Research on the parallel–batch scheduling with linearly lookahead model

Abstract

<p style='text-indent:20px;'>In this paper, we consider the new online scheduling model with linear lookahead intervals, which has the character that at any time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula>, one can foresee the jobs that will coming in the time interval <inline-formula><tex-math id="M2">\begin{document}$ (t, \lambda t+\beta ] $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \lambda\geq1, \beta\geq 0 $\end{document}</tex-math></inline-formula>. We consider online scheduling of unit length jobs on <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula> identical parallel-batch machines under this new lookahead model to minimize the maximum flowtime and the makespan, respectively. We give some lower bounds for these problems with the batch capacity <inline-formula><tex-math id="M5">\begin{document}$ b = \infty $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ b<\infty $\end{document}</tex-math></inline-formula>, respectively. And for the bounded model to minimize makespan, we give an online algorithm with competitive ratio <inline-formula><tex-math id="M7">\begin{document}$ 1+\alpha $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ 1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \frac{3}{2} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M10">\begin{document}$ \lambda\geq1, 0\leq\beta<1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M11">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> is the positive root of <inline-formula><tex-math id="M12">\begin{document}$ \lambda\alpha^2+(\lambda+\beta)\alpha+\beta-1 = 0 $\end{document}</tex-math></inline-formula>. The online algorithm is best possible when <inline-formula><tex-math id="M13">\begin{document}$ 1\leq \lambda <4/3, 0\leq \beta\leq \frac{4-3\lambda}{6} $\end{document}</tex-math></inline-formula>.</p>