Optimal algorithms for online scheduling on parallel machines to minimize the makespan with a periodic availability constraint

In this paper we investigate two online scheduling problems. The first one is online scheduling on m parallel machines with one machine periodically unavailable. The second problem is online scheduling on two uniform parallel machines where one machine is periodically unavailable. The online paradigm is that jobs arrive over list, i.e., when a job presents, we have to irrevocably assign it before the next one is seen. Preemption is not allowed. The objective is to minimize makespan. We suppose that the length of each available period is normalized to 1 and the length of each unavailable period is α>0. For the first problem, we give an optimal algorithm with competitive ratio 2. For the second problem, we assume that the speed of the periodically unavailable machine is normalized to 1, while the speed of the other one is s>0. In the case where s<1, we design an algorithm and show that it is optimal with competitive ratio 1+1s. Then we further give some lower bounds on competitive ratio in the case 0<s<1. We also study a special case and prove that LPT algorithm proposed in Xu et al. (2009) [7] is optimal with competitive ratio 32.