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A parallel related uniform machine system consists of m machines with different processing speeds. The speed of any machine is independent on jobs. In this paper, we consider online scheduling for jobs with arbitrary release times on the parallel uniform machine system. The jobs appear over list in terms of order. An order includes the processing size and releasing time of a job. For this model, an algorithm with competitive ratio of 12 is addressed in this paper.

For the online scheduling on a system of m uniform parallel machines, denoted by

proved that the algorithm LS has a competitive ratio

For

Cai and Yang [

Aspnes et al. [

Li and Shi [

bound

A generalization of the Grahamâ€™s classical on-line scheduling problem on m identical machines was proposed by Li and Huang [

Our task is to allocate a sequence of jobs to the machines in an on-line fashion, while minimizing the maximum completion time of the machines. In the following of this paper, m parallel uniform machines which have speeds of

The rest of the paper is organized as follows. In Section 2, some definitions are given. In Section 3, an algorithm U is addressed and its competitive ratio is analyzed.

In this section we will give some definitions.

Definition 1. We have m parallel machines with speeds

as the competitive ratio of algorithm A.

Definition 2. Suppose that

1) Machine

2)

It is obvious that if machine

In the following we consider m parallel uniform machines with speeds

Now we present the algorithm U by use of the notations given in the former section in the following:

Algorithm U:

Step 0. (*start the first phase*)

Step 1. If there is a new job

Step 2. If there is a machine

Step 3. Set

Step 4. (*start a new phase*)

Set

Now we begin to analyze the performance of algorithm U.

The following statement is obvious:

Lemma 1. Let

Proof: If

obvious that

It means that

we get:

That means:

This implies that there exists a job

By our assumptions, we have

This implies

But this means that the on-line algorithm should have placed job

Theorem 2. Algorithm achieves a competitive ratio of 12.

Proof: Let

Hence the total height generated by the assignment is:

The claim of the theorem is trivially true if

Therefore we have

In this paper, we consider on-line scheduling for jobs with arbitrary release times on uniform machines. An algorithm with the competitive ratio of 12 is given. It should be pointed out that more detailed consideration should be taken in order to improve the competitive ratio.

The authors would like to express their thanks to the National Natural Science Foundation of China for financially supporting under Grant No. 11471110 and No. 61271264.

Xiayan Cheng,Rongheng Li,Yunxia Zhou, (2016) On-Line Scheduling for Jobs with Arbitrary Release Times on Parallel Related Uniform Machines. Intelligent Information Management,08,98-102. doi: 10.4236/iim.2016.84008