structure of the steady-state queue length and the explicit expressions of the additional queue length distribution are presented. Additionally, the important relations between the steady-state queue length distributions at different time epochs $n^-$、$n$、$n^+$ and outside observer's are also reported. Furthermore, numerical examples are implemented to discuss the system capacity design based on the recursive formulas of the steady-state queue length distribution for calculating conveniently. Finally, employing the renewal reward theorem, the function of the long-run expected cost per unit time is derived under a given cost structure, and numerical calculation are provided to determine the optimal control policy for minimizing the long-run expected cost rate.

Key words: discrete-time queue, modified Min$(N, D)$-policy, multiple adaptive vacations, queue length distribution, system capacity optimum design, optimal control policy