Better Algorithm for Order On-Line Scheduling on Uniform Machines

In this paper, we consider online scheduling for jobs with arbitrary release times on the parallel uniform machine system. An algorithm with competitive ratio of 7.4641 is addressed, which is better than the best existing result of


Introduction
For the online scheduling on a system of m uniform parallel machines, denoted by Q m /online/C max , each machine ( ) has a speed s i , i.e., the time used for finishing a job with size p on M i is p/s i . Without loss of generality, we assume 1 2 m s s s < ≤ ≤  . Cho and Sahni [1] are the first to consider the online scheduling problem on m uniform machines. For Q 2 /online/C max , Epstein et al. [2] showed that LS has the competitive ratio Q 3 /online/C max was considered by Cai and Yang [3]. They showed that the algorithm LS is an optimal online algorithm when the speed ratios ( ) 1 [6]. It is proved that for 3 m ≤ LS is optimal when s m = 2 and they also developed an algorithm with a better competitive ratio than LS for m ≥ 4 and s m = s ≥ 1. For m ≥ 4 and 1 ≤ s ≤ 2, Cheng et al. [7] proposed an algorithm with a competitive ratio not greater than 2.45.
Motivated by air cargo import terminal problem, a generalization of the Graham's classical on-line scheduling problem was proposed by Li and Huang [8] [9]. They describe the requests of all jobs in terms of order, where for any job list , job J j is given as order with the information of a release time r j and a processing size of p j . More recent results can be found in the research by Li et al. [10] and Yin et al. [11].
Our task is to allocate an order sequence of jobs to m parallel uniform ma- than the existing result of 12 in Cheng et al. [12].
The rest of the paper is organized as follows. In Section 2, some definitions are given. In Section 3, an algorithm R is addressed and its competitive ratio is analyzed.

Some Definitions
In this section we will give some definitions. , T T satisfying the following two conditions: , T T and a job with release time T 2 has been assigned to machine M i to start at time T 2 .
2) 2 1 max , It is obvious that if machine M i has an idle time interval for job J j , then we can assign J j to machine M i in the idle interval. International Journal of Intelligence Science In the following we consider m parallel uniform machines with speeds 1 2 , , , m s s s  and a job list with information (r j , p j ) for each job j J L ∈ , where r i and p i represent its release time and processing size, respectively. For convenience, we assume that the sequence of machine speeds is

Algorithm R and Its Performance
Now we present the algorithm R by use of the notations given in the former section in the following: Algorithm R: Step 0. Step 1.
Let J j be a new job with release time r j and processing size p j given to the algorithm. If there is a machine M i which has an idle time interval for job J j , then we assign J j to machine M i in the idle interval and set : Step 2.  This means that it is true for all i. Now we will show the claim for (i, l) is true under the assumptions that the claim is true for (i−1, l) and (i, l−1). We prove it according to the following two cases: Case 1. c i > 0. In this case, any job J with release time r and size p satisfying By the inductive hypothesis for (i−1, l), we have The above two inequalities include the truth of the claim for (i, l). This includes that t m L is empty for every phase t.
Theorem 3. The competitive ratio of algorithm R is not greater than 7.4641.
Proof: Suppose that the algorithm ended at phase k. Then the optimal value is at least It is easy to see that the best value of r is 3 1 3 + and the performance ratio is 4 2 3 7.4641 + ≈ .

Conclusion
In this paper, on-line scheduling problem for jobs with arbitrary release times on uniform machines is considered. We developed an algorithm with the competitive ratio of 7.4641 which is better than existing result of 12. In order to improve the competitive ratio more detailed consideration should be taken in.