Maximizing Online Utilization with Commitment
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We investigate online scheduling with commitment for parallel identical machines. Our objective is to maximize the total processing time of accepted jobs. As soon as a job has been submitted, the commitment constraint forces us to decide immediately whether we accept or reject the job. Upon acceptance of a job, we must complete it before its deadline d that satisfies d ≥ (1+ϵ)· p + r, with p and r being the processing time and the submission time of the job, respectively while ϵ>0 is the slack of the system. Since the hard case typically arises for near-tight deadlines, we consider ε≤ 1. We use competitive analysis to evaluate our algorithms. Our first main contribution is a deterministic preemptive online algorithm with an almost tight competitive ratio on any number of machines. For a single machine, the competitive factor matches the optimal bound 1+ϵ/ϵ of the greedy acceptance policy. Then the competitive ratio improves with an increasing number of machines and approaches (1+ϵ)·1+ϵ/ϵ as the number of machines converges to infinity. This is an exponential improvement over the greedy acceptance policy for small ϵ. In the non-preemptive case, we present a deterministic algorithm on m machines with a competitive ratio of 1+m·(1+ϵ/ϵ)^1/m. This matches the optimal bound of 2+1/ϵ of the greedy acceptance policy for a single machine while it again guarantees an exponential improvement over the greedy acceptance policy for small ϵ and large m. In addition, we determine an almost tight lower bound that approaches m·(1/ϵ)^1/m for large m and small ϵ.
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