Abstract:

This paper studies the $M/G/1$ queueing system for single server vacation without interruption under the control of $D$-policy, in which when the server is transferred on vacation, the server starts service immediately if the total service times of waiting customers is no less than a given positive threshold $D$. Applying the total probability decomposition technique, renewal theory and the Laplace transform tool, the transient queue length distribution from any initial state is discussed. Both the expressions of the Laplace transformation of the transient queue length distribution and the recursive expressions of the steady-state queue length distribution are derived. Meanwhile, the stochastic decomposition structure of the steady-state queue length and the explicit expression of the additional queue distribution are displayed. Furthermore, by employing the steady-state queue length distribution $\left\{ {{p}_{j}}, j=0, 1, 2, \cdots \right\}$, we discuss the optimization design of the system capacity and illustrate the important effect of the steady-state queue length distribution. Finally, the explicit expression of the long-run expected cost rate is derived under a given cost structure. And by numerical calculation, we determine the optimal control policy ${{D}^{*}}$ for minimizing the long-run expected cost per unit time as well as the combined control strategy $({{T}^{*}}, {{D}^{*}})$ when the vacation time is fixed duration $T(>0)$.

Key words: D-Policy, Single vacation without interruption, Total probability decomposition technique, System capacity design, Optimal control policy