Online scheduling on two uniform machines to minimize the makespan with a periodic availability constraint

We consider the problem of online scheduling on 2 uniform machines where one machine is periodically unavailable. The problem is online in the sense that when a job presents, we have to assign it to one of the 2 uniform machines before the next one is seen. Preemption is not allowed. The objective is to minimize makespan. Assume that the speed of the periodically unavailable machine is normalized to 1, while the speed of the other one is s. Given a constant number α > 0, we also suppose that T_u = αT _a, where T_u and T_a are the length of each unavailable time period and the length of the time interval between two consecutive unavailable time periods, respectively. In the case where s ≥ 1, we show a lower bound of the competitive ratio 1 + 1/s and prove that LS algorithm is optimal. We also show that for the problem P2,M1PU|online, T _u = αT_a|C_max, LS algorithm proposed in [7] is optimal with a competitive ratio 2. After that, we give some lower bounds of competitive ratio in the case 0 < s < 1. At last, we study a special case P2,M1PU|online, T_u = αT_a, non - increasing sequence|C_max, where non-increasing sequence means that jobs arrive in a non-increasing order of their processing times. We show that LS algorithm is optimal with a competitive ratio 3/2.