Online and semi-online scheduling of two machines under a grade of service provision

doi:10.1016/j.orl.2005.11.004

Operations Research Letters

Volume 34, Issue 6, November 2006, Pages 692-696

https://doi.org/10.1016/j.orl.2005.11.004 Get rights and content

Abstract

We consider the online scheduling of two machines under a grade of service (GoS) provision and its semi-online variant where the total processing time is known. Respectively for the online and semi-online problems, we develop algorithms with competitive ratios of $\frac{5}{3}$ and $\frac{3}{2}$ which are shown to be optimal.

Introduction

It is a common practice in any service industry to provide differentiated services to the customers based on their entitled privileges assigned according to their promised grade of service (GoS) levels. GoS is certainly a highly qualitative concept, yet it is often translated into the level of access privilege to service capacity. For instance, we can think of dual internet servers processing service requests assigned by a load balancer, where the service requests from preferred membership, hence, entitled to a higher GoS, are allowed to be processed by all available servers, whereas those from free-trial or regular membership, hence, entitled to a lower GoS, may be processed only by a designated server. In providing services under such schemes, we face the scheduling problem under a special kind of eligibility that we refer to as GoS eligibility.

Within an off-line context, the GoS eligibility was introduced and analyzed in [7]. In this paper, we analyze the online version with two machines and its semi-online variant where the total processing time is known in advance. The total processing time is often assumed to be known in semi-online scheduling literature for various reasons as stated in [3], [6], [8], [10].

In the study of online and semi-online scheduling, the performance of an algorithm is often evaluated by its competitive ratio which is defined to be the size of the worst case makespan of the online or semi-online schedule divided by the minimum makespan of the schedule generated for the off-line version of the problem [1], [2], [3], [4], [5], [6], [8], [9], [10]. Due to the very nature of online and semi-online scheduling, it is often possible to prove that there is a lower bound to the competitive ratio achievable by any deterministic algorithm. In such cases, we say that an algorithm is optimal if its competitive ratio can be shown to be equal to the lower bound [1], [6], [8], [9], [10].

The online scheduling under general eligibility was first analyzed by Azar et al., where it was shown that the best possible competitive ratio for any deterministic algorithm is $⌈ \log_{2} m ⌉$ (up to an additive 1) where m is the number of machines [1]. The case with two machines under general eligibility is, in fact, equivalent to Crescenzi et al.'s model developed for the online cellular wireless communication channel assignment when there are only two cells [5]. Nevertheless, the analysis of cases with two machines under general eligibility is rather trivial since we can show that any algorithm would be optimal by the following arguments. First, we note that the competitive ratio of any algorithm for two machines can never be more than 2, since the makespan of any schedule is at most the total processing time which cannot be more than twice the minimum makespan. Second, the competitive ratio can never be less than 2 due to an example with two jobs with an identical processing time, where the first job is eligible for both machines and the second job is eligible for only the machine to which the first job is assigned. Note that this example also shows that, for two machines under general eligibility, any algorithm would be optimal for the semi-online variant where the total processing time is known in advance.

The concept of temporariness of jobs adds another dimension to the analysis of online scheduling [2], [5]. A temporary job is the one that may leave the system after being processed for a certain duration. Hence, each temporary job is characterized by its weight and duration. Then the load of each machine at a given time is defined to be the sum of the weights of jobs being processed by the machine at the time. In cases with temporary jobs, the measure to be minimized is the maximum of the loads that machines may carry at any time within the planning horizon. Then, as illustrated by the following example, any algorithm would be optimal for online scheduling of two machines under GoS eligibility when temporary jobs are allowed. The example begins with two jobs with an identical weight that are eligible for both machines. If they are assigned to one of the two machines, the competitive ratio becomes 2. If they are assigned to different machines, force the job assigned to the second machine to leave and generate a job with the same weight that must be processed by the first machine, which again results in the competitive ratio of 2. This example also shows that any algorithm would be optimal for semi-online cases where the maximum total weight is known in advance.

The online scheduling under a class of the eligibility which certainly encompasses the cases of GoS eligibility was analyzed by Bar-Noy et al. [2]. However, for the case of two machines, if we present two jobs with weight of 1 that are eligible to both machines, Bar-Noy et al.'s algorithm would assign both jobs to the second machines resulting the competitive ratio of 2. Hence, the question that still remains to be answered is whether there are online and semi-online algorithms for two machines under GoS eligibility with competitive ratio strictly less than 2 for the cases with no temporary job. We answer this question by developing algorithms that are shown to be optimal with competitive ratios of $\frac{5}{3}$ and $\frac{3}{2}$ , respectively, for online and semi-online cases.

In Section 2, we develop formal notations and definitions of our problems. The lower bounds for the competitive ratios for both online and semi-online problems are established in Section 3. Then, we present an optimal online algorithm in Section 4 and conclude this paper with an optimal semi-online algorithm in Section 5.

Section snippets

Definitions

We are given two machines and a series of jobs arriving online that 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. Although we do not even know the number of jobs in advance, we denote the set $J = {1, \dots, n}$ as the set of all job indices arranged in the order of arrival. The jth arriving job is referred to as job j and its required processing time is denoted by $p_{j}$ . The GoS assigned to job j is denoted by $g_{j}$ ,

Lower bounds for competitive ratios

We establish bounds to the competitive ratios for online and semi-online cases.

Lemma 1

Any algorithm H for the online problem has $z^{H} / z^{*} ≧ \frac{5}{3}$ .

Proof

We begin with job 1 and 2 with $p_{1} = p_{2} = 1$ and $g_{1} = g_{2} = 2$ . If both jobs are scheduled on one of the two machines, $z^{H} / z^{*} = 2 > \frac{5}{3}$ . Otherwise, we further generate job 3 with $p_{3} = 1$ and $g_{3} = 2$ . If job 3 is scheduled on the first machine, we generate job 4 with $p_{4} = 3$ and $g_{4} = 1$ which yields $z^{H} / z^{*} = \frac{5}{3}$ . If job 3 is scheduled on the second machine, we generate job 4 with $p_{4} = 3$ and $g_{4} = 2$ . If

Optimal online algorithm

Fig. 1 shows the proposed online algorithm. At the arrival of each job, P and T are updated to become the maximum processing time and a half of the total processing times of all jobs arrived. Also, D is updated to be the total processing time of all arrived jobs with $g_{j} = 1$ . Then, clearly the optimum makespan $z^{*} ≧ L = \max (T, P, D)$ . Then, when job j with $g_{j} = 2$ arrives, the algorithm assigns it to $S_{2}$ as far as $t (S_{2}) + p_{j} ≦ \frac{5}{3} L$ and, otherwise, to $S_{1}$ .

For the analysis of the competitive ratio of this algorithm,

Optimal semi-online algorithm

Fig. 2 shows the proposed algorithm for the semi-online problem. When, job j with $g_{j} = 2$ arrives, the algorithm assigns it to $S_{2}$ if $t (S_{2}) + p_{j}$ does not exceed $\frac{3}{2} L$ and, otherwise, to $S_{1}$ . In order to facilitate the analysis of the algorithm, we define $S_{1}^{j}$ and $S_{2}^{j}$ as in the previous section and let $S_{1}^{0} = S_{2}^{0} = \emptyset$ .

The proof for the competitive ratio of the proposed semi-online algorithm is by contradiction. Hence, we suppose that there exists a problem instance that we call, the counter example, for which

Acknowledgements

We thank the reviewers for their numerous suggestions; especially for pointing out the importance of considering temporariness of jobs and constructing and generously allowing us to include the example with temporary jobs presented in Section 1.

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