Analysis of $ {{M}^{({{\lambda }_{1}}, {{\lambda }_{2}})}}/G/1 $ queue with uninterrupted single vacation and server's workload controlled <i>D</i>-policy

Abstract

<p style='text-indent:20px;'>This paper studies an <inline-formula><tex-math id="M1">\begin{document}$ {{M}^{({{\lambda }_{1}}, {{\lambda }_{2}})}}/G/1 $\end{document}</tex-math></inline-formula> queueing system with uninterrupted single vacation under the control of <inline-formula><tex-math id="M2">\begin{document}$ D $\end{document}</tex-math></inline-formula>-policy based on server's workload, in which customers arrive at the system in variable input rates according to the states of the server. Employing the renewal process, the total probability decomposition technique and the Laplace transform, we discuss the transient queue length distribution at any time <inline-formula><tex-math id="M3">\begin{document}$ t $\end{document}</tex-math></inline-formula> under any initial state, and derive the expressions of the Laplace transform of the transient queue length distribution with respect to time <inline-formula><tex-math id="M4">\begin{document}$ t $\end{document}</tex-math></inline-formula>. Based on the transient analysis, the explicit recursive formulas of the steady-state queue length distribution are obtained by using L'Hospital's rule. Also, the expressions of its probability generating function of the steady-state queue length distribution and the expected queue size are presented. Meanwhile, some special cases are also discussed. Furthermore, numerical experiments are provided to investigate the sensitivity of the system performance measures and the system capacity optimization design. Finally, applying the renewal reward theorem, the explicit expression of the long-run expected cost per unit time of the system is derived. To demonstrate the model's application, we consider a practical situation related to a production system. Numerical examples are provided to determine the optimal control policy <inline-formula><tex-math id="M5">\begin{document}$ {{D}^{*}} $\end{document}</tex-math></inline-formula> for economizing the system cost as well as the optimal two-dimensional control policy <inline-formula><tex-math id="M6">\begin{document}$ ({{D}^{*}}, {{T}^{*}}) $\end{document}</tex-math></inline-formula> when the vacation time is a fixed length <inline-formula><tex-math id="M7">\begin{document}$ T $\end{document}</tex-math></inline-formula>.</p>