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This paper investigates semi-online scheduling problems on two parallel machines under a grade of service (

Scheduling problem under a grade of service provision was first proposed by Hwang et al. [

For the offline scheduling on

For

We use

Our results indicate that competitive ratios of three algorithms are better than

The rest of this paper is organized as follows: in Section

We are given two machines and a series of jobs arriving online which are to be scheduled irrevocably at the time of their arrivals. The arrival of a new job occurs only after the current job is scheduled. We denote by the set

The scheduled can be seen as the partition of

The minimum makespan obtained by an optimal offline algorithm is denoted by

In this section, we prove a lower bound of competitive ratio and present an optimal algorithm for the semi-online version as

Any semi-online algorithm for

The theorem will be proved by adversary method. Let

Since we know

At the arrival of each job,

Scheduled

Let

Suppose that the incoming job is

Suppose the incoming job is

If

If

The competitive ratio of Algorithm

Suppose that Theorem

If

If

In this section, we prove a lower bound of competitive ratio and present an optimal algorithm for the semi-online version as

Any semi-online algorithm for

We will construct a job sequence with

Otherwise, if job

If job

Otherwise, if job

Therefore, job

In this subsection, we design an optimal algorithm with a competitive ratio of

Suppose that the incoming job is

Suppose that the incoming job is

(

(

(

If

Otherwise, scheduled job

If

Since

The proof is completed.

Based on Lemma

(1) If

Since Algorithm

By using Corollary

If

Based on Lemma

If job

If

The proof is completed.

If

If

If job

Let

If

Since there is no job with

The proof is completed.

Based on Lemma

The competitive ratio of Algorithm

In this section, we show a lower bound of competitive ratio and present an optimal algorithm for the semi-online version as

Any semi-online algorithm for

The theorem will be proved by adversary method. Let

In this subsection, we design an optimal algorithm with a competitive ratio of

Scheduled

Let

Suppose that the incoming job is

Suppose that the incoming job is

If

(

If

If

The competitive ratio of Algorithm

Since the job with

Assume job

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China under Grants 71071123, 60921003, and 71371129 and Program for Changjiang Scholars and Innovative Research Team in University under Grant IRT1173.