An asymptotically optimal online algorithm to minimize the total completion time on two multipurpose machines with unit processing times

In the majority of works on online scheduling on multipurpose machines the objective is to minimize the makespan. We, in contrast, consider the objective of minimizing the total completion time. For this purpose, we analyze an online‐list scheduling problem of $n$ jobs with unit processing times on a set of two machines working in parallel. Each job belongs to one of two sets of job types. Jobs belonging to the first set can be processed on either of the two machines while jobs belonging to the second set can only be processed on the second machine. We present an online algorithm with a competitive ratio of ${\rho }_{LB}+O\left(\frac{1}{n}\right)$ , where ${\rho }_{LB}$ is a lower bound on the competitive ratio of any online algorithm and is equal to $1+{\left(\frac{‐\alpha +\sqrt{4{\alpha }^3‐{\alpha }^2+2\alpha ‐1}}{2{\alpha }^2+1}\right)}^2$ where $\alpha =\frac{1}{3}+\frac{1}{6}{(116‐6\sqrt{78})}^{1/3}+\frac{{(58+3\sqrt{78})}^{1/3}}{3{\left(2\right)}^{2/3}}\approx 1.918$ . This result implies that our online algorithm is asymptotically optimal.